Logiweb codex of kvanti

Up Help

∃∗: ∗, ∗ ⇒ ∗, kvanti, UniqueMember, UniqueMember(Type), SameSeries, A4, SameMember, Qclosed(Addition), Qclosed(Multiplication), FromCartProd(1), 1rule fromCartProd(2), constantRationalSeries(∗), cartProd(∗), Power(∗), binaryUnion(∗, ∗), SetOfRationalSeries, IsSubset(∗, ∗), (p∗, ∗), (s∗), (⋯),

Objekt-var, Ex-var, Ph-var, Værdi, Variabel, Op(∗), Op(∗, ∗), ∗ ¨ == ∗,

ContainsEmpty(∗), Nat(∗), Dedu(∗, ∗), Dedu_{0}(∗, ∗), Dedu_{s}(∗, ∗, ∗), Dedu_{1}(∗, ∗, ∗), Dedu_{2}(∗, ∗, ∗), Dedu_{3}(∗, ∗, ∗, ∗), Dedu_{4}(∗, ∗, ∗, ∗), Dedu_{4}^{∗}(∗, ∗, ∗, ∗), Dedu_{5}(∗, ∗, ∗), Dedu_{6}(∗, ∗, ∗, ∗), Dedu_{6}^{∗}(∗, ∗, ∗, ∗), Dedu_{7}(∗), Dedu_{8}(∗, ∗), Dedu_{8}^{∗}(∗, ∗), Ex_{1}, Ex_{2}, Ex3, Ex_{10}, Ex_{20}, ∗_{Ex}, ∗^{Ex}, ⟨∗≡ ∗ | ∗:==∗⟩_{Ex}, ⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ex}, ⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ex}, ⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ex}, ph_{1}, ph_{2}, ph_{3}, ∗_{Ph}, ∗^{Ph}, ⟨∗≡ ∗ | ∗:==∗⟩_{Ph}, ⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ph}, ⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ph}, ⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ph}, ⟨∗≡ ∗ | ∗:==∗⟩_{Me}, ⟨∗≡^{1} ∗ | ∗:==∗⟩_{Me}, ⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Me}, bs, OBS, BS, Ø, SystemQ, MP, Gen, Repetition, Neg, Ded, ExistIntro, Extensionality, Ødef, PairDef, UnionDef, PowerDef, SeparationDef, AddDoubleNeg, RemoveDoubleNeg, AndCommutativity, AutoImply, Contrapositive, FirstConjunct, SecondConjunct, FromContradiction, FromDisjuncts, IffCommutativity, IffFirst, IffSecond, ImplyTransitivity, JoinConjuncts, MP2, MP3, MP4, MP5, MT, NegativeMT, Technicality, Weakening, WeakenOr1, WeakenOr2, Formula2Pair, Pair2Formula, Formula2Union, Union2Formula, Formula2Sep, Sep2Formula, Formula2Power, SubsetInPower, HelperPowerIsSub, PowerIsSub, (Switch)HelperPowerIsSub, (Switch)PowerIsSub, ToSetEquality, HelperToSetEquality(t), ToSetEquality(t), HelperFromSetEquality, FromSetEquality, HelperReflexivity, Reflexivity, HelperSymmetry, Symmetry, HelperTransitivity, Transitivity, ERisReflexive, ERisSymmetric, ERisTransitive, ØisSubset, HelperMemberNotØ, MemberNotØ, HelperUniqueØ, UniqueØ, ==Reflexivity, ==Symmetry, Helper==Transitivity, ==Transitivity, HelperTransferNotEq, TransferNotEq, HelperPairSubset, Helper(2)PairSubset, PairSubset, SamePair, SameSingleton, UnionSubset, SameUnion, SeparationSubset, SameSeparation, SameBinaryUnion, IntersectionSubset, SameIntersection, AutoMember, HelperEqSysNotØ, EqSysNotØ, HelperEqSubset, EqSubset, HelperEqNecessary, EqNecessary, HelperNoneEqNecessary, Helper(2)NoneEqNecessary, NoneEqNecessary, EqClassIsSubset, EqClassesAreDisjoint, AllDisjoint, AllDisjointImply, BSsubset, Union(BS/R)subset, UnionIdentity, EqSysIsPartition, (x1), (x2), (y1), (y2), (v1), (v2), (v3), (v4), (v2n), (m1), (m2), (n1), (n2), (n3), (ϵ), (ϵ)_{1}, (ϵ2), (fep), (fx), (fy), (fz), (fu), (fv), (fw), (rx), (ry), (rz), (ru), (sx), (sx1), (sy), (sy1), (sz), (sz1), (su), (su1), (fxs), (fys), (crs1), (f1), (f2), (f3), (f4), (op1), (op2), (r1), (s1), (s2), X_{1}, X_{2}, Y_{1}, Y_{2}, V_{1}, V_{2}, V_{3}, V_{4}, V_{2n}, M_{1}, M_{2}, N_{1}, N_{2}, N_{3}, ϵ, ϵ1, ϵ2, FX, FY, FZ, FU, FV, FW, FEP, RX, RY, RZ, RU, (SX), (SX1), (SY), (SY1), (SZ), (SZ1), (SU), (SU1), FXS, FYS, (F1), (F2), (F3), (F4), (OP1), (OP2), (R1), (S1), (S2), (EPob), (CRS1ob), (F1ob), (F2ob), (F3ob), (F4ob), (N1ob), (N2ob), (OP1ob), (OP2ob), (R1ob), (S1ob), (S2ob), ph_{4}, ph_{5}, ph_{6}, NAT, RATIONAL_{S}ERIES, SERIES, SetOfReals, SetOfFxs, N, Q, X, xs, xaF, ysF, us, usFoelge, 0, 1, (−1), 2, 3, 1/2, 1/3, 2/3, 0f, 1f, 00, 01, (− − 01), 02, 01//02, PlusAssociativity(R), PlusAssociativity(R)XX, Plus0(R), Negative(R), Times1(R), lessAddition(R), PlusCommutativity(R), LeqAntisymmetry(R), LeqTransitivity(R), leqAddition(R), Distribution(R), A4(Axiom), InductionAxiom, EqualityAxiom, EqLeqAxiom, EqAdditionAxiom, EqMultiplicationAxiom, QisClosed(Reciprocal)(Imply), QisClosed(Reciprocal), QisClosed(Negative)(Imply), QisClosed(Negative), leqReflexivity, leqAntisymmetryAxiom, leqTransitivityAxiom, leqTotality, leqAdditionAxiom, leqMultiplicationAxiom, plusAssociativity, plusCommutativity, Negative, plus0, timesAssociativity, timesCommutativity, ReciprocalAxiom, times1, Distribution, 0not1, lemma eqLeq(R), TimesAssociativity(R), TimesCommutativity(R), (Adgic)SameR, Separation2formula(1), Separation2formula(2), Cauchy, PlusF, ReciprocalF, From ==, To ==, FromInR, PlusR(Sym), ReciprocalR(Axiom), LessMinus1(N), Nonnegative(N), US0, NextXS(UpperBound), NextXS(NoUpperBound), NextUS(UpperBound), NextUS(NoUpperBound), ExpZero, ExpPositive, ExpZero(R), ExpPositive(R), BSzero, BSpositive, UStelescope(Zero), UStelescope(Positive), EqAddition(R), FromLimit, ToUpperBound, FromUpperBound, USisUpperBound, 0not1(R), ExpUnbounded(R), FromLeq(Advanced)(N), FromLeastUpperBound, ToLeastUpperBound, XSisNotUpperBound, ysFGreater, ysFLess, SmallInverse, NatType, RationalType, SeriesType, Max, Numerical, NumericalF, MemberOfSeries(Imply), JoinConjuncts(2conditions), prop lemma imply negation, TND, FromNegatedImply, ToNegatedImply, FromNegated(2 ∗ Imply), FromNegatedAnd, FromNegatedOr, ToNegatedOr, FromNegations, From3Disjuncts, From2 ∗ 2Disjuncts, NegateDisjunct1, NegateDisjunct2, ExpandDisjuncts, SENC1, SENC2, LessLeq(R), MemberOfSeries, memberOfSeries(Type), ∗(exp)∗, R(∗), − − R(∗), rec∗, ∗/∗,

∗ ∩ ∗, ∗[∗], ∪∗, ∗ ∪ ∗, P(∗), {∗}, StateExpand(∗, ∗, ∗), extractSeries(∗), SetOfSeries(∗), − − Macro(∗), ExpandList(∗, ∗, ∗), ∗ ∗ Macro(∗), + + Macro(∗), << Macro(∗), ||Macro(∗), 01//Macro(∗), UB(∗, ∗), LUB(∗, ∗), BS(∗, ∗), UStelescope(∗, ∗), (∗), |f ∗ |, |r ∗ |, Limit(∗, ∗), Union(∗), IsOrderedPair(∗, ∗, ∗), IsRelation(∗, ∗, ∗), isFunction(∗, ∗, ∗), IsSeries(∗, ∗), IsNatural(∗, ∗), OrderedPair(∗, ∗), TypeNat(∗), TypeNat0(∗), TypeRational(∗), TypeRational0(∗), TypeSeries(∗, ∗), Typeseries0(∗, ∗), {∗, ∗}, ⟨∗, ∗⟩, (−u∗), −_{f}∗, (− − ∗), 1f/∗, 01//temp∗, ∗(∗, ∗), ReflRel(∗, ∗), SymRel(∗, ∗), TransRel(∗, ∗), EqRel(∗, ∗), [∗ ∈ ∗]_{∗}, Partition(∗, ∗), (∗ ∗ ∗), ∗ ∗_{f} ∗, ∗ ∗ ∗∗, (∗ + ∗), (∗ − ∗), ∗ +_{f} ∗, ∗ −_{f} ∗, ∗ + +∗, R(∗) − −R(∗), ∗ ∈ ∗, | ∗ |, if(∗, ∗, ∗), Max(∗, ∗), Max(∗, ∗), ∗ = ∗, ∗ ≠ ∗, ∗ <= ∗, ∗ < ∗, ∗ <_{f} ∗, ∗ ≤_{f} ∗, SF(∗, ∗),

∗ == ∗, ∗!! == ∗, ∗ << ∗, ∗ <<== ∗, ∗==∗, ∗ ⊆ ∗, ˙¬ (∗)n, ∗ ∉ ∗, ∗ ≠ ∗, ∗ ˙∧ ∗, ∗ ˙∨ ∗, ∃∗: ∗, ∗ ˙⇔ ∗, {ph ∈ ∗ | ∗},

∃∗: ∗

[∃x: y →^{tex} “(AARRGGHH!-exist-bug!)”]

∗ ⇒ ∗

[x ⇒ y →^{tex} “(i#1.

\Rightarrow #2. )i”]

kvanti

[kvanti →^{prio} Preassociative [kvanti], [base], [bracket ∗ end bracket], [big bracket ∗ end bracket], [ $ ∗ $ ], [flush left [∗]], [x], [y], [z], [[∗ ▷◁ ∗]], [[∗ →^{∗} ∗]], [pyk], [tex], [name], [prio], [∗], [T], [if(∗, ∗, ∗)], [[∗ ⇒^{∗} ∗]], [val], [claim], [⊥], [f(∗)], [(∗)^{I}], [F], [\underline{0}], [\underline{1}], [\underline{2}], [\underline{3}], [\underline{4}], [\underline{5}], [\underline{6}], [\underline{7}], [\underline{8}], [\underline{9}], [0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [a], [b], [c], [d], [e], [f], [g], [h], [i], [j], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [(∗)^{M}], [If(∗, ∗, ∗)], [array{∗} ∗ end array], [l], [c], [r], [empty], [⟨∗ | ∗:= ∗⟩], [M(∗)], [Ũ(∗)], [U(∗)], [U^{M}(∗)], [apply(∗, ∗)], [apply_{1}(∗, ∗)], [identifier(∗)], [identifier_{1}(∗, ∗)], [arrayplus(∗, ∗)], [array-remove(∗, ∗, ∗)], [array-put(∗, ∗, ∗, ∗)], [array-add(∗, ∗, ∗, ∗, ∗)], [bit(∗, ∗)], [bit_{1}(∗, ∗)], [rack], ["vector"], ["bibliography"], ["dictionary"], ["body"], ["codex"], ["expansion"], ["code"], ["cache"], ["diagnose"], ["pyk"], ["tex"], ["texname"], ["value"], ["message"], ["macro"], ["definition"], ["unpack"], ["claim"], ["priority"], ["lambda"], ["apply"], ["true"], ["if"], ["quote"], ["proclaim"], ["define"], ["introduce"], ["hide"], ["pre"], ["post"], [E(∗, ∗, ∗)], [E_{2}(∗, ∗, ∗, ∗, ∗)], [E_{3}(∗, ∗, ∗, ∗)], [E_{4}(∗, ∗, ∗, ∗)], [lookup(∗, ∗, ∗)], [abstract(∗, ∗, ∗, ∗)], [⌈∗⌉], [M(∗, ∗, ∗)], [M_{2}(∗, ∗, ∗, ∗)], [M^{∗}(∗, ∗, ∗)], [macro], [s_{0}], [zip(∗, ∗)], [assoc_{1}(∗, ∗, ∗)], [(∗)^{p}], [self], [[∗ ¨= ∗]], [[∗ ˙= ∗]], [[∗ ´= ∗]], [[∗ =^{pyk} ∗]], [[∗ =^{tex} ∗]], [[∗ =^{name} ∗]], [Priority table[∗]], [M̃_{1}], [M̃_{2}(∗)], [M̃_{3}(∗)], [M̃_{4}(∗, ∗, ∗, ∗)], [M̃(∗, ∗, ∗)], [Q̃(∗, ∗, ∗)], [Q̃_{2}(∗, ∗, ∗)], [Q̃_{3}(∗, ∗, ∗, ∗)], [Q̃^{∗}(∗, ∗, ∗)], [(∗)], [(∗)], [display(∗)], [statement(∗)], [[∗]^{·}], [[∗]^{−}], [aspect(∗, ∗)], [aspect(∗, ∗, ∗)], [⟨∗⟩], [tuple_{1}(∗)], [tuple_{2}(∗)], [let_{2}(∗, ∗)], [let_{1}(∗, ∗)], [[∗ =^{claim} ∗]], [checker], [check(∗, ∗)], [check_{2}(∗, ∗, ∗)], [check_{3}(∗, ∗, ∗)], [check^{∗}(∗, ∗)], [check_{2}^{∗}(∗, ∗, ∗)], [[∗]^{·}], [[∗]^{−}], [[∗]^{∘}], [msg], [[∗ =^{msg} ∗]], [<stmt>], [stmt], [[∗ =^{stmt} ∗]], [HeadNil′], [HeadPair′], [Transitivity′], [⊥ ⊥], [Contra′], [T′_{E}], [L_{1}], [\underline{∗}], [A], [B], [C], [D], [E], [F], [G], [H], [I], [J], [K], [L], [M], [N], [O], [P], [Q], [R], [S], [T ], [U], [V], [W], [X], [Y], [Z], [⟨∗ | ∗:= ∗⟩], [⟨^{∗}∗ | ∗:= ∗⟩], [∅], [Remainder], [(∗)^{v}], [intro(∗, ∗, ∗, ∗)], [intro(∗, ∗, ∗)], [error(∗, ∗)], [error_{2}(∗, ∗)], [proof(∗, ∗, ∗)], [proof_{2}(∗, ∗)], [S(∗, ∗)], [S^{I}(∗, ∗)], [S^{⊳} (∗, ∗)], [S_{1}^{⊳} (∗, ∗, ∗)], [S^{E}(∗, ∗)], [S_{1}^{E}(∗, ∗, ∗)], [S^{+}(∗, ∗)], [S_{1}^{+}(∗, ∗, ∗)], [S^{−}(∗, ∗)], [S_{1}^{−}(∗, ∗, ∗)], [S^{∗}(∗, ∗)], [S_{1}^{∗}(∗, ∗, ∗)], [S_{2}^{∗}(∗, ∗, ∗, ∗)], [S^{@}(∗, ∗)], [S_{1}^{@}(∗, ∗, ∗)], [S^{⊢}(∗, ∗)], [S_{1}^{⊢}(∗, ∗, ∗, ∗)], [S⊢ ⊢(∗, ∗)],

[S⊢ ⊢ ; 1(∗, ∗, ∗, ∗)], [S^{i.e.}(∗, ∗)], [S_{1}^{i.e.}(∗, ∗, ∗, ∗)], [S_{2}^{i.e.}(∗, ∗, ∗, ∗, ∗)], [S^{∀}(∗, ∗)],

[S_{1}^{∀}(∗, ∗, ∗, ∗)], [S^{;}(∗, ∗)], [S_{1}^{;}(∗, ∗, ∗)], [S_{2}^{;}(∗, ∗, ∗, ∗)], [T (∗)], [claims(∗, ∗, ∗)], [claims_{2}(∗, ∗, ∗)], [<proof>], [proof], [[Lemma ∗: ∗]], [[Proof of ∗: ∗]], [[∗ lemma ∗: ∗]], [[∗ antilemma ∗: ∗]], [[∗ rule ∗: ∗]], [[∗ antirule ∗: ∗]], [verifier], [V_{1}(∗)], [V_{2}(∗, ∗)], [V_{3}(∗, ∗, ∗, ∗)], [V_{4}(∗, ∗)], [V_{5}(∗, ∗, ∗, ∗)], [V_{6}(∗, ∗, ∗, ∗)], [V_{7}(∗, ∗, ∗, ∗)], [Cut(∗, ∗)], [Head_{⊕}(∗)], [Tail_{⊕}(∗)], [rule_{1}(∗, ∗)], [rule(∗, ∗)], [Rule tactic], [Plus(∗, ∗)], [[Theory ∗]], [theory_{2}(∗, ∗)], [theory_{3}(∗, ∗)], [theory_{4}(∗, ∗, ∗)], [HeadNil′′], [HeadPair′′], [Transitivity′′], [Contra′′], [HeadNil], [HeadPair], [Transitivity], [Contra], [T_{E}], [ragged right], [ragged right expansion ], [parm(∗, ∗, ∗)], [parm^{∗}(∗, ∗, ∗)], [inst(∗, ∗)], [inst^{∗}(∗, ∗)], [occur(∗, ∗, ∗)], [occur^{∗}(∗, ∗, ∗)], [unify(∗ = ∗, ∗)], [unify^{∗}(∗ = ∗, ∗)], [unify_{2}(∗ = ∗, ∗)], [L_{a}], [L_{b}], [L_{c}], [L_{d}], [L_{e}], [L_{f}], [L_{g}], [L_{h}], [L_{i}], [L_{j}], [L_{k}], [L_{l}], [L_{m}], [L_{n}], [L_{o}], [L_{p}], [L_{q}], [L_{r}], [L_{s}], [L_{t}], [L_{u}], [L_{v}], [L_{w}], [L_{x}], [L_{y}], [L_{z}], [L_{A}], [L_{B}], [L_{C}], [L_{D}], [L_{E}], [L_{F}], [L_{G}], [L_{H}], [L_{I}], [L_{J}], [L_{K}], [L_{L}], [L_{M}], [L_{N}], [L_{O}], [L_{P}], [L_{Q}], [L_{R}], [L_{S}], [L_{T}], [L_{U}], [L_{V}], [L_{W}], [L_{X}], [L_{Y}], [L_{Z}], [L_{?}], [Reflexivity], [Reflexivity_{1}], [Commutativity], [Commutativity_{1}], [<tactic>], [tactic], [[∗ =^{tactic} ∗]], [P(∗, ∗, ∗)], [P^{∗}(∗, ∗, ∗)], [p_{0}], [conclude_{1}(∗, ∗)], [conclude_{2}(∗, ∗, ∗)], [conclude_{3}(∗, ∗, ∗, ∗)], [conclude_{4}(∗, ∗)], [check], [[∗ =^{∘} ∗]], [RootVisible(∗)], [A], [R], [C], [T], [L], [\{∗\}], [\overline{∗}], [a], [b], [c], [d], [e], [f ], [g], [h], [i], [j ], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [x], [y], [z], [⟨∗≡ ∗ | ∗:=∗⟩], [⟨∗≡^{0} ∗ | ∗:=∗⟩], [⟨∗≡^{1} ∗ | ∗:=∗⟩], [⟨∗≡^{∗} ∗ | ∗:=∗⟩], [Ded(∗, ∗)], [Ded_{0}(∗, ∗)], [Ded_{1}(∗, ∗, ∗)], [Ded_{2}(∗, ∗, ∗)], [Ded_{3}(∗, ∗, ∗, ∗)], [Ded_{4}(∗, ∗, ∗, ∗)], [Ded_{4}^{∗}(∗, ∗, ∗, ∗)], [Ded_{5}(∗, ∗, ∗)], [Ded_{6}(∗, ∗, ∗, ∗)], [Ded_{6}^{∗}(∗, ∗, ∗, ∗)], [Ded_{7}(∗)], [Ded_{8}(∗, ∗)], [Ded_{8}^{∗}(∗, ∗)], [S], [Neg], [MP], [Gen], [Ded], [S1], [S2], [S3], [S4], [S5], [S6], [S7], [S8], [S9], [Repetition], [A1′], [A2′], [A4′], [A5′], [Prop 3.2a], [Prop 3.2b], [Prop 3.2c], [Prop 3.2d], [Prop 3.2e_{1}], [Prop 3.2e_{2}], [Prop 3.2e], [Prop 3.2f_{1}], [Prop 3.2f_{2}], [Prop 3.2f], [Prop 3.2g_{1}], [Prop 3.2g_{2}], [Prop 3.2g], [Prop 3.2h_{1}], [Prop 3.2h_{2}], [Prop 3.2h], [Block_{1}(∗, ∗, ∗)], [Block_{2}(∗)], [UniqueMember], [UniqueMember(Type)], [SameSeries], [A4], [SameMember], [Qclosed(Addition)], [Qclosed(Multiplication)], [FromCartProd(1)], [1rule fromCartProd(2)], [constantRationalSeries(∗)], [cartProd(∗)], [Power(∗)], [binaryUnion(∗, ∗)], [SetOfRationalSeries], [IsSubset(∗, ∗)], [(p∗, ∗)], [(s∗)], [(⋯)], [Objekt-var], [Ex-var], [Ph-var], [Værdi], [Variabel], [Op(∗)], [Op(∗, ∗)], [∗ ¨ == ∗], [ContainsEmpty(∗)], [Nat(∗)], [Dedu(∗, ∗)], [Dedu_{0}(∗, ∗)], [Dedu_{s}(∗, ∗, ∗)], [Dedu_{1}(∗, ∗, ∗)], [Dedu_{2}(∗, ∗, ∗)], [Dedu_{3}(∗, ∗, ∗, ∗)], [Dedu_{4}(∗, ∗, ∗, ∗)], [Dedu_{4}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{5}(∗, ∗, ∗)], [Dedu_{6}(∗, ∗, ∗, ∗)], [Dedu_{6}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{7}(∗)], [Dedu_{8}(∗, ∗)], [Dedu_{8}^{∗}(∗, ∗)], [Ex_{1}], [Ex_{2}], [Ex3], [Ex_{10}], [Ex_{20}], [∗_{Ex}], [∗^{Ex}], [⟨∗≡ ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ex}], [ph_{1}], [ph_{2}], [ph_{3}], [∗_{Ph}], [∗^{Ph}], [⟨∗≡ ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡ ∗ | ∗:==∗⟩_{Me}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Me}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Me}], [bs], [OBS], [BS], [Ø], [SystemQ], [MP], [Gen], [Repetition], [Neg], [Ded], [ExistIntro], [Extensionality], [Ødef], [PairDef], [UnionDef], [PowerDef], [SeparationDef], [AddDoubleNeg], [RemoveDoubleNeg], [AndCommutativity], [AutoImply], [Contrapositive], [FirstConjunct], [SecondConjunct], [FromContradiction], [FromDisjuncts], [IffCommutativity], [IffFirst], [IffSecond], [ImplyTransitivity], [JoinConjuncts], [MP2], [MP3], [MP4], [MP5], [MT], [NegativeMT], [Technicality], [Weakening], [WeakenOr1], [WeakenOr2], [Formula2Pair], [Pair2Formula], [Formula2Union], [Union2Formula], [Formula2Sep], [Sep2Formula], [Formula2Power], [SubsetInPower], [HelperPowerIsSub], [PowerIsSub], [(Switch)HelperPowerIsSub], [(Switch)PowerIsSub], [ToSetEquality], [HelperToSetEquality(t)], [ToSetEquality(t)], [HelperFromSetEquality], [FromSetEquality], [HelperReflexivity], [Reflexivity], [HelperSymmetry], [Symmetry], [HelperTransitivity], [Transitivity], [ERisReflexive], [ERisSymmetric], [ERisTransitive], [ØisSubset], [HelperMemberNotØ], [MemberNotØ], [HelperUniqueØ], [UniqueØ], [==Reflexivity], [==Symmetry], [Helper==Transitivity], [==Transitivity], [HelperTransferNotEq], [TransferNotEq], [HelperPairSubset], [Helper(2)PairSubset], [PairSubset], [SamePair], [SameSingleton], [UnionSubset], [SameUnion], [SeparationSubset], [SameSeparation], [SameBinaryUnion], [IntersectionSubset], [SameIntersection], [AutoMember], [HelperEqSysNotØ], [EqSysNotØ], [HelperEqSubset], [EqSubset], [HelperEqNecessary], [EqNecessary], [HelperNoneEqNecessary], [Helper(2)NoneEqNecessary], [NoneEqNecessary], [EqClassIsSubset], [EqClassesAreDisjoint], [AllDisjoint], [AllDisjointImply], [BSsubset], [Union(BS/R)subset], [UnionIdentity], [EqSysIsPartition], [(x1)], [(x2)], [(y1)], [(y2)], [(v1)], [(v2)], [(v3)], [(v4)], [(v2n)], [(m1)], [(m2)], [(n1)], [(n2)], [(n3)], [(ϵ)], [(ϵ)_{1}], [(ϵ2)], [(fep)], [(fx)], [(fy)], [(fz)], [(fu)], [(fv)], [(fw)], [(rx)], [(ry)], [(rz)], [(ru)], [(sx)], [(sx1)], [(sy)], [(sy1)], [(sz)], [(sz1)], [(su)], [(su1)], [(fxs)], [(fys)], [(crs1)], [(f1)], [(f2)], [(f3)], [(f4)], [(op1)], [(op2)], [(r1)], [(s1)], [(s2)], [X_{1}], [X_{2}], [Y_{1}], [Y_{2}], [V_{1}], [V_{2}], [V_{3}], [V_{4}], [V_{2n}], [M_{1}], [M_{2}], [N_{1}], [N_{2}], [N_{3}], [ϵ], [ϵ1], [ϵ2], [FX], [FY], [FZ], [FU], [FV], [FW], [FEP], [RX], [RY], [RZ], [RU], [(SX)], [(SX1)], [(SY)], [(SY1)], [(SZ)], [(SZ1)], [(SU)], [(SU1)], [FXS], [FYS], [(F1)], [(F2)], [(F3)], [(F4)], [(OP1)], [(OP2)], [(R1)], [(S1)], [(S2)], [(EPob)], [(CRS1ob)], [(F1ob)], [(F2ob)], [(F3ob)], [(F4ob)], [(N1ob)], [(N2ob)], [(OP1ob)], [(OP2ob)], [(R1ob)], [(S1ob)], [(S2ob)], [ph_{4}], [ph_{5}], [ph_{6}], [NAT], [RATIONAL_{S}ERIES], [SERIES], [SetOfReals], [SetOfFxs], [N], [Q], [X], [xs], [xaF], [ysF], [us], [usFoelge], [0], [1], [(−1)], [2], [3], [1/2], [1/3], [2/3], [0f], [1f], [00], [01], [(− − 01)], [02], [01//02], [PlusAssociativity(R)], [PlusAssociativity(R)XX], [Plus0(R)], [Negative(R)], [Times1(R)], [lessAddition(R)], [PlusCommutativity(R)], [LeqAntisymmetry(R)], [LeqTransitivity(R)], [leqAddition(R)], [Distribution(R)], [A4(Axiom)], [InductionAxiom], [EqualityAxiom], [EqLeqAxiom], [EqAdditionAxiom], [EqMultiplicationAxiom], [QisClosed(Reciprocal)(Imply)], [QisClosed(Reciprocal)], [QisClosed(Negative)(Imply)], [QisClosed(Negative)], [leqReflexivity], [leqAntisymmetryAxiom], [leqTransitivityAxiom], [leqTotality], [leqAdditionAxiom], [leqMultiplicationAxiom], [plusAssociativity], [plusCommutativity], [Negative], [plus0], [timesAssociativity], [timesCommutativity], [ReciprocalAxiom], [times1], [Distribution], [0not1], [lemma eqLeq(R)], [TimesAssociativity(R)], [TimesCommutativity(R)], [(Adgic)SameR], [Separation2formula(1)], [Separation2formula(2)], [Cauchy], [PlusF], [ReciprocalF], [From ==], [To ==], [FromInR], [PlusR(Sym)], [ReciprocalR(Axiom)], [LessMinus1(N)], [Nonnegative(N)], [US0], [NextXS(UpperBound)], [NextXS(NoUpperBound)], [NextUS(UpperBound)], [NextUS(NoUpperBound)], [ExpZero], [ExpPositive], [ExpZero(R)], [ExpPositive(R)], [BSzero], [BSpositive], [UStelescope(Zero)], [UStelescope(Positive)], [EqAddition(R)], [FromLimit], [ToUpperBound], [FromUpperBound], [USisUpperBound], [0not1(R)], [ExpUnbounded(R)], [FromLeq(Advanced)(N)], [FromLeastUpperBound], [ToLeastUpperBound], [XSisNotUpperBound], [ysFGreater], [ysFLess], [SmallInverse], [NatType], [RationalType], [SeriesType], [Max], [Numerical], [NumericalF], [MemberOfSeries(Imply)], [JoinConjuncts(2conditions)], [prop lemma imply negation], [TND], [FromNegatedImply], [ToNegatedImply], [FromNegated(2 ∗ Imply)], [FromNegatedAnd], [FromNegatedOr], [ToNegatedOr], [FromNegations], [From3Disjuncts], [From2 ∗ 2Disjuncts], [NegateDisjunct1], [NegateDisjunct2], [ExpandDisjuncts], [SENC1], [SENC2], [LessLeq(R)], [MemberOfSeries], [memberOfSeries(Type)]; Preassociative [∗ {∗}], [∗/indexintro(∗, ∗, ∗, ∗)], [∗/intro(∗, ∗, ∗)], [∗/bothintro(∗, ∗, ∗, ∗, ∗)], [∗/nameintro(∗, ∗, ∗, ∗)], [∗′], [∗[ ∗]], [∗[∗→∗]], [∗[∗⇒∗]], [∗0], [∗1], [0b], [∗-color(∗)], [∗-color^{∗}(∗)], [∗^{H}], [∗^{T}], [∗^{U}], [∗^{h}], [∗^{t}], [∗^{s}], [∗^{c}], [∗^{d}], [∗^{a}], [∗^{C}], [∗^{M}], [∗^{B}], [∗^{r}], [∗^{i}], [∗^{d}], [∗^{R}], [∗^{0}], [∗^{1}], [∗^{2}], [∗^{3}], [∗^{4}], [∗^{5}], [∗^{6}], [∗^{7}], [∗^{8}], [∗^{9}], [∗^{E}], [∗^{V}], [∗^{C}], [∗^{C^{∗}}], [∗^{hide}]; Preassociative [“ ∗ ”], [], [(∗)^{t}], [string(∗) + ∗], [string(∗) ++ ∗], [ ∗], [ ∗], [!∗], ["∗], [#∗], [$∗], [%∗], [&∗], [’∗], [(∗], [)∗], [∗∗], [+∗], [, ∗], [-∗], [.∗], [/∗], [0∗], [1∗], [2∗], [3∗], [4∗], [5∗], [6∗], [7∗], [8∗], [9∗], [:∗], [; ∗], [<∗], [=∗], [>∗], [?∗], [@∗], [A∗], [B∗], [C∗], [D∗], [E∗], [F∗], [G∗], [H∗], [I∗], [J∗], [K∗], [L∗], [M∗], [N∗], [O∗], [P∗], [Q∗], [R∗], [S∗], [T∗], [U∗], [V∗], [W∗], [X∗], [Y∗], [Z∗], [[∗], [\∗], []∗], [ˆ ∗], [ ∗], [‘∗], [a∗], [b∗], [c∗], [d∗], [e∗], [f∗], [g∗], [h∗], [i∗], [j∗], [k∗], [l∗], [m∗], [n∗], [o∗], [p∗], [q∗], [r∗], [s∗], [t∗], [u∗], [v∗], [w∗], [x∗], [y∗], [z∗], [{∗], [|∗], [}∗], [˜ ∗], [Preassociative ∗; ∗], [Postassociative ∗; ∗], [[∗], ∗], [priority ∗ end], [newline ∗], [macro newline ∗], [MacroIndent(∗)]; Preassociative [∗ ’ ∗], [∗ ‘ ∗]; Preassociative [∗(exp)∗]; Preassociative [∗′], [R(∗)], [− − R(∗)], [rec∗]; Preassociative [∗/∗], [∗ ∩ ∗], [∗[∗]]; Preassociative [∪∗], [∗ ∪ ∗], [P(∗)]; Preassociative [{∗}], [StateExpand(∗, ∗, ∗)], [extractSeries(∗)], [SetOfSeries(∗)], [− − Macro(∗)], [ExpandList(∗, ∗, ∗)], [∗ ∗ Macro(∗)], [+ + Macro(∗)], [<< Macro(∗)], [||Macro(∗)], [01//Macro(∗)], [UB(∗, ∗)], [LUB(∗, ∗)], [BS(∗, ∗)], [UStelescope(∗, ∗)], [(∗)], [|f ∗ |], [|r ∗ |], [Limit(∗, ∗)], [Union(∗)], [IsOrderedPair(∗, ∗, ∗)], [IsRelation(∗, ∗, ∗)], [isFunction(∗, ∗, ∗)], [IsSeries(∗, ∗)], [IsNatural(∗, ∗)], [OrderedPair(∗, ∗)], [TypeNat(∗)], [TypeNat0(∗)], [TypeRational(∗)], [TypeRational0(∗)], [TypeSeries(∗, ∗)], [Typeseries0(∗, ∗)]; Preassociative [{∗, ∗}], [⟨∗, ∗⟩], [(−u∗)], [−_{f}∗], [(− − ∗)], [1f/∗], [01//temp∗]; Preassociative [∗(∗, ∗)], [ReflRel(∗, ∗)], [SymRel(∗, ∗)], [TransRel(∗, ∗)], [EqRel(∗, ∗)], [[∗ ∈ ∗]_{∗}], [Partition(∗, ∗)]; Preassociative [∗ · ∗], [∗ ·_{0} ∗], [(∗ ∗ ∗)], [∗ ∗_{f} ∗], [∗ ∗ ∗∗]; Preassociative [∗ + ∗], [∗ +_{0} ∗], [∗ +_{1} ∗], [∗ − ∗], [∗ −_{0} ∗], [∗ −_{1} ∗], [(∗ + ∗)], [(∗ − ∗)], [∗ +_{f} ∗], [∗ −_{f} ∗], [∗ + +∗], [R(∗) − −R(∗)]; Preassociative [∗ ∈ ∗]; Preassociative [| ∗ |], [if(∗, ∗, ∗)], [Max(∗, ∗)], [Max(∗, ∗)]; Preassociative [∗ = ∗], [∗ ≠ ∗], [∗ <= ∗], [∗ < ∗], [∗ <_{f} ∗], [∗ ≤_{f} ∗], [SF(∗, ∗)], [∗ == ∗], [∗!! == ∗], [∗ << ∗], [∗ <<== ∗]; Preassociative [∗ ∪ {∗}], [∗ ∪ ∗], [∗\{∗}]; Postassociative [∗ ˙. . ∗], [∗ \underline{˙. .} ∗], [∗ \underline{: :} ∗], [∗ \underline{+2∗} ∗], [∗ : : ∗], [∗ +2∗ ∗]; Postassociative [∗, ∗]; Preassociative

[∗ ≈^{B} ∗], [∗ ≈^{D} ∗], [∗ ≈^{C} ∗], [∗ ≈^{P} ∗], [∗ ≈ ∗], [∗ = ∗], [∗ →^{+} ∗], [∗ =^{t} ∗], [∗ =^{t^{∗}} ∗], [∗ =^{r} ∗],

[∗ ∈_{t} ∗], [∗ ⊆_{T} ∗], [∗ =^{T} ∗], [∗ =^{s} ∗], [∗ free in ∗], [∗ free in^{∗} ∗], [∗ free for ∗ in ∗], [∗ free for^{∗} ∗ in ∗], [∗ ∈_{c} ∗], [∗ < ∗], [∗ <′ ∗], [∗ ≤′ ∗], [∗ = ∗], [∗ ≠ ∗], [∗^{var}], [∗#^{0}∗], [∗#^{1}∗], [∗#^{∗}∗], [∗==∗], [∗ ⊆ ∗]; Preassociative [¬∗], [ ˙¬ (∗)n], [∗ ∉ ∗], [∗ ≠ ∗]; Preassociative [∗ ∧ ∗], [∗ ¨∧ ∗], [∗ ˜∧ ∗], [∗ ∧_{c} ∗], [∗ ˙∧ ∗]; Preassociative [∗ ∨ ∗], [∗ ∥ ∗], [∗ ¨∨ ∗]; Postassociative [∗ ˙∨ ∗]; Preassociative [∃∗: ∗], [∀∗: ∗], [∀_{obj}∗: ∗], [∃∗: ∗]; Postassociative [∗ ¨⇒ ∗], [∗ ⇒ ∗], [∗ ⇔ ∗], [∗ ˙⇔ ∗]; Preassociative [{ph ∈ ∗ | ∗}]; Postassociative [∗ : ∗], [∗ spy ∗], [∗!∗]; Preassociative

[∗ \{ \begin{matrix}∗ \\ ∗ \end{matrix} ];

Preassociative [λ ∗ .∗], [Λ ∗ .∗], [Λ∗], [if ∗ then ∗ else ∗], [let ∗ = ∗ in ∗], [let ∗ ¨= ∗ in ∗]; Preassociative [∗#∗]; Preassociative [∗^{I}], [∗^{⊳} ], [∗^{V}], [∗^{+}], [∗^{−}], [∗^{∗}]; Preassociative [∗ @ ∗], [∗ ⊳ ∗], [∗ ⊳ ⊳ ∗], [∗ ≫ ∗], [∗ ⊵ ∗]; Postassociative [∗ ⊢ ∗], [∗ ⊢⊢ ∗], [∗ i.e. ∗]; Preassociative [∀∗: ∗], [Π∗: ∗]; Postassociative [∗ ⊕ ∗]; Postassociative [∗; ∗]; Preassociative [∗ proves ∗]; Preassociative [∗ proof of ∗ : ∗], [Line ∗ : ∗ ≫ ∗; ∗], [Last line ∗ ≫ ∗ □], [Line ∗ : Premise ≫ ∗; ∗], [Line ∗ : Side-condition ≫ ∗; ∗], [Arbitrary ≫ ∗; ∗], [Local ≫ ∗ = ∗; ∗], [Begin ∗; ∗ : End; ∗], [Last block line ∗ ≫ ∗; ], [Arbitrary ≫ ∗; ∗]; Postassociative [∗ | ∗]; Postassociative [∗ , ∗], [∗[ ∗]∗]; Preassociative [∗&∗]; Preassociative [∗\\∗], [∗ linebreak[4] ∗], [∗\\∗]; ]

[kvanti →^{tex} “kvanti”]

[kvanti →^{pyk} “kvanti”]

UniqueMember

[UniqueMember →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⊢ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⊢ \underline{(sx)} = \underline{(sy)} ⊢ Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; Repetition ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; SecondConjunct ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ≫ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}; A4 @ \underline{(sx)} ⊳ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ≫ ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}; A4 @ \underline{(sx1)} ⊳ ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ≫ ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \overline{(f3)} ⇒ \underline{(sx1)} = \overline{(f4)}; A4 @ \underline{(sy)} ⊳ ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \overline{(f3)} ⇒ \underline{(sx1)} = \overline{(f4)} ≫ ∀_{obj}\overline{(f4)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⇒ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \underline{(sy)} ⇒ \underline{(sx1)} = \overline{(f4)}; A4 @ \underline{(sy1)} ⊳ ∀_{obj}\overline{(f4)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⇒ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \underline{(sy)} ⇒ \underline{(sx1)} = \overline{(f4)} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⇒ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \underline{(sy)} ⇒ \underline{(sx1)} = \underline{(sy1)}; MP3 ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⇒ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⇒ \underline{(sx)} = \underline{(sy)} ⇒ \underline{(sx1)} = \underline{(sy1)} ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⊳ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⊳ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sx1)} = \underline{(sy1)}⌉, p_{0}, c)]

[UniqueMember →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⊢ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⊢ \underline{(sx)} = \underline{(sy)} ⊢ \underline{(sx1)} = \underline{(sy1)}]

[UniqueMember →^{tex} “UniqueMember”]

[UniqueMember →^{pyk} “lemma uniqueMember”]

UniqueMember(Type)

[UniqueMember(Type) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sz)}⌉) ⊢⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⊢ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⊢ \underline{(sx)} = \underline{(sy)} ⊢ SeriesType ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sz)}⌉) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; UniqueMember ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⊳ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⊳ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sx1)} = \underline{(sy1)}⌉, p_{0}, c)]

[UniqueMember(Type) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sz)}⌉) ⊢⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} ∈ \underline{(fx)} ⊢ \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ∈ \underline{(fx)} ⊢ \underline{(sx)} = \underline{(sy)} ⊢ \underline{(sx1)} = \underline{(sy1)}]

[UniqueMember(Type) →^{tex} “UniqueMember(Type)”]

[UniqueMember(Type) →^{pyk} “lemma uniqueMember(Type)”]

SameSeries

[SameSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀\underline{(sy)}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ λc.TypeNat0(⌈\underline{n}⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ⊢⊢ \underline{m} = \underline{n} ⊢ memberOfSeries(Type) ⊳ ⊳ λc.TypeNat0(⌈\underline{m}⌉) ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}; memberOfSeries(Type) ⊳ ⊳ λc.TypeNat0(⌈\underline{n}⌉) ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ≫ \{\{\underline{n}, \underline{n}\}, \{\underline{n}, \underline{(fx)}[\underline{n}]\}\} ∈ \underline{(fx)}; UniqueMember(Type) ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)} ⊳ \{\{\underline{n}, \underline{n}\}, \{\underline{n}, \underline{(fx)}[\underline{n}]\}\} ∈ \underline{(fx)} ⊳ \underline{m} = \underline{n} ≫ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{n}]⌉, p_{0}, c)]

[SameSeries →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀\underline{(sy)}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ λc.TypeNat0(⌈\underline{n}⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ⊢⊢ \underline{m} = \underline{n} ⊢ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{n}]]

[SameSeries →^{tex} “SameSeries”]

[SameSeries →^{pyk} “lemma sameSeries”]

A4

[A4 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨\underline{a}≡\underline{b}|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: \underline{b} ⊢ A4(Axiom) ⊳ ⊳ ⟨\underline{a}≡\underline{b}|\underline{(v1)}:==\underline{x}⟩_{Me} ≫ ∀_{obj}\underline{(v1)}: \underline{b} ⇒ \underline{a}; MP ⊳ ∀_{obj}\underline{(v1)}: \underline{b} ⇒ \underline{a} ⊳ ∀_{obj}\underline{(v1)}: \underline{b} ≫ \underline{a}⌉, p_{0}, c)]

[A4 →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨\underline{a}≡\underline{b}|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: \underline{b} ⊢ \underline{a}]

[A4 →^{tex} “A4”]

[A4 →^{pyk} “lemma a4”]

SameMember

[SameMember →^{proof} Rule tactic]

[SameMember →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \underline{(sx)} = \underline{(sy)} ⊢ \underline{(sx)} ∈ \underline{(sz)} ⊢ \underline{(sy)} ∈ \underline{(sz)}]

[SameMember →^{tex} “SameMember”]

[SameMember →^{pyk} “lemma sameMember”]

Qclosed(Addition)

[Qclosed(Addition) →^{proof} Rule tactic]

[Qclosed(Addition) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} ∈ Q ⊢ \underline{y} ∈ Q ⊢ (\underline{x} + \underline{y}) ∈ Q]

[Qclosed(Addition) →^{tex} “Qclosed(Addition)”]

[Qclosed(Addition) →^{pyk} “1rule Qclosed(Addition)”]

Qclosed(Multiplication)

[Qclosed(Multiplication) →^{proof} Rule tactic] [Qclosed(Multiplication) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} ∈ Q ⊢ \underline{y} ∈ Q ⊢ (\underline{x} ∗ \underline{y}) ∈ Q]

[Qclosed(Multiplication) →^{tex} “Qclosed(Multiplication)”]

[Qclosed(Multiplication) →^{pyk} “1rule Qclosed(Multiplication)”]

FromCartProd(1)

[FromCartProd(1) →^{proof} Rule tactic] [FromCartProd(1) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ \underline{(sx)} ∈ \underline{(sx1)}]

[FromCartProd(1) →^{tex} “FromCartProd(1)”]

[FromCartProd(1) →^{pyk} “1rule fromCartProd(1)”]

1rule fromCartProd(2)

[1rule fromCartProd(2) →^{proof} Rule tactic] [1rule fromCartProd(2) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ \underline{(sy)} ∈ \underline{(sy1)}]

[1rule fromCartProd(2) →^{pyk} “1rule fromCartProd(2)”]

constantRationalSeries(∗)

[constantRationalSeries(x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[constantRationalSeries(x) ¨= {ph ∈ cartProd(N) | ∃(CRS1ob): ph_{3} = OrderedPair((CRS1ob), x)}]⌉)] [constantRationalSeries(x) →^{tex} “constantRationalSeries(#1. )”]

[constantRationalSeries(∗) →^{pyk} “constantRationalSeries( " )”]

cartProd(∗)

[cartProd((sx)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[cartProd((sx)) ¨= {ph ∈ Power(Power(binaryUnion((sx), (sy)))) | IsOrderedPair(ph_{1}, (sx), (sy))}]⌉)]

[cartProd(x) →^{tex} “cartProd(#1. )”]

[cartProd(∗) →^{pyk} “cartProd( " , " )”]

Power(∗)

[Power(x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Power(x) ¨= P(x)]⌉)]

[Power(x) →^{tex} “Power(#1. )”]

[Power(∗) →^{pyk} “P( " )”]

binaryUnion(∗, ∗)

[binaryUnion(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[binaryUnion(x, y) ¨= Union((px, y))]⌉)]

[binaryUnion(x, y) →^{tex} “binaryUnion(#1. , #2. )”]

[binaryUnion(∗, ∗) →^{pyk} “binaryUnion( " , " )”]

SetOfRationalSeries

[SetOfRationalSeries →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[SetOfRationalSeries ¨= {ph ∈ Power(cartProd(N)) | IsSeries(ph_{2}, Q)}]⌉)]

[SetOfRationalSeries →^{tex} “SetOfRationalSeries”]

[SetOfRationalSeries →^{pyk} “setOfRationalSeries”]

IsSubset(∗, ∗)

[IsSubset(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[IsSubset(x, y) ¨= x ⊆ y]⌉)]

[IsSubset(x, y) →^{tex} “IsSubset(#1. , #2. )”]

[IsSubset(∗, ∗) →^{pyk} “isSubset( " , " )”]

(p∗, ∗)

[(px, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(px, y) ¨= {x, y}]⌉)]

[(px, y) →^{tex} “(p#1.

, #2. )”]

[(p∗, ∗) →^{pyk} “(p " , " )”]

(s∗)

[(sx) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(sx) ¨= {x}]⌉)]

[(sx) →^{tex} “(s#1.

)”]

[(s∗) →^{pyk} “(s " )”]

(⋯)

[(⋯) →^{tex} “(\cdots{})”]

[(⋯) →^{pyk} “cdots”]

Objekt-var

[Objekt-var →^{tex} “\texttt{Objekt-var}”]

[Objekt-var →^{pyk} “object-var”]

Ex-var

[Ex-var →^{tex} “\texttt{Ex-var}”] [Ex-var →^{pyk} “ex-var”]

Ph-var

[Ph-var →^{tex} “\texttt{Ph-var}”] [Ph-var →^{pyk} “ph-var”]

Værdi

[Værdi →^{tex} “\texttt{V\ae{}rdi}”] [Værdi →^{pyk} “vaerdi”]

Variabel

[Variabel →^{tex} “\texttt{Variabel}”] [Variabel →^{pyk} “variabel”]

Op(∗)

[Op(x) →^{tex} “Op(#1. )”] [Op(∗) →^{pyk} “op " end op”]

Op(∗, ∗)

[Op(x, y) →^{tex} “Op(#1. , #2. )”] [Op(∗, ∗) →^{pyk} “op2 " comma " end op2”]

∗ ¨ == ∗

[x ¨ == y →^{tex} “#1.

\mathrel {\ddot {==}} #2.”]

[∗ ¨ == ∗ →^{pyk} “define-equal " comma " end equal”]

ContainsEmpty(∗)

[ContainsEmpty(x) →^{tex} “ContainsEmpty(#1. )”]

[ContainsEmpty(∗) →^{pyk} “contains-empty " end empty”]

Nat(∗)

[Nat(x) →^{tex} “Nat(#1. )”]

[Nat(∗) →^{pyk} “Nat( " )”]

Dedu(∗, ∗)

[Dedu(x, y) →^{tex} “ Dedu(#1. , #2. )”]

[Dedu(∗, ∗) →^{pyk} “1deduction " conclude " end 1deduction”]

Dedu_{0}(∗, ∗)

[Dedu_{0}(x, y) →^{tex} “ Dedu 0(#1. , #2. )”]

[Dedu_{0}(∗, ∗) →^{pyk} “1deduction zero " conclude " end 1deduction”]

Dedu_{s}(∗, ∗, ∗)

[Dedu_{s}(x, y, z) →^{tex} “Dedu {s}(#1. , #2. , #3. )”]

[Dedu_{s}(∗, ∗, ∗) →^{pyk} “1deduction side " conclude " condition " end 1deduction”]

Dedu_{1}(∗, ∗, ∗)

[Dedu_{1}(x, y, z) →^{tex} “ Dedu 1(#1. , #2. , #3. )”]

[Dedu_{1}(∗, ∗, ∗) →^{pyk} “1deduction one " conclude " condition " end 1deduction”]

Dedu_{2}(∗, ∗, ∗)

[Dedu_{2}(x, y, z) →^{tex} “ Dedu 2(#1. , #2. , #3. )”]

[Dedu_{2}(∗, ∗, ∗) →^{pyk} “1deduction two " conclude " condition " end 1deduction”]

Dedu_{3}(∗, ∗, ∗, ∗)

[Dedu_{3}(x, y, z, u) →^{tex} “ Dedu 3(#1. , #2. , #3. , #4. )”]

[Dedu_{3}(∗, ∗, ∗, ∗) →^{pyk} “1deduction three " conclude " condition " bound " end 1deduction”]

Dedu_{4}(∗, ∗, ∗, ∗)

[Dedu_{4}(x, y, z, u) →^{tex} “ Dedu 4(#1. , #2. , #3. , #4. )”]

[Dedu_{4}(∗, ∗, ∗, ∗) →^{pyk} “1deduction four " conclude " condition " bound " end 1deduction”]

Dedu_{4}^{∗}(∗, ∗, ∗, ∗)

[Dedu_{4}^{∗}(x, y, z, u) →^{tex} “ Dedu 4ˆ ∗(#1. , #2. , #3. , #4. )”]

[Dedu_{4}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “1deduction four star " conclude " condition " bound " end 1deduction”]

Dedu_{5}(∗, ∗, ∗)

[Dedu_{5}(x, y, z) →^{tex} “ Dedu 5(#1. , #2. , #3. )”]

[Dedu_{5}(∗, ∗, ∗) →^{pyk} “1deduction five " condition " bound " end 1deduction”]

Dedu_{6}(∗, ∗, ∗, ∗)

[Dedu_{6}(p, c, e, b) →^{tex} “ Dedu 6(#1. , #2. , #3. , #4.

)”]

[Dedu_{6}(∗, ∗, ∗, ∗) →^{pyk} “1deduction six " conclude " exception " bound " end 1deduction”]

Dedu_{6}^{∗}(∗, ∗, ∗, ∗)

[Dedu_{6}^{∗}(p, c, e, b) →^{tex} “ Dedu 6ˆ ∗(#1. , #2. , #3. , #4. )”]

[Dedu_{6}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “1deduction six star " conclude " exception " bound " end 1deduction”]

Dedu_{7}(∗)

[Dedu_{7}(p) →^{tex} “ Dedu 7(#1. )”]

[Dedu_{7}(∗) →^{pyk} “1deduction seven " end 1deduction”]

Dedu_{8}(∗, ∗)

[Dedu_{8}(p, b) →^{tex} “ Dedu 8(#1. , #2. )”]

[Dedu_{8}(∗, ∗) →^{pyk} “1deduction eight " bound " end 1deduction”]

Dedu_{8}^{∗}(∗, ∗)

[Dedu_{8}^{∗}(p, b) →^{tex} “ Dedu 8ˆ ∗(#1. , #2. )”] [Dedu_{8}^{∗}(∗, ∗) →^{pyk} “1deduction eight star " bound " end 1deduction”]

Ex_{1}

[Ex_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{1} ¨= a_{Ex}]⌉)]

[Ex_{1} →^{tex} “Ex {1}”]

[Ex_{1} →^{pyk} “ex1”]

Ex_{2}

[Ex_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{2} ¨= b_{Ex}]⌉)]

[Ex_{2} →^{tex} “Ex {2}”]

[Ex_{2} →^{pyk} “ex2”]

Ex3

[Ex3 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex3 ¨= c_{Ex}]⌉)]

[Ex3 →^{tex} “Ex3”]

[Ex3 →^{pyk} “ex3”]

Ex_{10}

[Ex_{10} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{10} ¨= j_{Ex}]⌉)]

[Ex_{10} →^{tex} “Ex {10}”]

[Ex_{10} →^{pyk} “ex10”]

Ex_{20}

[Ex_{20} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Ex_{20} ¨= t_{Ex}]⌉)]

[Ex_{20} →^{tex} “Ex {20}”]

[Ex_{20} →^{pyk} “ex20”]

∗_{Ex}

[x_{Ex} →^{tex} “#1.

{Ex}”]

[∗_{Ex} →^{pyk} “existential var " end var”]

∗^{Ex}

[x^{Ex} →^{val} x =^{r} ⌈x_{Ex}⌉]

[x^{Ex} →^{tex} “#1.

ˆ {Ex}”]

[∗^{Ex} →^{pyk} “" is existential var”]

⟨∗≡ ∗ | ∗:==∗⟩_{Ex}

[⟨a≡b|x:==t⟩_{Ex} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨a≡b|x:==t⟩_{Ex} ¨= ⟨⌈a⌉≡^{0}⌈b⌉|⌈x⌉:==⌈t⌉⟩_{Ex}]⌉)]

[⟨x≡y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv} #2. | #3. {:==} #4. \rangle {Ex} ”]

[⟨∗≡ ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub " is " where " is " end sub”]

⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ex}

[⟨a≡^{0}b|x:==t⟩_{Ex} →^{val} λc.x^{Ex} ∧ ⟨a≡^{1}b|x:==t⟩_{Ex}]

[⟨x≡^{0}y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ 0 #2. | #3. {:==} #4. \rangle {Ex} ”]

[⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub0 " is " where " is " end sub”]

⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ex}

[⟨a≡^{1}b|x:==t⟩_{Ex} →^{val} a!x!t! If(b =^{r} ⌈∀_{obj}u: v⌉, F, If(b^{Ex} ∧ b =^{t} x, a =^{t} t, If( a =^{r} b, ⟨a^{t}≡^{∗}b^{t}|x:==t⟩_{Ex}, F)))]

[⟨x≡^{1}y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ 1 #2. | #3. {:==} #4. \rangle {Ex} ”]

[⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub1 " is " where " is " end sub”]

⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ex}

[⟨a≡^{∗}b|x:==t⟩_{Ex} →^{val} b!x!t!If(a, T, If(⟨a^{h}≡^{1}b^{h}|x:==t⟩_{Ex}, ⟨a^{t}≡^{∗}b^{t}|x:==t⟩_{Ex}, F))]

[⟨x≡^{∗}y|z:==u⟩_{Ex} →^{tex} “\langle #1. {\equiv}ˆ ∗ #2. | #3. {:==} #4. \rangle {Ex} ”]

[⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ex} →^{pyk} “exist-sub∗ " is " where " is " end sub”]

ph_{1}

[ph_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{1} ¨= a_{Ph}]⌉)]

[ph_{1} →^{tex} “ph {1}”]

[ph_{1} →^{pyk} “ph1”]

ph_{2}

[ph_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{2} ¨= b_{Ph}]⌉)]

[ph_{2} →^{tex} “ph {2}”]

[ph_{2} →^{pyk} “ph2”]

ph_{3}

[ph_{3} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{3} ¨= c_{Ph}]⌉)]

[ph_{3} →^{tex} “ph {3}”]

[ph_{3} →^{pyk} “ph3”]

∗_{Ph}

[∗_{Ph} →^{tex} “#1.

{Ph} ”]

[∗_{Ph} →^{pyk} “placeholder-var " end var”]

∗^{Ph}

[x^{Ph} →^{tex} “#1.

ˆ {Ph}”]

[∗^{Ph} →^{pyk} “" is placeholder-var”]

⟨∗≡ ∗ | ∗:==∗⟩_{Ph}

[⟨x≡y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv} #2. | #3. {:==} #4. \rangle {Ph} ”]

[⟨∗≡ ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub " is " where " is " end sub”]

⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ph}

[⟨x≡^{0}y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ 0 #2. | #3. {:==} #4. \rangle {Ph} ”]

[⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub0 " is " where " is " end sub”]

⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ph}

[⟨x≡^{1}y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ 1 #2. | #3. {:==} #4. \rangle {Ph} ”]

[⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub1 " is " where " is " end sub”]

⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ph}

[⟨x≡^{∗}y|z:==u⟩_{Ph} →^{tex} “\langle #1. {\equiv}ˆ ∗ #2. | #3. {:==} #4. \rangle {Ph} ”]

[⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ph} →^{pyk} “ph-sub∗ " is " where " is " end sub”]

⟨∗≡ ∗ | ∗:==∗⟩_{Me}

[⟨x≡y|z:==u⟩_{Me} →^{tex} “\langle #1. {\equiv} #2. | #3. {:==} #4. \rangle {Me}”]

[⟨∗≡ ∗ | ∗:==∗⟩_{Me} →^{pyk} “meta-sub " is " where " is " end sub”]

⟨∗≡^{1} ∗ | ∗:==∗⟩_{Me}

[⟨x≡^{1}y|z:==u⟩_{Me} →^{tex} “\langle #1. {\equiv}ˆ 1 #2. | #3. {:==} #4. \rangle {Me} ”]

[⟨∗≡^{1} ∗ | ∗:==∗⟩_{Me} →^{pyk} “meta-sub1 " is " where " is " end sub”]

⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Me}

[⟨x≡^{∗}y|z:==u⟩_{Me} →^{tex} “\langle #1. {\equiv}ˆ ∗ #2. | #3. {:==} #4. \rangle {Me} ”] [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Me} →^{pyk} “meta-sub∗ " is " where " is " end sub”]

bs

[bs →^{tex} “\mathsf {bs}”] [bs →^{pyk} “var big set”]

OBS

[OBS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[OBS ¨= \overline{bs}]⌉)] [OBS →^{tex} “ \mathsf {OBS}”] [OBS →^{pyk} “object big set”]

BS

[BS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[BS ¨= \underline{bs}]⌉)] [BS →^{tex} “{\cal BS}”] [BS →^{pyk} “meta big set”]

Ø

[Ø →^{tex} “\mathrm{\O}”] [Ø →^{pyk} “zermelo empty set”]

SystemQ

[SystemQ →^{stmt} ∀\underline{m}: UB(us[\underline{m}], SetOfFxs) ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]∗\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]∗\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊕ ∀\underline{m}: UB(\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals) ⊢ xs[(\underline{m} + 1)] = xs[\underline{m}] ⊕ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \underline{x} ⊢ \underline{b} ⊢ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y} ⊕ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ (\underline{s}==\underline{x})n ⇒ \underline{s}==\underline{y} ⇒ ˙¬ ( ˙¬ (\underline{s}==\underline{x})n ⇒ \underline{s}==\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\})n)n ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: \underline{m} ∈ N ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)} ⊕ ∀\underline{m}: ∀\underline{n}: \underline{n} = 0 ⊢ BS(\underline{m}, \underline{n}) = rec(1 + 1)(exp)\underline{m} ⊕ ∀\underline{(sx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)} ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fy)} ⇒ \overline{(s1)} ∈ \underline{(fx)} ⊢ \underline{(fx)} = \underline{(fy)} ⊕ ∀\underline{x}: (\underline{x} + 0) = \underline{x} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⊕ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: ˙¬ (ysF[\underline{m}] <= (xs[\underline{m}] + rec\underline{m}) ⇒ ˙¬ ( ˙¬ (ysF[\underline{m}] = (xs[\underline{m}] + rec\underline{m}))n)n)n ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} ∈ Q ⊢ \underline{y} ∈ Q ⊢ (\underline{x} ∗ \underline{y}) ∈ Q ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: (\underline{x} ∗ 1) = \underline{x} ⊕ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sz)} ⊢ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sz)}\} ⊕ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b} ⊕ ∀\underline{(fx)}: ∀\underline{(sy)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ⊢⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(ry)} = \underline{(rx)} ⊕ ∀\underline{m}: ∀\underline{x}: \underline{m} = 0 ⊢ \underline{x}(exp)\underline{m} = 1 ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⊕ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx)} = \underline{(sx1)} ⊢ \underline{(sy)} = \underline{(sy1)} ⊢ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx1)}, \underline{(sy1)}\} ⊕ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ \underline{b} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ LUB(\underline{(fx)}, \underline{(fys)}) ⊢ ˙¬ (UB(\underline{(fx)}, \underline{(fys)}) ⇒ ˙¬ (UB(\underline{(fz)}, \underline{(fys)}) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fz)})n)n ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fxs)}: \underline{(fx)} ∈ \underline{(fxs)} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fy)} ⊢ UB(\underline{(fy)}, \underline{(fxs)}) ⊕ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ˙¬ (0 = 1)n ⊕ ∀\underline{m}: Nat(\underline{m}) ⊢⊢ 0 <= \underline{m} ⊕ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ˙¬ (∀_{obj}\underline{(sy)}: ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sz)})n)n)n)n ⊢ \underline{(sx)} ∈ Union(\underline{(sz)}) ⊕ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x}==\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ (\overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ (\overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x})n)n ⇒ ˙¬ (∀_{obj}\overline{s}: ˙¬ (\overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ (\overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x})n)n ⇒ \underline{x}==\underline{y})n)n ⊕ ∀\underline{x}: ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(ry)} = \underline{(rz)} ⊢ \underline{(rx)} = \underline{(rz)} ⊕ ∀\underline{m}: ∀\underline{(fx)}: \underline{m} = 0 ⊢ \underline{(fx)}(exp)\underline{m} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: ∀\underline{y}: (\underline{x} + \underline{y}) = (\underline{y} + \underline{x}) ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] = (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]) ⊕ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ⟨\underline{b}≡\underline{a}|\underline{(v1)}:==0⟩_{Me} ⊢⊢ ⟨\underline{c}≡\underline{a}|\underline{(v1)}:==(\underline{(v1)} + 1)⟩_{Me} ⊢⊢ \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ \underline{c} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ⊕ ∀\underline{m}: ˙¬ (UB(xs[\underline{m}], SetOfFxs))n ⊕ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \underline{(sx)} = \underline{(sy)} ⊢ \underline{(sx)} ∈ \underline{(sz)} ⊢ \underline{(sy)} ∈ \underline{(sz)} ⊕ ∀\underline{m}: ∀\underline{n}: \underline{n} = 0 ⊢ UStelescope(\underline{m}, \underline{n}) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))| ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: ∀\underline{y}: (\underline{x} ∗ \underline{y}) = (\underline{y} ∗ \underline{x}) ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∈ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⊕ ∀\underline{a}: \underline{a} ⊢ \underline{a} ⊕ ∀\underline{m}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ \underline{m} ∈ N ⊕ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊕ ∀\underline{m}: UB(\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals) ⊢ us[(\underline{m} + 1)] = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(s1)} ∈ \underline{(sy)} ⊢ \underline{(sx)} ∈ P(\underline{(sy)}) ⊕ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x} ⊕ ∀\underline{s}: ∀\underline{x}: ˙¬ (\underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ (∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}))n)n ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} = \underline{(fy)} ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n)n ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fxs)}: UB(\underline{(fy)}, \underline{(fxs)}) ⊢ \underline{(fx)} ∈ \underline{(fxs)} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fy)} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊕ us[0] = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: \underline{x} <= \underline{x} ⊕ ∀\underline{s}: ˙¬ (\underline{s} ∈ Ø)n ⊕ ∀\underline{(fx)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)} ⇒ ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\})n)n ⊕ ∀\underline{m}: ∀\underline{(fx)}: ˙¬ (0 <= \underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{m})n)n)n ⊢ \underline{(fx)}(exp)\underline{m} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] ∗ y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] ∗ y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: (\underline{x} + (−u\underline{x})) = 0 ⊕ ∀\underline{m}: ∀\underline{(fx)}: \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = if(\underline{(fx)}[\underline{m}] = 0, 0, rec\underline{(fx)}[\underline{m}]) ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z} ⊕ ∀\underline{m}: ˙¬ (xs[\underline{m}] <= ysF[\underline{m}] ⇒ ˙¬ ( ˙¬ (xs[\underline{m}] = ysF[\underline{m}])n)n)n ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} ∈ Q ⊢ \underline{y} ∈ Q ⊢ (\underline{x} + \underline{y}) ∈ Q ⊕ ∀\underline{m}: ∀\underline{n}: ˙¬ (0 <= \underline{n} ⇒ ˙¬ ( ˙¬ (0 = \underline{n})n)n)n ⊢ UStelescope(\underline{m}, \underline{n}) = (|(us[(\underline{m} + \underline{n})] + (−uus[(\underline{m} + (\underline{n} + 1))]))| + UStelescope(\underline{m}, (\underline{n} + (−u1)))) ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⇒ (\underline{x} ∗ rec\underline{x}) = 1 ⊕ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⇒ \underline{x} ∈ Q ⇒ rec\underline{x} ∈ Q ⊕ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{b})n ⇒ \underline{a} ⊢ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n ⊢ \underline{b} ⊕ ∀\underline{x}: λc.TypeRational0(⌈\underline{x}⌉) ⊢⊢ \underline{x} ∈ Q ⊕ ∀\underline{(rx)}: \underline{(rx)} = \underline{(rx)} ⊕ ∀\underline{m}: ˙¬ (UB(\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals))n ⊢ us[(\underline{m} + 1)] = us[\underline{m}] ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⊕ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{z}: \underline{p}^{Ph} ∧ ⟨\underline{b}≡\underline{a}|\underline{p}:==\underline{z}⟩_{Ph} ⊢⊢ ˙¬ (\underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ (\underline{z} ∈ \underline{x} ⇒ ˙¬ (\underline{b})n)n ⇒ ˙¬ ( ˙¬ (\underline{z} ∈ \underline{x} ⇒ ˙¬ (\underline{b})n)n ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\})n)n ⊕ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ \underline{y} ∈ \underline{x} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{n}: \underline{m} <= \underline{n} ⊢ ˙¬ (∀_{obj}\underline{(m1)}: ˙¬ ((\underline{m} + \underline{(m1)}) = \underline{n})n)n ⊕ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fep)}: ∀\underline{(fx)}: ∀\underline{(fys)}: Limit(\underline{(fx)}, \underline{(fys)}) ⊢ ∀_{obj}\underline{(fep)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ \underline{m} <= \underline{n} ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ |f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−ux[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n)n ⊕ ∀\underline{m}: ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} ∗ (\underline{y} + \underline{z})) = ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ \underline{z})) ⊕ ∀\underline{m}: ∀\underline{n}: Nat(\underline{m}) ⊢⊢ Nat(\underline{n}) ⊢⊢ ˙¬ (\underline{m} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{m} = (\underline{n} + 1))n)n)n ⊢ \underline{m} <= \underline{n} ⊕ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sz)}\} ⊢ ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sz)} ⊕ ∀\underline{x}: \underline{x} ∈ Q ⇒ (−u\underline{x}) ∈ Q ⊕ ∀\underline{x}: ∀\underline{t}: ∀\underline{a}: ∀\underline{b}: ⟨⌈\underline{a}⌉≡^{0}⌈\underline{b}⌉|⌈\underline{x}⌉:==⌈\underline{t}⌉⟩_{Ex} ⊢⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n ⊕ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{m})n)n)n ⊢ \underline{x}(exp)\underline{m} = (\underline{x} ∗ \underline{x}(exp)(\underline{m} + (−u1))) ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + (\underline{y} + \underline{z})) ⊕ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} = \underline{(sy)} ⊢ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sy)}, \underline{(sy)}\} ⊕ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ⊕ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨\underline{a}≡\underline{b}|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: \underline{b} ⇒ \underline{a} ⊕ ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(fys)}: UB(\underline{(fx)}, \underline{(fys)}) ⊢ UB(\underline{(fz)}, \underline{(fys)}) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fz)} ⊢ LUB(\underline{(fx)}, \underline{(fys)}) ⊕ ∀\underline{m}: ∀\underline{n}: ˙¬ (0 <= \underline{n} ⇒ ˙¬ ( ˙¬ (0 = \underline{n})n)n)n ⊢ BS(\underline{m}, \underline{n}) = (rec(1 + 1)(exp)(\underline{m} + \underline{n}) + BS(\underline{m}, (\underline{n} + (−u1)))) ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = (\underline{x} ∗ (\underline{y} ∗ \underline{z})) ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⊕ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a} ⊕ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (rec\underline{m} <= \underline{x} ⇒ ˙¬ ( ˙¬ (rec\underline{m} = \underline{x})n)n)n)n)n ⊕ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ \underline{(sx)} ∈ \underline{(sx1)} ⊕ ∀\underline{m}: ˙¬ (UB(\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals))n ⊢ xs[(\underline{m} + 1)] = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ ˙¬ (\underline{y} ∈ \underline{x} ⇒ ˙¬ (\underline{b})n)n ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z} ⊕ ∀\underline{s}: ∀\underline{x}: ˙¬ (\underline{s} ∈ ∪\underline{x} ⇒ ˙¬ (\underline{s} ∈ j_{Ex} ⇒ ˙¬ (j_{Ex} ∈ \underline{x})n)n ⇒ ˙¬ ( ˙¬ (\underline{s} ∈ j_{Ex} ⇒ ˙¬ (j_{Ex} ∈ \underline{x})n)n ⇒ \underline{s} ∈ ∪\underline{x})n)n ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊕ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n]

[SystemQ →^{tex} “SystemQ”]

[SystemQ →^{pyk} “system Q”]

MP

[MP →^{proof} Rule tactic] [MP →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b}]

[MP →^{tex} “MP”]

[MP →^{pyk} “1rule mp”]

Gen

[Gen →^{proof} Rule tactic] [Gen →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a}] [Gen →^{tex} “Gen”] [Gen →^{pyk} “1rule gen”]

Repetition

[Repetition →^{proof} Rule tactic] [Repetition →^{stmt} SystemQ ⊢ ∀\underline{a}: \underline{a} ⊢ \underline{a}] [Repetition →^{tex} “Repetition”] [Repetition →^{pyk} “1rule repetition”]

Neg

[Neg →^{proof} Rule tactic] [Neg →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{b})n ⇒ \underline{a} ⊢ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n ⊢ \underline{b}] [Neg →^{tex} “Neg”] [Neg →^{pyk} “1rule ad absurdum”]

Ded

[Ded →^{proof} Rule tactic] [Ded →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b}] [Ded →^{tex} “Ded”] [Ded →^{pyk} “1rule deduction”]

ExistIntro

[ExistIntro →^{proof} Rule tactic] [ExistIntro →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{t}: ∀\underline{a}: ∀\underline{b}: ⟨⌈\underline{a}⌉≡^{0}⌈\underline{b}⌉|⌈\underline{x}⌉:==⌈\underline{t}⌉⟩_{Ex} ⊢⊢ \underline{a} ⊢ \underline{b}] [ExistIntro →^{tex} “ExistIntro”] [ExistIntro →^{pyk} “1rule exist intro”]

Extensionality

[Extensionality →^{proof} Rule tactic] [Extensionality →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x}==\underline{y} ⇒ ∀_{obj}\overline{s}: ˙¬ (\overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ (\overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x})n)n ⇒ ˙¬ (∀_{obj}\overline{s}: ˙¬ (\overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ (\overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x})n)n ⇒ \underline{x}==\underline{y})n)n] [Extensionality →^{tex} “Extensionality”] [Extensionality →^{pyk} “axiom extensionality”]

Ødef

[Ødef →^{proof} Rule tactic] [Ødef →^{stmt} SystemQ ⊢ ∀\underline{s}: ˙¬ (\underline{s} ∈ Ø)n] [Ødef →^{tex} “\O{}def”] [Ødef →^{pyk} “axiom empty set”]

PairDef

[PairDef →^{proof} Rule tactic] [PairDef →^{stmt} SystemQ ⊢ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ (\underline{s}==\underline{x})n ⇒ \underline{s}==\underline{y} ⇒ ˙¬ ( ˙¬ (\underline{s}==\underline{x})n ⇒ \underline{s}==\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\})n)n] [PairDef →^{tex} “PairDef”] [PairDef →^{pyk} “axiom pair definition”]

UnionDef

[UnionDef →^{proof} Rule tactic]

[UnionDef →^{stmt} SystemQ ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ (\underline{s} ∈ ∪\underline{x} ⇒ ˙¬ (\underline{s} ∈ j_{Ex} ⇒ ˙¬ (j_{Ex} ∈ \underline{x})n)n ⇒ ˙¬ ( ˙¬ (\underline{s} ∈ j_{Ex} ⇒ ˙¬ (j_{Ex} ∈ \underline{x})n)n ⇒ \underline{s} ∈ ∪\underline{x})n)n]

[UnionDef →^{tex} “UnionDef”]

[UnionDef →^{pyk} “axiom union definition”]

PowerDef

[PowerDef →^{proof} Rule tactic]

[PowerDef →^{stmt} SystemQ ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ (\underline{s} ∈ P(\underline{x}) ⇒ ∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ ˙¬ (∀_{obj}\overline{s}: \overline{s} ∈ \underline{s} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{s} ∈ P(\underline{x}))n)n]

[PowerDef →^{tex} “PowerDef”]

[PowerDef →^{pyk} “axiom power definition”]

SeparationDef

[SeparationDef →^{proof} Rule tactic]

[SeparationDef →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{p}: ∀\underline{x}: ∀\underline{z}: \underline{p}^{Ph} ∧ ⟨\underline{b}≡\underline{a}|\underline{p}:==\underline{z}⟩_{Ph} ⊢⊢ ˙¬ (\underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⇒ ˙¬ (\underline{z} ∈ \underline{x} ⇒ ˙¬ (\underline{b})n)n ⇒ ˙¬ ( ˙¬ (\underline{z} ∈ \underline{x} ⇒ ˙¬ (\underline{b})n)n ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\})n)n]

[SeparationDef →^{tex} “SeparationDef”]

[SeparationDef →^{pyk} “axiom separation definition”]

AddDoubleNeg

[AddDoubleNeg →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ˙¬ ( ˙¬ ( ˙¬ (\underline{a})n)n)n ⊢ RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{a})n)n)n ≫ ˙¬ (\underline{a})n; ∀\underline{a}: Ded⊳∀\underline{a}: ˙¬ ( ˙¬ ( ˙¬ (\underline{a})n)n)n ⊢ ˙¬ (\underline{a})n ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ ˙¬ (\underline{a})n; \underline{a} ⊢ Weakening ⊳ \underline{a} ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ \underline{a}; Neg⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ \underline{a}⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ ˙¬ (\underline{a})n ≫ ˙¬ ( ˙¬ (\underline{a})n)n⌉, p_{0}, c)]

[AddDoubleNeg →^{stmt} SystemQ ⊢ ∀\underline{a}: \underline{a} ⊢ ˙¬ ( ˙¬ (\underline{a})n)n] [AddDoubleNeg →^{tex} “AddDoubleNeg”]

[AddDoubleNeg →^{pyk} “prop lemma add double neg”]

RemoveDoubleNeg

[RemoveDoubleNeg →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ˙¬ ( ˙¬ (\underline{a})n)n ⊢ Weakening ⊳ ˙¬ ( ˙¬ (\underline{a})n)n ≫ ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{a})n)n; AutoImply ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n; Neg ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{a})n)n ≫ \underline{a}⌉, p_{0}, c)]

[RemoveDoubleNeg →^{stmt} SystemQ ⊢ ∀\underline{a}: ˙¬ ( ˙¬ (\underline{a})n)n ⊢ \underline{a}]

[RemoveDoubleNeg →^{tex} “RemoveDoubleNeg”]

[RemoveDoubleNeg →^{pyk} “prop lemma remove double neg”]

AndCommutativity

[AndCommutativity →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ ˙¬ (\underline{a})n ⊢ \underline{a} ⊢ AddDoubleNeg ⊳ \underline{a} ≫ ˙¬ ( ˙¬ (\underline{a})n)n; MT ⊳ \underline{b} ⇒ ˙¬ (\underline{a})n ⊳ ˙¬ ( ˙¬ (\underline{a})n)n ≫ ˙¬ (\underline{b})n; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ ˙¬ (\underline{a})n ⊢ \underline{a} ⊢ ˙¬ (\underline{b})n ≫ \underline{b} ⇒ ˙¬ (\underline{a})n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n; ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⊢ Repetition ≫ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n; MT ⊳ \underline{b} ⇒ ˙¬ (\underline{a})n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n ⊳ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ≫ ˙¬ (\underline{b} ⇒ ˙¬ (\underline{a})n)n; Repetition ⊳ ˙¬ (\underline{b} ⇒ ˙¬ (\underline{a})n)n ≫ ˙¬ (\underline{b} ⇒ ˙¬ (\underline{a})n)n⌉, p_{0}, c)]

[AndCommutativity →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⊢ ˙¬ (\underline{b} ⇒ ˙¬ (\underline{a})n)n]

[AndCommutativity →^{tex} “AndCommutativity”]

[AndCommutativity →^{pyk} “prop lemma and commutativity”]

AutoImply

[AutoImply →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: \underline{a} ⊢ Repetition ⊳ \underline{a} ≫ \underline{a}; ∀\underline{a}: Ded ⊳ ∀\underline{a}: \underline{a} ⊢ \underline{a} ≫ \underline{a} ⇒ \underline{a}⌉, p_{0}, c)]

[AutoImply →^{stmt} SystemQ ⊢ ∀\underline{a}: \underline{a} ⇒ \underline{a}]

[AutoImply →^{tex} “AutoImply”]

[AutoImply →^{pyk} “prop lemma auto imply”]

Contrapositive

[Contrapositive →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ MT ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a})n; ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ (\underline{a})n ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n; MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n ⊳ \underline{a} ⇒ \underline{b} ≫ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n⌉, p_{0}, c)]

[Contrapositive →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n]

[Contrapositive →^{tex} “Contrapositive”]

[Contrapositive →^{pyk} “prop lemma contrapositive”]

FirstConjunct

[FirstConjunct →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⊢ AndCommutativity ⊳ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ≫ ˙¬ (\underline{b} ⇒ ˙¬ (\underline{a})n)n; SecondConjunct ⊳ ˙¬ (\underline{b} ⇒ ˙¬ (\underline{a})n)n ≫ \underline{a}⌉, p_{0}, c)]

[FirstConjunct →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⊢ \underline{a}]

[FirstConjunct →^{tex} “FirstConjunct”]

[FirstConjunct →^{pyk} “prop lemma first conjunct”]

SecondConjunct

[SecondConjunct →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{b})n ⊢ Weakening ⊳ ˙¬ (\underline{b})n ≫ \underline{a} ⇒ ˙¬ (\underline{b})n; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{b})n ⊢ \underline{a} ⇒ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{b})n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n; ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⊢ Repetition⊳ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ≫ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n; NegativeMT ⊳ ˙¬ (\underline{b})n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n ⊳ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ≫ \underline{b}⌉, p_{0}, c)]

[SecondConjunct →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⊢ \underline{b}]

[SecondConjunct →^{tex} “SecondConjunct”]

[SecondConjunct →^{pyk} “prop lemma second conjunct”]

FromContradiction

[FromContradiction →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{a})n ⊢ Weakening ⊳ \underline{a} ≫ ˙¬ (\underline{b})n ⇒ \underline{a}; Weakening ⊳ ˙¬ (\underline{a})n ≫ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n; Neg ⊳ ˙¬ (\underline{b})n ⇒ \underline{a} ⊳ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n ≫ \underline{b}⌉, p_{0}, c)]

[FromContradiction →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{a})n ⊢ \underline{b}] [FromContradiction →^{tex} “FromContradiction”]

[FromContradiction →^{pyk} “prop lemma from contradiction”]

FromDisjuncts

[FromDisjuncts →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{c} ⊢ Repetition ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ≫ ˙¬ (\underline{a})n ⇒ \underline{b}; Contrapositive ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ≫ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{a})n)n; Technicality ⊳ \underline{a} ⇒ \underline{c} ≫ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{c}; ImplyTransitivity ⊳ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{a})n)n ⊳ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{c} ≫ ˙¬ (\underline{b})n ⇒ \underline{c}; Contrapositive ⊳ ˙¬ (\underline{b})n ⇒ \underline{c} ≫ ˙¬ (\underline{c})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n; Contrapositive ⊳ \underline{b} ⇒ \underline{c} ≫ ˙¬ (\underline{c})n ⇒ ˙¬ (\underline{b})n; Neg ⊳ ˙¬ (\underline{c})n ⇒ ˙¬ (\underline{b})n ⊳ ˙¬ (\underline{c})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ≫ \underline{c}⌉, p_{0}, c)]

[FromDisjuncts →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{c}]

[FromDisjuncts →^{tex} “FromDisjuncts”]

[FromDisjuncts →^{pyk} “prop lemma from disjuncts”]

IffCommutativity

[IffCommutativity →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ⊢ Repetition ⊳ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ≫ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n; AndCommutativity ⊳ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ≫ ˙¬ (\underline{b} ⇒ \underline{a} ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n)n; Repetition ⊳ ˙¬ (\underline{b} ⇒ \underline{a} ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n)n ≫ ˙¬ (\underline{b} ⇒ \underline{a} ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n)n⌉, p_{0}, c)]

[IffCommutativity →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ⊢ ˙¬ (\underline{b} ⇒ \underline{a} ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n)n]

[IffCommutativity →^{tex} “IffCommutativity”]

[IffCommutativity →^{pyk} “prop lemma iff commutativity”]

IffFirst

[IffFirst →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ⊢ \underline{b} ⊢ SecondConjunct ⊳ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ≫ \underline{b} ⇒ \underline{a}; MP ⊳ \underline{b} ⇒ \underline{a} ⊳ \underline{b} ≫ \underline{a}⌉, p_{0}, c)]

[IffFirst →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ⊢ \underline{b} ⊢ \underline{a}]

[IffFirst →^{tex} “IffFirst”]

[IffFirst →^{pyk} “prop lemma iff first”]

IffSecond

[IffSecond →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ⊢ \underline{a} ⊢ FirstConjunct⊳ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ≫ \underline{a} ⇒ \underline{b}; MP⊳\underline{a} ⇒ \underline{b}⊳\underline{a} ≫ \underline{b}⌉, p_{0}, c)]

[IffSecond →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{b} ⇒ \underline{a})n)n ⊢ \underline{a} ⊢ \underline{b}]

[IffSecond →^{tex} “IffSecond”]

[IffSecond →^{pyk} “prop lemma iff second”]

ImplyTransitivity

[ImplyTransitivity →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; MP ⊳ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ≫ \underline{c}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{c} ≫ \underline{a} ⇒ \underline{b} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{c}; MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{c} ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{b} ⇒ \underline{c} ≫ \underline{a} ⇒ \underline{c}⌉, p_{0}, c)]

[ImplyTransitivity →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⊢ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{c}]

[ImplyTransitivity →^{tex} “ImplyTransitivity”]

[ImplyTransitivity →^{pyk} “prop lemma imply transitivity”]

JoinConjuncts

[JoinConjuncts →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{a} ⇒ ˙¬ (\underline{b})n ⊢ MP ⊳ \underline{a} ⇒ ˙¬ (\underline{b})n ⊳ \underline{a} ≫ ˙¬ (\underline{b})n; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{a} ⇒ ˙¬ (\underline{b})n ⊢ ˙¬ (\underline{b})n ≫ \underline{a} ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{b})n; \underline{a} ⊢ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{b})n ⊳ \underline{a} ≫ \underline{a} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{b})n; AddDoubleNeg ⊳ \underline{b} ≫ ˙¬ ( ˙¬ (\underline{b})n)n; MT ⊳ \underline{a} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{b})n ⊳ ˙¬ ( ˙¬ (\underline{b})n)n ≫ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n; Repetition ⊳ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ≫ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n⌉, p_{0}, c)]

[JoinConjuncts →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b} ⊢ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n]

[JoinConjuncts →^{tex} “JoinConjuncts”]

[JoinConjuncts →^{pyk} “prop lemma join conjuncts”]

MP2

[MP2 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊳ \underline{a} ≫ \underline{b} ⇒ \underline{c}; MP ⊳ \underline{b} ⇒ \underline{c} ⊳ \underline{b} ≫ \underline{c}⌉, p_{0}, c)]

[MP2 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c}]

[MP2 →^{tex} “MP2”]

[MP2 →^{pyk} “prop lemma mp2”]

MP3

[MP3 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{d}; MP ⊳ \underline{c} ⇒ \underline{d} ⊳ \underline{c} ≫ \underline{d}⌉, p_{0}, c)]

[MP3 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d}]

[MP3 →^{tex} “MP3”]

[MP3 →^{pyk} “prop lemma mp3”]

MP4

[MP4 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{d} ⇒ \underline{e}; MP2 ⊳ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{c} ⊳ \underline{d} ≫ \underline{e}⌉, p_{0}, c)]

[MP4 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e}]

[MP4 →^{tex} “MP4”]

[MP4 →^{pyk} “prop lemma mp4”]

MP5

[MP5 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ∀\underline{f}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e} ⊢ MP3 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊳ \underline{a} ⊳ \underline{b} ⊳ \underline{c} ≫ \underline{d} ⇒ \underline{e} ⇒ \underline{f}; MP2 ⊳ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊳ \underline{d} ⊳ \underline{e} ≫ \underline{f}⌉, p_{0}, c)]

[MP5 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ∀\underline{f}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{f} ⊢ \underline{a} ⊢ \underline{b} ⊢ \underline{c} ⊢ \underline{d} ⊢ \underline{e} ⊢ \underline{f}]

[MP5 →^{tex} “MP5”]

[MP5 →^{pyk} “prop lemma mp5”]

MT

[MT →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ Technicality ≫ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{b}; NegativeMT ⊳ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a})n⌉, p_{0}, c)] [MT →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ (\underline{a})n]

[MT →^{tex} “MT”]

[MT →^{pyk} “prop lemma mt”]

NegativeMT

[NegativeMT →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ Weakening ⊳ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n; Neg ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ≫ \underline{a}⌉, p_{0}, c)] [NegativeMT →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ \underline{a}] [NegativeMT →^{tex} “NegativeMT”] [NegativeMT →^{pyk} “prop lemma negative mt”]

Technicality

[Technicality →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ( ˙¬ (\underline{a})n)n ⊢ RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{a})n)n ≫ \underline{a}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ( ˙¬ (\underline{a})n)n ⊢ \underline{b} ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{b}; \underline{a} ⇒ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{b} ⊳ \underline{a} ⇒ \underline{b} ≫ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{b}⌉, p_{0}, c)] [Technicality →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ ( ˙¬ (\underline{a})n)n ⇒ \underline{b}] [Technicality →^{tex} “Technicality”] [Technicality →^{pyk} “prop lemma technicality”]

Weakening

[Weakening →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⊢ Repetition ⊳ \underline{b} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⊢ \underline{b} ≫ \underline{b} ⇒ \underline{a} ⇒ \underline{b}; \underline{b} ⊢ MP ⊳ \underline{b} ⇒ \underline{a} ⇒ \underline{b} ⊳ \underline{b} ≫ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)] [Weakening →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⇒ \underline{b}] [Weakening →^{tex} “Weakening”] [Weakening →^{pyk} “prop lemma weakening”]

WeakenOr1

[WeakenOr1 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ Weakening ⊳ \underline{b} ≫ ˙¬ (\underline{a})n ⇒ \underline{b}; Repetition ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ≫ ˙¬ (\underline{a})n ⇒ \underline{b}⌉, p_{0}, c)]

[WeakenOr1 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ ˙¬ (\underline{a})n ⇒ \underline{b}]

[WeakenOr1 →^{tex} “WeakenOr1”]

[WeakenOr1 →^{pyk} “prop lemma weaken or first”]

WeakenOr2

[WeakenOr2 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{a})n ⊢ FromContradiction ⊳ \underline{a} ⊳ ˙¬ (\underline{a})n ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{a})n ⊢ \underline{b} ≫ \underline{a} ⇒ ˙¬ (\underline{a})n ⇒ \underline{b}; \underline{a} ⊢ MP ⊳ \underline{a} ⇒ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ \underline{a} ≫ ˙¬ (\underline{a})n ⇒ \underline{b}; Repetition ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ≫ ˙¬ (\underline{a})n ⇒ \underline{b}⌉, p_{0}, c)]

[WeakenOr2 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{a})n ⇒ \underline{b}]

[WeakenOr2 →^{tex} “WeakenOr2”]

[WeakenOr2 →^{pyk} “prop lemma weaken or second”]

Formula2Pair

[Formula2Pair →^{proof} Rule tactic]

[Formula2Pair →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sz)} ⊢ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sz)}\}]

[Formula2Pair →^{tex} “Formula2Pair”]

[Formula2Pair →^{pyk} “lemma formula2pair”]

Pair2Formula

[Pair2Formula →^{proof} Rule tactic]

[Pair2Formula →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sz)}\} ⊢ ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sz)}]

[Pair2Formula →^{tex} “Pair2Formula”]

[Pair2Formula →^{pyk} “lemma pair2formula”]

Formula2Union

[Formula2Union →^{proof} Rule tactic] [Formula2Union →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ˙¬ (∀_{obj}\underline{(sy)}: ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sz)})n)n)n)n ⊢ \underline{(sx)} ∈ Union(\underline{(sz)})] [Formula2Union →^{tex} “Formula2Union”] [Formula2Union →^{pyk} “lemma formula2union”]

Union2Formula

[Union2Formula →^{tex} “Union2Formula”] [Union2Formula →^{pyk} “lemma union2formula”]

Formula2Sep

[Formula2Sep →^{proof} Rule tactic] [Formula2Sep →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \underline{x} ⊢ \underline{b} ⊢ \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\}] [Formula2Sep →^{tex} “Formula2Sep”] [Formula2Sep →^{pyk} “lemma formula2separation”]

Sep2Formula

[Sep2Formula →^{proof} Rule tactic] [Sep2Formula →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ ˙¬ (\underline{y} ∈ \underline{x} ⇒ ˙¬ (\underline{b})n)n] [Sep2Formula →^{tex} “Sep2Formula”] [Sep2Formula →^{pyk} “lemma separation2formula”]

Formula2Power

[Formula2Power →^{proof} Rule tactic] [Formula2Power →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(s1)} ∈ \underline{(sy)} ⊢ \underline{(sx)} ∈ P(\underline{(sy)})] [Formula2Power →^{tex} “Formula2Power”] [Formula2Power →^{pyk} “lemma formula2power”]

SubsetInPower

[SubsetInPower →^{tex} “SubsetInPower”] [SubsetInPower →^{pyk} “lemma subset in power set”]

HelperPowerIsSub

[HelperPowerIsSub →^{tex} “HelperPowerIsSub”] [HelperPowerIsSub →^{pyk} “lemma power set is subset0”]

PowerIsSub

[PowerIsSub →^{tex} “PowerIsSub”] [PowerIsSub →^{pyk} “lemma power set is subset”]

(Switch)HelperPowerIsSub

[(Switch)HelperPowerIsSub →^{tex} “(Switch)HelperPowerIsSub”] [(Switch)HelperPowerIsSub →^{pyk} “lemma power set is subset0-switch”]

(Switch)PowerIsSub

[(Switch)PowerIsSub →^{tex} “(Switch)PowerIsSub”] [(Switch)PowerIsSub →^{pyk} “lemma power set is subset-switch”]

ToSetEquality

[ToSetEquality →^{proof} Rule tactic] [ToSetEquality →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)} ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fy)} ⇒ \overline{(s1)} ∈ \underline{(fx)} ⊢ \underline{(fx)} = \underline{(fy)}] [ToSetEquality →^{tex} “ToSetEquality”] [ToSetEquality →^{pyk} “lemma set equality suff condition”]

HelperToSetEquality(t)

[HelperToSetEquality(t) →^{tex} “HelperToSetEquality(t)”] [HelperToSetEquality(t) →^{pyk} “lemma set equality suff condition(t)0”]

ToSetEquality(t)

[ToSetEquality(t) →^{tex} “ToSetEquality(t)”] [ToSetEquality(t) →^{pyk} “lemma set equality suff condition(t)”]

HelperFromSetEquality

[HelperFromSetEquality →^{tex} “HelperFromSetEquality”] [HelperFromSetEquality →^{pyk} “lemma set equality skip quantifier”]

FromSetEquality

[FromSetEquality →^{tex} “FromSetEquality”] [FromSetEquality →^{pyk} “lemma set equality nec condition”]

HelperReflexivity

[HelperReflexivity →^{tex} “HelperReflexivity”] [HelperReflexivity →^{pyk} “lemma reflexivity0”]

Reflexivity

[Reflexivity →^{tex} “Reflexivity”] [Reflexivity →^{pyk} “lemma reflexivity”]

HelperSymmetry

[HelperSymmetry →^{tex} “HelperSymmetry”] [HelperSymmetry →^{pyk} “lemma symmetry0”]

Symmetry

[Symmetry →^{tex} “Symmetry”] [Symmetry →^{pyk} “lemma symmetry”]

HelperTransitivity

[HelperTransitivity →^{tex} “HelperTransitivity”] [HelperTransitivity →^{pyk} “lemma transitivity0”]

Transitivity

[Transitivity →^{tex} “Transitivity”] [Transitivity →^{pyk} “lemma transitivity”]

ERisReflexive

[ERisReflexive →^{tex} “ERisReflexive”] [ERisReflexive →^{pyk} “lemma er is reflexive”]

ERisSymmetric

[ERisSymmetric →^{tex} “ERisSymmetric”] [ERisSymmetric →^{pyk} “lemma er is symmetric”]

ERisTransitive

[ERisTransitive →^{tex} “ERisTransitive”] [ERisTransitive →^{pyk} “lemma er is transitive”]

ØisSubset

[ØisSubset →^{tex} “\O{}isSubset”] [ØisSubset →^{pyk} “lemma empty set is subset”]

HelperMemberNotØ

[HelperMemberNotØ →^{tex} “HelperMemberNot\O{}”] [HelperMemberNotØ →^{pyk} “lemma member not empty0”]

MemberNotØ

[MemberNotØ →^{tex} “MemberNot\O{}”] [MemberNotØ →^{pyk} “lemma member not empty”]

HelperUniqueØ

[HelperUniqueØ →^{tex} “HelperUnique\O{}”] [HelperUniqueØ →^{pyk} “lemma unique empty set0”]

UniqueØ

[UniqueØ →^{tex} “Unique\O{}”] [UniqueØ →^{pyk} “lemma unique empty set”]

==Reflexivity

[==Reflexivity →^{proof} Rule tactic] [==Reflexivity →^{stmt} SystemQ ⊢ ∀\underline{(rx)}: \underline{(rx)} = \underline{(rx)}] [==Reflexivity →^{tex} “==\!{}Reflexivity”] [==Reflexivity →^{pyk} “lemma ==Reflexivity”]

==Symmetry

[==Symmetry →^{proof} Rule tactic] [==Symmetry →^{stmt} SystemQ ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(ry)} = \underline{(rx)}] [==Symmetry →^{tex} “==\!{}Symmetry”] [==Symmetry →^{pyk} “lemma ==Symmetry”]

Helper==Transitivity

[Helper==Transitivity →^{tex} “Helper\!{}==\!{}Transitivity”] [Helper==Transitivity →^{pyk} “lemma ==Transitivity0”]

==Transitivity

[==Transitivity →^{proof} Rule tactic] [==Transitivity →^{stmt} SystemQ ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(ry)} = \underline{(rz)} ⊢ \underline{(rx)} = \underline{(rz)}] [==Transitivity →^{tex} “\!{}==\!{}Transitivity”] [==Transitivity →^{pyk} “lemma ==Transitivity”]

HelperTransferNotEq

[HelperTransferNotEq →^{tex} “HelperTransferNotEq”] [HelperTransferNotEq →^{pyk} “lemma transfer ˜ is0”]

TransferNotEq

[TransferNotEq →^{tex} “TransferNotEq”] [TransferNotEq →^{pyk} “lemma transfer ˜ is”]

HelperPairSubset

[HelperPairSubset →^{tex} “HelperPairSubset”] [HelperPairSubset →^{pyk} “lemma pair subset0”]

Helper(2)PairSubset

[Helper(2)PairSubset →^{tex} “Helper(2)PairSubset”] [Helper(2)PairSubset →^{pyk} “lemma pair subset1”]

PairSubset

[PairSubset →^{tex} “PairSubset”] [PairSubset →^{pyk} “lemma pair subset”]

SamePair

[SamePair →^{proof} Rule tactic] [SamePair →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx)} = \underline{(sx1)} ⊢ \underline{(sy)} = \underline{(sy1)} ⊢ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx1)}, \underline{(sy1)}\}] [SamePair →^{tex} “SamePair”] [SamePair →^{pyk} “lemma same pair”]

SameSingleton

[SameSingleton →^{proof} Rule tactic] [SameSingleton →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} = \underline{(sy)} ⊢ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sy)}, \underline{(sy)}\}] [SameSingleton →^{tex} “SameSingleton”] [SameSingleton →^{pyk} “lemma same singleton”]

UnionSubset

[UnionSubset →^{tex} “UnionSubset”] [UnionSubset →^{pyk} “lemma union subset”]

SameUnion

[SameUnion →^{tex} “SameUnion”] [SameUnion →^{pyk} “lemma same union”]

SeparationSubset

[SeparationSubset →^{tex} “SeparationSubset”] [SeparationSubset →^{pyk} “lemma separation subset”]

SameSeparation

[SameSeparation →^{tex} “SameSeparation”] [SameSeparation →^{pyk} “lemma same separation”]

SameBinaryUnion

[SameBinaryUnion →^{tex} “SameBinaryUnion”] [SameBinaryUnion →^{pyk} “lemma same binary union”]

IntersectionSubset

[IntersectionSubset →^{tex} “IntersectionSubset”] [IntersectionSubset →^{pyk} “lemma intersection subset”]

SameIntersection

[SameIntersection →^{tex} “SameIntersection”] [SameIntersection →^{pyk} “lemma same intersection”]

AutoMember

[AutoMember →^{tex} “AutoMember”] [AutoMember →^{pyk} “lemma auto member”]

HelperEqSysNotØ

[HelperEqSysNotØ →^{tex} “HelperEqSysNot\O{}”] [HelperEqSysNotØ →^{pyk} “lemma eq-system not empty0”]

EqSysNotØ

[EqSysNotØ →^{tex} “EqSysNot\O{}”] [EqSysNotØ →^{pyk} “lemma eq-system not empty”]

HelperEqSubset

[HelperEqSubset →^{tex} “HelperEqSubset”] [HelperEqSubset →^{pyk} “lemma eq subset0”]

EqSubset

[EqSubset →^{tex} “EqSubset”] [EqSubset →^{pyk} “lemma eq subset”]

HelperEqNecessary

[HelperEqNecessary →^{tex} “HelperEqNecessary”] [HelperEqNecessary →^{pyk} “lemma equivalence nec condition0”]

EqNecessary

[EqNecessary →^{tex} “EqNecessary”] [EqNecessary →^{pyk} “lemma equivalence nec condition”]

HelperNoneEqNecessary

[HelperNoneEqNecessary →^{tex} “HelperNoneEqNecessary”] [HelperNoneEqNecessary →^{pyk} “lemma none-equivalence nec condition0”]

Helper(2)NoneEqNecessary

[Helper(2)NoneEqNecessary →^{tex} “Helper(2)NoneEqNecessary”] [Helper(2)NoneEqNecessary →^{pyk} “lemma none-equivalence nec condition1”]

NoneEqNecessary

[NoneEqNecessary →^{tex} “NoneEqNecessary”] [NoneEqNecessary →^{pyk} “lemma none-equivalence nec condition”]

EqClassIsSubset

[EqClassIsSubset →^{tex} “EqClassIsSubset”] [EqClassIsSubset →^{pyk} “lemma equivalence class is subset”]

EqClassesAreDisjoint

[EqClassesAreDisjoint →^{tex} “EqClassesAreDisjoint”] [EqClassesAreDisjoint →^{pyk} “lemma equivalence classes are disjoint”]

AllDisjoint

[AllDisjoint →^{tex} “AllDisjoint”] [AllDisjoint →^{pyk} “lemma all disjoint”]

AllDisjointImply

[AllDisjointImply →^{tex} “AllDisjointImply”] [AllDisjointImply →^{pyk} “lemma all disjoint-imply”]

BSsubset

[BSsubset →^{tex} “BSsubset”] [BSsubset →^{pyk} “lemma bs subset union(bs/r)”]

Union(BS/R)subset

[Union(BS/R)subset →^{tex} “Union(BS/R)subset”] [Union(BS/R)subset →^{pyk} “lemma union(bs/r) subset bs”]

UnionIdentity

[UnionIdentity →^{tex} “UnionIdentity”] [UnionIdentity →^{pyk} “lemma union(bs/r) is bs”]

EqSysIsPartition

[EqSysIsPartition →^{tex} “EqSysIsPartition”] [EqSysIsPartition →^{pyk} “theorem eq-system is partition”]

(x1)

[(x1) →^{tex} “(x1)”]

[(x1) →^{pyk} “var x1”]

(x2)

[(x2) →^{tex} “(x2)”]

[(x2) →^{pyk} “var x2”]

(y1)

[(y1) →^{tex} “(y1)”]

[(y1) →^{pyk} “var y1”]

(y2)

[(y2) →^{tex} “(y2)”]

[(y2) →^{pyk} “var y2”]

(v1)

[(v1) →^{tex} “(v1)”]

[(v1) →^{pyk} “var v1”]

(v2)

[(v2) →^{tex} “(v2)”]

[(v2) →^{pyk} “var v2”]

(v3)

[(v3) →^{tex} “(v3)”]

[(v3) →^{pyk} “var v3”]

(v4)

[(v4) →^{tex} “(v4)”]

[(v4) →^{pyk} “var v4”]

(v2n)

[(v2n) →^{tex} “(v2n)”]

[(v2n) →^{pyk} “var v2n”]

(m1)

[(m1) →^{tex} “(m1)”]

[(m1) →^{pyk} “var m1”]

(m2)

[(m2) →^{tex} “(m2)”]

[(m2) →^{pyk} “var m2”]

(n1)

[(n1) →^{tex} “(n1)”]

[(n1) →^{pyk} “var n1”]

(n2)

[(n2) →^{tex} “(n2)”]

[(n2) →^{pyk} “var n2”]

(n3)

[(n3) →^{tex} “(n3)”]

[(n3) →^{pyk} “var n3”]

(ϵ)

[(ϵ) →^{tex} “(\epsilon)”] [(ϵ) →^{pyk} “var ep”]

(ϵ)_{1}

[(ϵ)_{1} →^{tex} “(\epsilon) {1}”] [(ϵ)_{1} →^{pyk} “var ep1”]

(ϵ2)

[(ϵ2) →^{tex} “(\epsilon 2)”] [(ϵ2) →^{pyk} “var ep2”]

(fep)

[(fep) →^{tex} “(fep)”] [(fep) →^{pyk} “var fep”]

(fx)

[(fx) →^{tex} “(fx)”]

[(fx) →^{pyk} “var fx”]

(fy)

[(fy) →^{tex} “(fy)”]

[(fy) →^{pyk} “var fy”]

(fz)

[(fz) →^{tex} “(fz)”]

[(fz) →^{pyk} “var fz”]

(fu)

[(fu) →^{tex} “(fu)”]

[(fu) →^{pyk} “var fu”]

(fv)

[(fv) →^{tex} “(fv)”]

[(fv) →^{pyk} “var fv”]

(fw)

[(fw) →^{tex} “(fw)”]

[(fw) →^{pyk} “var fw”]

(rx)

[(rx) →^{tex} “(rx)”]

[(rx) →^{pyk} “var rx”]

(ry)

[(ry) →^{tex} “(ry)”]

[(ry) →^{pyk} “var ry”]

(rz)

[(rz) →^{tex} “(rz)”]

[(rz) →^{pyk} “var rz”]

(ru)

[(ru) →^{tex} “(ru)”]

[(ru) →^{pyk} “var ru”]

(sx)

[(sx) →^{tex} “(sx)”]

[(sx) →^{pyk} “var sx”]

(sx1)

[(sx1) →^{tex} “(sx1)”]

[(sx1) →^{pyk} “var sx1”]

(sy)

[(sy) →^{tex} “(sy)”]

[(sy) →^{pyk} “var sy”]

(sy1)

[(sy1) →^{tex} “(sy1)”]

[(sy1) →^{pyk} “var sy1”]

(sz)

[(sz) →^{tex} “(sz)”]

[(sz) →^{pyk} “var sz”]

(sz1)

[(sz1) →^{tex} “(sz1)”]

[(sz1) →^{pyk} “var sz1”]

(su)

[(su) →^{tex} “(su)”]

[(su) →^{pyk} “var su”]

(su1)

[(su1) →^{tex} “(su1)”]

[(su1) →^{pyk} “var su1”]

(fxs)

[(fxs) →^{tex} “(fxs)”] [(fxs) →^{pyk} “var fxs”]

(fys)

[(fys) →^{tex} “(fys)”] [(fys) →^{pyk} “var fys”]

(crs1)

[(crs1) →^{tex} “(crs1)”] [(crs1) →^{pyk} “var crs1”]

(f1)

[(f1) →^{tex} “(f1)”]

[(f1) →^{pyk} “var f1”]

(f2)

[(f2) →^{tex} “(f2)”]

[(f2) →^{pyk} “var f2”]

(f3)

[(f3) →^{tex} “(f3)”]

[(f3) →^{pyk} “var f3”]

(f4)

[(f4) →^{tex} “(f4)”]

[(f4) →^{pyk} “var f4”]

(op1)

[(op1) →^{tex} “(op1)”]

[(op1) →^{pyk} “var op1”]

(op2)

[(op2) →^{tex} “(op2)”]

[(op2) →^{pyk} “var op2”]

(r1)

[(r1) →^{tex} “(r1)”]

[(r1) →^{pyk} “var r1”]

(s1)

[(s1) →^{tex} “(s1)”]

[(s1) →^{pyk} “var s1”]

(s2)

[(s2) →^{tex} “(s2)”]

[(s2) →^{pyk} “var s2”]

X_{1}

[X_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[X_{1} ¨= \underline{(x1)}]⌉)]

[X_{1} →^{tex} “X {1}”]

[X_{1} →^{pyk} “meta x1”]

X_{2}

[X_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[X_{2} ¨= \underline{(x2)}]⌉)]

[X_{2} →^{tex} “X {2}”]

[X_{2} →^{pyk} “meta x2”]

Y_{1}

[Y_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Y_{1} ¨= \underline{(y1)}]⌉)]

[Y_{1} →^{tex} “Y {1}”]

[Y_{1} →^{pyk} “meta y1”]

Y_{2}

[Y_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Y_{2} ¨= \underline{(y2)}]⌉)]

[Y_{2} →^{tex} “Y {2}”]

[Y_{2} →^{pyk} “meta y2”]

V_{1}

[V_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[V_{1} ¨= \underline{(v1)}]⌉)]

[V_{1} →^{tex} “V {1}”]

[V_{1} →^{pyk} “meta v1”]

V_{2}

[V_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[V_{2} ¨= \underline{(v2)}]⌉)]

[V_{2} →^{tex} “V {2}”]

[V_{2} →^{pyk} “meta v2”]

V_{3}

[V_{3} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[V_{3} ¨= \underline{(v3)}]⌉)]

[V_{3} →^{tex} “V {3}”]

[V_{3} →^{pyk} “meta v3”]

V_{4}

[V_{4} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[V_{4} ¨= \underline{(v4)}]⌉)]

[V_{4} →^{tex} “V {4}”]

[V_{4} →^{pyk} “meta v4”]

V_{2n}

[V_{2n} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[V_{2n} ¨= \underline{(v2n)}]⌉)]

[V_{2n} →^{tex} “V {2n}”]

[V_{2n} →^{pyk} “meta v2n”]

M_{1}

[M_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[M_{1} ¨= \underline{(m1)}]⌉)]

[M_{1} →^{tex} “M {1}”]

[M_{1} →^{pyk} “meta m1”]

M_{2}

[M_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[M_{2} ¨= \underline{(m2)}]⌉)]

[M_{2} →^{tex} “M {2}”]

[M_{2} →^{pyk} “meta m2”]

N_{1}

[N_{1} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[N_{1} ¨= \underline{(n1)}]⌉)]

[N_{1} →^{tex} “N {1} ”]

[N_{1} →^{pyk} “meta n1”]

N_{2}

[N_{2} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[N_{2} ¨= \underline{(n2)}]⌉)]

[N_{2} →^{tex} “N {2} ”]

[N_{2} →^{pyk} “meta n2”]

N_{3}

[N_{3} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[N_{3} ¨= \underline{(n3)}]⌉)]

[N_{3} →^{tex} “N {3} ”]

[N_{3} →^{pyk} “meta n3”]

ϵ

[ϵ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ϵ ¨= \underline{(ϵ)}]⌉)]

[ϵ →^{tex} “\epsilon”]

[ϵ →^{pyk} “meta ep”]

ϵ1

[ϵ1 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ϵ1 ¨= \underline{(ϵ)_{1}}]⌉)]

[ϵ1 →^{tex} “\epsilon 1”]

[ϵ1 →^{pyk} “meta ep1”]

ϵ2

[ϵ2 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ϵ2 ¨= \underline{(ϵ2)}]⌉)]

[ϵ2 →^{tex} “\epsilon 2”] [ϵ2 →^{pyk} “meta ep2”]

FX

[FX →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FX ¨= \underline{(fx)}]⌉)]

[FX →^{tex} “FX”]

[FX →^{pyk} “meta fx”]

FY

[FY →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FY ¨= \underline{(fy)}]⌉)]

[FY →^{tex} “FY”]

[FY →^{pyk} “meta fy”]

FZ

[FZ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FZ ¨= \underline{(fz)}]⌉)]

[FZ →^{tex} “FZ”]

[FZ →^{pyk} “meta fz”]

FU

[FU →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FU ¨= \underline{(fu)}]⌉)]

[FU →^{tex} “FU”]

[FU →^{pyk} “meta fu”]

FV

[FV →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FV ¨= \underline{(fv)}]⌉)]

[FV →^{tex} “FV”]

[FV →^{pyk} “meta fv”]

FW

[FW →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FW ¨= \underline{(fw)}]⌉)]

[FW →^{tex} “FW”]

[FW →^{pyk} “meta fw”]

FEP

[FEP →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FEP ¨= \underline{(fep)}]⌉)] [FEP →^{tex} “FEP”] [FEP →^{pyk} “meta fep”]

RX

[RX →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RX ¨= \underline{(rx)}]⌉)]

[RX →^{tex} “RX”]

[RX →^{pyk} “meta rx”]

RY

[RY →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RY ¨= \underline{(ry)}]⌉)]

[RY →^{tex} “RY”]

[RY →^{pyk} “meta ry”]

RZ

[RZ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RZ ¨= \underline{(rz)}]⌉)]

[RZ →^{tex} “RZ”]

[RZ →^{pyk} “meta rz”]

RU

[RU →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[RU ¨= \underline{(ru)}]⌉)]

[RU →^{tex} “RU”]

[RU →^{pyk} “meta ru”]

(SX)

[(SX) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SX) ¨= \underline{(sx)}]⌉)]

[(SX) →^{tex} “(SX)”]

[(SX) →^{pyk} “meta sx”]

(SX1)

[(SX1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SX1) ¨= \underline{(sx1)}]⌉)]

[(SX1) →^{tex} “(SX1)”]

[(SX1) →^{pyk} “meta sx1”]

(SY)

[(SY) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SY) ¨= \underline{(sy)}]⌉)]

[(SY) →^{tex} “(SY)”]

[(SY) →^{pyk} “meta sy”]

(SY1)

[(SY1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SY1) ¨= \underline{(sy1)}]⌉)]

[(SY1) →^{tex} “(SY1)”]

[(SY1) →^{pyk} “meta sy1”]

(SZ)

[(SZ) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SZ) ¨= \underline{(sz)}]⌉)]

[(SZ) →^{tex} “(SZ)”]

[(SZ) →^{pyk} “meta sz”]

(SZ1)

[(SZ1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SZ1) ¨= \underline{(sz1)}]⌉)]

[(SZ1) →^{tex} “(SZ1)”]

[(SZ1) →^{pyk} “meta sz1”]

(SU)

[(SU) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SU) ¨= \underline{(su)}]⌉)]

[(SU) →^{tex} “(SU)”]

[(SU) →^{pyk} “meta su”]

(SU1)

[(SU1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(SU1) ¨= \underline{(su1)}]⌉)]

[(SU1) →^{tex} “(SU1)”]

[(SU1) →^{pyk} “meta su1”]

FXS

[FXS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FXS ¨= \underline{(fxs)}]⌉)] [FXS →^{tex} “FXS”] [FXS →^{pyk} “meta fxs”]

FYS

[FYS →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[FYS ¨= \underline{(fys)}]⌉)] [FYS →^{tex} “FYS”] [FYS →^{pyk} “meta fys”]

(F1)

[(F1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F1) ¨= \underline{(f1)}]⌉)]

[(F1) →^{tex} “(F1)”]

[(F1) →^{pyk} “meta f1”]

(F2)

[(F2) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F2) ¨= \underline{(f2)}]⌉)]

[(F2) →^{tex} “(F2)”]

[(F2) →^{pyk} “meta f2”]

(F3)

[(F3) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F3) ¨= \underline{(f3)}]⌉)]

[(F3) →^{tex} “(F3)”]

[(F3) →^{pyk} “meta f3”]

(F4)

[(F4) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F4) ¨= \underline{(f4)}]⌉)]

[(F4) →^{tex} “(F4)”]

[(F4) →^{pyk} “meta f4”]

(OP1)

[(OP1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(OP1) ¨= \underline{(op1)}]⌉)]

[(OP1) →^{tex} “(OP1)”]

[(OP1) →^{pyk} “meta op1”]

(OP2)

[(OP2) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(OP2) ¨= \underline{(op2)}]⌉)]

[(OP2) →^{tex} “(OP2)”]

[(OP2) →^{pyk} “meta op2”]

(R1)

[(R1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(R1) ¨= \underline{(r1)}]⌉)]

[(R1) →^{tex} “(R1)”]

[(R1) →^{pyk} “meta r1”]

(S1)

[(S1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(S1) ¨= \underline{(s1)}]⌉)]

[(S1) →^{tex} “(S1)”]

[(S1) →^{pyk} “meta s1”]

(S2)

[(S2) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(S2) ¨= \underline{(s2)}]⌉)]

[(S2) →^{tex} “(S2)”]

[(S2) →^{pyk} “meta s2”]

(EPob)

[(EPob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(EPob) ¨= \overline{(ϵ)}]⌉)] [(EPob) →^{tex} “(EPob)”] [(EPob) →^{pyk} “object ep”]

(CRS1ob)

[(CRS1ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(CRS1ob) ¨= \overline{(crs1)}]⌉)] [(CRS1ob) →^{tex} “(CRS1ob)”] [(CRS1ob) →^{pyk} “object crs1”]

(F1ob)

[(F1ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F1ob) ¨= \overline{(f1)}]⌉)] [(F1ob) →^{tex} “(F1ob)”] [(F1ob) →^{pyk} “object f1”]

(F2ob)

[(F2ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F2ob) ¨= \overline{(f2)}]⌉)] [(F2ob) →^{tex} “(F2ob)”] [(F2ob) →^{pyk} “object f2”]

(F3ob)

[(F3ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F3ob) ¨= \overline{(f3)}]⌉)] [(F3ob) →^{tex} “(F3ob)”] [(F3ob) →^{pyk} “object f3”]

(F4ob)

[(F4ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(F4ob) ¨= \overline{(f4)}]⌉)] [(F4ob) →^{tex} “(F4ob)”] [(F4ob) →^{pyk} “object f4”]

(N1ob)

[(N1ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(N1ob) ¨= \overline{(n1)}]⌉)] [(N1ob) →^{tex} “(N1ob)”] [(N1ob) →^{pyk} “object n1”]

(N2ob)

[(N2ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(N2ob) ¨= \overline{(n2)}]⌉)] [(N2ob) →^{tex} “(N2ob)”] [(N2ob) →^{pyk} “object n2”]

(OP1ob)

[(OP1ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(OP1ob) ¨= \overline{(op1)}]⌉)] [(OP1ob) →^{tex} “(OP1ob)”] [(OP1ob) →^{pyk} “object op1”]

(OP2ob)

[(OP2ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(OP2ob) ¨= \overline{(op2)}]⌉)] [(OP2ob) →^{tex} “(OP2ob)”] [(OP2ob) →^{pyk} “object op2”]

(R1ob)

[(R1ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(R1ob) ¨= \overline{(r1)}]⌉)] [(R1ob) →^{tex} “(R1ob)”] [(R1ob) →^{pyk} “object r1”]

(S1ob)

[(S1ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(S1ob) ¨= \overline{(s1)}]⌉)] [(S1ob) →^{tex} “(S1ob)”] [(S1ob) →^{pyk} “object s1”]

(S2ob)

[(S2ob) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(S2ob) ¨= \overline{(s2)}]⌉)] [(S2ob) →^{tex} “(S2ob)”] [(S2ob) →^{pyk} “object s2”]

ph_{4}

[ph_{4} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{4} ¨= d_{Ph}]⌉)]

[ph_{4} →^{tex} “ph {4}”]

[ph_{4} →^{pyk} “ph4”]

ph_{5}

[ph_{5} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{5} ¨= e_{Ph}]⌉)]

[ph_{5} →^{tex} “ph {5}”]

[ph_{5} →^{pyk} “ph5”]

ph_{6}

[ph_{6} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ph_{6} ¨= f_{Ph}]⌉)]

[ph_{6} →^{tex} “ph {6}”]

[ph_{6} →^{pyk} “ph6”]

NAT

[NAT →^{tex} “NAT”] [NAT →^{pyk} “NAT”]

RATIONAL_{S}ERIES

[RATIONAL_{S}ERIES →^{tex} “RATIONAL SERIES”] [RATIONAL_{S}ERIES →^{pyk} “RATIONAL SERIES”]

SERIES

[SERIES →^{tex} “SERIES”] [SERIES →^{pyk} “SERIES”]

SetOfReals

[SetOfReals →^{tex} “SetOfReals”] [SetOfReals →^{pyk} “setOfReals”]

SetOfFxs

[SetOfFxs →^{tex} “SetOfFxs”] [SetOfFxs →^{pyk} “setOfFxs”]

N

[N →^{tex} “N”]

[N →^{pyk} “N”]

Q

[Q →^{tex} “Q”]

[Q →^{pyk} “Q”]

X

[X →^{tex} “X”]

[X →^{pyk} “X”]

xs

[xs →^{tex} “xs”]

[xs →^{pyk} “xs”]

xaF

[xaF →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[xaF ¨= xs]⌉)] [xaF →^{tex} “xaF”] [xaF →^{pyk} “xsF”]

ysF

[ysF →^{tex} “ysF”] [ysF →^{pyk} “ysF”]

us

[us →^{tex} “us”]

[us →^{pyk} “us”]

usFoelge

[usFoelge →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[usFoelge ¨= us]⌉)] [usFoelge →^{tex} “usFoelge”] [usFoelge →^{pyk} “usF”]

0

[0 →^{tex} “0”]

[0 →^{pyk} “0”]

1

[1 →^{tex} “1”]

[1 →^{pyk} “1”]

(−1)

[(−1) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(−1) ¨= (−u1)]⌉)]

[(−1) →^{tex} “(-1)”]

[(−1) →^{pyk} “(-1)”]

[2 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[2 ¨= (1 + 1)]⌉)]

[2 →^{tex} “2”]

[2 →^{pyk} “2”]

3

[3 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[3 ¨= (2 + 1)]⌉)]

[3 →^{tex} “3”]

[3 →^{pyk} “3”]

1/2

[1/2 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[1/2 ¨= rec2]⌉)]

[1/2 →^{tex} “1/2”]

[1/2 →^{pyk} “1/2”]

1/3

[1/3 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[1/3 ¨= rec3]⌉)]

[1/3 →^{tex} “1/3”]

[1/3 →^{pyk} “1/3”]

2/3

[2/3 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[2/3 ¨= (2 ∗ 1/3)]⌉)]

[2/3 →^{tex} “2/3”]

[2/3 →^{pyk} “2/3”]

0f

[0f →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[0f ¨= constantRationalSeries(0)]⌉)]

[0f →^{tex} “0f”]

[0f →^{pyk} “0f”]

1f

[1f →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[1f ¨= constantRationalSeries(1)]⌉)]

[1f →^{pyk} “1f”]

00

[00 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[00 ¨= R(0f)]⌉)]

[00 →^{tex} “00”]

[00 →^{pyk} “00”]

01

[01 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[01 ¨= R(1f)]⌉)]

[01 →^{pyk} “01”]

(− − 01)

[(− − 01) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(− − 01) ¨= (− − 01)]⌉)]

[(− − 01) →^{tex} “(--01)”]

[(− − 01) →^{pyk} “(--01)”]

02

[02 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[02 ¨= 01 + +01]⌉)]

[02 →^{tex} “02”]

[02 →^{pyk} “02”]

01//02

[01//02 →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[01//02 ¨= 01//temp02]⌉)]

[01//02 →^{tex} “01//02”]

[01//02 →^{pyk} “01//02”]

PlusAssociativity(R)

[PlusAssociativity(R) →^{proof} Rule tactic]

[PlusAssociativity(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[PlusAssociativity(R) →^{tex} “PlusAssociativity(R)”]

[PlusAssociativity(R) →^{pyk} “lemma plusAssociativity(R)”]

PlusAssociativity(R)XX

[PlusAssociativity(R)XX →^{proof} Rule tactic]

[PlusAssociativity(R)XX →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[PlusAssociativity(R)XX →^{tex} “PlusAssociativity(R)XX”]

[PlusAssociativity(R)XX →^{pyk} “lemma plusAssociativity(R)XX”]

Plus0(R)

[Plus0(R) →^{proof} Rule tactic] [Plus0(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Plus0(R) →^{tex} “Plus0(R)”]

[Plus0(R) →^{pyk} “lemma plus0(R)”]

Negative(R)

[Negative(R) →^{proof} Rule tactic] [Negative(R) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Negative(R) →^{tex} “Negative(R)”] [Negative(R) →^{pyk} “lemma negative(R)”]

Times1(R)

[Times1(R) →^{proof} Rule tactic]

[Times1(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Times1(R) →^{tex} “Times1(R)”] [Times1(R) →^{pyk} “lemma times1(R)”]

lessAddition(R)

[lessAddition(R) →^{proof} Rule tactic]

[lessAddition(R) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[lessAddition(R) →^{tex} “lessAddition(R)”]

[lessAddition(R) →^{pyk} “lemma lessAddition(R)”]

PlusCommutativity(R)

[PlusCommutativity(R) →^{proof} Rule tactic]

[PlusCommutativity(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[PlusCommutativity(R) →^{tex} “PlusCommutativity(R)”]

[PlusCommutativity(R) →^{pyk} “lemma plusCommutativity(R)”]

LeqAntisymmetry(R)

[LeqAntisymmetry(R) →^{proof} Rule tactic]

[LeqAntisymmetry(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[LeqAntisymmetry(R) →^{tex} “LeqAntisymmetry(R)”]

[LeqAntisymmetry(R) →^{pyk} “lemma leqAntisymmetry(R)”]

LeqTransitivity(R)

[LeqTransitivity(R) →^{proof} Rule tactic]

[LeqTransitivity(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[LeqTransitivity(R) →^{tex} “LeqTransitivity(R)”]

[LeqTransitivity(R) →^{pyk} “lemma leqTransitivity(R)”]

leqAddition(R)

[leqAddition(R) →^{proof} Rule tactic]

[leqAddition(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}] [leqAddition(R) →^{tex} “leqAddition(R)”]

[leqAddition(R) →^{pyk} “lemma leqAddition(R)”]

Distribution(R)

[Distribution(R) →^{proof} Rule tactic]

[Distribution(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Distribution(R) →^{tex} “Distribution(R)”]

[Distribution(R) →^{pyk} “lemma distribution(R)”]

A4(Axiom)

[A4(Axiom) →^{proof} Rule tactic]

[A4(Axiom) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨\underline{a}≡\underline{b}|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: \underline{b} ⇒ \underline{a}]

[A4(Axiom) →^{tex} “A4(Axiom)”]

[A4(Axiom) →^{pyk} “axiom a4”]

InductionAxiom

[InductionAxiom →^{proof} Rule tactic]

[InductionAxiom →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ⟨\underline{b}≡\underline{a}|\underline{(v1)}:==0⟩_{Me} ⊢⊢ ⟨\underline{c}≡\underline{a}|\underline{(v1)}:==(\underline{(v1)} + 1)⟩_{Me} ⊢⊢ \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ \underline{c} ⇒ ∀_{obj}\underline{(v1)}: \underline{a}]

[InductionAxiom →^{tex} “InductionAxiom”] [InductionAxiom →^{pyk} “axiom induction”]

EqualityAxiom

[EqualityAxiom →^{proof} Rule tactic] [EqualityAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z}] [EqualityAxiom →^{tex} “EqualityAxiom”] [EqualityAxiom →^{pyk} “axiom equality”]

EqLeqAxiom

[EqLeqAxiom →^{proof} Rule tactic] [EqLeqAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y}] [EqLeqAxiom →^{tex} “EqLeqAxiom”] [EqLeqAxiom →^{pyk} “axiom eqLeq”]

EqAdditionAxiom

[EqAdditionAxiom →^{proof} Rule tactic] [EqAdditionAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z})] [EqAdditionAxiom →^{tex} “EqAdditionAxiom”] [EqAdditionAxiom →^{pyk} “axiom eqAddition”]

EqMultiplicationAxiom

[EqMultiplicationAxiom →^{proof} Rule tactic] [EqMultiplicationAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z})] [EqMultiplicationAxiom →^{tex} “EqMultiplicationAxiom”] [EqMultiplicationAxiom →^{pyk} “axiom eqMultiplication”]

QisClosed(Reciprocal)(Imply)

[QisClosed(Reciprocal)(Imply) →^{proof} Rule tactic] [QisClosed(Reciprocal)(Imply) →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⇒ \underline{x} ∈ Q ⇒ rec\underline{x} ∈ Q] [QisClosed(Reciprocal)(Imply) →^{tex} “QisClosed(Reciprocal)(Imply)”] [QisClosed(Reciprocal)(Imply) →^{pyk} “axiom QisClosed(reciprocal)”]

QisClosed(Reciprocal)

[QisClosed(Reciprocal) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⊢ \underline{x} ∈ Q ⊢ QisClosed(Reciprocal)(Imply) ≫ ˙¬ (\underline{x} = 0)n ⇒ \underline{x} ∈ Q ⇒ rec\underline{x} ∈ Q; MP2⊳ ˙¬ (\underline{x} = 0)n ⇒ \underline{x} ∈ Q ⇒ rec\underline{x} ∈ Q ⊳ ˙¬ (\underline{x} = 0)n ⊳ \underline{x} ∈ Q ≫ rec\underline{x} ∈ Q⌉, p_{0}, c)] [QisClosed(Reciprocal) →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⊢ \underline{x} ∈ Q ⊢ rec\underline{x} ∈ Q] [QisClosed(Reciprocal) →^{tex} “QisClosed(Reciprocal)”] [QisClosed(Reciprocal) →^{pyk} “lemma QisClosed(reciprocal)”]

QisClosed(Negative)(Imply)

[QisClosed(Negative)(Imply) →^{proof} Rule tactic] [QisClosed(Negative)(Imply) →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x} ∈ Q ⇒ (−u\underline{x}) ∈ Q] [QisClosed(Negative)(Imply) →^{tex} “QisClosed(Negative)(Imply)”] [QisClosed(Negative)(Imply) →^{pyk} “axiom QisClosed(negative)”]

QisClosed(Negative)

[QisClosed(Negative) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: \underline{x} ∈ Q ⊢ QisClosed(Negative)(Imply) ≫ \underline{x} ∈ Q ⇒ (−u\underline{x}) ∈ Q; MP ⊳ \underline{x} ∈ Q ⇒ (−u\underline{x}) ∈ Q ⊳ \underline{x} ∈ Q ≫ (−u\underline{x}) ∈ Q⌉, p_{0}, c)] [QisClosed(Negative) →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x} ∈ Q ⊢ (−u\underline{x}) ∈ Q] [QisClosed(Negative) →^{tex} “QisClosed(Negative)”] [QisClosed(Negative) →^{pyk} “lemma QisClosed(negative)”]

leqReflexivity

[leqReflexivity →^{proof} Rule tactic] [leqReflexivity →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x} <= \underline{x}] [leqReflexivity →^{tex} “leqReflexivity”] [leqReflexivity →^{pyk} “axiom leqReflexivity”]

leqAntisymmetryAxiom

[leqAntisymmetryAxiom →^{proof} Rule tactic] [leqAntisymmetryAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y}] [leqAntisymmetryAxiom →^{tex} “leqAntisymmetryAxiom”] [leqAntisymmetryAxiom →^{pyk} “axiom leqAntisymmetry”]

leqTransitivityAxiom

[leqTransitivityAxiom →^{proof} Rule tactic] [leqTransitivityAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z}] [leqTransitivityAxiom →^{tex} “leqTransitivityAxiom”] [leqTransitivityAxiom →^{pyk} “axiom leqTransitivity”]

leqTotality

[leqTotality →^{proof} Rule tactic] [leqTotality →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x}] [leqTotality →^{tex} “leqTotality”] [leqTotality →^{pyk} “axiom leqTotality”]

leqAdditionAxiom

[leqAdditionAxiom →^{proof} Rule tactic] [leqAdditionAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z})] [leqAdditionAxiom →^{tex} “leqAdditionAxiom”] [leqAdditionAxiom →^{pyk} “axiom leqAddition”]

leqMultiplicationAxiom

[leqMultiplicationAxiom →^{proof} Rule tactic] [leqMultiplicationAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z})] [leqMultiplicationAxiom →^{tex} “leqMultiplicationAxiom”] [leqMultiplicationAxiom →^{pyk} “axiom leqMultiplication”]

plusAssociativity

[plusAssociativity →^{proof} Rule tactic] [plusAssociativity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + (\underline{y} + \underline{z}))] [plusAssociativity →^{tex} “plusAssociativity”] [plusAssociativity →^{pyk} “axiom plusAssociativity”]

plusCommutativity

[plusCommutativity →^{proof} Rule tactic] [plusCommutativity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: (\underline{x} + \underline{y}) = (\underline{y} + \underline{x})] [plusCommutativity →^{tex} “plusCommutativity”] [plusCommutativity →^{pyk} “axiom plusCommutativity”]

Negative

[Negative →^{proof} Rule tactic] [Negative →^{stmt} SystemQ ⊢ ∀\underline{x}: (\underline{x} + (−u\underline{x})) = 0] [Negative →^{tex} “Negative”] [Negative →^{pyk} “axiom negative”]

plus0

[plus0 →^{proof} Rule tactic] [plus0 →^{stmt} SystemQ ⊢ ∀\underline{x}: (\underline{x} + 0) = \underline{x}] [plus0 →^{tex} “plus0”] [plus0 →^{pyk} “axiom plus0”]

timesAssociativity

[timesAssociativity →^{proof} Rule tactic] [timesAssociativity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = (\underline{x} ∗ (\underline{y} ∗ \underline{z}))] [timesAssociativity →^{tex} “timesAssociativity”] [timesAssociativity →^{pyk} “axiom timesAssociativity”]

timesCommutativity

[timesCommutativity →^{proof} Rule tactic] [timesCommutativity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: (\underline{x} ∗ \underline{y}) = (\underline{y} ∗ \underline{x})] [timesCommutativity →^{tex} “timesCommutativity”] [timesCommutativity →^{pyk} “axiom timesCommutativity”]

ReciprocalAxiom

[ReciprocalAxiom →^{proof} Rule tactic] [ReciprocalAxiom →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⇒ (\underline{x} ∗ rec\underline{x}) = 1] [ReciprocalAxiom →^{tex} “ReciprocalAxiom”] [ReciprocalAxiom →^{pyk} “axiom reciprocal”]

times1

[times1 →^{proof} Rule tactic]

[times1 →^{stmt} SystemQ ⊢ ∀\underline{x}: (\underline{x} ∗ 1) = \underline{x}]

[times1 →^{tex} “times1”]

[times1 →^{pyk} “axiom times1”]

Distribution

[Distribution →^{proof} Rule tactic]

[Distribution →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} ∗ (\underline{y} + \underline{z})) = ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ \underline{z}))]

[Distribution →^{tex} “Distribution”]

[Distribution →^{pyk} “axiom distribution”]

0not1

[0not1 →^{proof} Rule tactic]

[0not1 →^{stmt} SystemQ ⊢ ˙¬ (0 = 1)n]

[0not1 →^{tex} “0not1”]

[0not1 →^{pyk} “axiom 0not1”]

lemma eqLeq(R)

[lemma eqLeq(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ WeakenOr1 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[lemma eqLeq(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[lemma eqLeq(R) →^{pyk} “lemma eqLeq(R)”]

TimesAssociativity(R)

[TimesAssociativity(R) →^{proof} Rule tactic]

[TimesAssociativity(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]∗\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]∗\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[TimesAssociativity(R) →^{tex} “TimesAssociativity(R)”]

[TimesAssociativity(R) →^{pyk} “lemma timesAssociativity(R)”]

TimesCommutativity(R)

[TimesCommutativity(R) →^{proof} Rule tactic] [TimesCommutativity(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[TimesCommutativity(R) →^{tex} “TimesCommutativity(R)”]

[TimesCommutativity(R) →^{pyk} “lemma timesCommutativity(R)”]

(Adgic)SameR

[(Adgic)SameR →^{proof} Rule tactic] [(Adgic)SameR →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} = \underline{(fy)} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[(Adgic)SameR →^{tex} “(Adgic)SameR”]

[(Adgic)SameR →^{pyk} “1rule adhoc sameR”]

Separation2formula(1)

[Separation2formula(1) →^{proof} Rule tactic]

[Separation2formula(1) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ \underline{y} ∈ \underline{x}]

[Separation2formula(1) →^{tex} “Separation2formula(1)”]

[Separation2formula(1) →^{pyk} “lemma separation2formula(1)”]

Separation2formula(2)

[Separation2formula(2) →^{proof} Rule tactic]

[Separation2formula(2) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{x}: ∀\underline{y}: \underline{y} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊢ \underline{b}] [Separation2formula(2) →^{tex} “Separation2formula(2)”]

[Separation2formula(2) →^{pyk} “lemma separation2formula(2)”]

Cauchy

[Cauchy →^{proof} Rule tactic]

[Cauchy →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n]

[Cauchy →^{tex} “Cauchy”]

[Cauchy →^{pyk} “axiom cauchy”]

PlusF

[PlusF →^{proof} Rule tactic]

[PlusF →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] = (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])]

[PlusF →^{tex} “PlusF”]

[PlusF →^{pyk} “axiom plusF”]

ReciprocalF

[ReciprocalF →^{proof} Rule tactic]

[ReciprocalF →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = if(\underline{(fx)}[\underline{m}] = 0, 0, rec\underline{(fx)}[\underline{m}])]

[ReciprocalF →^{tex} “ReciprocalF”]

[ReciprocalF →^{pyk} “axiom reciprocalF”]

From ==

[From == →^{proof} Rule tactic]

[From == →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n]

[From ==→^{tex} “From==”]

[From == →^{pyk} “1rule from==”]

To ==

[To == →^{proof} Rule tactic]

[To == →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[To ==→^{tex} “To==”]

[To == →^{pyk} “1rule to==”]

FromInR

[FromInR →^{proof} Rule tactic]

[FromInR →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∈ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n]

[FromInR →^{tex} “FromInR”]

[FromInR →^{pyk} “1rule fromInR”]

PlusR(Sym)

[PlusR(Sym) →^{tex} “PlusR(Sym)”] [PlusR(Sym) →^{pyk} “lemma plusR(Sym)”]

ReciprocalR(Axiom)

[ReciprocalR(Axiom) →^{tex} “ReciprocalR(Axiom)”] [ReciprocalR(Axiom) →^{pyk} “axiom reciprocalR”]

LessMinus1(N)

[LessMinus1(N) →^{proof} Rule tactic] [LessMinus1(N) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: Nat(\underline{m}) ⊢⊢ Nat(\underline{n}) ⊢⊢ ˙¬ (\underline{m} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{m} = (\underline{n} + 1))n)n)n ⊢ \underline{m} <= \underline{n}] [LessMinus1(N) →^{tex} “LessMinus1(N)”] [LessMinus1(N) →^{pyk} “1rule lessMinus1(N)”]

Nonnegative(N)

[Nonnegative(N) →^{proof} Rule tactic] [Nonnegative(N) →^{stmt} SystemQ ⊢ ∀\underline{m}: Nat(\underline{m}) ⊢⊢ 0 <= \underline{m}] [Nonnegative(N) →^{tex} “Nonnegative(N)”] [Nonnegative(N) →^{pyk} “axiom nonnegative(N)”]

US0

[US0 →^{proof} Rule tactic] [US0 →^{stmt} SystemQ ⊢ us[0] = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[US0 →^{tex} “US0”]

[US0 →^{pyk} “axiom US0”]

NextXS(UpperBound)

[NextXS(UpperBound) →^{proof} Rule tactic]

[NextXS(UpperBound) →^{stmt} SystemQ ⊢ ∀\underline{m}: UB(\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals) ⊢ xs[(\underline{m} + 1)] = xs[\underline{m}]]

[NextXS(UpperBound) →^{tex} “NextXS(UpperBound)”]

[NextXS(UpperBound) →^{pyk} “1rule nextXS(upperBound)”]

NextXS(NoUpperBound)

[NextXS(NoUpperBound) →^{proof} Rule tactic]

[NextXS(NoUpperBound) →^{stmt} SystemQ ⊢ ∀\underline{m}: ˙¬ (UB(\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals))n ⊢ xs[(\underline{m} + 1)] = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NextXS(NoUpperBound) →^{tex} “NextXS(NoUpperBound)”] [NextXS(NoUpperBound) →^{pyk} “1rule nextXS(noUpperBound)”]

NextUS(UpperBound)

[NextUS(UpperBound) →^{proof} Rule tactic]

[NextUS(UpperBound) →^{stmt} SystemQ ⊢ ∀\underline{m}: UB(\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals) ⊢ us[(\underline{m} + 1)] = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NextUS(UpperBound) →^{tex} “NextUS(UpperBound)”]

[NextUS(UpperBound) →^{pyk} “1rule nextUS(upperBound)”]

NextUS(NoUpperBound)

[NextUS(NoUpperBound) →^{proof} Rule tactic]

[NextUS(NoUpperBound) →^{stmt} SystemQ ⊢ ∀\underline{m}: ˙¬ (UB(\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}, SetOfReals))n ⊢ us[(\underline{m} + 1)] = us[\underline{m}]]

[NextUS(NoUpperBound) →^{tex} “NextUS(NoUpperBound)”]

[NextUS(NoUpperBound) →^{pyk} “1rule nextUS(noUpperBound)”]

ExpZero

[ExpZero →^{proof} Rule tactic]

[ExpZero →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: \underline{m} = 0 ⊢ \underline{x}(exp)\underline{m} = 1]

[ExpZero →^{tex} “ExpZero”]

[ExpZero →^{pyk} “1rule expZero”]

ExpPositive

[ExpPositive →^{proof} Rule tactic]

[ExpPositive →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{m})n)n)n ⊢ \underline{x}(exp)\underline{m} = (\underline{x} ∗ \underline{x}(exp)(\underline{m} + (−u1)))]

[ExpPositive →^{tex} “ExpPositive”] [ExpPositive →^{pyk} “1rule expPositive”]

ExpZero(R)

[ExpZero(R) →^{proof} Rule tactic]

[ExpZero(R) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: \underline{m} = 0 ⊢ \underline{(fx)}(exp)\underline{m} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[ExpZero(R) →^{tex} “ExpZero(R)”]

[ExpZero(R) →^{pyk} “1rule expZero(R)”]

ExpPositive(R)

[ExpPositive(R) →^{proof} Rule tactic]

[ExpPositive(R) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ˙¬ (0 <= \underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{m})n)n)n ⊢ \underline{(fx)}(exp)\underline{m} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] ∗ y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] ∗ y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}] [ExpPositive(R) →^{tex} “ExpPositive(R)”] [ExpPositive(R) →^{pyk} “1rule expPositive(R)”]

BSzero

[BSzero →^{proof} Rule tactic] [BSzero →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: \underline{n} = 0 ⊢ BS(\underline{m}, \underline{n}) = rec(1 + 1)(exp)\underline{m}] [BSzero →^{tex} “BSzero”] [BSzero →^{pyk} “1rule base(1/2)Sum zero”]

BSpositive

[BSpositive →^{proof} Rule tactic] [BSpositive →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ˙¬ (0 <= \underline{n} ⇒ ˙¬ ( ˙¬ (0 = \underline{n})n)n)n ⊢ BS(\underline{m}, \underline{n}) = (rec(1 + 1)(exp)(\underline{m} + \underline{n}) + BS(\underline{m}, (\underline{n} + (−u1))))] [BSpositive →^{tex} “BSpositive”] [BSpositive →^{pyk} “1rule base(1/2)Sum positive”]

UStelescope(Zero)

[UStelescope(Zero) →^{proof} Rule tactic] [UStelescope(Zero) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: \underline{n} = 0 ⊢ UStelescope(\underline{m}, \underline{n}) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))|] [UStelescope(Zero) →^{tex} “UStelescope(Zero)”] [UStelescope(Zero) →^{pyk} “1rule UStelescope zero”]

UStelescope(Positive)

[UStelescope(Positive) →^{proof} Rule tactic]

[UStelescope(Positive) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ˙¬ (0 <= \underline{n} ⇒ ˙¬ ( ˙¬ (0 = \underline{n})n)n)n ⊢ UStelescope(\underline{m}, \underline{n}) = (|(us[(\underline{m} + \underline{n})] + (−uus[(\underline{m} + (\underline{n} + 1))]))| + UStelescope(\underline{m}, (\underline{n} + (−u1))))]

[UStelescope(Positive) →^{tex} “UStelescope(Positive)”]

[UStelescope(Positive) →^{pyk} “1rule UStelescope positive”]

EqAddition(R)

[EqAddition(R) →^{proof} Rule tactic]

[EqAddition(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[EqAddition(R) →^{tex} “EqAddition(R)”]

[EqAddition(R) →^{pyk} “1rule adhoc eqAddition(R)”]

FromLimit

[FromLimit →^{proof} Rule tactic]

[FromLimit →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fep)}: ∀\underline{(fx)}: ∀\underline{(fys)}: Limit(\underline{(fx)}, \underline{(fys)}) ⊢ ∀_{obj}\underline{(fep)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ \underline{m} <= \underline{n} ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ |f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−ux[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n)n]

[FromLimit →^{tex} “FromLimit”]

[FromLimit →^{pyk} “1rule fromLimit”]

ToUpperBound

[ToUpperBound →^{proof} Rule tactic]

[ToUpperBound →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fxs)}: \underline{(fx)} ∈ \underline{(fxs)} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fy)} ⊢ UB(\underline{(fy)}, \underline{(fxs)})]

[ToUpperBound →^{tex} “ToUpperBound”]

[ToUpperBound →^{pyk} “1rule toUpperBound”]

FromUpperBound

[FromUpperBound →^{proof} Rule tactic]

[FromUpperBound →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fxs)}: UB(\underline{(fy)}, \underline{(fxs)}) ⊢ \underline{(fx)} ∈ \underline{(fxs)} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fy)}]

[FromUpperBound →^{tex} “FromUpperBound”]

[FromUpperBound →^{pyk} “1rule fromUpperBound”]

USisUpperBound

[USisUpperBound →^{proof} Rule tactic] [USisUpperBound →^{stmt} SystemQ ⊢ ∀\underline{m}: UB(us[\underline{m}], SetOfFxs)]

[USisUpperBound →^{tex} “USisUpperBound”]

[USisUpperBound →^{pyk} “axiom USisUpperBound”]

0not1(R)

[0not1(R) →^{proof} Rule tactic]

[0not1(R) →^{stmt} SystemQ ⊢ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n] [0not1(R) →^{tex} “0not1(R)”] [0not1(R) →^{pyk} “axiom 0not1(R)”]

ExpUnbounded(R)

[ExpUnbounded(R) →^{proof} Rule tactic] [ExpUnbounded(R) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}]+(−u\overline{(ϵ)})))n)n)n)n)n)n)n)n] [ExpUnbounded(R) →^{tex} “ExpUnbounded(R)”] [ExpUnbounded(R) →^{pyk} “1rule expUnbounded”]

FromLeq(Advanced)(N)

[FromLeq(Advanced)(N) →^{proof} Rule tactic] [FromLeq(Advanced)(N) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{n}: \underline{m} <= \underline{n} ⊢ ˙¬ (∀_{obj}\underline{(m1)}: ˙¬ ((\underline{m} + \underline{(m1)}) = \underline{n})n)n] [FromLeq(Advanced)(N) →^{tex} “FromLeq(Advanced)(N)”] [FromLeq(Advanced)(N) →^{pyk} “1rule fromLeq(Advanced)(N)”]

FromLeastUpperBound

[FromLeastUpperBound →^{proof} Rule tactic] [FromLeastUpperBound →^{stmt} SystemQ ⊢ LUB(\underline{(fx)}, \underline{(fys)}) ⊢ ˙¬ (UB(\underline{(fx)}, \underline{(fys)}) ⇒ ˙¬ (UB(\underline{(fz)}, \underline{(fys)}) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fz)})n)n] [FromLeastUpperBound →^{tex} “FromLeastUpperBound”] [FromLeastUpperBound →^{pyk} “1rule fromLeastUpperBound”]

ToLeastUpperBound

[ToLeastUpperBound →^{proof} Rule tactic] [ToLeastUpperBound →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(fys)}: UB(\underline{(fx)}, \underline{(fys)}) ⊢ UB(\underline{(fz)}, \underline{(fys)}) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \underline{(fx)} = \underline{(fz)} ⊢ LUB(\underline{(fx)}, \underline{(fys)})] [ToLeastUpperBound →^{tex} “ToLeastUpperBound”] [ToLeastUpperBound →^{pyk} “1rule toLeastUpperBound”]

XSisNotUpperBound

[XSisNotUpperBound →^{proof} Rule tactic] [XSisNotUpperBound →^{stmt} SystemQ ⊢ ∀\underline{m}: ˙¬ (UB(xs[\underline{m}], SetOfFxs))n] [XSisNotUpperBound →^{tex} “XSisNotUpperBound”] [XSisNotUpperBound →^{pyk} “axiom XSisNotUpperBound”]

ysFGreater

[ysFGreater →^{proof} Rule tactic] [ysFGreater →^{stmt} SystemQ ⊢ ∀\underline{m}: ˙¬ (xs[\underline{m}] <= ysF[\underline{m}] ⇒ ˙¬ ( ˙¬ (xs[\underline{m}] = ysF[\underline{m}])n)n)n] [ysFGreater →^{tex} “ysFGreater”] [ysFGreater →^{pyk} “axiom ysFGreater”]

ysFLess

[ysFLess →^{proof} Rule tactic] [ysFLess →^{stmt} SystemQ ⊢ ∀\underline{m}: ˙¬ (ysF[\underline{m}] <= (xs[\underline{m}] + rec\underline{m}) ⇒ ˙¬ ( ˙¬ (ysF[\underline{m}] = (xs[\underline{m}] + rec\underline{m}))n)n)n] [ysFLess →^{tex} “ysFLess”] [ysFLess →^{pyk} “axiom ysFLess”]

SmallInverse

[SmallInverse →^{proof} Rule tactic] [SmallInverse →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (rec\underline{m} <= \underline{x} ⇒ ˙¬ ( ˙¬ (rec\underline{m} = \underline{x})n)n)n)n)n] [SmallInverse →^{tex} “SmallInverse”] [SmallInverse →^{pyk} “1rule smallInverse”]

NatType

[NatType →^{proof} Rule tactic] [NatType →^{stmt} SystemQ ⊢ ∀\underline{m}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ \underline{m} ∈ N] [NatType →^{tex} “NatType”] [NatType →^{pyk} “axiom natType”]

RationalType

[RationalType →^{proof} Rule tactic] [RationalType →^{stmt} SystemQ ⊢ ∀\underline{x}: λc.TypeRational0(⌈\underline{x}⌉) ⊢⊢ \underline{x} ∈ Q] [RationalType →^{tex} “RationalType”] [RationalType →^{pyk} “axiom rationalType”]

SeriesType

[SeriesType →^{proof} Rule tactic] [SeriesType →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sy)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ⊢⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n] [SeriesType →^{tex} “SeriesType”] [SeriesType →^{pyk} “axiom seriesType”]

Max

[Max →^{proof} Rule tactic]

[Max →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n]

[Max →^{tex} “Max”]

[Max →^{pyk} “axiom max”]

Numerical

[Numerical →^{proof} Rule tactic]

[Numerical →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n]

[Numerical →^{tex} “Numerical”]

[Numerical →^{pyk} “axiom numerical”]

NumericalF

[NumericalF →^{proof} Rule tactic]

[NumericalF →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)} ⇒ ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\})n)n]

[NumericalF →^{tex} “NumericalF”]

[NumericalF →^{pyk} “axiom numericalF”]

MemberOfSeries(Imply)

[MemberOfSeries(Imply) →^{proof} Rule tactic] [MemberOfSeries(Imply) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: \underline{m} ∈ N ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}]

[MemberOfSeries(Imply) →^{tex} “MemberOfSeries(Imply)”]

[MemberOfSeries(Imply) →^{pyk} “axiom memberOfSeries”]

JoinConjuncts(2conditions)

[JoinConjuncts(2conditions) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c}; MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{d}; JoinConjuncts ⊳ \underline{c} ⊳ \underline{d} ≫ ˙¬ (\underline{c} ⇒ ˙¬ (\underline{d})n)n; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ⊢ \underline{a} ⊢ \underline{b} ⊢ ˙¬ (\underline{c} ⇒ ˙¬ (\underline{d})n)n ≫ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{c} ⇒ ˙¬ (\underline{d})n)n; \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{c} ⇒ ˙¬ (\underline{d})n)n ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{c} ⇒ ˙¬ (\underline{d})n)n⌉, p_{0}, c)]

[JoinConjuncts(2conditions) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{c} ⇒ ˙¬ (\underline{d})n)n]

[JoinConjuncts(2conditions) →^{tex} “JoinConjuncts(2conditions)”]

[JoinConjuncts(2conditions) →^{pyk} “prop lemma doubly conditioned join conjuncts”]

prop lemma imply negation

[prop lemma imply negation →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: \underline{a} ⇒ ˙¬ (\underline{a})n ⊢ AutoImply ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n; TND ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n; FromDisjuncts ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n ⊳ \underline{a} ⇒ ˙¬ (\underline{a})n ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n ≫ ˙¬ (\underline{a})n⌉, p_{0}, c)]

[prop lemma imply negation →^{stmt} SystemQ ⊢ ∀\underline{a}: \underline{a} ⇒ ˙¬ (\underline{a})n ⊢ ˙¬ (\underline{a})n]

[prop lemma imply negation →^{pyk} “prop lemma imply negation”]

TND

[TND →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: AutoImply ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n; Repetition ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n⌉, p_{0}, c)]

[TND →^{stmt} SystemQ ⊢ ∀\underline{a}: ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n]

[TND →^{tex} “TND”]

[TND →^{pyk} “prop lemma tertium non datur”]

FromNegatedImply

[FromNegatedImply →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊳ \underline{a} ≫ ˙¬ ( ˙¬ (\underline{b})n)n; RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{b})n)n ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊢ \underline{a} ⊢ \underline{b} ≫ \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⇒ \underline{a} ⇒ \underline{b}; ˙¬ (\underline{a} ⇒ \underline{b})n ⊢ MT ⊳ \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⇒ \underline{a} ⇒ \underline{b} ⊳ ˙¬ (\underline{a} ⇒ \underline{b})n ≫ ˙¬ (\underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n; Repetition ⊳ ˙¬ (\underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ≫ ˙¬ (\underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n⌉, p_{0}, c)]

[FromNegatedImply →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a} ⇒ \underline{b})n ⊢ ˙¬ (\underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n]

[FromNegatedImply →^{tex} “FromNegatedImply”]

[FromNegatedImply →^{pyk} “prop lemma from negated imply”]

ToNegatedImply

[ToNegatedImply →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⊢ RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ≫ \underline{a} ⇒ \underline{b}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; FromContradiction ⊳ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a} ⇒ \underline{b})n; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⊢ ˙¬ (\underline{a} ⇒ \underline{b})n ≫ \underline{a} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n; \underline{a} ⊢ ˙¬ (\underline{b})n ⊢ MP2 ⊳ \underline{a} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n ⊳ \underline{a} ⊳ ˙¬ (\underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n; AutoImply ≫ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⇒ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n; Neg ⊳ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⇒ ˙¬ (\underline{a} ⇒ \underline{b})n ⊳ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ⇒ ˙¬ ( ˙¬ (\underline{a} ⇒ \underline{b})n)n ≫ ˙¬ (\underline{a} ⇒ \underline{b})n⌉, p_{0}, c)]

[ToNegatedImply →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ (\underline{a} ⇒ \underline{b})n]

[ToNegatedImply →^{tex} “ToNegatedImply”]

[ToNegatedImply →^{pyk} “prop lemma to negated imply”]

FromNegated(2 ∗ Imply)

[FromNegated(2 ∗ Imply) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n ⊢ FromNegatedImply ⊳ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n ≫ ˙¬ (\underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b} ⇒ \underline{c})n)n)n; FirstConjunct ⊳ ˙¬ (\underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b} ⇒ \underline{c})n)n)n ≫ \underline{a}; SecondConjunct ⊳ ˙¬ (\underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b} ⇒ \underline{c})n)n)n ≫ ˙¬ (\underline{b} ⇒ \underline{c})n; FromNegatedImply ⊳ ˙¬ (\underline{b} ⇒ \underline{c})n ≫ ˙¬ (\underline{b} ⇒ ˙¬ ( ˙¬ (\underline{c})n)n)n; FirstConjunct ⊳ ˙¬ (\underline{b} ⇒ ˙¬ ( ˙¬ (\underline{c})n)n)n ≫ \underline{b}; SecondConjunct ⊳ ˙¬ (\underline{b} ⇒ ˙¬ ( ˙¬ (\underline{c})n)n)n ≫ ˙¬ (\underline{c})n; JoinConjuncts ⊳ \underline{a} ⊳ \underline{b} ≫ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n; JoinConjuncts ⊳ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⊳ ˙¬ (\underline{c})n ≫ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⇒ ˙¬ ( ˙¬ (\underline{c})n)n)n⌉, p_{0}, c)]

[FromNegated(2 ∗ Imply) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n ⊢ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n ⇒ ˙¬ ( ˙¬ (\underline{c})n)n)n]

[FromNegated(2 ∗ Imply) →^{tex} “FromNegated(2∗Imply)”]

[FromNegated(2 ∗ Imply) →^{pyk} “prop lemma from negated double imply”]

FromNegatedAnd

[FromNegatedAnd →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ⊢ \underline{a} ⊢ Repetition ⊳ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ≫ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n; RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ≫ \underline{a} ⇒ ˙¬ (\underline{b})n; MP ⊳ \underline{a} ⇒ ˙¬ (\underline{b})n ⊳ \underline{a} ≫ ˙¬ (\underline{b})n⌉, p_{0}, c)]

[FromNegatedAnd →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ⊢ \underline{a} ⊢ ˙¬ (\underline{b})n]

[FromNegatedAnd →^{tex} “FromNegatedAnd”]

[FromNegatedAnd →^{pyk} “prop lemma from negated and”]

FromNegatedOr

[FromNegatedOr →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n ⊢ Repetition ⊳ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n; FromNegatedImply ⊳ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n⌉, p_{0}, c)]

[FromNegatedOr →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n ⊢ ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n]

[FromNegatedOr →^{tex} “FromNegatedOr”]

[FromNegatedOr →^{pyk} “prop lemma from negated or”]

ToNegatedOr

[ToNegatedOr →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ⊢ ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ FirstConjunct ⊳ ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ≫ ˙¬ (\underline{a})n; SecondConjunct ⊳ ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ≫ ˙¬ (\underline{b})n; NegateDisjunct1 ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{a})n ≫ \underline{b}; FromContradiction ⊳ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ⊢ ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ⇒ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n; ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ⊢ MP ⊳ ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ⇒ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n ⊳ ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ≫ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n; prop lemma imply negation ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n⌉, p_{0}, c)]

[ToNegatedOr →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n)n ⊢ ˙¬ ( ˙¬ (\underline{a})n ⇒ \underline{b})n]

[ToNegatedOr →^{tex} “ToNegatedOr”]

[ToNegatedOr →^{pyk} “prop lemma to negated or”]

FromNegations

[FromNegations →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ TND ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n; FromDisjuncts ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{a})n ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ≫ \underline{b}⌉, p_{0}, c)]

[FromNegations →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ \underline{b}]

[FromNegations →^{tex} “FromNegations”]

[FromNegations →^{pyk} “prop lemma from negations”]

From3Disjuncts

[From3Disjuncts →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ ˙¬ (\underline{a})n ⊢ Repetition ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c}; MP ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⊳ ˙¬ (\underline{a})n ≫ ˙¬ (\underline{b})n ⇒ \underline{c}; FromDisjuncts ⊳ ˙¬ (\underline{b})n ⇒ \underline{c} ⊳ \underline{b} ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{d} ≫ \underline{d}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ ˙¬ (\underline{a})n ⊢ \underline{d} ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{c} ⇒ \underline{d} ⇒ ˙¬ (\underline{a})n ⇒ \underline{d}; AutoImply ≫ \underline{a} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{d}; ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ MP3 ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{c} ⇒ \underline{d} ⇒ ˙¬ (\underline{a})n ⇒ \underline{d} ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⊳ \underline{b} ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{d} ≫ ˙¬ (\underline{a})n ⇒ \underline{d}; MP ⊳ \underline{a} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{d} ⊳ \underline{a} ⇒ \underline{d} ≫ \underline{a} ⇒ \underline{d}; FromNegations ⊳ \underline{a} ⇒ \underline{d} ⊳ ˙¬ (\underline{a})n ⇒ \underline{d} ≫ \underline{d}⌉, p_{0}, c)]

[From3Disjuncts →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{b})n ⇒ \underline{c} ⊢ \underline{a} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{d} ⊢ \underline{c} ⇒ \underline{d} ⊢ \underline{d}]

[From3Disjuncts →^{tex} “From3Disjuncts”]

[From3Disjuncts →^{pyk} “prop lemma from three disjuncts”]

From2 ∗ 2Disjuncts

[From2 ∗ 2Disjuncts →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{a} ≫ \underline{c} ⇒ \underline{e}; MP ⊳ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{a} ≫ \underline{d} ⇒ \underline{e}; FromDisjuncts ⊳ ˙¬ (\underline{c})n ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{e} ⊳ \underline{d} ⇒ \underline{e} ≫ \underline{e}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ ˙¬ (\underline{a})n ⊢ NegateDisjunct1 ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{a})n ≫ \underline{b}; MP ⊳ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{b} ≫ \underline{c} ⇒ \underline{e}; MP ⊳ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊳ \underline{b} ≫ \underline{d} ⇒ \underline{e}; FromDisjuncts ⊳ ˙¬ (\underline{c})n ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{e} ⊳ \underline{d} ⇒ \underline{e} ≫ \underline{e}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{e} ≫ ˙¬ (\underline{c})n ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{e}; Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ ˙¬ (\underline{a})n ⊢ \underline{e} ≫ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ (\underline{c})n ⇒ \underline{d} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⇒ ˙¬ (\underline{a})n ⇒ \underline{e}; ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ MP3 ⊳ ˙¬ (\underline{c})n ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{e} ⊳ ˙¬ (\underline{c})n ⇒ \underline{d} ⊳ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ≫ \underline{a} ⇒ \underline{e}; MP4 ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ (\underline{c})n ⇒ \underline{d} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⇒ ˙¬ (\underline{a})n ⇒ \underline{e} ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{c})n ⇒ \underline{d} ⊳ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ≫ ˙¬ (\underline{a})n ⇒ \underline{e}; FromNegations ⊳ \underline{a} ⇒ \underline{e} ⊳ ˙¬ (\underline{a})n ⇒ \underline{e} ≫ \underline{e}⌉, p_{0}, c)]

[From2 ∗ 2Disjuncts →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{e}]

[From2 ∗ 2Disjuncts →^{tex} “From2∗2Disjuncts”]

[From2 ∗ 2Disjuncts →^{pyk} “prop lemma from two times two disjuncts”]

NegateDisjunct1

[NegateDisjunct1 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{a})n ⊢ Repetition ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ≫ ˙¬ (\underline{a})n ⇒ \underline{b}; MP ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{a})n ≫ \underline{b}⌉, p_{0}, c)]

[NegateDisjunct1 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{a})n ⊢ \underline{b}]

[NegateDisjunct1 →^{tex} “NegateDisjunct1”]

[NegateDisjunct1 →^{pyk} “prop lemma negate first disjunct”]

NegateDisjunct2

[NegateDisjunct2 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ Repetition ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ≫ ˙¬ (\underline{a})n ⇒ \underline{b}; NegativeMT ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ \underline{a}⌉, p_{0}, c)]

[NegateDisjunct2 →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{b})n ⊢ \underline{a}]

[NegateDisjunct2 →^{tex} “NegateDisjunct2”]

[NegateDisjunct2 →^{pyk} “prop lemma negate second disjunct”]

ExpandDisjuncts

[ExpandDisjuncts →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ (\underline{d})n ⊢ NegateDisjunct2 ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ \underline{a}; NegateDisjunct2 ⊳ ˙¬ (\underline{c})n ⇒ \underline{d} ⊳ ˙¬ (\underline{d})n ≫ \underline{c}; JoinConjuncts ⊳ \underline{a} ⊳ \underline{c} ≫ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ (\underline{d})n ⊢ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n ≫ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ (\underline{c})n ⇒ \underline{d} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{d})n ⇒ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n; ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ MP2 ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⇒ ˙¬ (\underline{c})n ⇒ \underline{d} ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{d})n ⇒ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n ⊳ ˙¬ (\underline{a})n ⇒ \underline{b} ⊳ ˙¬ (\underline{c})n ⇒ \underline{d} ≫ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{d})n ⇒ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n; Repetition ⊳ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{d})n ⇒ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n ≫ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{d})n ⇒ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n⌉, p_{0}, c)]

[ExpandDisjuncts →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ (\underline{a})n ⇒ \underline{b} ⊢ ˙¬ (\underline{c})n ⇒ \underline{d} ⊢ ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{d})n ⇒ ˙¬ (\underline{a} ⇒ ˙¬ (\underline{c})n)n]

[ExpandDisjuncts →^{tex} “ExpandDisjuncts”]

[ExpandDisjuncts →^{pyk} “prop lemma expand disjuncts”]

SENC1

[SENC1 →^{proof} Rule tactic]

[SENC1 →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fx)} ∈ \underline{(ry)}]

[SENC1 →^{tex} “SENC1”]

[SENC1 →^{pyk} “lemma set equality nec condition(1)”]

SENC2

[SENC2 →^{proof} Rule tactic]

[SENC2 →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} = \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)}]

[SENC2 →^{tex} “SENC2”]

[SENC2 →^{pyk} “lemma set equality nec condition(2)”]

LessLeq(R)

[LessLeq(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ WeakenOr2 ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[LessLeq(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[LessLeq(R) →^{tex} “LessLeq(R)”]

[LessLeq(R) →^{pyk} “lemma lessLeq(R)”]

MemberOfSeries

[MemberOfSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: \underline{m} ∈ N ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ MemberOfSeries(Imply) ≫ \underline{m} ∈ N ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}; MP2 ⊳ \underline{m} ∈ N ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)} ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}⌉, p_{0}, c)]

[MemberOfSeries →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: \underline{m} ∈ N ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}]

[MemberOfSeries →^{tex} “MemberOfSeries”]

[MemberOfSeries →^{pyk} “lemma memberOfSeries”]

memberOfSeries(Type)

[memberOfSeries(Type) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ⊢⊢ NatType ⊳ ⊳ λc.TypeNat0(⌈\underline{m}⌉) ≫ \underline{m} ∈ N; SeriesType ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; MemberOfSeries ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}⌉, p_{0}, c)]

[memberOfSeries(Type) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sy)}⌉) ⊢⊢ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}]

[memberOfSeries(Type) →^{tex} “memberOfSeries(Type)”] [memberOfSeries(Type) →^{pyk} “lemma memberOfSeries(Type)”]

∗(exp)∗

[x(exp)y →^{tex} “ #1. (exp) #2.”]

[∗(exp)∗ →^{pyk} “" ˆ "”]

R(∗)

[R((fx)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[R((fx)) ¨= {ph ∈ Power(SetOfSeries(Q)) | SF((fx), ph_{4})}]⌉)]

[R(x) →^{tex} “R(#1.

)”]

[R(∗) →^{pyk} “R( " )”]

− − R(∗)

[− − R(x) →^{tex} “--R(#1.

)”]

[− − R(∗) →^{pyk} “--R( " )”]

rec∗

[recx →^{tex} “rec#1.”]

[rec∗ →^{pyk} “1/ "”]

∗/∗

[bs/r →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[bs/r ¨= {ph ∈ P(bs) | Ex_{20} ∈ bs ˙∧ [Ex_{20} ∈ bs]_{r} ==ph_{2}}]⌉)]

[x/y →^{tex} “#1.

/ #2.”]

[∗/∗ →^{pyk} “eq-system of " modulo "”]

∗ ∩ ∗

[x ∩ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∩ y ¨= {ph ∈ x ∪ y | ph_{3} ∈ x ˙∧ ph_{3} ∈ y}]⌉)]

[x ∩ y →^{tex} “#1.

\cap #2.”]

[∗ ∩ ∗ →^{pyk} “intersection " comma " end intersection”]

∗[∗]

[x[y] →^{tex} “#1.

[#2. ]”]

[∗[∗] →^{pyk} “[ " ; " ]”]

∪∗

[∪x →^{tex} “\cup #1.”]

[∪∗ →^{pyk} “union " end union”]

∗ ∪ ∗

[x ∪ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∪ y ¨= ∪{{x}, {y}}]⌉)]

[x ∪ y →^{tex} “#1.

\mathrel{\cup} #2.”]

[∗ ∪ ∗ →^{pyk} “binary-union " comma " end union”]

P(∗)

[P(x) →^{tex} “P(#1.

)”]

[P(∗) →^{pyk} “power " end power”]

{∗}

[{x} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[{x} ¨= {x, x}]⌉)]

[{x} →^{tex} “\{#1.

\}”]

[{∗} →^{pyk} “zermelo singleton " end singleton”]

StateExpand(∗, ∗, ∗)

[StateExpand(t, s, c) →^{val} t!s!c!U^{M}(s^{h} ‘ t ‘ s ‘ c)]

[StateExpand(t, s, c) →^{tex} “StateExpand(#1. , #2. , #3. )”]

[StateExpand(∗, ∗, ∗) →^{pyk} “stateExpand( " , " , " )”]

extractSeries(∗)

[extractSeries(t) →^{val} t^{2}^{2}^{1}^{2}^{1}^{2}^{2}^{2}^{1}^{1}^{1}^{1}^{1}^{1}]

[extractSeries(t) →^{tex} “extractSeries(#1. )”]

[extractSeries(∗) →^{pyk} “extractSeries( " )”]

SetOfSeries(∗)

[SetOfSeries((sx)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[SetOfSeries((sx)) ¨= {ph ∈ Power(cartProd(N)) | IsSeries(ph_{6}, (sx))}]⌉)]

[SetOfSeries(x) →^{tex} “SetOfSeries(#1. )”]

[SetOfSeries(∗) →^{pyk} “setOfSeries( " )”]

− − Macro(∗)

[− − Macro(t) →^{val} Q̃(t, ⌈{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−ux[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−ux[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, ⌈x⌉ : : extractSeries(t^{1}) : : T)]

[− − Macro(x) →^{tex} “--Macro(#1. )”]

[− − Macro(∗) →^{pyk} “--Macro( " )”]

ExpandList(∗, ∗, ∗)

[ExpandList(t, s, c) →^{val} t!s!c!If(t^{a}, T, StateExpand(t^{h}, s, c) : : ExpandList(t^{t}, s, c))]

[ExpandList(x, y, z) →^{tex} “ExpandList(#1. , #2. , #3. )”]

[ExpandList(∗, ∗, ∗) →^{pyk} “expandList( " , " , " )”]

∗ ∗ Macro(∗)

[∗ ∗ Macro(t) →^{val} Q̃(t, ⌈{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] ∗ y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] ∗ y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, ⌈x⌉ : : extractSeries(t^{1}) : : ⌈y⌉ : : extractSeries(t^{2}) : : T)]

[∗ ∗ Macro(x) →^{tex} “∗∗Macro(#1. )”]

[∗ ∗ Macro(∗) →^{pyk} “∗∗Macro( " )”]

+ + Macro(∗)

[+ + Macro(t) →^{val} Q̃(t, ⌈{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (x[\underline{m}] + y[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, ⌈x⌉ : : extractSeries(t^{1}) : : ⌈y⌉ : : extractSeries(t^{2}) : : T)]

[+ + Macro(x) →^{tex} “++Macro(#1.

)”]

[+ + Macro(∗) →^{pyk} “++Macro( " )”]

<< Macro(∗)

[<< Macro(t) →^{val} Q̃(t, ⌈ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ x[\overline{m}] <= (y[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, ⌈x⌉ : : extractSeries(t^{1}) : : ⌈y⌉ : : extractSeries(t^{2}) : : T)]

[<< Macro(x) →^{tex} “<<Macro(#1. )”]

[<< Macro(∗) →^{pyk} “<<Macro( " )”]

||Macro(∗)

[||Macro(t) →^{val} Q̃(t, ⌈{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|fx|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|fx|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, ⌈x⌉ : : extractSeries(t^{1}) : : T)]

[||Macro(x) →^{tex} “||Macro(#1. )”]

[||Macro(∗) →^{pyk} “||Macro( " )”]

01//Macro(∗)

[01//Macro(t) →^{val} Q̃(t, ⌈{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (x[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, recx[\underline{m}]\}\})n)n)n ⇒ ˙¬ (x[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (x[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, recx[\underline{m}]\}\})n)n)n ⇒ ˙¬ (x[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, ⌈x⌉ : : extractSeries(t^{1}) : : T)]

[01//Macro(x) →^{tex} “01//Macro(#1. )”]

[01//Macro(∗) →^{pyk} “01//Macro( " )”]

UB(∗, ∗)

[UB(x, y) →^{tex} “UB(#1. , #2. )”]

[UB(∗, ∗) →^{pyk} “upperBound( " , " )”]

LUB(∗, ∗)

[LUB(x, y) →^{tex} “LUB(#1. , #2. )”]

[LUB(∗, ∗) →^{pyk} “leastUpperBound( " , " )”]

BS(∗, ∗)

[BS(x, y) →^{tex} “BS(#1. , #2. )”] [BS(∗, ∗) →^{pyk} “base(1/2)Sum( " , " )”]

UStelescope(∗, ∗)

[UStelescope(x, y) →^{tex} “UStelescope(#1. , #2. )”]

[UStelescope(∗, ∗) →^{pyk} “UStelescope( " , " )”]

(∗)

[(x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(x) ¨= (x)]⌉)]

[(x) →^{tex} “(#1.

)”]

[(∗) →^{pyk} “( " )”]

|f ∗ |

[|fx| →^{tex} “|f#1.

|”]

[|f ∗ | →^{pyk} “|f " |”]

|r ∗ |

[|rx| →^{macro} λt.λs.λc.||Macro(t^{h} : : ExpandList(t^{t}, s, c))]

[|rx| →^{tex} “|r#1.

|”]

[|r ∗ | →^{pyk} “|r " |”]

Limit(∗, ∗)

[Limit(x, y) →^{tex} “Limit(#1. , #2. )”] [Limit(∗, ∗) →^{pyk} “limit( " , " )”]

Union(∗)

[Union(x) →^{tex} “Union(#1. )”]

[Union(∗) →^{pyk} “U( " )”]

IsOrderedPair(∗, ∗, ∗)

[IsOrderedPair((sx), (sy), (sz)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[IsOrderedPair((sx), (sy), (sz)) ¨= ∃(OP1ob): ∃(OP2ob): (OP1ob) ∈ (sy) ˙∧ (OP2ob) ∈ (sz) ˙∧ (sx) = OrderedPair((OP1ob), (OP2ob))]⌉)]

[IsOrderedPair(x, y, z) →^{tex} “IsOrderedPair(#1. , #2. , #3. )”]

[IsOrderedPair(∗, ∗, ∗) →^{pyk} “isOrderedPair( " , " , " )”]

IsRelation(∗, ∗, ∗)

[IsRelation((sx), (sy), (sz)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[IsRelation((sx), (sy), (sz)) ¨= ∀(R1ob): ((R1ob) ∈ (sx) ⇒ IsOrderedPair((R1ob), (sy), (sz)))]⌉)]

[IsRelation(x, y, z) →^{tex} “IsRelation(#1. , #2. , #3. )”]

[IsRelation(∗, ∗, ∗) →^{pyk} “isRelation( " , " , " )”]

isFunction(∗, ∗, ∗)

[isFunction((sx), (sy), (sz)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[isFunction((sx), (sy), (sz)) ¨= IsRelation((sx), (sy), (sz)) ˙∧ ∀(F1ob), (F2ob), (F3ob), (F4ob): (OrderedPair((F1ob), (F2ob)) ∈ (sx) ⇒ OrderedPair((F3ob), (F4ob)) ∈ (sx) ⇒ (F1ob) = (F3ob) ⇒ (F2ob) = (F4ob))]⌉)]

[isFunction(x, y, z) →^{tex} “isFunction(#1. , #2. , #3. )”]

[isFunction(∗, ∗, ∗) →^{pyk} “isFunction( " , " , " )”]

IsSeries(∗, ∗)

[IsSeries((fx), (fy)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[IsSeries((fx), (fy)) ¨= isFunction((fx), N, (fy)) ˙∧ ∀(S1ob): ((S1ob) ∈ N ⇒ ∃(S2ob): OrderedPair((S1ob), (S2ob)) ∈ (fx))]⌉)]

[IsSeries(x, y) →^{tex} “IsSeries(#1. , #2. )”]

[IsSeries(∗, ∗) →^{pyk} “isSeries( " , " )”]

IsNatural(∗, ∗)

[IsNatural(xy, ∗) →^{tex} “IsNatural(#1. , #2. )”]

[IsNatural(∗, ∗) →^{pyk} “isNatural( " )”]

OrderedPair(∗, ∗)

[OrderedPair(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[OrderedPair(x, y) ¨= ⟨x, y⟩]⌉)]

[OrderedPair(x, y) →^{tex} “OrderedPair(#1. , #2. )”]

[OrderedPair(∗, ∗) →^{pyk} “(o " , " )”]

TypeNat(∗)

[TypeNat(x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[TypeNat(x) ¨= λc.TypeNat0(⌈x⌉)]⌉)]

[TypeNat(x) →^{tex} “TypeNat(#1. )”]

[TypeNat(∗) →^{pyk} “typeNat( " )”]

TypeNat0(∗)

[TypeNat0(x) →^{val} x ∈_{t} ⌈0⌉ : : ⌈\underline{(v2n)}⌉ : : ⌈\underline{m}⌉ : : ⌈\underline{n}⌉ : : ⌈(\underline{n} + 1)⌉ : : ⌈(\underline{m} + 0)⌉ : : ⌈(\underline{m} + \underline{n})⌉ : : ⌈\underline{o}⌉ : : ⌈\underline{p}⌉ : : ⌈((\underline{m} + \underline{n}) + 1)⌉ : : ⌈(\underline{m} + \underline{(m1)})⌉ : : ⌈(\underline{m} + (\underline{n} + 1))⌉ : : ⌈\underline{(m1)}⌉ : : ⌈\underline{(m2)}⌉ : : ⌈\underline{(n1)}⌉ : : ⌈\underline{(n2)}⌉ : : ⌈\overline{m}⌉ : : ⌈\overline{n}⌉ : : T]

[TypeNat0(x) →^{tex} “TypeNat0(#1. )”]

[TypeNat0(∗) →^{pyk} “typeNat0( " )”]

TypeRational(∗)

[TypeRational(x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[TypeRational(x) ¨= λc.TypeRational0(⌈x⌉)]⌉)]

[TypeRational(x) →^{tex} “TypeRational(#1. )”]

[TypeRational(∗) →^{pyk} “typeRational( " )”]

TypeRational0(∗)

[TypeRational0(x) →^{val} x ∈_{t} ⌈\underline{x}⌉ : : ⌈\underline{y}⌉ : : ⌈\underline{z}⌉ : : ⌈0⌉ : : ⌈1⌉ : : T]

[TypeRational0(x) →^{tex} “TypeRational0(#1. )”]

[TypeRational0(∗) →^{pyk} “typeRational0( " )”]

TypeSeries(∗, ∗)

[TypeSeries(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[TypeSeries(x, y) ¨= λc.Typeseries0(⌈x⌉, ⌈y⌉)]⌉)]

[TypeSeries(x, y) →^{tex} “TypeSeries(#1. , #2. )”]

[TypeSeries(∗, ∗) →^{pyk} “typeSeries( " , " )”]

Typeseries0(∗, ∗)

[Typeseries0(x, y) →^{val} y!x ∈_{t} ⌈\underline{(fx)}⌉ : : ⌈\underline{(fy)}⌉ : : ⌈\underline{(fz)}⌉ : : ⌈us⌉ : : ⌈\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}⌉ : : ⌈\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}⌉ : : ⌈\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}⌉ : : ⌈\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}⌉ : : ⌈\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}⌉ : : T]

[Typeseries0(x, y) →^{tex} “Typeseries0(#1. , #2. )”]

[Typeseries0(∗, ∗) →^{pyk} “typeSeries0( " , " )”]

{∗, ∗}

[{x, y} →^{tex} “\{#1.

, #2. \}”]

[{∗, ∗} →^{pyk} “zermelo pair " comma " end pair”]

⟨∗, ∗⟩

[⟨x, y⟩ →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨x, y⟩ ¨= {{x}, {x, y}}]⌉)]

[⟨x, y⟩ →^{tex} “\langle #1. , #2. \rangle”]

[⟨∗, ∗⟩ →^{pyk} “zermelo ordered pair " comma " end pair”]

(−u∗)

[(−ux) →^{tex} “(-u#1.

)”]

[(−u∗) →^{pyk} “- "”]

−_{f}∗

[−_{f}(fx) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[−_{f}(fx) ¨= {ph ∈ cartProd(N) | ∃M: ph_{6} = OrderedPair(M, (−u(fx)[M]))}]⌉)]

[−_{f}x →^{tex} “- {f}#1.”]

[−_{f}∗ →^{pyk} “-f "”]

(− − ∗)

[(− − x) →^{macro} λt.λs.λc. − −Macro(t^{h} : : ExpandList(t^{t}, s, c))]

[(− − x) →^{tex} “(--#1.

)”]

[(− − ∗) →^{pyk} “-- "”]

1f/∗

[1f/(fx) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[1f/(fx) ¨= {ph ∈ cartProd(N) | ∃M: ((fx)[M] ≠ 0 ˙∧ ph_{6} = OrderedPair(M, rec(fx)[M])) ˙∨ ((fx)[M] = 0 ˙∧ ph_{6} = OrderedPair(M, 0))}]⌉)]

[1f/x →^{tex} “1f/#1.”]

[1f/∗ →^{pyk} “1f/ "”]

01//temp∗

[01//tempx →^{macro} λt.λs.λc.01//Macro(t^{h} : : ExpandList(t^{t}, s, c))]

[01//tempx →^{tex} “01//temp#1.”]

[01//temp∗ →^{pyk} “01// "”]

∗(∗, ∗)

[r(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[r(x, y) ¨= ⟨x, y⟩ ∈ r]⌉)]

[z(x, y) →^{tex} “#3.

(#1. , #2. )”]

[∗(∗, ∗) →^{pyk} “" is related to " under "”]

ReflRel(∗, ∗)

[ReflRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[ReflRel(r, x) ¨= ∀s: (s ∈ x ⇒ r(s, s))]⌉)]

[ReflRel(r, x) →^{tex} “ReflRel(#1. , #2. )”]

[ReflRel(∗, ∗) →^{pyk} “" is reflexive relation in "”]

SymRel(∗, ∗)

[SymRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[SymRel(r, x) ¨= ∀s, t: (s ∈ x ⇒ t ∈ x ⇒ r(s, t) ⇒ r(t, s))]⌉)] [SymRel(r, x) →^{tex} “SymRel(#1. , #2. )”]

[SymRel(∗, ∗) →^{pyk} “" is symmetric relation in "”]

TransRel(∗, ∗)

[TransRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[TransRel(r, x) ¨= ∀s, t, u: (s ∈ x ⇒ t ∈ x ⇒ u ∈ x ⇒ r(s, t) ⇒ r(t, u) ⇒ r(s, u))]⌉)]

[TransRel(r, x) →^{tex} “TransRel(#1. , #2. )”]

[TransRel(∗, ∗) →^{pyk} “" is transitive relation in "”]

EqRel(∗, ∗)

[EqRel(r, x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[EqRel(r, x) ¨= ReflRel(r, x) ˙∧ SymRel(r, x) ˙∧ TransRel(r, x)]⌉)]

[EqRel(r, x) →^{tex} “EqRel(#1. , #2. )”]

[EqRel(∗, ∗) →^{pyk} “" is equivalence relation in "”]

[∗ ∈ ∗]_{∗}

[[x ∈ bs]_{r} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[[x ∈ bs]_{r} ¨= {ph ∈ bs | r(ph_{1}, x)}]⌉)]

[[x ∈ bs]_{r} →^{tex} “[#1.

\mathrel{\in} #2. ] {#3. }”]

[[∗ ∈ ∗]_{∗} →^{pyk} “equivalence class of " in " modulo "”]

Partition(∗, ∗)

[Partition(p, bs) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Partition(p, bs) ¨= (∀s: (s ∈ p ⇒ s ≠ Ø)) ˙∧ (∀s, t: (s ∈ p ⇒ t ∈ p ⇒ s ≠ t ⇒ s ∩ t ==Ø)) ˙∧ ∪ p==bs]⌉)]

[Partition(x, y) →^{tex} “Partition(#1. , #2. )”]

[Partition(∗, ∗) →^{pyk} “" is partition of "”]

(∗ ∗ ∗)

[(x ∗ y) →^{tex} “(#1.

∗#2. )”]

[(∗ ∗ ∗) →^{pyk} “" ∗ "”]

∗ ∗_{f} ∗

[(fx) ∗_{f} (fy) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(fx) ∗_{f} (fy) ¨= {ph ∈ cartProd(N) | ∃M: ph_{5} = OrderedPair(M, ((fx)[M] ∗ (fy)[M]))}]⌉)]

[(fx) ∗_{f} (fy) →^{tex} “#1.

∗ {f}#2.”]

[∗ ∗_{f} ∗ →^{pyk} “" ∗f "”]

∗ ∗ ∗∗

[x ∗ ∗y →^{macro} λt.λs.λc. ∗ ∗Macro(t^{h} : : ExpandList(t^{t}, s, c))]

[x ∗ ∗y →^{tex} “#1.

∗∗#2.”]

[∗ ∗ ∗∗ →^{pyk} “" ∗∗ "”]

(∗ + ∗)

[(x + y) →^{tex} “(#1.

+#2. )”]

[(∗ + ∗) →^{pyk} “" + "”]

(∗ − ∗)

[(x − y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(x − y) ¨= (x + (−uy))]⌉)]

[(x − y) →^{tex} “(#1.

-#2. )”]

[(∗ − ∗) →^{pyk} “" - "”]

∗ +_{f} ∗

[(fx) +_{f} (fy) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(fx) +_{f} (fy) ¨= {ph ∈ cartProd(N) | ∃M: ph_{4} = OrderedPair(M, ((fx)[M] + (fy)[M]))}]⌉)]

[(fx) +_{f} (fy) →^{tex} “#1. + {f}#2.”]

[∗ +_{f} ∗ →^{pyk} “" +f "”]

∗ −_{f} ∗

[(fx) −_{f} (fy) →^{tex} “#1.

- {f}#2.”]

[∗ −_{f} ∗ →^{pyk} “" -f "”]

∗ + +∗

[x + +y →^{macro} λt.λs.λc. + +Macro(t^{h} : : ExpandList(t^{t}, s, c))]

[x + +y →^{tex} “#1.

++#2.”]

[∗ + +∗ →^{pyk} “" ++ "”]

R(∗) − −R(∗)

[R((fx)) − −R((fy)) →^{tex} “R(#1. ) -- R(#2. )”]

[R(∗) − −R(∗) →^{pyk} “R( " ) -- R( " )”]

∗ ∈ ∗

[x ∈ y →^{tex} “#1.

\mathrel{\in} #2.”]

[∗ ∈ ∗ →^{pyk} “" in0 "”]

| ∗ |

[|x| →^{tex} “|#1.

|”]

[| ∗ | →^{pyk} “| " |”]

if(∗, ∗, ∗)

[if(x, y, z) →^{tex} “if(#1. , #2. , #3. )”]

[if(∗, ∗, ∗) →^{pyk} “if( " , " , " )”]

Max(∗, ∗)

[Max(x, y) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Max(x, y) ¨= if(y <= x, x, y)]⌉)]

[Max(x, y) →^{tex} “Max(#1. , #2. )”]

[Max(∗, ∗) →^{pyk} “max( " , " )”]

Max(∗, ∗)

[Max(x, y) →^{tex} “Max(#1. , #2. )”]

[Max(∗, ∗) →^{pyk} “maxR( " , " )”]

∗ = ∗

[x = y →^{tex} “#1.

= #2.”]

[∗ = ∗ →^{pyk} “" = "”]

∗ ≠ ∗

[x ≠ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ≠ y ¨= ˙¬ (x = y)n]⌉)]

[x ≠ y →^{tex} “#1.

\neq #2.”]

[∗ ≠ ∗ →^{pyk} “" != "”]

∗ <= ∗

[x <= y →^{tex} “#1.

<= #2.”]

[∗ <= ∗ →^{pyk} “" <= "”]

∗ < ∗

[x < y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x < y ¨= x <= y ˙∧ x ≠ y]⌉)]

[x < y →^{tex} “#1.

< #2.”]

[∗ < ∗ →^{pyk} “" < "”]

∗ <_{f} ∗

[(fx) <_{f} (fy) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(fx) <_{f} (fy) ¨= ∃(EPob): ∃n: ∀m: 0 < (EPob) ˙∧ (n <= m ⇒ (fx)[m] <= ((fy)[m] − (EPob)))]⌉)]

[x <_{f} y →^{tex} “#1.

< {f}#2.”]

[∗ <_{f} ∗ →^{pyk} “" <f "”]

∗ ≤_{f} ∗

[(fx) ≤_{f} (fy) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[(fx) ≤_{f} (fy) ¨= (fx) <_{f} (fy) ˙∨ SF((fx), (fy))]⌉)]

[x ≤_{f} y →^{tex} “#1.

\leq {f}#2.”]

[∗ ≤_{f} ∗ →^{pyk} “" <=f "”]

SF(∗, ∗)

[SF((fx), (fy)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[SF((fx), (fy)) ¨= ∀(EPob): ∃n: ∀m: (0 < (EPob) ⇒ n <= m ⇒ |((fx)[m] − (fy)[m])| < (EPob))]⌉)]

[SF(x, y) →^{tex} “SF(#1. , #2. )”]

[SF(∗, ∗) →^{pyk} “" sameF "”]

∗ == ∗

[x == y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x == y ¨= x = y]⌉)]

[x == y →^{tex} “#1.

== #2.”]

[∗ == ∗ →^{pyk} “" == "”]

∗!! == ∗

[x!! == y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x!! == y ¨= ˙¬ (x == y)n]⌉)]

[x!! == y →^{tex} “#1.

!!== #2.”]

[∗!! == ∗ →^{pyk} “" !!== "”]

∗ << ∗

[x << y →^{macro} λt.λs.λc. << Macro(t^{h} : : ExpandList(t^{t}, s, c))]

[x << y →^{tex} “#1.

<< #2.”]

[∗ << ∗ →^{pyk} “" << "”]

∗ <<== ∗

[x <<== y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x <<== y ¨= x << y ˙∨ x == y]⌉)]

[x <<== y →^{tex} “#1.

<<== #2.”]

[∗ <<== ∗ →^{pyk} “" <<== "”]

∗==∗

[x==y →^{tex} “#1.

\!\mathrel{==}\! #2.”]

[∗==∗ →^{pyk} “" zermelo is "”]

∗ ⊆ ∗

[x ⊆ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ⊆ y ¨= ∀(S1ob): ((S1ob) ∈ x ⇒ (S1ob) ∈ y)]⌉)]

[x ⊆ y →^{tex} “#1.

\mathrel{\subseteq} #2.”]

[∗ ⊆ ∗ →^{pyk} “" is subset of "”]

˙¬ (∗)n

[ ˙¬ (x)n →^{tex} “\dot{\neg}\, (#1. )n”]

[ ˙¬ (∗)n →^{pyk} “not0 "”]

∗ ∉ ∗

[x ∉ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ∉ y ¨= ˙¬ (x ∈ y)n]⌉)]

[x ∉ y →^{tex} “#1.

\mathrel{\notin} #2.”]

[∗ ∉ ∗ →^{pyk} “" zermelo ˜ in "”]

∗ ≠ ∗

[x ≠ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ≠ y ¨= ˙¬ (x==y)n]⌉)]

[x ≠ y →^{tex} “#1.

\mathrel{\neq} #2.”]

[∗ ≠ ∗ →^{pyk} “" zermelo ˜ is "”]

∗ ˙∧ ∗

[x ˙∧ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙∧ y ¨= ˙¬ ((x ⇒ ˙¬ (y)n))n]⌉)]

[x ˙∧ y →^{tex} “#1.

\mathrel{\dot{\wedge}} #2.”]

[∗ ˙∧ ∗ →^{pyk} “" and0 "”]

∗ ˙∨ ∗

[x ˙∨ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙∨ y ¨= ˙¬ (x)n ⇒ y]⌉)]

[x ˙∨ y →^{tex} “#1.

\mathrel{\dot{\vee}} #2.”]

[∗ ˙∨ ∗ →^{pyk} “" or0 "”]

∃∗: ∗

[∃(v1): a →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[∃(v1): a ¨= ˙¬ (∀(v1): ˙¬ (a)n)n]⌉)]

[∃x: y →^{tex} “

\exists #1. \colon #2.”]

[∃∗: ∗ →^{pyk} “exist0 " indeed "”]

∗ ˙⇔ ∗

[x ˙⇔ y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ˙⇔ y ¨= (x ⇒ y) ˙∧ (y ⇒ x)]⌉)]

[x ˙⇔ y →^{tex} “#1.

\mathrel{\dot{\Leftrightarrow}} #2.”]

[∗ ˙⇔ ∗ →^{pyk} “" iff "”]

{ph ∈ ∗ | ∗}

[{ph ∈ x | a} →^{tex} “ \{ ph \mathrel{\in} #1. \mid #2. \}”]

[{ph ∈ ∗ | ∗} →^{pyk} “the set of ph in " such that " end set”]

The pyk compiler, version 0.grue.20060417+ by Klaus Grue GRD-2006-12-15.UTC:00:32:42.052453 = MJD-54084.TAI:00:33:15.052453 = LGT-4672859595052453e-6