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am, (· · ·), Objekt-var, Ex-var, Ph-var, Værdi, Variabel, Op(∗), Op(∗, ∗),

∗ ¨ == ∗, ContainsEmpty(∗), 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}, bs, OBS, BS, Ø, ZFsub, 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, 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, (ϵ), (fx), (fy), (fz), (fv), var fv, (rx), (ry), (rz), (ru), ϵ, FX, FY, FZ, FU, FV, RX, RY, RZ, RU, 0, 1, (−1), 2, 1/2, 0f, 1f, 00, 01, leqReflexivity, leqAntisymmetryAxiom, leqTransitivityAxiom, leqTotality, leqAdditionAxiom, leqMultiplicationAxiom, plusAssociativity, plusCommutativity, Negative, plus0, timesAssociativity, timesCommutativity, ReciprocalAxiom, times1, Distribution, 0not1, equalityAxiom, eqLeqAxiom, eqAdditionAxiom, eqMultiplicationAxiom, SENC1, SENC2, IfThenElse(T), IfThenElse(F), From = f, To = f, From < f, To < f, PlusF, TimesF, MinusF, 0f, 1f, FromSF, ToSF, To == XX, From ==, To ==, From << XX, From << (1), From << (2), to << XX, From <<, To <<, FromInR, PlusR, TimesR, leqAntisymmetry, leqTransitivity, leqAddition, leqMultiplication, Reciprocal, Equality, eqLeq, eqAddition, eqMultiplication, ToNegatedImply, TND, ImplyNegation, FromNegations, From3Disjuncts, From2 ∗ 2Disjuncts, NegateDisjunct1, NegateDisjunct2, ExpandDisjuncts, eqReflexivity, eqSymmetry, eqTransitivity, eqTransitivity4, eqTransitivity5, eqTransitivity6, plus0Left, times1Left, lemma eqAdditionLeft, EqMultiplicationLeft, DistributionOut, Three2twoTerms, Three2threeTerms, Three2threeFactors, AddEquations, SubtractEquations, SubtractEquationsLeft, EqNegated, PositiveToRight(Eq), PositiveToLeft(Eq)(1term), NegativeToLeft(Eq), LessNeq, NeqSymmetry, NeqNegated, SubNeqRight, SubNeqLeft, NeqAddition, NeqMultiplication, UniqueNegative, DoubleMinus, LeqLessEq, LessLeq, FromLeqGeq, subLeqRight, subLeqLeft, Leq + 1, PositiveToRight(Leq), PositiveToRight(Leq)(1term), negativeToLeft(Leq), LeqAdditionLeft, leqSubtraction, leqSubtractionLeft, thirdGeq, LeqNegated, AddEquations(Leq), ThirdGeqSeries, LeqNeqLess, FromLess, ToLess, fromNotLess, toNotLess, NegativeLessPositive, leqLessTransitivity, LessLeqTransitivity, LessTransitivity, LessTotality, SubLessRight, SubLessLeft, LessAddition, LessAdditionLeft, LessMultiplication, LessMultiplicationLeft, LessDivision, AddEquations(Less), LessNegated, PositiveNegated, NonpositiveNegated, NegativeNegated, NonnegativeNegated, PositiveHalved, NonnegativeNumerical, NegativeNumerical, PositiveNumerical, lemma nonpositiveNumerical, |0| = 0, 0 <= |x|, SameNumerical, SignNumerical(+), SignNumerical, NumericalDifference, SplitNumericalSumHelper, splitNumericalSum(++), splitNumericalSum(−−), splitNumericalSum(+ − small), splitNumericalSum(+ − big), splitNumericalSum(+−), splitNumericalSum(−+), splitNumericalSum, insertMiddleTerm(Numerical), x + y = zBackwards, x ∗ y = zBackwards, x = x + (y − y), x = x + y − y, , insertMiddleTerm(Sum), insertMiddleTerm(Difference), x ∗ 0 + x = x, x ∗ 0 = 0, (−1) ∗ (−1) + (−1) ∗ 1 = 0, (−1) ∗ (−1) = 1, 0 < 1Helper, 0 < 1, 0 < 2, 0 < 1/2, TwoWholes, TwoHalves, −x − y = −(x + y), MinusNegated, Times(−1), Times(−1)Left, −0 = 0, SFsymmetry, SFtransitivity, = fToSameF, PlusF(Sym), TimesF(Sym), f2R(Plus), f2R(Times), PlusR(Sym), TimesR(Sym), LessLeq(R), eqLeq(R), SubLessRight(R), SubLessLeft(R), << TransitivityHelper(Q), << Transitivity, <<== Reflexivity, <<== AntisymmetryHelper(Q), <<== Antisymmetry, <<== Transitivity, Plus0f, Plus00, == Addition, == AdditionLeft, << Addition, <<== Addition, PlusAssociativity(F), PlusAssociativity(R), Negative(R), PlusCommutativity(F), PlusCommutativity(R), TimesAssociativity(F), TimesAssociativity(R), Times1f, Times01, TimesCommutativity(F), TimesCommutativity(R), Distribution(F), Distribution(R), R(∗), − − R(∗), rec∗, ∗/∗, ∗ ∩ ∗, ∗[∗], ∪∗, ∗ ∪ ∗, P(∗), {∗}, {∗, ∗}, ⟨∗, ∗⟩, −∗, −_{f}∗, ∗ ∈ ∗, ∗(∗, ∗), ReflRel(∗, ∗), SymRel(∗, ∗), TransRel(∗, ∗), EqRel(∗, ∗), [∗ ∈ ∗]_{∗}, Partition(∗, ∗), ∗ ∗ ∗, ∗ ∗_{f} ∗, ∗ ∗ ∗∗, ∗ + ∗, ∗ − ∗, ∗ +_{f} ∗, ∗ −_{f} ∗, ∗ + +∗, R(∗) − −R(∗), | ∗ |, if(∗, ∗, ∗), ∗ = ∗, ∗ ≠ ∗,

∗ <= ∗, ∗ < ∗, ∗ =_{f} ∗, ∗ <_{f} ∗, SF(∗, ∗), ∗ == ∗, ∗ << ∗, ∗ <<== ∗, ∗==∗,

∗ ⊆ ∗, ˙¬ ∗, ∗ ∉ ∗, ∗ ≠ ∗, ∗ ˙∧ ∗, ∗ ˙∨ ∗, ∗ ˙⇔ ∗, {ph ∈ ∗ | ∗},

am

[am →^{prio}

Preassociative [am], [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}(∗)], [(⋯)], [Objekt-var], [Ex-var], [Ph-var], [Værdi], [Variabel], [Op(∗)], [Op(∗, ∗)], [∗ ¨ == ∗], [ContainsEmpty(∗)], [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}], [bs], [OBS], [BS], [Ø], [ZFsub], [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], [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], [(ϵ)], [(fx)], [(fy)], [(fz)], [(fv)], [var fv], [(rx)], [(ry)], [(rz)], [(ru)], [ϵ], [FX], [FY], [FZ], [FU], [FV], [RX], [RY], [RZ], [RU], [0], [1], [(−1)], [2], [1/2], [0f], [1f], [00], [01], [leqReflexivity], [leqAntisymmetryAxiom], [leqTransitivityAxiom], [leqTotality], [leqAdditionAxiom], [leqMultiplicationAxiom], [plusAssociativity], [plusCommutativity], [Negative], [plus0], [timesAssociativity], [timesCommutativity], [ReciprocalAxiom], [times1], [Distribution], [0not1], [equalityAxiom], [eqLeqAxiom], [eqAdditionAxiom], [eqMultiplicationAxiom], [SENC1], [SENC2], [IfThenElse(T)], [IfThenElse(F)], [From = f], [To = f], [From < f], [To < f], [PlusF], [TimesF], [MinusF], [0f], [1f], [FromSF], [ToSF], [To == XX], [From ==], [To ==], [From << XX], [From << (1)], [From << (2)], [to << XX], [From <<], [To <<], [FromInR], [PlusR], [TimesR], [leqAntisymmetry], [leqTransitivity], [leqAddition], [leqMultiplication], [Reciprocal], [Equality], [eqLeq], [eqAddition], [eqMultiplication], [ToNegatedImply], [TND], [ImplyNegation], [FromNegations], [From3Disjuncts], [From2 ∗ 2Disjuncts], [NegateDisjunct1], [NegateDisjunct2], [ExpandDisjuncts], [eqReflexivity], [eqSymmetry], [eqTransitivity], [eqTransitivity4], [eqTransitivity5], [eqTransitivity6], [plus0Left], [times1Left], [lemma eqAdditionLeft], [EqMultiplicationLeft], [DistributionOut], [Three2twoTerms], [Three2threeTerms], [Three2threeFactors], [AddEquations], [SubtractEquations], [SubtractEquationsLeft], [EqNegated], [PositiveToRight(Eq)], [PositiveToLeft(Eq)(1term)], [NegativeToLeft(Eq)], [LessNeq], [NeqSymmetry], [NeqNegated], [SubNeqRight], [SubNeqLeft], [NeqAddition], [NeqMultiplication], [UniqueNegative], [DoubleMinus], [LeqLessEq], [LessLeq], [FromLeqGeq], [subLeqRight], [subLeqLeft], [Leq + 1], [PositiveToRight(Leq)], [PositiveToRight(Leq)(1term)], [negativeToLeft(Leq)], [LeqAdditionLeft], [leqSubtraction], [leqSubtractionLeft], [thirdGeq], [LeqNegated], [AddEquations(Leq)], [ThirdGeqSeries], [LeqNeqLess], [FromLess], [ToLess], [fromNotLess], [toNotLess], [NegativeLessPositive], [leqLessTransitivity], [LessLeqTransitivity], [LessTransitivity], [LessTotality], [SubLessRight], [SubLessLeft], [LessAddition], [LessAdditionLeft], [LessMultiplication], [LessMultiplicationLeft], [LessDivision], [AddEquations(Less)], [LessNegated], [PositiveNegated], [NonpositiveNegated], [NegativeNegated], [NonnegativeNegated], [PositiveHalved], [NonnegativeNumerical], [NegativeNumerical], [PositiveNumerical], [lemma nonpositiveNumerical], [|0| = 0], [0 <= |x|], [SameNumerical], [SignNumerical(+)], [SignNumerical], [NumericalDifference], [SplitNumericalSumHelper], [splitNumericalSum(++)], [splitNumericalSum(−−)], [splitNumericalSum(+ − small)], [splitNumericalSum(+ − big)], [splitNumericalSum(+−)], [splitNumericalSum(−+)], [splitNumericalSum], [insertMiddleTerm(Numerical)], [x + y = zBackwards], [x ∗ y = zBackwards], [x = x + (y − y)], [x = x + y − y], [], [insertMiddleTerm(Sum)], [insertMiddleTerm(Difference)], [x ∗ 0 + x = x], [x ∗ 0 = 0], [(−1) ∗ (−1) + (−1) ∗ 1 = 0], [(−1) ∗ (−1) = 1], [0 < 1Helper], [0 < 1], [0 < 2], [0 < 1/2], [TwoWholes], [TwoHalves], [−x − y = −(x + y)], [MinusNegated], [Times(−1)], [Times(−1)Left], [−0 = 0], [SFsymmetry], [SFtransitivity], [= fToSameF], [PlusF(Sym)], [TimesF(Sym)], [f2R(Plus)], [f2R(Times)], [PlusR(Sym)], [TimesR(Sym)], [LessLeq(R)], [eqLeq(R)], [SubLessRight(R)], [SubLessLeft(R)], [<< TransitivityHelper(Q)], [<< Transitivity], [<<== Reflexivity], [<<== AntisymmetryHelper(Q)], [<<== Antisymmetry], [<<== Transitivity], [Plus0f], [Plus00], [== Addition], [== AdditionLeft], [<< Addition], [<<== Addition], [PlusAssociativity(F)], [PlusAssociativity(R)], [Negative(R)], [PlusCommutativity(F)], [PlusCommutativity(R)], [TimesAssociativity(F)], [TimesAssociativity(R)], [Times1f], [Times01], [TimesCommutativity(F)], [TimesCommutativity(R)], [Distribution(F)], [Distribution(R)]; 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 [∗′], [R(∗)], [− − R(∗)], [rec∗]; Preassociative [∗/∗], [∗ ∩ ∗], [∗[∗]]; Preassociative [∪∗], [∗ ∪ ∗], [P(∗)]; Preassociative [{∗}]; Preassociative [{∗, ∗}], [⟨∗, ∗⟩], [−∗], [−_{f}∗]; Preassociative [∗ ∈ ∗], [∗(∗, ∗)], [ReflRel(∗, ∗)], [SymRel(∗, ∗)], [TransRel(∗, ∗)], [EqRel(∗, ∗)], [[∗ ∈ ∗]_{∗}], [Partition(∗, ∗)]; Preassociative [∗ · ∗], [∗ ·_{0} ∗], [∗ ∗ ∗], [∗ ∗_{f} ∗], [∗ ∗ ∗∗]; Preassociative [∗ + ∗], [∗ +_{0} ∗], [∗ +_{1} ∗], [∗ − ∗], [∗ −_{0} ∗], [∗ −_{1} ∗], [∗ + ∗], [∗ − ∗], [∗ +_{f} ∗], [∗ −_{f} ∗], [∗ + +∗], [R(∗) − −R(∗)]; Preassociative [| ∗ |], [if(∗, ∗, ∗)]; 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 [¬∗], [ ˙¬ ∗], [∗ ∉ ∗], [∗ ≠ ∗]; 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] ∗], [∗\\∗]; ]

[am →^{tex} “am”]

[am →^{pyk} “am”]

(⋯)

[(⋯) →^{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”]

Dedu(∗, ∗)

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

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

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

Dedu_{0}(∗, ∗)

[Dedu_{0}(p, c) →^{val} c!If(Dedu_{8}(p, T), Dedu_{s}(Dedu_{7}(p), c, T), F)]

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

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

Dedu_{s}(∗, ∗, ∗)

[Dedu_{s}(p, c, s) →^{val} If(p =^{r} ⌈x ⊢⊢ y⌉, c =^{r} ⌈x ⊢⊢ y⌉ ∧ p^{1} =^{t} c^{1} ∧ Dedu_{s}(p^{2}, c^{2}, c^{1} : : s), Dedu_{1}(p, c, 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}(p, c, s) →^{val} If(c =^{r} ⌈x ⊢⊢ y⌉, Dedu_{1}(p, c^{2}, c^{1} : : s), Dedu_{2}(p, c, s))]

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

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

Dedu_{2}(∗, ∗, ∗)

[Dedu_{2}(p, c, s) →^{val} s!p =^{r} ⌈x ⊢ y⌉ ∧ c =^{r} ⌈x ⇒

y⌉ \{ \begin{matrix}Dedu_{3}(p^{1}, c^{1}, s, T) ∧ Dedu_{2}(p^{2}, c^{2}, s) \\ Dedu_{4}(p, c, s, Dedu_{6}(p, c, T, T)) \end{matrix} ]

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

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

Dedu_{3}(∗, ∗, ∗, ∗)

[Dedu_{3}(p, c, s, b) →^{val} If(¬c =^{r} ⌈∀_{obj}x: y⌉, Dedu_{4}(p, c, s, b), If(p =^{r} ⌈∀_{obj}x: y⌉ ∧ p^{1} =^{t} c^{1}, Dedu_{4}(p, c, s, b), Dedu_{3}(p, c^{2}, s, c^{1} : : c^{1} : : b)))] [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}(p, c, s, b) →^{val} s!b!If(p =^{r} ⌈\overline{x}⌉, lookup(p, b, T) =^{t} c, If(¬p =^{r} c, F, If(p =^{r} ⌈∀_{obj}x: y⌉, p^{1} =^{t} c^{1} ∧ Dedu_{4}(p^{2}, c^{2}, s, p^{1} : : p^{1} : : b), If(¬p =^{r} ⌈\underline{x}⌉, Dedu_{4}^{∗}(p^{t}, c^{t}, s, b), p^{1} =^{t} c^{1} ∧ Dedu_{5}(p, s, b)))))]

[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}^{∗}(p, c, s, b) →^{val} c!s!b!If(p, T, Dedu_{4}(p^{h}, c^{h}, s, b) ∧ Dedu_{4}^{∗}(p^{t}, c^{t}, s, b))]

[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}(p, s, b) →^{val} p!s!If(b, T, ⌈⌈x⌉#^{0}⌈y⌉⌉^{h} : : ⌈⌈∗⌉⌉^{h} : : b^{h}^{h} : : T : : ⌈⌈x⌉⌉^{h} : : p : : T : : T ∈_{t} s ∧ Dedu_{5}(p, s, b^{t}))]

[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) →^{val} p!c!b!e!If(p =^{r} ⌈\overline{x}⌉, p ∈_{t} e \{ \begin{matrix}b \\ p : : c : : b \end{matrix} , If(¬p =^{r} c, T,

If(p =^{r} ⌈\underline{a}⌉, b, If(p =^{r} ⌈∀_{obj}x: y⌉, Dedu_{6}(p^{2}, c^{2}, c^{1} : : e, b), Dedu_{6}^{∗}(p^{t}, c^{t}, e, b)))))]

[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) →^{val} p!c!b!e!If(p, b, Dedu_{6}^{∗}(p^{t}, c^{t}, e, Dedu_{6}(p^{h}, c^{h}, e, b)))]

[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) →^{val} p =^{r} ⌈∀x: y⌉ \{ \begin{matrix}Dedu_{7}(p^{2}) \\ p \end{matrix} ]

[Dedu_{7}(p) →^{tex} “ Dedu 7(#1. )”]

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

Dedu_{8}(∗, ∗)

[Dedu_{8}(p, b) →^{val} If(p =^{r} ⌈∀x: y⌉, Dedu_{8}(p^{2}, p^{1} : : b), If(p =^{r} ⌈\underline{a}⌉, p ∈_{t} b, Dedu_{8}^{∗}(p^{t}, b)))]

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

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

Dedu_{8}^{∗}(∗, ∗)

[Dedu_{8}^{∗}(p, b) →^{val} b!If(p, T, If(Dedu_{8}(p^{h}, b), Dedu_{8}^{∗}(p^{t}, b), F))]

[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} →^{tex} “ph {1}”]

[ph_{1} →^{pyk} “placeholder-var1”]

ph_{2}

[ph_{2} →^{tex} “ph {2}”]

[ph_{2} →^{pyk} “placeholder-var2”]

ph_{3}

[ph_{3} →^{tex} “ph {3}”]

[ph_{3} →^{pyk} “placeholder-var3”]

∗_{Ph}

[x_{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”]

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”]

ZFsub

[ZFsub →^{stmt} ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} ∗ \underline{z} = \underline{x} ∗ \underline{y} ∗ \underline{z} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ \underline{(fx)} <_{f} \underline{(fy)} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ \underline{(rx)} == \underline{(rz)} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z} ⊕ ∀\underline{x}: ∀\underline{t}: ∀\underline{a}: ∀\underline{b}: ⟨⌈\underline{a}⌉≡^{0}⌈\underline{b}⌉|⌈\underline{x}⌉:==⌈\underline{t}⌉⟩_{Ex} ⊢⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊢ SF(\underline{(fx)}, \underline{(fy)}) ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} + \underline{z} = \underline{x}+\underline{y} +\underline{z} ⊕ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊕ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{x} ⊕ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a} ⊕ ∀\underline{m}: ∀\underline{(fx)}: −_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}] ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} + \underline{z} = \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} ⊕ ∀\underline{x}: \underline{x} + −\underline{x} = 0 ⊕ ∀\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} ⇒ ˙¬ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\} ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ t_{Ex} ∈ \underline{(ry)} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rx)} ⊕ ∀\underline{x}: \underline{x} <= \underline{x} ⊕ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: 1f[\underline{m}] = 1 ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⇒ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊕ ∀\underline{s}: ˙¬ \underline{s} ∈ Ø ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)}) ⊕ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∈ R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)}) ⊕ ∀\underline{x}: ˙¬ \underline{x} = 0 ⇒ \underline{x} ∗ rec\underline{x} = 1 ⊕ ∀\underline{x}: \underline{x} + 0 = \underline{x} ⊕ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(rx)} << \underline{(ry)} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \underline{(fx)} =_{f} \underline{(fy)} ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y} ⊕ ∀\underline{a}: ∀\underline{b}: λx.Dedu_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ⊢⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⊕ ∀\underline{s}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⇒ ˙¬ \underline{s}==\underline{x} ⇒ \underline{s}==\underline{y} ⇒ ˙¬ ˙¬ \underline{s}==\underline{x} ⇒ \underline{s}==\underline{y} ⇒ \underline{s} ∈ \{\underline{x}, \underline{y}\} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ R(\underline{(fx)}) == R(\underline{(fy)}) ⊕ ∀\underline{(fx)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊕ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b} ⊕ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}) ⊕ ∀\underline{x}: \underline{x} ∗ 1 = \underline{x} ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} + \underline{y} = \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}) ⊕ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ j_{Ex} ∈ \underline{(rx)} ⊕ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{y} ⊕ ∀\underline{(rx)}: \underline{(rx)} == \underline{(rx)} ⊕ ∀\underline{a}: \underline{a} ⊢ \underline{a} ⊕ ∀\underline{m}: 0f[\underline{m}] = 0 ⊕ ˙¬ 0 = 1 ⊕ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊕ ∀\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} ⇒ ˙¬ ∀_{obj}\overline{s}: ˙¬ \overline{s} ∈ \underline{x} ⇒ \overline{s} ∈ \underline{y} ⇒ ˙¬ \overline{s} ∈ \underline{y} ⇒ \overline{s} ∈ \underline{x} ⇒ \underline{x}==\underline{y} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ SF(\underline{(fx)}, \underline{(fy)}) ⊢ \underline{(rx)} == \underline{(ry)} ⊕ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊕ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(fx)} <_{f} \underline{(fy)} ⊕ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ R(\underline{(fx)}) << R(\underline{(fy)}) ⊕ ∀\underline{x}: ∀\underline{y}: \underline{x} ∗ \underline{y} = \underline{y} ∗ \underline{x}] [ZFsub →^{tex} “ZFsub”] [ZFsub →^{pyk} “system Q”]

MP

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

Neg

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

Ded

[Ded →^{proof} Rule tactic] [Ded →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: λx.Dedu_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ⊢⊢ \underline{a} ⊢ \underline{b}] [Ded →^{tex} “Ded”] [Ded →^{pyk} “1rule deduction”]

ExistIntro

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

Ødef

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

PairDef

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

UnionDef

[UnionDef →^{proof} Rule tactic] [UnionDef →^{stmt} ZFsub ⊢ ∀\underline{s}: ∀\underline{x}: ˙¬ \underline{s} ∈ ∪\underline{x} ⇒ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ ˙¬ ˙¬ \underline{s} ∈ j_{Ex} ⇒ ˙¬ j_{Ex} ∈ \underline{x} ⇒ \underline{s} ∈ ∪\underline{x}] [UnionDef →^{tex} “UnionDef”] [UnionDef →^{pyk} “axiom union definition”]

PowerDef

[PowerDef →^{proof} Rule tactic] [PowerDef →^{stmt} ZFsub ⊢ ∀\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})] [PowerDef →^{tex} “PowerDef”] [PowerDef →^{pyk} “axiom power definition”]

SeparationDef

[SeparationDef →^{proof} Rule tactic] [SeparationDef →^{stmt} ZFsub ⊢ ∀\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} ⇒ ˙¬ ˙¬ \underline{z} ∈ \underline{x} ⇒ ˙¬ \underline{b} ⇒ \underline{z} ∈ \{ph ∈ \underline{x} | \underline{a}\}] [SeparationDef →^{tex} “SeparationDef”] [SeparationDef →^{pyk} “axiom separation definition”]

AddDoubleNeg

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

[AddDoubleNeg →^{stmt} ZFsub ⊢ ∀\underline{a}: \underline{a} ⊢ ˙¬ ˙¬ \underline{a}]

[AddDoubleNeg →^{tex} “AddDoubleNeg”]

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

RemoveDoubleNeg

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

[RemoveDoubleNeg →^{stmt} ZFsub ⊢ ∀\underline{a}: ˙¬ ˙¬ \underline{a} ⊢ \underline{a}]

[RemoveDoubleNeg →^{tex} “RemoveDoubleNeg”]

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

AndCommutativity

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

[AndCommutativity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}]

[AndCommutativity →^{tex} “AndCommutativity”]

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

AutoImply

[AutoImply →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\underline{a}: \underline{a} ⇒ \underline{a}]

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

Contrapositive

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

[Contrapositive →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⇒ ˙¬ \underline{a}]

[Contrapositive →^{tex} “Contrapositive”]

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

FirstConjunct

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

[FirstConjunct →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ \underline{a}]

[FirstConjunct →^{tex} “FirstConjunct”]

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

SecondConjunct

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

[SecondConjunct →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⊢ \underline{b}]

[SecondConjunct →^{tex} “SecondConjunct”]

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

FromContradiction

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

[FromContradiction →^{tex} “FromContradiction”]

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

FromDisjuncts

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

[FromDisjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ˙¬ \underline{a} ⇒ \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(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a}; AndCommutativity ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; Repetition ⊳ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

[IffCommutativity →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ ˙¬ \underline{b} ⇒ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}]

[IffCommutativity →^{tex} “IffCommutativity”]

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

IffFirst

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

[IffFirst →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{b} ⊢ \underline{a}]

[IffFirst →^{tex} “IffFirst”]

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

IffSecond

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

[IffSecond →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{b} ⇒ \underline{a} ⊢ \underline{a} ⊢ \underline{b}]

[IffSecond →^{tex} “IffSecond”]

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

ImplyTransitivity

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

[JoinConjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b} ⊢ ˙¬ \underline{a} ⇒ ˙¬ \underline{b}]

[JoinConjuncts →^{tex} “JoinConjuncts”]

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

MP2

[MP2 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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(⌈ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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(⌈ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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(⌈ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ Technicality ≫ ˙¬ ˙¬ \underline{a} ⇒ \underline{b}; NegativeMT ⊳ ˙¬ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{b} ≫ ˙¬ \underline{a}⌉, p_{0}, c)] [MT →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a}]

[MT →^{tex} “MT”]

[MT →^{pyk} “prop lemma mt”]

NegativeMT

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

Technicality

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

Weakening

[Weakening →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ \underline{a} ⇒ \underline{b}] [Weakening →^{tex} “Weakening”] [Weakening →^{pyk} “prop lemma weakening”]

WeakenOr1

[WeakenOr1 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ Weakening ⊳ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)] [WeakenOr1 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b}] [WeakenOr1 →^{tex} “WeakenOr1”] [WeakenOr1 →^{pyk} “prop lemma weaken or first”]

WeakenOr2

[WeakenOr2 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ FromContradiction ⊳ \underline{a} ⊳ ˙¬ \underline{a} ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⊢ \underline{b} ≫ \underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}; \underline{a} ⊢ MP⊳\underline{a} ⇒ ˙¬ \underline{a} ⇒ \underline{b}⊳\underline{a} ≫ ˙¬ \underline{a} ⇒ \underline{b}; Repetition⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)] [WeakenOr2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{a} ⇒ \underline{b}] [WeakenOr2 →^{tex} “WeakenOr2”] [WeakenOr2 →^{pyk} “prop lemma weaken or second”]

Formula2Pair

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

Pair2Formula

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

Formula2Union

[Formula2Union →^{tex} “Formula2Union”] [Formula2Union →^{pyk} “lemma formula2union”]

Union2Formula

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

Formula2Sep

[Formula2Sep →^{tex} “Formula2Sep”] [Formula2Sep →^{pyk} “lemma formula2separation”]

Sep2Formula

[Sep2Formula →^{tex} “Sep2Formula”] [Sep2Formula →^{pyk} “lemma separation2formula”]

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 →^{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} ZFsub ⊢ ∀\underline{(rx)}: \underline{(rx)} == \underline{(rx)}] [==Reflexivity →^{tex} “==\!{}Reflexivity”] [==Reflexivity →^{pyk} “lemma ==Reflexivity”]

==Symmetry

[==Symmetry →^{proof} Rule tactic] [==Symmetry →^{stmt} ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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 →^{tex} “SamePair”] [SamePair →^{pyk} “lemma same pair”]

SameSingleton

[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”]

(ϵ)

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

(fx)

[(fx) →^{tex} “(fx)”]

[(fx) →^{pyk} “var fx”]

(fy)

[(fy) →^{tex} “(fy)”]

[(fy) →^{pyk} “var fy”]

(fz)

[(fz) →^{tex} “(fz)”]

[(fz) →^{pyk} “var fz”]

(fv)

[(fv) →^{tex} “(fv)”]

[(fv) →^{pyk} “var fu”]

var fv

[var fv →^{pyk} “var fv”]

(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”]

ϵ

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

[ϵ →^{tex} “\epsilon”] [ϵ →^{pyk} “meta ep”]

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{(fv)}]⌉)]

[FU →^{tex} “FU”]

[FU →^{pyk} “meta fu”]

FV

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

[FV →^{tex} “FV”]

[FV →^{pyk} “meta fv”]

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”]

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) ¨= −1]⌉)]

[(−1) →^{tex} “(-1)”]

[(−1) →^{pyk} “(-1)”]

2

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

[2 →^{tex} “2”]

[2 →^{pyk} “2”]

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”]

0f

[0f →^{tex} “0f”]

[0f →^{pyk} “0f”]

1f

[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”]

leqReflexivity

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

leqAntisymmetryAxiom

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

leqAdditionAxiom

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

plus0

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

timesAssociativity

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

times1

[times1 →^{proof} Rule tactic] [times1 →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ 1 = \underline{x}] [times1 →^{tex} “times1”] [times1 →^{pyk} “axiom times1”]

Distribution

[Distribution →^{proof} Rule tactic] [Distribution →^{stmt} ZFsub ⊢ ∀\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} ZFsub ⊢ ˙¬ 0 = 1] [0not1 →^{tex} “0not1”] [0not1 →^{pyk} “axiom 0not1”]

equalityAxiom

[equalityAxiom →^{proof} Rule tactic] [equalityAxiom →^{stmt} ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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”]

SENC1

[SENC1 →^{proof} Rule tactic] [SENC1 →^{stmt} ZFsub ⊢ ∀\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} ZFsub ⊢ ∀\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)”]

IfThenElse(T)

[IfThenElse(T) →^{proof} Rule tactic] [IfThenElse(T) →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{x}] [IfThenElse(T) →^{tex} “IfThenElse(T)”] [IfThenElse(T) →^{pyk} “1rule ifThenElse true”]

IfThenElse(F)

[IfThenElse(F) →^{proof} Rule tactic] [IfThenElse(F) →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{a} ⊢ if(\underline{a}, \underline{x}, \underline{y}) = \underline{y}] [IfThenElse(F) →^{tex} “IfThenElse(F)”] [IfThenElse(F) →^{pyk} “1rule ifThenElse false”]

From = f

[From = f →^{proof} Rule tactic] [From = f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]] [From = f →^{tex} “From=f”] [From = f →^{pyk} “1rule from=f”]

To = f

[To = f →^{proof} Rule tactic] [To = f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \underline{(fx)} =_{f} \underline{(fy)}] [To = f →^{tex} “To=f”] [To = f →^{pyk} “1rule to=f”]

From < f

[From < f →^{proof} Rule tactic] [From < f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}] [From < f →^{tex} “From<f”] [From < f →^{pyk} “1rule from<f”]

To < f

[To < f →^{proof} Rule tactic] [To < f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(fx)} <_{f} \underline{(fy)}] [To < f →^{tex} “To<f”] [To < f →^{pyk} “1rule to<f”]

PlusF

[PlusF →^{proof} Rule tactic] [PlusF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]] [PlusF →^{tex} “PlusF”] [PlusF →^{pyk} “axiom plusF”]

TimesF

[TimesF →^{proof} Rule tactic] [TimesF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]] [TimesF →^{tex} “TimesF”] [TimesF →^{pyk} “axiom timesF”]

MinusF

[MinusF →^{proof} Rule tactic] [MinusF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: −_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}]] [MinusF →^{tex} “MinusF”] [MinusF →^{pyk} “axiom minusF”]

0f

[0f →^{proof} Rule tactic] [0f →^{stmt} ZFsub ⊢ ∀\underline{m}: 0f[\underline{m}] = 0]

[0f →^{tex} “0f”]

[0f →^{pyk} “axiom 0f”]

1f

[1f →^{proof} Rule tactic] [1f →^{stmt} ZFsub ⊢ ∀\underline{m}: 1f[\underline{m}] = 1]

[1f →^{tex} “1f”]

[1f →^{pyk} “axiom 1f”]

FromSF

[FromSF →^{proof} Rule tactic] [FromSF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}] [FromSF →^{tex} “FromSF”] [FromSF →^{pyk} “1rule fromSameF”]

ToSF

[ToSF →^{proof} Rule tactic] [ToSF →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊢ SF(\underline{(fx)}, \underline{(fy)})] [ToSF →^{tex} “ToSF”] [ToSF →^{pyk} “1rule toSameF”]

To == XX

[To == XX →^{proof} Rule tactic] [To == XX →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ SF(\underline{(fx)}, \underline{(fy)}) ⊢ \underline{(rx)} == \underline{(ry)}] [To == XX →^{tex} “To==XX”] [To == XX →^{pyk} “1rule to==XX”]

From ==

[From == →^{proof} Rule tactic] [From == →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)})] [From ==→^{tex} “From==”] [From == →^{pyk} “1rule from==”]

To ==

[To == →^{proof} Rule tactic] [To == →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ R(\underline{(fx)}) == R(\underline{(fy)})]

[To ==→^{tex} “To==”]

[To == →^{pyk} “1rule to==”]

From << XX

[From << XX →^{proof} Rule tactic] [From << XX →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}] [From << XX →^{tex} “From<<XX”] [From << XX →^{pyk} “1rule from<<XX”]

From << (1)

[From << (1) →^{proof} Rule tactic]

[From << (1) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ j_{Ex} ∈ \underline{(rx)}]

[From << (1) →^{tex} “From<<(1)”]

[From << (1) →^{pyk} “1rule from<<XX(1)”]

From << (2)

[From << (2) →^{proof} Rule tactic]

[From << (2) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ t_{Ex} ∈ \underline{(ry)}]

[From << (2) →^{tex} “From<<(2)”]

[From << (2) →^{pyk} “1rule from<<XX(2)”]

to << XX

[to << XX →^{proof} Rule tactic]

[to << XX →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊢ \underline{(rx)} << \underline{(ry)}]

[to << XX →^{tex} “to<<XX”]

[to << XX →^{pyk} “1rule to<<XX”]

From <<

[From << →^{proof} Rule tactic]

[From << →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ \underline{(fx)} <_{f} \underline{(fy)}]

[From <<→^{tex} “From<<”]

[From << →^{pyk} “1rule from<<”]

To <<

[To << →^{proof} Rule tactic] [To << →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} <_{f} \underline{(fy)} ⊢ R(\underline{(fx)}) << R(\underline{(fy)})]

[To <<→^{tex} “To<<”]

[To << →^{pyk} “1rule to<<”]

FromInR

[FromInR →^{proof} Rule tactic] [FromInR →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} ∈ R(\underline{(fy)}) ⊢ SF(\underline{(fx)}, \underline{(fy)})] [FromInR →^{tex} “FromInR”] [FromInR →^{pyk} “1rule fromInR”]

PlusR

[PlusR →^{proof} Rule tactic] [PlusR →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)})] [PlusR →^{tex} “PlusR”] [PlusR →^{pyk} “axiom plusR”]

TimesR

[TimesR →^{proof} Rule tactic] [TimesR →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)})] [TimesR →^{tex} “TimesR”] [TimesR →^{pyk} “axiom timesR”]

leqAntisymmetry

[leqAntisymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ leqAntisymmetryAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y}; MP2 ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}⌉, p_{0}, c)] [leqAntisymmetry →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ \underline{x} = \underline{y}] [leqAntisymmetry →^{tex} “leqAntisymmetry”] [leqAntisymmetry →^{pyk} “lemma leqAntisymmetry”]

leqTransitivity

[leqTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ leqTransitivityAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z}; MP2 ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z} ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}⌉, p_{0}, c)] [leqTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ \underline{x} <= \underline{z}] [leqTransitivity →^{tex} “leqTransitivity”] [leqTransitivity →^{pyk} “lemma leqTransitivity”]

leqAddition

[leqAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ leqAdditionAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; MP ⊳ \underline{x} <= \underline{y} ⇒ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}⌉, p_{0}, c)] [leqAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{x} + \underline{z} <= \underline{y} + \underline{z}] [leqAddition →^{tex} “leqAddition”] [leqAddition →^{pyk} “lemma leqAddition”]

leqMultiplication

[leqMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ leqMultiplicationAxiom ≫ 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}; MP2 ⊳ 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⊳ 0 <= \underline{z} ⊳ \underline{x} <= \underline{y} ≫ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}⌉, p_{0}, c)] [leqMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}] [leqMultiplication →^{tex} “leqMultiplication”] [leqMultiplication →^{pyk} “lemma leqMultiplication”]

Reciprocal

[Reciprocal →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} = 0 ⊢ ReciprocalAxiom ≫ ˙¬ \underline{x} = 0 ⇒ \underline{x} ∗ rec\underline{x} = 1; MP ⊳ ˙¬ \underline{x} = 0 ⇒ \underline{x} ∗ rec\underline{x} = 1 ⊳ ˙¬ \underline{x} = 0 ≫ \underline{x} ∗ rec\underline{x} = 1⌉, p_{0}, c)]

[Reciprocal →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} = 0 ⊢ \underline{x} ∗ rec\underline{x} = 1]

[Reciprocal →^{tex} “Reciprocal”]

[Reciprocal →^{pyk} “lemma reciprocal”]

Equality

[Equality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} = \underline{z} ⊢ equalityAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z}; MP2 ⊳ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z} ⊳ \underline{x} = \underline{y} ⊳ \underline{x} = \underline{z} ≫ \underline{y} = \underline{z}⌉, p_{0}, c)]

[Equality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} = \underline{z} ⊢ \underline{y} = \underline{z}]

[Equality →^{tex} “Equality”]

[Equality →^{pyk} “lemma equality”]

eqLeq

[eqLeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ eqLeqAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y}; MP ⊳ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⊳ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[eqLeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y}]

[eqLeq →^{tex} “eqLeq”]

[eqLeq →^{pyk} “lemma eqLeq”]

eqAddition

[eqAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqAdditionAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z}; MP ⊳ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z}⌉, p_{0}, c)]

[eqAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{z}]

[eqAddition →^{tex} “eqAddition”]

[eqAddition →^{pyk} “lemma eqAddition”]

eqMultiplication

[eqMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqMultiplicationAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}; MP ⊳ \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊳ \underline{x} = \underline{y} ≫ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}⌉, p_{0}, c)]

[eqMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}]

[eqMultiplication →^{tex} “eqMultiplication”]

[eqMultiplication →^{pyk} “lemma eqMultiplication”]

ToNegatedImply

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

[ToNegatedImply →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ ˙¬ \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b}]

[ToNegatedImply →^{tex} “ToNegatedImply”]

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

TND

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

[TND →^{stmt} ZFsub ⊢ ∀\underline{a}: ˙¬ \underline{a} ⇒ ˙¬ \underline{a}]

[TND →^{tex} “TND”]

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

ImplyNegation

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

[ImplyNegation →^{tex} “ImplyNegation”]

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

FromNegations

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

[FromNegations →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⇒ \underline{b} ⊢ \underline{b}]

[FromNegations →^{tex} “FromNegations”]

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

From3Disjuncts

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

[From3Disjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ˙¬ \underline{a} ⇒ ˙¬ \underline{b} ⇒ \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(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{c} ⇒ \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} ⇒ \underline{d} ⊳ \underline{c} ⇒ \underline{e} ⊳ \underline{d} ⇒ \underline{e} ≫ \underline{e}; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ ˙¬ \underline{a} ⊢ NegateDisjunct1 ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{a} ≫ \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} ⇒ \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} ⇒ \underline{d} ⊢ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⊢ \underline{a} ⊢ \underline{e} ≫ ˙¬ \underline{c} ⇒ \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} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \underline{d} ⊢ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊢ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⊢ ˙¬ \underline{a} ⊢ \underline{e} ≫ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{c} ⇒ \underline{d} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⇒ ˙¬ \underline{a} ⇒ \underline{e}; ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \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} ⇒ \underline{d} ⇒ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ⇒ \underline{a} ⇒ \underline{e} ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⊳ \underline{a} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{a} ⇒ \underline{d} ⇒ \underline{e} ≫ \underline{a} ⇒ \underline{e}; MP4 ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⇒ ˙¬ \underline{c} ⇒ \underline{d} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⇒ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ⇒ ˙¬ \underline{a} ⇒ \underline{e} ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{c} ⇒ \underline{d} ⊳ \underline{b} ⇒ \underline{c} ⇒ \underline{e} ⊳ \underline{b} ⇒ \underline{d} ⇒ \underline{e} ≫ ˙¬ \underline{a} ⇒ \underline{e}; FromNegations ⊳ \underline{a} ⇒ \underline{e} ⊳ ˙¬ \underline{a} ⇒ \underline{e} ≫ \underline{e}⌉, p_{0}, c)]

[From2 ∗ 2Disjuncts →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ∀\underline{e}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{c} ⇒ \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(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⊢ Repetition ⊳ ˙¬ \underline{a} ⇒ \underline{b} ≫ ˙¬ \underline{a} ⇒ \underline{b}; MP ⊳ ˙¬ \underline{a} ⇒ \underline{b} ⊳ ˙¬ \underline{a} ≫ \underline{b}⌉, p_{0}, c)]

[NegateDisjunct1 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{a} ⊢ \underline{b}]

[NegateDisjunct1 →^{tex} “NegateDisjunct1”]

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

NegateDisjunct2

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

[NegateDisjunct2 →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ \underline{a} ⇒ \underline{b} ⊢ ˙¬ \underline{b} ⊢ \underline{a}]

[NegateDisjunct2 →^{tex} “NegateDisjunct2”]

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

ExpandDisjuncts

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

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

[ExpandDisjuncts →^{tex} “ExpandDisjuncts”]

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

eqReflexivity

[eqReflexivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: leqReflexivity ≫ \underline{x} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{x} ⊳ \underline{x} <= \underline{x} ≫ \underline{x} = \underline{x}⌉, p_{0}, c)]

[eqReflexivity →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} = \underline{x}]

[eqReflexivity →^{tex} “eqReflexivity”]

[eqReflexivity →^{pyk} “lemma eqReflexivity”]

eqSymmetry

[eqSymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ eqReflexivity ≫ \underline{x} = \underline{x}; Equality ⊳ \underline{x} = \underline{y} ⊳ \underline{x} = \underline{x} ≫ \underline{y} = \underline{x}⌉, p_{0}, c)]

[eqSymmetry →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{x}]

[eqSymmetry →^{tex} “eqSymmetry”]

[eqSymmetry →^{pyk} “lemma eqSymmetry”]

eqTransitivity

[eqTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; Equality ⊳ \underline{y} = \underline{x} ⊳ \underline{y} = \underline{z} ≫ \underline{x} = \underline{z}⌉, p_{0}, c)]

[eqTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z}]

[eqTransitivity →^{tex} “eqTransitivity”]

[eqTransitivity →^{pyk} “lemma eqTransitivity”]

eqTransitivity4

[eqTransitivity4 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ eqTransitivity ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ≫ \underline{x} = \underline{z}; eqTransitivity ⊳ \underline{x} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ \underline{x} = \underline{u}⌉, p_{0}, c)]

[eqTransitivity4 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{x} = \underline{u}]

[eqTransitivity4 →^{tex} “eqTransitivity4”]

[eqTransitivity4 →^{pyk} “lemma eqTransitivity4”]

eqTransitivity5

[eqTransitivity5 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ eqTransitivity4 ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ \underline{x} = \underline{u}; eqTransitivity ⊳ \underline{x} = \underline{u} ⊳ \underline{u} = \underline{v} ≫ \underline{x} = \underline{v}⌉, p_{0}, c)]

[eqTransitivity5 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{x} = \underline{v}]

[eqTransitivity5 →^{tex} “eqTransitivity5”]

[eqTransitivity5 →^{pyk} “lemma eqTransitivity5”]

eqTransitivity6

[eqTransitivity6 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ∀\underline{w}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{v} = \underline{w} ⊢ eqTransitivity5 ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ⊳ \underline{z} = \underline{u} ⊳ \underline{u} = \underline{v} ≫ \underline{x} = \underline{v}; eqTransitivity ⊳ \underline{x} = \underline{v} ⊳ \underline{v} = \underline{w} ≫ \underline{x} = \underline{w}⌉, p_{0}, c)]

[eqTransitivity6 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ∀\underline{w}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{v} = \underline{w} ⊢ \underline{x} = \underline{w}]

[eqTransitivity6 →^{tex} “eqTransitivity6”]

[eqTransitivity6 →^{pyk} “lemma eqTransitivity6”]

plus0Left

[plus0Left →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: plus0 ≫ \underline{x} + 0 = \underline{x}; plusCommutativity ≫ 0 + \underline{x} = \underline{x} + 0; eqTransitivity ⊳ 0 + \underline{x} = \underline{x} + 0 ⊳ \underline{x} + 0 = \underline{x} ≫ 0 + \underline{x} = \underline{x}⌉, p_{0}, c)]

[plus0Left →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 + \underline{x} = \underline{x}] [plus0Left →^{tex} “plus0Left”]

[plus0Left →^{pyk} “lemma plus0Left”]

times1Left

[times1Left →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: times1 ≫ \underline{x} ∗ 1 = \underline{x}; timesCommutativity ≫ 1 ∗ \underline{x} = \underline{x} ∗ 1; eqTransitivity ⊳ 1 ∗ \underline{x} = \underline{x} ∗ 1 ⊳ \underline{x} ∗ 1 = \underline{x} ≫ 1 ∗ \underline{x} = \underline{x}⌉, p_{0}, c)]

[times1Left →^{stmt} ZFsub ⊢ ∀\underline{x}: 1 ∗ \underline{x} = \underline{x}]

[times1Left →^{tex} “times1Left”]

[times1Left →^{pyk} “lemma times1Left”]

lemma eqAdditionLeft

[lemma eqAdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqAddition ⊳ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z}; plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; plusCommutativity ≫ \underline{y} + \underline{z} = \underline{z} + \underline{y}; eqTransitivity4 ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{y} + \underline{z} = \underline{z} + \underline{y} ≫ \underline{z} + \underline{x} = \underline{z} + \underline{y}⌉, p_{0}, c)]

[lemma eqAdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} + \underline{x} = \underline{z} + \underline{y}]

[lemma eqAdditionLeft →^{pyk} “lemma eqAdditionLeft”]

EqMultiplicationLeft

[EqMultiplicationLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqMultiplication ⊳ \underline{x} = \underline{y} ≫ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}; timesCommutativity ≫ \underline{z} ∗ \underline{x} = \underline{x} ∗ \underline{z}; timesCommutativity ≫ \underline{y} ∗ \underline{z} = \underline{z} ∗ \underline{y}; eqTransitivity4 ⊳ \underline{z} ∗ \underline{x} = \underline{x} ∗ \underline{z} ⊳ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊳ \underline{y} ∗ \underline{z} = \underline{z} ∗ \underline{y} ≫ \underline{z} ∗ \underline{x} = \underline{z} ∗ \underline{y}⌉, p_{0}, c)]

[EqMultiplicationLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} ∗ \underline{x} = \underline{z} ∗ \underline{y}]

[EqMultiplicationLeft →^{tex} “EqMultiplicationLeft”]

[EqMultiplicationLeft →^{pyk} “lemma eqMultiplicationLeft”]

DistributionOut

[DistributionOut →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Distribution ≫ \underline{x} ∗ \underline{y} + \underline{z} = \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z}; eqSymmetry ⊳ \underline{x} ∗ \underline{y} + \underline{z} = \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} ≫ \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} = \underline{x} ∗ \underline{y} + \underline{z}⌉, p_{0}, c)]

[DistributionOut →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} + \underline{x} ∗ \underline{z} = \underline{x} ∗ \underline{y} + \underline{z}]

[DistributionOut →^{tex} “DistributionOut”]

[DistributionOut →^{pyk} “lemma distributionOut”]

Three2twoTerms

[Three2twoTerms →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} + \underline{z} = \underline{u} ⊢ lemma eqAdditionLeft ⊳ \underline{y} + \underline{z} = \underline{u} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u}; plusAssociativity ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{y} + \underline{z}; eqTransitivity ⊳ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{y} + \underline{z} ⊳ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u}⌉, p_{0}, c)]

[Three2twoTerms →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} + \underline{z} = \underline{u} ⊢ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{u}]

[Three2twoTerms →^{tex} “Three2twoTerms”]

[Three2twoTerms →^{pyk} “lemma three2twoTerms”]

Three2threeTerms

[Three2threeTerms →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: plusCommutativity ≫ \underline{y} + \underline{z} = \underline{z} + \underline{y}; Three2twoTerms ⊳ \underline{y} + \underline{z} = \underline{z} + \underline{y} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y}; plusAssociativity ≫ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y}; eqSymmetry ⊳ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y} ≫ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y}; eqTransitivity ⊳ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y} ⊳ \underline{x} + \underline{z} + \underline{y} = \underline{x} + \underline{z} + \underline{y} ≫ \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y}⌉, p_{0}, c)]

[Three2threeTerms →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} + \underline{z} = \underline{x} + \underline{z} + \underline{y}]

[Three2threeTerms →^{tex} “Three2threeTerms”]

[Three2threeTerms →^{pyk} “lemma three2threeTerms”]

Three2threeFactors

[Three2threeFactors →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} ∗ \underline{z} = \underline{u} ⊢ EqMultiplicationLeft⊳\underline{y}∗\underline{z} = \underline{u} ≫ \underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{u}; timesAssociativity ≫ \underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{y}∗\underline{z}; eqTransitivity⊳\underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{y}∗\underline{z}⊳\underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{u} ≫ \underline{x}∗\underline{y}∗\underline{z} = \underline{x}∗\underline{u}⌉, p_{0}, c)]

[Three2threeFactors →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{y} ∗ \underline{z} = \underline{u} ⊢ \underline{x} ∗ \underline{y} ∗ \underline{z} = \underline{x} ∗ \underline{u}] [Three2threeFactors →^{tex} “Three2threeFactors”]

[Three2threeFactors →^{pyk} “lemma three2twoFactors”]

AddEquations

[AddEquations →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{z} = \underline{u} ⊢ eqAddition ⊳ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z}; lemma eqAdditionLeft ⊳ \underline{z} = \underline{u} ≫ \underline{y} + \underline{z} = \underline{y} + \underline{u}; eqTransitivity ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{y} + \underline{z} = \underline{y} + \underline{u} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{u}⌉, p_{0}, c)]

[AddEquations →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{z} = \underline{u} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{u}]

[AddEquations →^{tex} “AddEquations”]

[AddEquations →^{pyk} “lemma addEquations”]

SubtractEquations

[SubtractEquations →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊢ \underline{z} = \underline{u} ⊢ eqAddition ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{u} ≫ \underline{x} + \underline{z} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z}; plus0Left ≫ 0 + \underline{z} = \underline{z}; eqTransitivity ⊳ 0 + \underline{z} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ 0 + \underline{z} = \underline{u}; PositiveToRight(Eq) ⊳ 0 + \underline{z} = \underline{u} ≫ 0 = \underline{u} + −\underline{z}; eqSymmetry ⊳ 0 = \underline{u} + −\underline{z} ≫ \underline{u} + −\underline{z} = 0; lemma eqAdditionLeft ⊳ \underline{u} + −\underline{z} = 0 ≫ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + 0; plusAssociativity ≫ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z}; plus0 ≫ \underline{y} + 0 = \underline{y}; eqTransitivity4 ⊳ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z} ⊳ \underline{y} + \underline{u} + −\underline{z} = \underline{y} + 0 ⊳ \underline{y} + 0 = \underline{y} ≫ \underline{y} + \underline{u} + −\underline{z} = \underline{y}; x = x + y − y ≫ \underline{x} = \underline{x} + \underline{z} + −\underline{z}; eqTransitivity4 ⊳ \underline{x} = \underline{x} + \underline{z} + −\underline{z} ⊳ \underline{x} + \underline{z} + −\underline{z} = \underline{y} + \underline{u} + −\underline{z} ⊳ \underline{y} + \underline{u} + −\underline{z} = \underline{y} ≫ \underline{x} = \underline{y}⌉, p_{0}, c)]

[SubtractEquations →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊢ \underline{z} = \underline{u} ⊢ \underline{x} = \underline{y}]

[SubtractEquations →^{tex} “SubtractEquations”]

[SubtractEquations →^{pyk} “lemma subtractEquations”]

SubtractEquationsLeft

[SubtractEquationsLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊢ \underline{x} = \underline{y} ⊢ plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; plusCommutativity ≫ \underline{y} + \underline{u} = \underline{u} + \underline{y}; eqTransitivity4 ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{u} ⊳ \underline{y} + \underline{u} = \underline{u} + \underline{y} ≫ \underline{z} + \underline{x} = \underline{u} + \underline{y}; SubtractEquations ⊳ \underline{z} + \underline{x} = \underline{u} + \underline{y} ⊳ \underline{x} = \underline{y} ≫ \underline{z} = \underline{u}⌉, p_{0}, c)]

[SubtractEquationsLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x}+\underline{z} = \underline{y}+\underline{u} ⊢ \underline{x} = \underline{y} ⊢ \underline{z} = \underline{u}]

[SubtractEquationsLeft →^{tex} “SubtractEquationsLeft”] [SubtractEquationsLeft →^{pyk} “lemma subtractEquationsLeft”]

EqNegated

[EqNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ Negative ≫ \underline{x} + −\underline{x} = 0; Negative ≫ \underline{y} + −\underline{y} = 0; eqSymmetry ⊳ \underline{y} + −\underline{y} = 0 ≫ 0 = \underline{y} + −\underline{y}; eqTransitivity ⊳ \underline{x} + −\underline{x} = 0 ⊳ 0 = \underline{y} + −\underline{y} ≫ \underline{x} + −\underline{x} = \underline{y} + −\underline{y}; SubtractEquationsLeft ⊳ \underline{x} + −\underline{x} = \underline{y} + −\underline{y} ⊳ \underline{x} = \underline{y} ≫ −\underline{x} = −\underline{y}⌉, p_{0}, c)]

[EqNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ −\underline{x} = −\underline{y}]

[EqNegated →^{tex} “EqNegated”]

[EqNegated →^{pyk} “lemma eqNegated”]

PositiveToRight(Eq)

[PositiveToRight(Eq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ eqAddition ⊳ \underline{x} + \underline{y} = \underline{z} ≫ \underline{x} + \underline{y} + −\underline{y} = \underline{z} + −\underline{y}; x = x + y − y ≫ \underline{x} = \underline{x}+\underline{y}+−\underline{y}; eqTransitivity⊳\underline{x} = \underline{x}+\underline{y}+−\underline{y}⊳\underline{x}+\underline{y}+−\underline{y} = \underline{z}+−\underline{y} ≫ \underline{x} = \underline{z}+−\underline{y}⌉, p_{0}, c)]

[PositiveToRight(Eq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z} + −\underline{y}]

[PositiveToRight(Eq) →^{tex} “PositiveToRight(Eq)”]

[PositiveToRight(Eq) →^{pyk} “lemma positiveToRight(Eq)”]

PositiveToLeft(Eq)(1term)

[PositiveToLeft(Eq)(1term) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ eqAddition ⊳ \underline{x} = \underline{y} ≫ \underline{x} + −\underline{y} = \underline{y} + −\underline{y}; Negative ≫ \underline{y} + −\underline{y} = 0; eqTransitivity ⊳ \underline{x} + −\underline{y} = \underline{y} + −\underline{y} ⊳ \underline{y} + −\underline{y} = 0 ≫ \underline{x} + −\underline{y} = 0⌉, p_{0}, c)]

[PositiveToLeft(Eq)(1term) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{x} + −\underline{y} = 0]

[PositiveToLeft(Eq)(1term) →^{tex} “PositiveToLeft(Eq)(1 term)”]

[PositiveToLeft(Eq)(1term) →^{pyk} “lemma positiveToLeft(Eq)(1 term)”]

NegativeToLeft(Eq)

[NegativeToLeft(Eq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} + −\underline{z} ⊢ eqAddition ⊳ \underline{x} = \underline{y} + −\underline{z} ≫ \underline{x} + \underline{z} = \underline{y} + −\underline{z} + \underline{z}; Three2threeTerms ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + \underline{z} + −\underline{z}; x = x + y − y ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y}; eqTransitivity4 ⊳ \underline{x} + \underline{z} = \underline{y} + −\underline{z} + \underline{z} ⊳ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + \underline{z} + −\underline{z} ⊳ \underline{y} + \underline{z} + −\underline{z} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y}⌉, p_{0}, c)]

[NegativeToLeft(Eq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} + −\underline{z} ⊢ \underline{x} + \underline{z} = \underline{y}]

[NegativeToLeft(Eq) →^{tex} “NegativeToLeft(Eq)”]

[NegativeToLeft(Eq) →^{pyk} “lemma negativeToLeft(Eq)”]

LessNeq

[LessNeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ Repetition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; SecondConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}⌉, p_{0}, c)]

[LessNeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{y}]

[LessNeq →^{tex} “LessNeq”]

[LessNeq →^{pyk} “lemma lessNeq”]

NeqSymmetry

[NeqSymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} = \underline{x} ⊢ eqSymmetry ⊳ \underline{y} = \underline{x} ≫ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} = \underline{x} ⊢ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x} ⇒ \underline{x} = \underline{y}; ˙¬ \underline{x} = \underline{y} ⊢ MT ⊳ \underline{y} = \underline{x} ⇒ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[NeqSymmetry →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} = \underline{x}]

[NeqSymmetry →^{tex} “NeqSymmetry”]

[NeqSymmetry →^{pyk} “lemma neqSymmetry”]

NeqNegated

[NeqNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ −\underline{x} = −\underline{y} ⊢ EqNegated ⊳ −\underline{x} = −\underline{y} ≫ − − \underline{x} = − − \underline{y}; DoubleMinus ≫ − − \underline{x} = \underline{x}; eqSymmetry ⊳ − − \underline{x} = \underline{x} ≫ \underline{x} = − − \underline{x}; DoubleMinus ≫ − − \underline{y} = \underline{y}; eqTransitivity4 ⊳ \underline{x} = − − \underline{x} ⊳ − − \underline{x} = − − \underline{y} ⊳ − − \underline{y} = \underline{y} ≫ \underline{x} = \underline{y}; FromContradiction⊳\underline{x} = \underline{y}⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ −\underline{x} = −\underline{y}; ∀\underline{x}: ∀\underline{y}: Ded⊳∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ −\underline{x} = −\underline{y} ⊢ ˙¬ − \underline{x} = −\underline{y} ≫ ˙¬ \underline{x} = \underline{y} ⇒ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y}; ˙¬ \underline{x} = \underline{y} ⊢ MP ⊳ ˙¬ \underline{x} = \underline{y} ⇒ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y}; ImplyNegation ⊳ −\underline{x} = −\underline{y} ⇒ ˙¬ − \underline{x} = −\underline{y} ≫ ˙¬ − \underline{x} = −\underline{y}⌉, p_{0}, c)]

[NeqNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ − \underline{x} = −\underline{y}]

[NeqNegated →^{tex} “NeqNegated”]

[NeqNegated →^{pyk} “lemma neqNegated”]

SubNeqRight

[SubNeqRight →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} = \underline{x} ⊢ NeqSymmetry ⊳ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{x} = \underline{z}; SubNeqLeft ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}; NeqSymmetry ⊳ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{z} = \underline{y}⌉, p_{0}, c)]

[SubNeqRight →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} = \underline{x} ⊢ ˙¬ \underline{z} = \underline{y}]

[SubNeqRight →^{tex} “SubNeqRight”]

[SubNeqRight →^{pyk} “lemma subNeqRight”]

SubNeqLeft

[SubNeqLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{z} ⊢ equalityAxiom ≫ \underline{y} = \underline{x} ⇒ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z}; eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; MP ⊳ \underline{y} = \underline{x} ⇒ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ⊳ \underline{y} = \underline{x} ≫ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z}; Contrapositive ⊳ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z} ⇒ ˙¬ \underline{y} = \underline{z}; MP ⊳ ˙¬ \underline{x} = \underline{z} ⇒ ˙¬ \underline{y} = \underline{z} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}⌉, p_{0}, c)]

[SubNeqLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{z} ⊢ ˙¬ \underline{y} = \underline{z}]

[SubNeqLeft →^{tex} “SubNeqLeft”]

[SubNeqLeft →^{pyk} “lemma subNeqLeft”]

NeqAddition

[NeqAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊢ eqReflexivity ≫ \underline{z} = \underline{z}; SubtractEquations ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ \underline{z} = \underline{z} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊢ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; ˙¬ \underline{x} = \underline{y} ⊢ MP ⊳ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⊳ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; ImplyNegation ⊳ \underline{x} + \underline{z} = \underline{y} + \underline{z} ⇒ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}⌉, p_{0}, c)]

[NeqAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}]

[NeqAddition →^{tex} “NeqAddition”]

[NeqAddition →^{pyk} “lemma neqAddition”]

NeqMultiplication

[NeqMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊢ ⊳ ˙¬ \underline{z} = 0 ≫ \underline{x} = \underline{x} ∗ \underline{z} ∗ rec\underline{z}; eqMultiplication ⊳ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ≫ \underline{x} ∗ \underline{z} ∗ rec\underline{z} = \underline{y}∗\underline{z}∗rec\underline{z}; ⊳ ˙¬ \underline{z} = 0 ≫ \underline{y} = \underline{y}∗\underline{z}∗rec\underline{z}; eqSymmetry⊳\underline{y} = \underline{y}∗\underline{z}∗rec\underline{z} ≫ \underline{y}∗\underline{z}∗rec\underline{z} = \underline{y}; eqTransitivity4 ⊳ \underline{x} = \underline{x} ∗ \underline{z} ∗ rec\underline{z} ⊳ \underline{x} ∗ \underline{z} ∗ rec\underline{z} = \underline{y} ∗ \underline{z} ∗ rec\underline{z} ⊳ \underline{y} ∗ \underline{z} ∗ rec\underline{z} = \underline{y} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} ∗ \underline{z} = \underline{y}∗\underline{z}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded⊳∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ⊢ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{z} = 0 ⇒ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ⇒ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}; ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ MP2⊳ ˙¬ \underline{z} = 0 ⇒ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⇒ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊳ ˙¬ \underline{z} = 0 ⊳ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⇒ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}; ImplyNegation⊳\underline{x}∗\underline{z} = \underline{y}∗\underline{z} ⇒ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}⌉, p_{0}, c)]

[NeqMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{z} = 0 ⊢ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}]

[NeqMultiplication →^{tex} “NeqMultiplication”]

[NeqMultiplication →^{pyk} “lemma neqMultiplication”]

UniqueNegative

[UniqueNegative →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = 0 ⊢ \underline{x} + \underline{z} = 0 ⊢ plusCommutativity ≫ \underline{y} + \underline{x} = \underline{x} + \underline{y}; eqTransitivity ⊳ \underline{y} + \underline{x} = \underline{x} + \underline{y} ⊳ \underline{x} + \underline{y} = 0 ≫ \underline{y} + \underline{x} = 0; PositiveToRight(Eq) ⊳ \underline{y} + \underline{x} = 0 ≫ \underline{y} = 0 + −\underline{x}; plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; eqTransitivity ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{x} + \underline{z} = 0 ≫ \underline{z} + \underline{x} = 0; PositiveToRight(Eq) ⊳ \underline{z} + \underline{x} = 0 ≫ \underline{z} = 0 + −\underline{x}; eqSymmetry ⊳ \underline{z} = 0 + −\underline{x} ≫ 0 + −\underline{x} = \underline{z}; eqTransitivity ⊳ \underline{y} = 0 + −\underline{x} ⊳ 0 + −\underline{x} = \underline{z} ≫ \underline{y} = \underline{z}⌉, p_{0}, c)]

[UniqueNegative →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = 0 ⊢ \underline{x} + \underline{z} = 0 ⊢ \underline{y} = \underline{z}]

[UniqueNegative →^{tex} “UniqueNegative”]

[UniqueNegative →^{pyk} “lemma uniqueNegative”]

DoubleMinus

[DoubleMinus →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: Negative ≫ −\underline{x} + − − \underline{x} = 0; x + y = zBackwards ⊳ −\underline{x} + − − \underline{x} = 0 ≫ 0 = − − \underline{x} + −\underline{x}; NegativeToLeft(Eq) ⊳ 0 = − − \underline{x} + −\underline{x} ≫ 0 + \underline{x} = − − \underline{x}; plus0Left ≫ 0 + \underline{x} = \underline{x}; Equality ⊳ 0 + \underline{x} = − − \underline{x} ⊳ 0 + \underline{x} = \underline{x} ≫ − − \underline{x} = \underline{x}⌉, p_{0}, c)]

[DoubleMinus →^{stmt} ZFsub ⊢ ∀\underline{x}: − − \underline{x} = \underline{x}]

[DoubleMinus →^{tex} “DoubleMinus”]

[DoubleMinus →^{pyk} “lemma doubleMinus”]

LeqLessEq

[LeqLessEq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ fromNotLess ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{y} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}; \underline{x} <= \underline{y} ⊢ MP ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}; Repetition ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}⌉, p_{0}, c)]

[LeqLessEq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{y}]

[LeqLessEq →^{tex} “LeqLessEq”]

[LeqLessEq →^{pyk} “lemma leqLessEq”]

LessLeq

[LessLeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ Repetition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[LessLeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y}]

[LessLeq →^{tex} “LessLeq”]

[LessLeq →^{pyk} “lemma lessLeq”]

FromLeqGeq

[FromLeqGeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{a} ⊢ \underline{y} <= \underline{x} ⇒ \underline{a} ⊢ leqTotality ≫ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ \underline{a} ⊳ \underline{y} <= \underline{x} ⇒ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[FromLeqGeq →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{a} ⊢ \underline{y} <= \underline{x} ⇒ \underline{a} ⊢ \underline{a}]

[FromLeqGeq →^{tex} “FromLeqGeq”]

[FromLeqGeq →^{pyk} “lemma from leqGeq”]

subLeqRight

[subLeqRight →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} <= \underline{x} ⊢ eqLeq ⊳ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; leqTransitivity ⊳ \underline{z} <= \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{z} <= \underline{y}⌉, p_{0}, c)]

[subLeqRight →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} <= \underline{x} ⊢ \underline{z} <= \underline{y}]

[subLeqRight →^{tex} “subLeqRight”]

[subLeqRight →^{pyk} “lemma subLeqRight”]

subLeqLeft

[subLeqLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{z} ⊢ eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; eqLeq ⊳ \underline{y} = \underline{x} ≫ \underline{y} <= \underline{x}; leqTransitivity ⊳ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{z} ≫ \underline{y} <= \underline{z}⌉, p_{0}, c)]

[subLeqLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{z} ⊢ \underline{y} <= \underline{z}]

[subLeqLeft →^{tex} “subLeqLeft”]

[subLeqLeft →^{pyk} “lemma subLeqLeft”]

Leq + 1

[Leq + 1 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; LessAdditionLeft ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ ˙¬ \underline{y} + 0 <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} + 0 = \underline{y} + 1; plus0 ≫ \underline{y} + 0 = \underline{y}; SubLessLeft ⊳ \underline{y} + 0 = \underline{y} ⊳ ˙¬ \underline{y} + 0 <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} + 0 = \underline{y} + 1 ≫ ˙¬ \underline{y} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} = \underline{y} + 1; leqLessTransitivity ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{y} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{y} = \underline{y} + 1 ≫ ˙¬ \underline{x} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} + 1⌉, p_{0}, c)]

[Leq + 1 →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} <= \underline{y} + 1 ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} + 1] [Leq + 1 →^{tex} “Leq+1”]

[Leq + 1 →^{pyk} “lemma leqPlus1”]

PositiveToRight(Leq)

[PositiveToRight(Leq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} <= \underline{z} ⊢ leqAddition ⊳ \underline{x} + \underline{y} <= \underline{z} ≫ \underline{x} + \underline{y} + −\underline{y} <= \underline{z} + −\underline{y}; x = x + y − y ≫ \underline{x} = \underline{x} + \underline{y} + −\underline{y}; eqSymmetry ⊳ \underline{x} = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} + \underline{y} + −\underline{y} = \underline{x}; subLeqLeft ⊳ \underline{x} + \underline{y} + −\underline{y} = \underline{x} ⊳ \underline{x} + \underline{y} + −\underline{y} <= \underline{z} + −\underline{y} ≫ \underline{x} <= \underline{z} + −\underline{y}⌉, p_{0}, c)]

[PositiveToRight(Leq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} <= \underline{z} ⊢ \underline{x} <= \underline{z} + −\underline{y}]

[PositiveToRight(Leq) →^{tex} “PositiveToRight(Leq)”]

[PositiveToRight(Leq) →^{pyk} “lemma positiveToRight(Leq)”]

PositiveToRight(Leq)(1term)

[PositiveToRight(Leq)(1term) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{y}: ∀\underline{z}: \underline{y} <= \underline{z} ⊢ plus0Left ≫ 0 + \underline{y} = \underline{y}; eqSymmetry ⊳ 0 + \underline{y} = \underline{y} ≫ \underline{y} = 0 + \underline{y}; subLeqLeft ⊳ \underline{y} = 0 + \underline{y} ⊳ \underline{y} <= \underline{z} ≫ 0 + \underline{y} <= \underline{z}; PositiveToRight(Leq) ⊳ 0 + \underline{y} <= \underline{z} ≫ 0 <= \underline{z} + −\underline{y}⌉, p_{0}, c)]

[PositiveToRight(Leq)(1term) →^{stmt} ZFsub ⊢ ∀\underline{y}: ∀\underline{z}: \underline{y} <= \underline{z} ⊢ 0 <= \underline{z} + −\underline{y}]

[PositiveToRight(Leq)(1term) →^{tex} “PositiveToRight(Leq)(1 term)”]

[PositiveToRight(Leq)(1term) →^{pyk} “lemma positiveToRight(Leq)(1 term)”]

negativeToLeft(Leq)

[negativeToLeft(Leq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} + −\underline{z} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} + −\underline{z} ≫ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z}; x = x + y − y ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; Three2threeTerms ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y} + −\underline{z} + \underline{z}; eqTransitivity ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ⊳ \underline{y} + \underline{z} + −\underline{z} = \underline{y} + −\underline{z} + \underline{z} ≫ \underline{y} = \underline{y} + −\underline{z} + \underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + −\underline{z} + \underline{z} ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y}; subLeqRight ⊳ \underline{y} + −\underline{z} + \underline{z} = \underline{y} ⊳ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z} ≫ \underline{x} + \underline{z} <= \underline{y}⌉, p_{0}, c)]

[negativeToLeft(Leq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} + −\underline{z} ⊢ \underline{x} + \underline{z} <= \underline{y}]

[negativeToLeft(Leq) →^{tex} “negativeToLeft(Leq)”]

[negativeToLeft(Leq) →^{pyk} “lemma negativeToLeft(Leq)”]

LeqAdditionLeft

[LeqAdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; plusCommutativity ≫ \underline{x} + \underline{z} = \underline{z} + \underline{x}; plusCommutativity ≫ \underline{y}+\underline{z} = \underline{z}+\underline{y}; subLeqLeft⊳\underline{x}+\underline{z} = \underline{z}+\underline{x}⊳\underline{x}+\underline{z} <= \underline{y}+\underline{z} ≫ \underline{z}+\underline{x} <= \underline{y}+\underline{z}; subLeqRight⊳\underline{y}+\underline{z} = \underline{z}+\underline{y}⊳\underline{z}+\underline{x} <= \underline{y}+\underline{z} ≫ \underline{z}+\underline{x} <= \underline{z}+\underline{y}⌉, p_{0}, c)]

[LeqAdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{z} + \underline{x} <= \underline{z} + \underline{y}]

[LeqAdditionLeft →^{tex} “LeqAdditionLeft”]

[LeqAdditionLeft →^{pyk} “lemma leqAdditionLeft”]

leqSubtraction

[leqSubtraction →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊢ leqAddition ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ≫ \underline{x} + \underline{z} + −\underline{z} <= \underline{y} + \underline{z} + −\underline{z}; x = x + y − y ≫ \underline{x} = \underline{x} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{x} = \underline{x} + \underline{z} + −\underline{z} ≫ \underline{x} + \underline{z} + −\underline{z} = \underline{x}; x = x + y − y ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y}; subLeqLeft ⊳ \underline{x} + \underline{z} + −\underline{z} = \underline{x} ⊳ \underline{x} + \underline{z} + −\underline{z} <= \underline{y} + \underline{z} + −\underline{z} ≫ \underline{x} <= \underline{y} + \underline{z} + −\underline{z}; subLeqRight ⊳ \underline{y} + \underline{z} + −\underline{z} = \underline{y} ⊳ \underline{x} <= \underline{y} + \underline{z} + −\underline{z} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[leqSubtraction →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊢ \underline{x} <= \underline{y}]

[leqSubtraction →^{tex} “leqSubtraction”]

[leqSubtraction →^{pyk} “lemma leqSubtraction”]

leqSubtractionLeft

[leqSubtractionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⊢ plusCommutativity ≫ \underline{z} + \underline{x} = \underline{x} + \underline{z}; plusCommutativity ≫ \underline{z} + \underline{y} = \underline{y} + \underline{z}; subLeqLeft ⊳ \underline{z} + \underline{x} = \underline{x} + \underline{z} ⊳ \underline{z} + \underline{x} <= \underline{z} + \underline{y} ≫ \underline{x} + \underline{z} <= \underline{z} + \underline{y}; subLeqRight ⊳ \underline{z} + \underline{y} = \underline{y} + \underline{z} ⊳ \underline{x} + \underline{z} <= \underline{z} + \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; leqSubtraction ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[leqSubtractionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⊢ \underline{x} <= \underline{y}]

[leqSubtractionLeft →^{tex} “leqSubtractionLeft”]

[leqSubtractionLeft →^{pyk} “lemma leqSubtractionLeft”]

thirdGeq

[thirdGeq →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ leqReflexivity ≫ \underline{y} <= \underline{y}; JoinConjuncts ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{y}; ExistIntro @ c_{Ex} @ \underline{y} ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{y} ≫ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ leqReflexivity ≫ \underline{x} <= \underline{x}; JoinConjuncts ⊳ \underline{x} <= \underline{x} ⊳ \underline{y} <= \underline{x} ≫ ˙¬ \underline{x} <= \underline{x} ⇒ ˙¬ \underline{y} <= \underline{x}; ExistIntro @ c_{Ex} @ \underline{x} ⊳ ˙¬ \underline{x} <= \underline{x} ⇒ ˙¬ \underline{y} <= \underline{x} ≫ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ≫ \underline{y} <= \underline{x} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}; leqTotality ≫ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex} ≫ ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}⌉, p_{0}, c)]

[thirdGeq →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= c_{Ex} ⇒ ˙¬ \underline{y} <= c_{Ex}]

[thirdGeq →^{tex} “thirdGeq”]

[thirdGeq →^{pyk} “lemma thirdGeq”]

LeqNegated

[LeqNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + −\underline{x} <= \underline{y} + −\underline{x}; Negative ≫ \underline{x} + −\underline{x} = 0; subLeqLeft ⊳ \underline{x} + −\underline{x} = 0 ⊳ \underline{x} + −\underline{x} <= \underline{y} + −\underline{x} ≫ 0 <= \underline{y} + −\underline{x}; plusCommutativity ≫ \underline{y} + −\underline{x} = −\underline{x} + \underline{y}; subLeqRight ⊳ \underline{y} + −\underline{x} = −\underline{x} + \underline{y} ⊳ 0 <= \underline{y} + −\underline{x} ≫ 0 <= −\underline{x} + \underline{y}; leqAddition ⊳ 0 <= −\underline{x} + \underline{y} ≫ 0 + −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y}; plus0Left ≫ 0 + −\underline{y} = −\underline{y}; x = x + y − y ≫ −\underline{x} = −\underline{x} + \underline{y} + −\underline{y}; eqSymmetry ⊳ −\underline{x} = −\underline{x} + \underline{y} + −\underline{y} ≫ −\underline{x} + \underline{y} + −\underline{y} = −\underline{x}; subLeqLeft ⊳ 0 + −\underline{y} = −\underline{y} ⊳ 0 + −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y} ≫ −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y}; subLeqRight ⊳ −\underline{x} + \underline{y} + −\underline{y} = −\underline{x} ⊳ −\underline{y} <= −\underline{x} + \underline{y} + −\underline{y} ≫ −\underline{y} <= −\underline{x}⌉, p_{0}, c)]

[LeqNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ −\underline{y} <= −\underline{x}]

[LeqNegated →^{tex} “LeqNegated”]

[LeqNegated →^{pyk} “lemma leqNegated”]

AddEquations(Leq)

[AddEquations(Leq) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; LeqAdditionLeft ⊳ \underline{z} <= \underline{u} ≫ \underline{y} + \underline{z} <= \underline{y} + \underline{u}; leqTransitivity ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊳ \underline{y} + \underline{z} <= \underline{y} + \underline{u} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{u}⌉, p_{0}, c)] [AddEquations(Leq) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ \underline{x} + \underline{z} <= \underline{y} + \underline{u}]

[AddEquations(Leq) →^{tex} “AddEquations(Leq)”]

[AddEquations(Leq) →^{pyk} “lemma addEquations(Leq)”]

ThirdGeqSeries

[ThirdGeqSeries →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ∀\underline{(ru)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(rz)} << \underline{(ru)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fv)} ∈ \underline{(ru)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From << XX ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(fx)} ∈ \underline{(rx)} ⊳ \underline{(fy)} ∈ \underline{(ry)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; From << XX ⊳ \underline{(rz)} << \underline{(ru)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ \underline{(fv)} ∈ \underline{(ru)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}; ExistIntro @ b_{Ex} @ a_{Ex} ⊳ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)} ≫ b_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}; thirdGeq ≫ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex}; FirstConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ a_{Ex} <= c_{Ex}; leqTransitivity ⊳ a_{Ex} <= c_{Ex} ⊳ c_{Ex} <= \underline{m} ≫ a_{Ex} <= \underline{m}; MP ⊳ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊳ a_{Ex} <= \underline{m} ≫ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; SecondConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ b_{Ex} <= c_{Ex}; leqTransitivity ⊳ b_{Ex} <= c_{Ex} ⊳ c_{Ex} <= \underline{m} ≫ b_{Ex} <= \underline{m}; MP ⊳ b_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)} ⊳ b_{Ex} <= \underline{m} ≫ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}; JoinConjuncts ⊳ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)} ≫ ˙¬ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}⌉, p_{0}, c)]

[ThirdGeqSeries →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ∀\underline{(ru)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(rz)} << \underline{(ru)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fv)} ∈ \underline{(ru)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ \underline{(fz)}[\underline{m}] <= \underline{(fv)}[\underline{m}] + −\underline{(ϵ)}]

[ThirdGeqSeries →^{tex} “ThirdGeqSeries”]

[ThirdGeqSeries →^{pyk} “lemma thirdGeqSeries”]

LeqNeqLess

[LeqNeqLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ JoinConjuncts ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; Repetition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}⌉, p_{0}, c)]

[LeqNeqLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}] [LeqNeqLess →^{tex} “LeqNeqLess”]

[LeqNeqLess →^{pyk} “lemma leqNeqLess”]

FromLess

[FromLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ toNotLess ⊳ \underline{y} <= \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ AddDoubleNeg ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; MT ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ ˙¬ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{y} <= \underline{x}⌉, p_{0}, c)]

[FromLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{x}]

[FromLess →^{tex} “FromLess”]

[FromLess →^{pyk} “lemma fromLess”]

ToLess

[ToLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⊢ fromNotLess ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ \underline{x} <= \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⊢ \underline{x} <= \underline{y} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⇒ \underline{x} <= \underline{y}; ˙¬ \underline{x} <= \underline{y} ⊢ NegativeMT ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⇒ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{x} <= \underline{y} ≫ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[ToLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}]

[ToLess →^{tex} “ToLess”]

[ToLess →^{pyk} “lemma toLess”]

fromNotLess

[fromNotLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y} ⊢ Repetition ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; RemoveDoubleNeg ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; MP ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ≫ ˙¬ ˙¬ \underline{x} = \underline{y}; RemoveDoubleNeg ⊳ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} = \underline{y}; eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; eqLeq ⊳ \underline{y} = \underline{x} ≫ \underline{y} <= \underline{x}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ MP ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; AutoImply ≫ \underline{y} <= \underline{x} ⇒ \underline{y} <= \underline{x}; leqTotality ≫ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{y} <= \underline{x} ⇒ \underline{y} <= \underline{x} ≫ \underline{y} <= \underline{x}⌉, p_{0}, c)]

[fromNotLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{x}]

[fromNotLess →^{tex} “fromNotLess”]

[fromNotLess →^{pyk} “lemma fromNotLess”]

toNotLess

[toNotLess →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ leqAntisymmetry ⊳ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{y} = \underline{x}; AddDoubleNeg ⊳ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} = \underline{x}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; \underline{x} <= \underline{y} ⊢ MP ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; AddDoubleNeg ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; Repetition ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; Repetition ⊳ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[toNotLess →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}]

[toNotLess →^{tex} “toNotLess”]

[toNotLess →^{pyk} “lemma toNotLess”]

NegativeLessPositive

[NegativeLessPositive →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ FirstConjunct ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ 0 <= \underline{x}; leqAddition ⊳ 0 <= \underline{x} ≫ 0 + −\underline{x} <= \underline{x} + −\underline{x}; plus0Left ≫ 0 + −\underline{x} = −\underline{x}; Negative ≫ \underline{x} + −\underline{x} = 0; subLeqLeft⊳0+−\underline{x} = −\underline{x}⊳0+−\underline{x} <= \underline{x}+−\underline{x} ≫ −\underline{x} <= \underline{x}+−\underline{x}; subLeqRight⊳ \underline{x} + −\underline{x} = 0 ⊳ −\underline{x} <= \underline{x} + −\underline{x} ≫ −\underline{x} <= 0; leqLessTransitivity ⊳ −\underline{x} <= 0 ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ − \underline{x} <= \underline{x} ⇒ ˙¬ ˙¬ − \underline{x} = \underline{x}⌉, p_{0}, c)]

[NegativeLessPositive →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ ˙¬ − \underline{x} <= \underline{x} ⇒ ˙¬ ˙¬ − \underline{x} = \underline{x}]

[NegativeLessPositive →^{tex} “NegativeLessPositive”]

[NegativeLessPositive →^{pyk} “lemma negativeLessPositive”]

leqLessTransitivity

[leqLessTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z} ⊢ FirstConjunct ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{y} <= \underline{z}; SecondConjunct⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}; subLeqLeft⊳\underline{x} = \underline{z}⊳\underline{x} <= \underline{y} ≫ \underline{z} <= \underline{y}; leqAntisymmetry⊳\underline{y} <= \underline{z}⊳\underline{z} <= \underline{y} ≫ \underline{y} = \underline{z}; FromContradiction⊳ \underline{y} = \underline{z} ⊳ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z} ⊢ ˙¬ \underline{x} = \underline{z} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z}; \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ MP2 ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z} ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z}; ImplyNegation ⊳ \underline{x} = \underline{z} ⇒ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z}; FirstConjunct ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{y} <= \underline{z}; leqTransitivity ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}; JoinConjuncts ⊳ \underline{x} <= \underline{z} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}⌉, p_{0}, c)]

[leqLessTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}]

[leqLessTransitivity →^{tex} “leqLessTransitivity”]

[leqLessTransitivity →^{pyk} “lemma leqLessTransitivity”]

LessLeqTransitivity

[LessLeqTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ \underline{z} = \underline{x} ⊢ FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; SecondConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; subLeqRight ⊳ \underline{z} = \underline{x} ⊳ \underline{y} <= \underline{z} ≫ \underline{y} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{z} = \underline{x}; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ \underline{z} = \underline{x} ⊢ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x}; ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ MP2 ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x}⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}⊳\underline{y} <= \underline{z} ≫ \underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x}; ImplyNegation⊳\underline{z} = \underline{x} ⇒ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} = \underline{x}; NeqSymmetry ⊳ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{x} = \underline{z}; FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; leqTransitivity ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}; JoinConjuncts ⊳ \underline{x} <= \underline{z} ⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}⌉, p_{0}, c)]

[LessLeqTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}]

[LessLeqTransitivity →^{tex} “LessLeqTransitivity”]

[LessLeqTransitivity →^{pyk} “lemma lessLeqTransitivity”]

LessTransitivity

[LessTransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ FirstConjunct ⊳ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ≫ \underline{y} <= \underline{z}; LessLeqTransitivity ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ \underline{y} <= \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}⌉, p_{0}, c)]

[LessTransitivity →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}]

[LessTransitivity →^{tex} “LessTransitivity”]

[LessTransitivity →^{pyk} “lemma lessTransitivity”]

LessTotality

[LessTotality →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ fromNotLess ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{y} <= \underline{x}; NeqSymmetry ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{y} = \underline{x}; LeqNeqLess ⊳ \underline{y} <= \underline{x} ⊳ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}; Repetition ⊳ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x} ≫ ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}⌉, p_{0}, c)]

[LessTotality →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{x} = \underline{y} ⇒ ˙¬ \underline{y} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{y} = \underline{x}]

[LessTotality →^{tex} “LessTotality”]

[LessTotality →^{pyk} “lemma lessTotality”]

SubLessRight

[SubLessRight →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ⊢ Repetition ⊳ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x}; FirstConjunct ⊳ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ≫ \underline{z} <= \underline{x}; subLeqRight ⊳ \underline{x} = \underline{y} ⊳ \underline{z} <= \underline{x} ≫ \underline{z} <= \underline{y}; SecondConjunct ⊳ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} = \underline{x}; SubNeqRight ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{z} = \underline{x} ≫ ˙¬ \underline{z} = \underline{y}; JoinConjuncts ⊳ \underline{z} <= \underline{y} ⊳ ˙¬ \underline{z} = \underline{y} ≫ ˙¬ \underline{z} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{z} = \underline{y}⌉, p_{0}, c)]

[SubLessRight →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{x} ⇒ ˙¬ ˙¬ \underline{z} = \underline{x} ⊢ ˙¬ \underline{z} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{z} = \underline{y}]

[SubLessRight →^{tex} “SubLessRight”] [SubLessRight →^{pyk} “lemma subLessRight”]

SubLessLeft

[SubLessLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ⊢ Repetition ⊳ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z}; FirstConjunct ⊳ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ≫ \underline{x} <= \underline{z}; subLeqLeft ⊳ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{z} ≫ \underline{y} <= \underline{z}; SecondConjunct⊳ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{x} = \underline{z}; SubNeqLeft⊳\underline{x} = \underline{y}⊳ ˙¬ \underline{x} = \underline{z} ≫ ˙¬ \underline{y} = \underline{z}; JoinConjuncts ⊳ \underline{y} <= \underline{z} ⊳ ˙¬ \underline{y} = \underline{z} ≫ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z}⌉, p_{0}, c)]

[SubLessLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} ⊢ ˙¬ \underline{y} <= \underline{z} ⇒ ˙¬ ˙¬ \underline{y} = \underline{z}]

[SubLessLeft →^{tex} “SubLessLeft”]

[SubLessLeft →^{pyk} “lemma subLessLeft”]

LessAddition

[LessAddition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessLeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; leqAddition ⊳ \underline{x} <= \underline{y} ≫ \underline{x} + \underline{z} <= \underline{y} + \underline{z}; LessNeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; NeqAddition ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; JoinConjuncts ⊳ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⊳ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}⌉, p_{0}, c)]

[LessAddition →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}]

[LessAddition →^{tex} “LessAddition”]

[LessAddition →^{pyk} “lemma lessAddition”]

LessAdditionLeft

[LessAdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessAddition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z}; plusCommutativity ≫ \underline{x} + \underline{z} = \underline{z} + \underline{x}; SubLessLeft ⊳ \underline{x} + \underline{z} = \underline{z} + \underline{x} ⊳ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{z} ≫ ˙¬ \underline{z} + \underline{x} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{y}+\underline{z}; plusCommutativity ≫ \underline{y}+\underline{z} = \underline{z}+\underline{y}; SubLessRight⊳\underline{y}+\underline{z} = \underline{z}+\underline{y}⊳ ˙¬ \underline{z}+\underline{x} <= \underline{y} + \underline{z} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{y} + \underline{z} ≫ ˙¬ \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{z} + \underline{y}⌉, p_{0}, c)]

[LessAdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} + \underline{x} <= \underline{z} + \underline{y} ⇒ ˙¬ ˙¬ \underline{z} + \underline{x} = \underline{z} + \underline{y}] [LessAdditionLeft →^{tex} “LessAdditionLeft”]

[LessAdditionLeft →^{pyk} “lemma lessAdditionLeft”]

LessMultiplication

[LessMultiplication →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessLeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; LessLeq ⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ≫ 0 <= \underline{z}; leqMultiplication ⊳ 0 <= \underline{z} ⊳ \underline{x} <= \underline{y} ≫ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z}; LessNeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; LessNeq ⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ≫ ˙¬ 0 = \underline{z}; NeqSymmetry ⊳ ˙¬ 0 = \underline{z} ≫ ˙¬ \underline{z} = 0; NeqMultiplication ⊳ ˙¬ \underline{z} = 0 ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}; LeqNeqLess ⊳ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⊳ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ≫ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}⌉, p_{0}, c)]

[LessMultiplication →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z}]

[LessMultiplication →^{tex} “LessMultiplication”]

[LessMultiplication →^{pyk} “lemma lessMultiplication”]

LessMultiplicationLeft

[LessMultiplicationLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessMultiplication⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z}⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x}∗\underline{z} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z}; timesCommutativity ≫ \underline{x}∗\underline{z} = \underline{z} ∗ \underline{x}; timesCommutativity ≫ \underline{y} ∗ \underline{z} = \underline{z} ∗ \underline{y}; SubLessLeft ⊳ \underline{x} ∗ \underline{z} = \underline{z} ∗ \underline{x} ⊳ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{x}∗\underline{z} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{z}∗\underline{x} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{z}∗\underline{x} = \underline{y}∗\underline{z}; SubLessRight⊳\underline{y}∗\underline{z} = \underline{z}∗\underline{y}⊳ ˙¬ \underline{z}∗\underline{x} <= \underline{y}∗\underline{z} ⇒ ˙¬ ˙¬ \underline{z}∗\underline{x} = \underline{y}∗\underline{z} ≫ ˙¬ \underline{z}∗\underline{x} <= \underline{z}∗\underline{y} ⇒ ˙¬ ˙¬ \underline{z}∗\underline{x} = \underline{z}∗\underline{y}⌉, p_{0}, c)]

[LessMultiplicationLeft →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} ∗ \underline{x} <= \underline{z} ∗ \underline{y} ⇒ ˙¬ ˙¬ \underline{z} ∗ \underline{x} = \underline{z} ∗ \underline{y}]

[LessMultiplicationLeft →^{tex} “LessMultiplicationLeft”]

[LessMultiplicationLeft →^{pyk} “lemma lessMultiplicationLeft”]

LessDivision

[LessDivision →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊢ FromLess ⊳ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ≫ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z}; leqMultiplicationAxiom ≫ 0 <= \underline{z} ⇒ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z}; MP ⊳ 0 <= \underline{z} ⇒ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ⊳ 0 <= \underline{z} ≫ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z}; Contrapositive ⊳ \underline{y} <= \underline{x} ⇒ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ≫ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ⇒ ˙¬ \underline{y} <= \underline{x}; MP ⊳ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ⇒ ˙¬ \underline{y} <= \underline{x} ⊳ ˙¬ \underline{y} ∗ \underline{z} <= \underline{x} ∗ \underline{z} ≫ ˙¬ \underline{y} <= \underline{x}; ToLess ⊳ ˙¬ \underline{y} <= \underline{x} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}⌉, p_{0}, c)]

[LessDivision →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ ˙¬ \underline{x} ∗ \underline{z} <= \underline{y} ∗ \underline{z} ⇒ ˙¬ ˙¬ \underline{x} ∗ \underline{z} = \underline{y} ∗ \underline{z} ⊢ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}]

[LessDivision →^{tex} “LessDivision”]

[LessDivision →^{pyk} “lemma lessDivision”]

AddEquations(Less)

[AddEquations(Less) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{u} ⇒ ˙¬ ˙¬ \underline{z} = \underline{u} ⊢ LessAddition ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} + \underline{z} <= \underline{y}+\underline{z} ⇒ ˙¬ ˙¬ \underline{x}+\underline{z} = \underline{y}+\underline{z}; LessAdditionLeft⊳ ˙¬ \underline{z} <= \underline{u} ⇒ ˙¬ ˙¬ \underline{z} = \underline{u} ≫ ˙¬ \underline{y}+\underline{z} <= \underline{y}+\underline{u} ⇒ ˙¬ ˙¬ \underline{y}+\underline{z} = \underline{y}+\underline{u}; LessTransitivity⊳ ˙¬ \underline{x}+\underline{z} <= \underline{y}+\underline{z} ⇒ ˙¬ ˙¬ \underline{x}+\underline{z} = \underline{y}+\underline{z}⊳ ˙¬ \underline{y}+\underline{z} <= \underline{y}+\underline{u} ⇒ ˙¬ ˙¬ \underline{y}+\underline{z} = \underline{y}+\underline{u} ≫ ˙¬ \underline{x}+\underline{z} <= \underline{y}+\underline{u} ⇒ ˙¬ ˙¬ \underline{x}+\underline{z} = \underline{y}+\underline{u}⌉, p_{0}, c)]

[AddEquations(Less) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ \underline{z} <= \underline{u} ⇒ ˙¬ ˙¬ \underline{z} = \underline{u} ⊢ ˙¬ \underline{x} + \underline{z} <= \underline{y} + \underline{u} ⇒ ˙¬ ˙¬ \underline{x} + \underline{z} = \underline{y} + \underline{u}]

[AddEquations(Less) →^{tex} “AddEquations(Less)”]

[AddEquations(Less) →^{pyk} “lemma addEquations(Less)”]

LessNegated

[LessNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ LessLeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; LeqNegated ⊳ \underline{x} <= \underline{y} ≫ −\underline{y} <= −\underline{x}; LessNeq ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ \underline{x} = \underline{y}; NeqNegated ⊳ ˙¬ \underline{x} = \underline{y} ≫ ˙¬ − \underline{x} = −\underline{y}; NeqSymmetry ⊳ ˙¬ − \underline{x} = −\underline{y} ≫ ˙¬ − \underline{y} = −\underline{x}; LeqNeqLess ⊳ −\underline{y} <= −\underline{x} ⊳ ˙¬ − \underline{y} = −\underline{x} ≫ ˙¬ − \underline{y} <= −\underline{x} ⇒ ˙¬ ˙¬ − \underline{y} = −\underline{x}⌉, p_{0}, c)]

[LessNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊢ ˙¬ − \underline{y} <= −\underline{x} ⇒ ˙¬ ˙¬ − \underline{y} = −\underline{x}]

[LessNegated →^{tex} “LessNegated”]

[LessNegated →^{pyk} “lemma lessNegated”]

PositiveNegated

[PositiveNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ LessNegated ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ − \underline{x} <= −0 ⇒ ˙¬ ˙¬ − \underline{x} = −0; −0 = 0 ≫ −0 = 0; SubLessRight ⊳ −0 = 0 ⊳ ˙¬ − \underline{x} <= −0 ⇒ ˙¬ ˙¬ − \underline{x} = −0 ≫ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0⌉, p_{0}, c)]

[PositiveNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0]

[PositiveNegated →^{tex} “PositiveNegated”]

[PositiveNegated →^{pyk} “lemma positiveNegated”]

NonpositiveNegated

[NonpositiveNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ LeqNegated ⊳ \underline{x} <= 0 ≫ −0 <= −\underline{x}; −0 = 0 ≫ −0 = 0; subLeqLeft ⊳ −0 = 0 ⊳ −0 <= −\underline{x} ≫ 0 <= −\underline{x}⌉, p_{0}, c)]

[NonpositiveNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ 0 <= −\underline{x}]

[NonpositiveNegated →^{tex} “NonpositiveNegated”]

[NonpositiveNegated →^{pyk} “lemma nonpositiveNegated”]

NegativeNegated

[NegativeNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ LessNegated ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ − 0 <= −\underline{x} ⇒ ˙¬ ˙¬ − 0 = −\underline{x}; −0 = 0 ≫ −0 = 0; SubLessLeft ⊳ −0 = 0 ⊳ ˙¬ − 0 <= −\underline{x} ⇒ ˙¬ ˙¬ − 0 = −\underline{x} ≫ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}⌉, p_{0}, c)]

[NegativeNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}]

[NegativeNegated →^{tex} “NegativeNegated”]

[NegativeNegated →^{pyk} “lemma negativeNegated”]

NonnegativeNegated

[NonnegativeNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ LeqNegated ⊳ 0 <= \underline{x} ≫ −\underline{x} <= −0; −0 = 0 ≫ −0 = 0; subLeqRight ⊳ −0 = 0 ⊳ −\underline{x} <= −0 ≫ −\underline{x} <= 0⌉, p_{0}, c)]

[NonnegativeNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ −\underline{x} <= 0]

[NonnegativeNegated →^{tex} “NonnegativeNegated”]

[NonnegativeNegated →^{pyk} “lemma nonnegativeNegated”]

PositiveHalved

[PositiveHalved →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ 0 < 1/2 ≫ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1; LessMultiplicationLeft ⊳ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ rec1 + 1 ∗ 0 <= rec1 + 1 ∗ \underline{x} ⇒ ˙¬ ˙¬ rec1 + 1 ∗ 0 = rec1 + 1 ∗ \underline{x}; x ∗ 0 = 0 ≫ rec1 + 1 ∗ 0 = 0; SubLessLeft ⊳ rec1 + 1 ∗ 0 = 0 ⊳ ˙¬ rec1 + 1 ∗ 0 <= rec1 + 1 ∗ \underline{x} ⇒ ˙¬ ˙¬ rec1+1∗0 = rec1+1∗\underline{x} ≫ ˙¬ 0 <= rec1+1∗\underline{x} ⇒ ˙¬ ˙¬ 0 = rec1+1∗\underline{x}⌉, p_{0}, c)]

[PositiveHalved →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ ˙¬ 0 <= rec1 + 1 ∗ \underline{x} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{x}]

[PositiveHalved →^{tex} “PositiveHalved”]

[PositiveHalved →^{pyk} “lemma positiveHalved”]

NonnegativeNumerical

[NonnegativeNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ IfThenElse(T) ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; Repetition ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}⌉, p_{0}, c)]

[NonnegativeNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}]

[NonnegativeNumerical →^{tex} “NonnegativeNumerical”]

[NonnegativeNumerical →^{pyk} “lemma nonnegativeNumerical”]

NegativeNumerical

[NegativeNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ FromLess ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ 0 <= \underline{x}; IfThenElse(F) ⊳ ˙¬ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; Repetition ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}⌉, p_{0}, c)]

[NegativeNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}] [NegativeNumerical →^{tex} “NegativeNumerical”]

[NegativeNumerical →^{pyk} “lemma negativeNumerical”]

PositiveNumerical

[PositiveNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ LessLeq ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ 0 <= \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}⌉, p_{0}, c)]

[PositiveNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}]

[PositiveNumerical →^{tex} “PositiveNumerical”]

[PositiveNumerical →^{pyk} “lemma positiveNumerical”]

lemma nonpositiveNumerical

[lemma nonpositiveNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ NegativeNumerical ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; ∀\underline{x}: \underline{x} = 0 ⊢ eqSymmetry ⊳ \underline{x} = 0 ≫ 0 = \underline{x}; eqLeq ⊳ 0 = \underline{x} ≫ 0 <= \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; −0 = 0 ≫ −0 = 0; eqSymmetry ⊳ −0 = 0 ≫ 0 = −0; EqNegated ⊳ 0 = \underline{x} ≫ −0 = −\underline{x}; eqTransitivity5⊳if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}⊳\underline{x} = 0⊳0 = −0⊳−0 = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; ∀\underline{x}: Ded ⊳ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; Ded ⊳ ∀\underline{x}: \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; \underline{x} <= 0 ⊢ LeqLessEq ⊳ \underline{x} <= 0 ≫ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ \underline{x} = 0; FromDisjuncts ⊳ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ \underline{x} = 0 ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ⊳ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}⌉, p_{0}, c)]

[lemma nonpositiveNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}]

[lemma nonpositiveNumerical →^{pyk} “lemma nonpositiveNumerical”]

|0| = 0

[|0| = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ leqReflexivity ≫ 0 <= 0; NonnegativeNumerical ⊳ 0 <= 0 ≫ if(0 <= 0, 0, −0) = 0⌉, p_{0}, c)]

[|0| = 0 →^{stmt} ZFsub ⊢ if(0 <= 0, 0, −0) = 0]

[|0| = 0 →^{tex} “|0|=0”]

[|0| = 0 →^{pyk} “lemma |0|=0”]

0 <= |x|

[0 <= |x| →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ≫ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}); subLeqRight ⊳ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ 0 <= \underline{x} ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); ∀\underline{x}: ˙¬ 0 <= \underline{x} ⊢ ToLess ⊳ ˙¬ 0 <= \underline{x} ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0; NegativeNumerical ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ≫ −\underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}); NegativeNegated ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}; LessLeq ⊳ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x} ≫ 0 <= −\underline{x}; subLeqRight ⊳ −\underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ 0 <= −\underline{x} ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); ∀\underline{x}: Ded ⊳ ∀\underline{x}: 0 <= \underline{x} ⊢ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); Ded ⊳ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⊢ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ ˙¬ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); FromNegations ⊳ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ ˙¬ 0 <= \underline{x} ⇒ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x})⌉, p_{0}, c)]

[0 <= |x| →^{stmt} ZFsub ⊢ ∀\underline{x}: 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x})]

[0 <= |x| →^{tex} “0<=|x|”]

[0 <= |x| →^{pyk} “lemma 0<=|x|”]

SameNumerical

[SameNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; subLeqRight ⊳ \underline{x} = \underline{y} ⊳ 0 <= \underline{x} ≫ 0 <= \underline{y}; NonnegativeNumerical ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y}; eqSymmetry ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y} ≫ \underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}); eqTransitivity4 ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: ˙¬ 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ ToLess ⊳ ˙¬ 0 <= \underline{x} ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0; NegativeNumerical ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x}; SubLessLeft ⊳ \underline{x} = \underline{y} ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ \underline{y} <= 0 ⇒ ˙¬ ˙¬ \underline{y} = 0; NegativeNumerical ⊳ ˙¬ \underline{y} <= 0 ⇒ ˙¬ ˙¬ \underline{y} = 0 ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y}; eqSymmetry ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y} ≫ −\underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}); EqNegated ⊳ \underline{x} = \underline{y} ≫ −\underline{x} = −\underline{y}; eqTransitivity4 ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = −\underline{x} ⊳ −\underline{x} = −\underline{y} ⊳ −\underline{y} = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromNegations ⊳ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ ˙¬ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); MP ⊳ \underline{x} = \underline{y} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{x} = \underline{y} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[SameNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[SameNumerical →^{tex} “SameNumerical”]

[SameNumerical →^{pyk} “lemma sameNumerical”]

SignNumerical(+)

[SignNumerical(+) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ PositiveNumerical ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; PositiveNegated ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0; NegativeNumerical ⊳ ˙¬ − \underline{x} <= 0 ⇒ ˙¬ ˙¬ − \underline{x} = 0 ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = −−\underline{x}; DoubleMinus ≫ −−\underline{x} = \underline{x}; eqTransitivity⊳if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) = −−\underline{x}⊳ −−\underline{x} = \underline{x} ≫ if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) = \underline{x}; eqSymmetry⊳if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) = \underline{x} ≫ \underline{x} = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); eqTransitivity ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ \underline{x} = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})⌉, p_{0}, c)]

[SignNumerical(+) →^{stmt} ZFsub ⊢ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})]

[SignNumerical(+) →^{tex} “SignNumerical(+)”]

[SignNumerical(+) →^{pyk} “lemma signNumerical(+)”]

SignNumerical

[SignNumerical →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ NegativeNegated ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ≫ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x}; SignNumerical(+) ⊳ ˙¬ 0 <= −\underline{x} ⇒ ˙¬ ˙¬ 0 = −\underline{x} ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= −−\underline{x}, −−\underline{x}, −−−\underline{x}); DoubleMinus ≫ −−\underline{x} = \underline{x}; SameNumerical⊳−−\underline{x} = \underline{x} ≫ if(0 <= − − \underline{x}, − − \underline{x}, − − −\underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}); eqTransitivity ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= − − \underline{x}, − − \underline{x}, − − −\underline{x}) ⊳ if(0 <= − − \underline{x}, − − \underline{x}, − − −\underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}); eqSymmetry ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); ∀\underline{x}: \underline{x} = 0 ⊢ EqNegated ⊳ \underline{x} = 0 ≫ −\underline{x} = −0; −0 = 0 ≫ −0 = 0; eqSymmetry ⊳ \underline{x} = 0 ≫ 0 = \underline{x}; eqTransitivity4 ⊳ −\underline{x} = −0 ⊳ −0 = 0 ⊳ 0 = \underline{x} ≫ −\underline{x} = \underline{x}; eqSymmetry ⊳ −\underline{x} = \underline{x} ≫ \underline{x} = −\underline{x}; SameNumerical ⊳ \underline{x} = −\underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ SignNumerical(+) ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); ∀\underline{x}: Ded ⊳ ∀\underline{x}: ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); Ded ⊳ ∀\underline{x}: \underline{x} = 0 ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); Ded ⊳ ∀\underline{x}: ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊢ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); LessTotality ≫ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ ˙¬ \underline{x} = 0 ⇒ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x}; From3Disjuncts ⊳ ˙¬ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ ˙¬ \underline{x} = 0 ⇒ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⊳ ˙¬ \underline{x} <= 0 ⇒ ˙¬ ˙¬ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ \underline{x} = 0 ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ ˙¬ 0 <= \underline{x} ⇒ ˙¬ ˙¬ 0 = \underline{x} ⇒ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})⌉, p_{0}, c)]

[SignNumerical →^{stmt} ZFsub ⊢ ∀\underline{x}: if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x})]

[SignNumerical →^{tex} “SignNumerical”]

[SignNumerical →^{pyk} “lemma signNumerical”]

NumericalDifference

[NumericalDifference →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: SignNumerical ≫ if(0 <= \underline{x}+−\underline{y}, \underline{x}+−\underline{y}, −\underline{x}+−\underline{y}) = if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+−\underline{y}, −−\underline{x}+−\underline{y}); MinusNegated ≫ −\underline{x}+−\underline{y} = \underline{y}+−\underline{x}; SameNumerical⊳−\underline{x}+−\underline{y} = \underline{y}+−\underline{x} ≫ if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+ −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x}); eqTransitivity ⊳ if(0 <= \underline{x} + −\underline{y}, \underline{x} + −\underline{y}, −\underline{x} + −\underline{y}) = if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x}) ≫ if(0 <= \underline{x} + −\underline{y}, \underline{x} + −\underline{y}, −\underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x})⌉, p_{0}, c)]

[NumericalDifference →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{x} + −\underline{y}, \underline{x} + −\underline{y}, −\underline{x} + −\underline{y}) = if(0 <= \underline{y} + −\underline{x}, \underline{y} + −\underline{x}, −\underline{y} + −\underline{x})]

[NumericalDifference →^{tex} “NumericalDifference”]

[NumericalDifference →^{pyk} “lemma numericalDifference”]

SplitNumericalSumHelper

[SplitNumericalSumHelper →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+−\underline{y}, −−\underline{x}+−\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, −−\underline{x})+if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}) ⊢ SignNumerical ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); SignNumerical ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); AddEquations ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); −x − y = −(x + y) ≫ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}; SameNumerical ⊳ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y} ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}); SignNumerical ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{x}+\underline{y}, −\underline{x}+\underline{y}, −−\underline{x}+\underline{y}) ≫ if(0 <= −\underline{x}+\underline{y}, −\underline{x}+\underline{y}, −−\underline{x}+\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); eqTransitivity ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) ⊳ if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); subLeqRight ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); subLeqLeft ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[SplitNumericalSumHelper →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= −\underline{x}+−\underline{y}, −\underline{x}+−\underline{y}, −−\underline{x}+−\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, −−\underline{x})+if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}) ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[SplitNumericalSumHelper →^{tex} “SplitNumericalSumHelper”]

[SplitNumericalSumHelper →^{pyk} “lemma splitNumericalSumHelper”]

splitNumericalSum(++)

[splitNumericalSum(++) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ AddEquations(Leq) ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ 0 + 0 <= \underline{x} + \underline{y}; plus0 ≫ 0 + 0 = 0; subLeqLeft⊳0+0 = 0⊳0+0 <= \underline{x}+\underline{y} ≫ 0 <= \underline{x}+\underline{y}; NonnegativeNumerical⊳ 0 <= \underline{x}+\underline{y} ≫ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) = \underline{x}+\underline{y}; NonnegativeNumerical⊳0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y}; AddEquations ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y}; eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y} ≫ \underline{x} + \underline{y} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); eqTransitivity ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = \underline{x} + \underline{y} ⊳ \underline{x} + \underline{y} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); eqLeq ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(++) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})] [splitNumericalSum(++) →^{tex} “splitNumericalSum(++)”]

[splitNumericalSum(++) →^{pyk} “lemma splitNumericalSum(++)”]

splitNumericalSum(−−)

[splitNumericalSum(−−) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ NonpositiveNegated ⊳ \underline{x} <= 0 ≫ 0 <= −\underline{x}; NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= −\underline{y}; splitNumericalSum(++) ⊳ 0 <= −\underline{x} ⊳ 0 <= −\underline{y} ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SplitNumericalSumHelper ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(−−) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(−−) →^{tex} “splitNumericalSum(--)”]

[splitNumericalSum(−−) →^{pyk} “lemma splitNumericalSum(--)”]

splitNumericalSum(+ − small)

[splitNumericalSum(+ − small) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ LeqAdditionLeft ⊳ \underline{y} <= 0 ≫ \underline{x} + \underline{y} <= \underline{x} + 0; plus0 ≫ \underline{x} + 0 = \underline{x}; subLeqRight ⊳ \underline{x} + 0 = \underline{x} ⊳ \underline{x} + \underline{y} <= \underline{x} + 0 ≫ \underline{x} + \underline{y} <= \underline{x}; PositiveToRight(Leq)(1term) ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}); lemma nonpositiveNumerical ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y}; EqNegated ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = −\underline{y} ≫ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = − − \underline{y}; DoubleMinus ≫ − − \underline{y} = \underline{y}; eqTransitivity ⊳ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = − − \underline{y} ⊳ −−\underline{y} = \underline{y} ≫ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y}; NonnegativeNumerical⊳0 <= \underline{x} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x}; AddEquations ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ⊳ −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{y} ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y}; subLeqRight ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}) = \underline{x} + \underline{y} ⊳ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + −if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} + \underline{y}; NonnegativeNumerical ⊳ 0 <= \underline{x} + \underline{y} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = \underline{x} + \underline{y}; eqSymmetry ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = \underline{x} + \underline{y} ≫ \underline{x} + \underline{y} = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = \underline{x} ≫ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}); subLeqLeft ⊳ \underline{x} + \underline{y} = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ \underline{x} + \underline{y} <= \underline{x} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= \underline{x}; subLeqRight ⊳ \underline{x} = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= \underline{x} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})⌉, p_{0}, c)]

[splitNumericalSum(+ − small) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})]

[splitNumericalSum(+ − small) →^{tex} “splitNumericalSum(+-small)”]

[splitNumericalSum(+ − small) →^{pyk} “lemma splitNumericalSum(+-, smallNegative)”]

splitNumericalSum(+ − big)

[splitNumericalSum(+ − big) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊢ NonnegativeNegated ⊳ 0 <= \underline{x} ≫ −\underline{x} <= 0; NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= −\underline{y}; SignNumerical ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}); SubLessLeft ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) ⊳ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); SignNumerical ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SubLessRight ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ⊳ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}); LessLeq⊳ ˙¬ if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{x}, −\underline{x}, −−\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, −−\underline{y}); splitNumericalSum(+−small)⊳0 <= −\underline{y} ⊳ −\underline{x} <= 0 ⊳ if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SignNumerical ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}); −x − y = −(x + y) ≫ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}; plusCommutativity ≫ −\underline{x} + −\underline{y} = −\underline{y} + −\underline{x}; Equality ⊳ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y} ⊳ −\underline{x} + −\underline{y} = −\underline{y} + −\underline{x} ≫ −\underline{x} + \underline{y} = −\underline{y}+−\underline{x}; SameNumerical⊳−\underline{x}+\underline{y} = −\underline{y}+−\underline{x} ≫ if(0 <= −\underline{x}+\underline{y}, −\underline{x}+\underline{y}, −−\underline{x}+\underline{y}) = if(0 <= −\underline{y} +−\underline{x}, −\underline{y} +−\underline{x}, −−\underline{y} +−\underline{x}); eqTransitivity ⊳if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+ \underline{y}) = if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) ⊳ if(0 <= −\underline{x} + \underline{y}, −\underline{x} + \underline{y}, − − \underline{x} + \underline{y}) = if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= −\underline{y}+−\underline{x}, −\underline{y}+−\underline{x}, −−\underline{y}+−\underline{x}); eqSymmetry⊳if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) = if(0 <= −\underline{y}+−\underline{x}, −\underline{y}+−\underline{x}, −−\underline{y}+−\underline{x}) ≫ if(0 <= −\underline{y}+−\underline{x}, −\underline{y}+−\underline{x}, −−\underline{y}+−\underline{x}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); subLeqLeft ⊳ if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ if(0 <= −\underline{y} + −\underline{x}, −\underline{y} + −\underline{x}, − − \underline{y} + −\underline{x}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); subLeqRight ⊳ if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(+ − big) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(+ − big) →^{tex} “splitNumericalSum(+-big)”]

[splitNumericalSum(+ − big) →^{pyk} “lemma splitNumericalSum(+-, bigNegative)”]

splitNumericalSum(+−)

[splitNumericalSum(+−) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(+ − small) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); 0 <= |x| ≫ 0 <= if(0 <= \underline{y}, \underline{y}, −\underline{y}); LeqAdditionLeft ⊳ 0 <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); plus0 ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 = if(0 <= \underline{x}, \underline{x}, −\underline{x}); subLeqLeft ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 = if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) + 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); leqTransitivity ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊳ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ToLess ⊳ ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); splitNumericalSum(+ − big) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ⊳ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⇒ ˙¬ ˙¬ if(0 <= \underline{x}, \underline{x}, −\underline{x}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}); 0 <= |x| ≫ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}); leqAddition ⊳ 0 <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ≫ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); plus0Left ≫ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}); subLeqLeft ⊳ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) = if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ 0 + if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); leqTransitivity ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ FromNegations ⊳ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ ˙¬ if(0 <= \underline{y}, \underline{y}, −\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); MP2 ⊳ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(+−) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(+−) →^{tex} “splitNumericalSum(+-)”]

[splitNumericalSum(+−) →^{pyk} “lemma splitNumericalSum(+-)”]

splitNumericalSum(−+)

[splitNumericalSum(−+) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ NonpositiveNegated ⊳ \underline{x} <= 0 ≫ 0 <= −\underline{x}; NonnegativeNegated ⊳ 0 <= \underline{y} ≫ −\underline{y} <= 0; splitNumericalSum(+−) ⊳ 0 <= −\underline{x} ⊳ −\underline{y} <= 0 ≫ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}); SplitNumericalSumHelper ⊳ if(0 <= −\underline{x} + −\underline{y}, −\underline{x} + −\underline{y}, − − \underline{x} + −\underline{y}) <= if(0 <= −\underline{x}, −\underline{x}, − − \underline{x}) + if(0 <= −\underline{y}, −\underline{y}, − − \underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum(−+) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum(−+) →^{tex} “splitNumericalSum(-+)”]

[splitNumericalSum(−+) →^{pyk} “lemma splitNumericalSum(-+)”]

splitNumericalSum

[splitNumericalSum →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ splitNumericalSum(++) ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(+−) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ splitNumericalSum(−+) ⊳ \underline{x} <= 0 ⊳ 0 <= \underline{y} ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(−−) ⊳ \underline{x} <= 0 ⊳ \underline{y} <= 0 ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded⊳∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); Ded⊳∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ 0 <= \underline{y} ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ \underline{y} <= 0 ⇒ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x})+if(0 <= \underline{y}, \underline{y}, −\underline{y}); FromLeqGeq⊳0 <= \underline{y} ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ⊳ \underline{y} <= 0 ⇒ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y}) ≫ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})⌉, p_{0}, c)]

[splitNumericalSum →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) <= if(0 <= \underline{x}, \underline{x}, −\underline{x}) + if(0 <= \underline{y}, \underline{y}, −\underline{y})]

[splitNumericalSum →^{tex} “splitNumericalSum”]

[splitNumericalSum →^{pyk} “lemma splitNumericalSum”]

insertMiddleTerm(Numerical)

[insertMiddleTerm(Numerical) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: splitNumericalSum ≫ if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}) <= if(0 <= \underline{x} + −\underline{z}, \underline{x} + −\underline{z}, −\underline{x} + −\underline{z}) + if(0 <= \underline{z} + \underline{y}, \underline{z} + \underline{y}, −\underline{z} + \underline{y}); insertMiddleTerm(Sum) ≫ \underline{x} + \underline{y} = \underline{x}+−\underline{z}+\underline{z}+\underline{y}; SameNumerical⊳\underline{x}+\underline{y} = \underline{x}+−\underline{z}+\underline{z}+\underline{y} ≫ if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) = if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}); eqSymmetry ⊳ if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) = if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}) ≫ if(0 <= \underline{x}+−\underline{z}+\underline{z}+\underline{y}, \underline{x}+−\underline{z}+\underline{z}+\underline{y}, −\underline{x}+−\underline{z}+\underline{z}+\underline{y}) = if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+ \underline{y}); subLeqLeft⊳if(0 <= \underline{x}+−\underline{z}+\underline{z}+\underline{y}, \underline{x}+−\underline{z}+\underline{z}+\underline{y}, −\underline{x}+−\underline{z}+\underline{z}+\underline{y}) = if(0 <= \underline{x} + \underline{y}, \underline{x} + \underline{y}, −\underline{x} + \underline{y}) ⊳ if(0 <= \underline{x} + −\underline{z} + \underline{z} + \underline{y}, \underline{x} + −\underline{z} + \underline{z} + \underline{y}, −\underline{x} + −\underline{z} + \underline{z} + \underline{y}) <= if(0 <= \underline{x}+−\underline{z}, \underline{x}+−\underline{z}, −\underline{x}+−\underline{z})+if(0 <= \underline{z}+\underline{y}, \underline{z}+\underline{y}, −\underline{z}+\underline{y}) ≫ if(0 <= \underline{x}+\underline{y}, \underline{x}+ \underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}+−\underline{z}, \underline{x}+−\underline{z}, −\underline{x}+−\underline{z})+if(0 <= \underline{z}+\underline{y}, \underline{z}+\underline{y}, −\underline{z}+\underline{y})⌉, p_{0}, c)]

[insertMiddleTerm(Numerical) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: if(0 <= \underline{x}+\underline{y}, \underline{x}+\underline{y}, −\underline{x}+\underline{y}) <= if(0 <= \underline{x}+−\underline{z}, \underline{x}+−\underline{z}, −\underline{x}+−\underline{z})+if(0 <= \underline{z}+\underline{y}, \underline{z}+\underline{y}, −\underline{z}+\underline{y})]

[insertMiddleTerm(Numerical) →^{tex} “insertMiddleTerm(Numerical)”]

[insertMiddleTerm(Numerical) →^{pyk} “lemma insertMiddleTerm(Numerical)”]

x + y = zBackwards

[x + y = zBackwards →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ plusCommutativity ≫ \underline{x} + \underline{y} = \underline{y} + \underline{x}; Equality ⊳ \underline{x} + \underline{y} = \underline{z} ≫ \underline{z} = \underline{y} + \underline{x}⌉, p_{0}, c)]

[x + y = zBackwards →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{z} ⊢ \underline{z} = \underline{y} + \underline{x}]

[x + y = zBackwards →^{tex} “x+y=zBackwards”]

[x + y = zBackwards →^{pyk} “lemma x+y=zBackwards”]

x ∗ y = zBackwards

[x ∗ y = zBackwards →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} = \underline{z} ⊢ timesCommutativity ≫ \underline{x} ∗ \underline{y} = \underline{y} ∗ \underline{x}; Equality ⊳ \underline{x} ∗ \underline{y} = \underline{z} ≫ \underline{z} = \underline{y} ∗ \underline{x}⌉, p_{0}, c)]

[x ∗ y = zBackwards →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} ∗ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{y} ∗ \underline{x}]

[x ∗ y = zBackwards →^{tex} “x∗y=zBackwards”]

[x ∗ y = zBackwards →^{pyk} “lemma x∗y=zBackwards”]

x = x + (y − y)

[x = x + (y − y) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: plus0 ≫ \underline{x} + 0 = \underline{x}; Negative ≫ \underline{y} + −\underline{y} = 0; eqSymmetry ⊳ \underline{y} + −\underline{y} = 0 ≫ 0 = \underline{y} + −\underline{y}; lemma eqAdditionLeft ⊳ 0 = \underline{y} + −\underline{y} ≫ \underline{x} + 0 = \underline{x} + \underline{y} + −\underline{y}; Equality ⊳ \underline{x} + 0 = \underline{x} ⊳ \underline{x} + 0 = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} = \underline{x} + \underline{y} + −\underline{y}⌉, p_{0}, c)]

[x = x + (y − y) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{x} + \underline{y} + −\underline{y}]

[x = x + (y − y) →^{tex} “x=x+(y-y)”]

[x = x + (y − y) →^{pyk} “lemma x=x+(y-y)”]

x = x + y − y

[x = x + y − y →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: x = x + (y − y) ≫ \underline{x} = \underline{x}+\underline{y}+−\underline{y}; plusAssociativity ≫ \underline{x}+\underline{y}+−\underline{y} = \underline{x}+\underline{y}+−\underline{y}; eqSymmetry⊳\underline{x}+\underline{y}+−\underline{y} = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} + \underline{y} + −\underline{y} = \underline{x} + \underline{y} + −\underline{y}; eqTransitivity ⊳ \underline{x} = \underline{x} + \underline{y} + −\underline{y} ⊳ \underline{x} + \underline{y} + −\underline{y} = \underline{x} + \underline{y} + −\underline{y} ≫ \underline{x} = \underline{x} + \underline{y} + −\underline{y}⌉, p_{0}, c)]

[x = x + y − y →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{x} + \underline{y} + −\underline{y}] [x = x + y − y →^{tex} “x=x+y-y”]

[x = x + y − y →^{pyk} “lemma x=x+y-y”]

[ →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{y} = 0 ⊢ times1 ≫ \underline{x} ∗ 1 = \underline{x}; Reciprocal ⊳ ˙¬ \underline{y} = 0 ≫ \underline{y} ∗ rec\underline{y} = 1; Three2threeFactors ⊳ \underline{y} ∗ rec\underline{y} = 1 ≫ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x} ∗ 1; eqTransitivity ⊳ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x} ∗ 1 ⊳ \underline{x} ∗ 1 = \underline{x} ≫ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x}; eqSymmetry ⊳ \underline{x} ∗ \underline{y} ∗ rec\underline{y} = \underline{x} ≫ \underline{x} = \underline{x} ∗ \underline{y} ∗ rec\underline{y}⌉, p_{0}, c)]

[ →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ \underline{y} = 0 ⊢ \underline{x} = \underline{x} ∗ \underline{y} ∗ rec\underline{y}]

[→^{tex} “ ”]

[ →^{pyk} “lemma x=x∗y∗(1/y)”]

insertMiddleTerm(Sum)

[insertMiddleTerm(Sum) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: x = x + y − y ≫ \underline{x} = \underline{x} + \underline{z} + −\underline{z}; Three2threeTerms ≫ \underline{x} + \underline{z} + −\underline{z} = \underline{x} + −\underline{z} + \underline{z}; eqTransitivity ⊳ \underline{x} = \underline{x} + \underline{z} + −\underline{z} ⊳ \underline{x} + \underline{z} + −\underline{z} = \underline{x} + −\underline{z} + \underline{z} ≫ \underline{x} = \underline{x} + −\underline{z} + \underline{z}; eqAddition ⊳ \underline{x} = \underline{x} + −\underline{z} + \underline{z} ≫ \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}; plusAssociativity ≫ \underline{x} + −\underline{z} + \underline{z} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}; eqTransitivity ⊳ \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y} ⊳ \underline{x} + −\underline{z} + \underline{z} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y} ≫ \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}⌉, p_{0}, c)]

[insertMiddleTerm(Sum) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + \underline{y} = \underline{x} + −\underline{z} + \underline{z} + \underline{y}]

[insertMiddleTerm(Sum) →^{tex} “insertMiddleTerm(Sum)”]

[insertMiddleTerm(Sum) →^{pyk} “lemma insertMiddleTerm(Sum)”]

insertMiddleTerm(Difference)

[insertMiddleTerm(Difference) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: insertMiddleTerm(Sum) ≫ \underline{x} + −\underline{y} = \underline{x}+−−\underline{z}+−\underline{z}+−\underline{y}; DoubleMinus ≫ −−\underline{z} = \underline{z}; lemma eqAdditionLeft⊳−−\underline{z} = \underline{z} ≫ \underline{x} + − − \underline{z} = \underline{x} + \underline{z}; plusCommutativity ≫ −\underline{z} + −\underline{y} = −\underline{y} + −\underline{z}; −x − y = −(x + y) ≫ −\underline{y} + −\underline{z} = −\underline{y} + \underline{z}; eqTransitivity ⊳ −\underline{z} + −\underline{y} = −\underline{y} + −\underline{z} ⊳ −\underline{y} + −\underline{z} = −\underline{y} + \underline{z} ≫ −\underline{z} + −\underline{y} = −\underline{y} + \underline{z}; AddEquations ⊳ \underline{x} + − − \underline{z} = \underline{x} + \underline{z} ⊳ −\underline{z} + −\underline{y} = −\underline{y}+\underline{z} ≫ \underline{x}+−−\underline{z}+−\underline{z}+−\underline{y} = \underline{x}+\underline{z}+−\underline{y}+\underline{z}; eqTransitivity⊳\underline{x}+−\underline{y} = \underline{x}+−−\underline{z}+ −\underline{z}+−\underline{y} ⊳\underline{x}+−−\underline{z}+−\underline{z}+−\underline{y} = \underline{x}+\underline{z}+−\underline{y} +\underline{z} ≫ \underline{x}+−\underline{y} = \underline{x}+\underline{z}+−\underline{y} +\underline{z}⌉, p_{0}, c)]

[insertMiddleTerm(Difference) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} + −\underline{y} = \underline{x} + \underline{z} + −\underline{y} + \underline{z}] [insertMiddleTerm(Difference) →^{tex} “insertMiddleTerm(Difference)”]

[insertMiddleTerm(Difference) →^{pyk} “lemma insertMiddleTerm(Difference)”]

x ∗ 0 + x = x

[x∗0+x = x →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: times1 ≫ \underline{x}∗1 = \underline{x}; eqSymmetry⊳\underline{x}∗1 = \underline{x} ≫ \underline{x} = \underline{x} ∗ 1; lemma eqAdditionLeft ⊳ \underline{x} = \underline{x} ∗ 1 ≫ \underline{x} ∗ 0 + \underline{x} = \underline{x} ∗ 0 + \underline{x} ∗ 1; Distribution ≫ \underline{x} ∗ 0 + 1 = \underline{x} ∗ 0 + \underline{x} ∗ 1; eqSymmetry ⊳ \underline{x} ∗ 0 + 1 = \underline{x} ∗ 0 + \underline{x} ∗ 1 ≫ \underline{x} ∗ 0 + \underline{x} ∗ 1 = \underline{x} ∗ 0 + 1; plus0Left ≫ 0 + 1 = 1; EqMultiplicationLeft⊳0+1 = 1 ≫ \underline{x}∗0+1 = \underline{x}∗1; eqTransitivity5⊳\underline{x}∗0+\underline{x} = \underline{x}∗0+\underline{x}∗1⊳\underline{x}∗0+\underline{x}∗1 = \underline{x}∗0+1⊳\underline{x}∗0+1 = \underline{x}∗1⊳\underline{x}∗1 = \underline{x} ≫ \underline{x}∗0+\underline{x} = \underline{x}⌉, p_{0}, c)]

[x ∗ 0 + x = x →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ 0 + \underline{x} = \underline{x}]

[x ∗ 0 + x = x →^{tex} “x∗0+x=x”]

[x ∗ 0 + x = x →^{pyk} “lemma x∗0+x=x”]

x ∗ 0 = 0

[x ∗ 0 = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: x = x + (y − y) ≫ \underline{x} ∗ 0 = \underline{x} ∗ 0 + \underline{x} + −\underline{x}; plusAssociativity ≫ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x} ∗ 0 + \underline{x} + −\underline{x}; eqSymmetry ⊳ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x} ∗ 0 + \underline{x} + −\underline{x} ≫ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x}∗0+\underline{x}+−\underline{x}; x∗0+x = x ≫ \underline{x}∗0+\underline{x} = \underline{x}; eqAddition⊳\underline{x}∗0+\underline{x} = \underline{x} ≫ \underline{x}∗0+\underline{x}+−\underline{x} = \underline{x}+−\underline{x}; Negative ≫ \underline{x}+−\underline{x} = 0; eqTransitivity5⊳\underline{x}∗0 = \underline{x}∗0+\underline{x}+−\underline{x}⊳\underline{x}∗0+\underline{x}+−\underline{x} = \underline{x} ∗ 0 + \underline{x} + −\underline{x} ⊳ \underline{x} ∗ 0 + \underline{x} + −\underline{x} = \underline{x} + −\underline{x} ⊳ \underline{x} + −\underline{x} = 0 ≫ \underline{x} ∗ 0 = 0⌉, p_{0}, c)]

[x ∗ 0 = 0 →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ 0 = 0]

[x ∗ 0 = 0 →^{tex} “x∗0=0”]

[x ∗ 0 = 0 →^{pyk} “lemma x∗0=0”]

(−1) ∗ (−1) + (−1) ∗ 1 = 0

[(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ DistributionOut ≫ −1 ∗ −1 + −1 ∗ 1 = −1 ∗ −1 + 1; Negative ≫ 1 + −1 = 0; plusCommutativity ≫ −1 + 1 = 1 + −1; eqTransitivity ⊳ −1 + 1 = 1 + −1 ⊳ 1 + −1 = 0 ≫ −1 + 1 = 0; EqMultiplicationLeft ⊳ −1 + 1 = 0 ≫ −1 ∗ −1 + 1 = −1 ∗ 0; x ∗ 0 = 0 ≫ −1 ∗ 0 = 0; eqTransitivity4 ⊳ −1 ∗ −1 + −1 ∗ 1 = −1 ∗ −1 + 1 ⊳ −1 ∗ −1 + 1 = −1 ∗ 0 ⊳ −1 ∗ 0 = 0 ≫ −1 ∗ −1 + −1 ∗ 1 = 0⌉, p_{0}, c)]

[(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{stmt} ZFsub ⊢ −1 ∗ −1 + −1 ∗ 1 = 0]

[(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{tex} “(-1)∗(-1)+(-1)∗1=0”]

[(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{pyk} “lemma (-1)∗(-1)+(-1)∗1=0”]

(−1) ∗ (−1) = 1

[(−1) ∗ (−1) = 1 →^{proof} λc.λx.P(⌈ZFsub ⊢ x = x + (y − y) ≫ −1 ∗ −1 = −1 ∗ −1 + 1 + −1; times1 ≫ −1 ∗ 1 = −1; eqSymmetry ⊳ −1 ∗ 1 = −1 ≫ −1 = −1 ∗ 1; lemma eqAdditionLeft ⊳ −1 = −1 ∗ 1 ≫ 1 + −1 = 1 + −1 ∗ 1; lemma eqAdditionLeft ⊳ 1 + −1 = 1 + −1 ∗ 1 ≫ −1 ∗ −1 + 1 + −1 = −1 ∗ −1 + 1 + −1 ∗ 1; plusCommutativity ≫ 1 + −1 ∗ 1 = −1∗1+1; lemma eqAdditionLeft⊳1+−1∗1 = −1∗1+1 ≫ −1∗−1+1+−1∗1 = −1 ∗ −1 + −1 ∗ 1 + 1; plusAssociativity ≫ −1 ∗ −1 + −1 ∗ 1 + 1 = −1∗−1+−1∗1+1; eqSymmetry⊳−1∗−1+−1∗1+1 = −1∗−1+−1∗1+1 ≫ −1 ∗ −1 + −1 ∗ 1 + 1 = −1 ∗ −1 + −1 ∗ 1 + 1; (−1) ∗ (−1) + (−1) ∗ 1 = 0 ≫ −1 ∗ −1 + −1 ∗ 1 = 0; eqAddition ⊳ −1 ∗ −1 + −1 ∗ 1 = 0 ≫ −1 ∗ −1 + −1 ∗ 1 + 1 = 0 + 1; plus0Left ≫ 0 + 1 = 1; eqTransitivity5 ⊳ −1 ∗ −1 = −1∗−1+1+−1⊳−1∗−1+1+−1 = −1∗−1+1+−1∗1⊳−1∗−1+1+−1∗1 = −1 ∗ −1 + −1 ∗ 1 + 1 ⊳ −1 ∗ −1 + −1 ∗ 1 + 1 = −1 ∗ −1 + −1 ∗ 1 + 1 ≫ −1 ∗ −1 = −1 ∗ −1 + −1 ∗ 1 + 1; eqTransitivity4 ⊳ −1 ∗ −1 = −1∗−1+−1∗1+1⊳−1∗−1+−1∗1+1 = 0+1⊳0+1 = 1 ≫ −1∗−1 = 1⌉, p_{0}, c)]

[(−1) ∗ (−1) = 1 →^{stmt} ZFsub ⊢ −1 ∗ −1 = 1]

[(−1) ∗ (−1) = 1 →^{tex} “(-1)∗(-1)=1”]

[(−1) ∗ (−1) = 1 →^{pyk} “lemma (-1)∗(-1)=1”]

0 < 1Helper

[0 < 1Helper →^{proof} λc.λx.P(⌈ZFsub ⊢ 1 <= 0 ⊢ leqAddition ⊳ 1 <= 0 ≫ 1+−1 <= 0+−1; Negative ≫ 1+−1 = 0; subLeqLeft⊳1+−1 = 0⊳1+−1 <= 0 + −1 ≫ 0 <= 0 + −1; plus0Left ≫ 0 + −1 = −1; subLeqRight ⊳ 0 + −1 = −1 ⊳ 0 <= 0 + −1 ≫ 0 <= −1; leqMultiplication ⊳ 0 <= −1 ⊳ 0 <= −1 ≫ 0 ∗ −1 <= −1 ∗ −1; x ∗ 0 = 0 ≫ −1 ∗ 0 = 0; timesCommutativity ≫ 0 ∗ −1 = −1 ∗ 0; eqTransitivity ⊳ 0 ∗ −1 = −1 ∗ 0 ⊳ −1 ∗ 0 = 0 ≫ 0 ∗ −1 = 0; subLeqLeft ⊳ 0 ∗ −1 = 0 ⊳ 0 ∗ −1 <= −1 ∗ −1 ≫ 0 <= −1 ∗ −1; (−1) ∗ (−1) = 1 ≫ −1 ∗ −1 = 1; subLeqRight ⊳ −1 ∗ −1 = 1 ⊳ 0 <= −1 ∗ −1 ≫ 0 <= 1; Ded ⊳ 1 <= 0 ⊢ 0 <= 1 ≫ 1 <= 0 ⇒ 0 <= 1⌉, p_{0}, c)]

[0 < 1Helper →^{stmt} ZFsub ⊢ 1 <= 0 ⇒ 0 <= 1]

[0 < 1Helper →^{tex} “0<1Helper”]

[0 < 1Helper →^{pyk} “lemma 0<1Helper”]

0 < 1

[0 < 1 →^{proof} λc.λx.P(⌈ZFsub ⊢ leqTotality ≫ ˙¬ 0 <= 1 ⇒ 1 <= 0; AutoImply ≫ 0 <= 1 ⇒ 0 <= 1; 0 < 1Helper ≫ 1 <= 0 ⇒ 0 <= 1; FromDisjuncts ⊳ ˙¬ 0 <= 1 ⇒ 1 <= 0 ⊳ 0 <= 1 ⇒ 0 <= 1 ⊳ 1 <= 0 ⇒ 0 <= 1 ≫ 0 <= 1; 0not1 ≫ ˙¬ 0 = 1; JoinConjuncts ⊳ 0 <= 1 ⊳ ˙¬ 0 = 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1⌉, p_{0}, c)]

[0 < 1 →^{stmt} ZFsub ⊢ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1]

[0 < 1 →^{tex} “0<1”]

[0 < 1 →^{pyk} “lemma 0<1”]

0 < 2

[0 < 2 →^{proof} λc.λx.P(⌈ZFsub ⊢ 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; LessAddition ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ ˙¬ 0 + 1 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 + 1 = 1 + 1; plus0Left ≫ 0 + 1 = 1; SubLessLeft ⊳ 0 + 1 = 1 ⊳ ˙¬ 0 + 1 <= 1 + 1 ⇒ ˙¬ ˙¬ 0+1 = 1+1 ≫ ˙¬ 1 <= 1+1 ⇒ ˙¬ ˙¬ 1 = 1+1; LessTransitivity⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1⊳ ˙¬ 1 <= 1+1 ⇒ ˙¬ ˙¬ 1 = 1+1 ≫ ˙¬ 0 <= 1+1 ⇒ ˙¬ ˙¬ 0 = 1+1⌉, p_{0}, c)]

[0 < 2 →^{stmt} ZFsub ⊢ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1]

[0 < 2 →^{tex} “0<2”]

[0 < 2 →^{pyk} “lemma 0<2”]

0 < 1/2

[0 < 1/2 →^{proof} λc.λx.P(⌈ZFsub ⊢ 0 < 2 ≫ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1; FirstConjunct ⊳ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1 ≫ 0 <= 1 + 1; SecondConjunct ⊳ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1 ≫ ˙¬ 0 = 1 + 1; NeqSymmetry ⊳ ˙¬ 0 = 1 + 1 ≫ ˙¬ 1 + 1 = 0; 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; x ∗ 0 = 0 ≫ 1 + 1 ∗ 0 = 0; x ∗ y = zBackwards ⊳ 1 + 1 ∗ 0 = 0 ≫ 0 = 0 ∗ 1 + 1; SubLessLeft ⊳ 0 = 0 ∗ 1 + 1 ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ ˙¬ 0 ∗ 1 + 1 <= 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = 1; Reciprocal ⊳ ˙¬ 1 + 1 = 0 ≫ 1 + 1 ∗ rec1 + 1 = 1; x ∗ y = zBackwards ⊳ 1 + 1 ∗ rec1 + 1 = 1 ≫ 1 = rec1 + 1 ∗ 1 + 1; SubLessRight ⊳ 1 = rec1 + 1 ∗ 1 + 1 ⊳ ˙¬ 0 ∗ 1 + 1 <= 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = 1 ≫ ˙¬ 0 ∗ 1 + 1 <= rec1 + 1 ∗ 1 + 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = rec1 + 1 ∗ 1 + 1; LessDivision ⊳ 0 <= 1 + 1 ⊳ ˙¬ 0 ∗ 1 + 1 <= rec1 + 1 ∗ 1 + 1 ⇒ ˙¬ ˙¬ 0 ∗ 1 + 1 = rec1 + 1 ∗ 1 + 1 ≫ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1⌉, p_{0}, c)]

[0 < 1/2 →^{stmt} ZFsub ⊢ ˙¬ 0 <= rec1 + 1 ⇒ ˙¬ ˙¬ 0 = rec1 + 1]

[0 < 1/2 →^{tex} “0<1/2”]

[0 < 1/2 →^{pyk} “lemma 0<1/2”]

TwoWholes

[TwoWholes →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: times1 ≫ \underline{x} ∗ 1 = \underline{x}; eqSymmetry ≫ \underline{x} = \underline{x} ∗ 1; lemma eqAdditionLeft ⊳ \underline{x} = \underline{x} ∗ 1 ≫ \underline{x} + \underline{x} = \underline{x} + \underline{x} ∗ 1; eqAddition ⊳ \underline{x} = \underline{x} ∗ 1 ≫ \underline{x} + \underline{x} ∗ 1 = \underline{x} ∗ 1 + \underline{x} ∗ 1; eqTransitivity ⊳ \underline{x} + \underline{x} = \underline{x} + \underline{x} ∗ 1 ⊳ \underline{x} + \underline{x} ∗ 1 = \underline{x} ∗ 1 + \underline{x} ∗ 1 ≫ \underline{x} + \underline{x} = \underline{x} ∗ 1 + \underline{x} ∗ 1; DistributionOut ≫ \underline{x} ∗ 1 + \underline{x} ∗ 1 = \underline{x} ∗ 1 + 1; Repetition ⊳ \underline{x} ∗ 1 + \underline{x} ∗ 1 = \underline{x} ∗ 1 + 1 ≫ \underline{x} ∗ 1 + \underline{x} ∗ 1 = \underline{x} ∗ 1 + 1; timesCommutativity ≫ \underline{x} ∗ 1 + 1 = 1 + 1 ∗ \underline{x}; eqTransitivity4 ⊳ \underline{x} + \underline{x} = \underline{x}∗1+\underline{x}∗1⊳\underline{x}∗1+\underline{x}∗1 = \underline{x}∗1+1⊳\underline{x}∗1+1 = 1+1∗\underline{x} ≫ \underline{x}+\underline{x} = 1+1∗\underline{x}⌉, p_{0}, c)]

[TwoWholes →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} + \underline{x} = 1 + 1 ∗ \underline{x}]

[TwoWholes →^{tex} “TwoWholes”]

[TwoWholes →^{pyk} “lemma x+x=2∗x”]

TwoHalves

[TwoHalves →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: 0 < 2 ≫ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1; LessNeq ⊳ ˙¬ 0 <= 1 + 1 ⇒ ˙¬ ˙¬ 0 = 1 + 1 ≫ ˙¬ 0 = 1 + 1; NeqSymmetry ⊳ ˙¬ 0 = 1 + 1 ≫ ˙¬ 1 + 1 = 0; TwoWholes ≫ rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x}; timesAssociativity ≫ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x}; eqSymmetry ⊳ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1+1∗rec1+1∗\underline{x} ≫ 1+1∗rec1+1∗\underline{x} = 1+1∗rec1+1∗\underline{x}; Reciprocal⊳ ˙¬ 1+1 = 0 ≫ 1 + 1 ∗ rec1 + 1 = 1; eqMultiplication ⊳ 1 + 1 ∗ rec1 + 1 = 1 ≫ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 ∗ \underline{x}; times1Left ≫ 1 ∗ \underline{x} = \underline{x}; eqTransitivity5 ⊳ rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x} ⊳ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 + 1 ∗ rec1 + 1 ∗ \underline{x} ⊳ 1 + 1 ∗ rec1 + 1 ∗ \underline{x} = 1 ∗ \underline{x} ⊳ 1 ∗ \underline{x} = \underline{x} ≫ rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = \underline{x}⌉, p_{0}, c)]

[TwoHalves →^{stmt} ZFsub ⊢ ∀\underline{x}: rec1 + 1 ∗ \underline{x} + rec1 + 1 ∗ \underline{x} = \underline{x}]

[TwoHalves →^{tex} “TwoHalves”]

[TwoHalves →^{pyk} “lemma (1/2)x+(1/2)x=x”]

−x − y = −(x + y)

[−x − y = −(x + y) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: Times(−1)Left ≫ −1 ∗ \underline{x} = −\underline{x}; Times(−1)Left ≫ −1∗\underline{y} = −\underline{y}; AddEquations⊳−1∗\underline{x} = −\underline{x}⊳−1∗\underline{y} = −\underline{y} ≫ −1∗\underline{x}+−1∗\underline{y} = −\underline{x}+−\underline{y}; eqSymmetry⊳−1∗\underline{x}+−1∗\underline{y} = −\underline{x}+−\underline{y} ≫ −\underline{x}+−\underline{y} = −1∗\underline{x}+−1∗\underline{y}; DistributionOut ≫ −1∗\underline{x}+−1∗\underline{y} = −1∗\underline{x}+\underline{y}; Times(−1)Left ≫ −1∗\underline{x}+\underline{y} = −\underline{x}+\underline{y}; eqTransitivity4⊳−\underline{x}+−\underline{y} = −1∗\underline{x}+−1∗\underline{y}⊳−1∗\underline{x}+−1∗\underline{y} = −1 ∗ \underline{x} + \underline{y} ⊳ −1 ∗ \underline{x} + \underline{y} = −\underline{x} + \underline{y} ≫ −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}⌉, p_{0}, c)]

[−x − y = −(x + y) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: −\underline{x} + −\underline{y} = −\underline{x} + \underline{y}]

[−x − y = −(x + y) →^{tex} “-x-y=-(x+y)”]

[−x − y = −(x + y) →^{pyk} “lemma -x-y=-(x+y)”]

MinusNegated

[MinusNegated →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: DoubleMinus ≫ − − \underline{y} = \underline{y}; eqAddition ⊳ − − \underline{y} = \underline{y} ≫ − − \underline{y} + −\underline{x} = \underline{y} + −\underline{x}; eqSymmetry ⊳ − − \underline{y} + −\underline{x} = \underline{y} + −\underline{x} ≫ \underline{y} + −\underline{x} = − − \underline{y} + −\underline{x}; −x − y = −(x + y) ≫ − − \underline{y} + −\underline{x} = − − \underline{y} + \underline{x}; plusCommutativity ≫ −\underline{y} + \underline{x} = \underline{x} + −\underline{y}; EqNegated ⊳ −\underline{y} + \underline{x} = \underline{x} + −\underline{y} ≫ − − \underline{y} + \underline{x} = −\underline{x} + −\underline{y}; eqTransitivity4 ⊳ \underline{y} + −\underline{x} = − − \underline{y} + −\underline{x} ⊳ − − \underline{y} + −\underline{x} = − − \underline{y} + \underline{x} ⊳ − − \underline{y} + \underline{x} = −\underline{x} + −\underline{y} ≫ \underline{y} + −\underline{x} = −\underline{x} + −\underline{y}; eqSymmetry ⊳ \underline{y} + −\underline{x} = −\underline{x} + −\underline{y} ≫ −\underline{x} + −\underline{y} = \underline{y} + −\underline{x}⌉, p_{0}, c)]

[MinusNegated →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: −\underline{x} + −\underline{y} = \underline{y} + −\underline{x}]

[MinusNegated →^{tex} “MinusNegated”]

[MinusNegated →^{pyk} “lemma minusNegated”]

Times(−1)

[Times(−1) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: Negative ≫ 1 + −1 = 0; plusCommutativity ≫ −1 + 1 = 1 + −1; eqTransitivity ⊳ −1 + 1 = 1 + −1 ⊳ 1 + −1 = 0 ≫ −1 + 1 = 0; EqMultiplicationLeft ⊳ −1 + 1 = 0 ≫ \underline{x} ∗ −1 + 1 = \underline{x} ∗ 0; x ∗ 0 = 0 ≫ \underline{x} ∗ 0 = 0; eqTransitivity ⊳ \underline{x} ∗ −1 + 1 = \underline{x} ∗ 0 ⊳ \underline{x} ∗ 0 = 0 ≫ \underline{x} ∗ −1 + 1 = 0; Distribution ≫ \underline{x} ∗ −1 + 1 = \underline{x} ∗ −1 + \underline{x} ∗ 1; eqSymmetry ⊳ \underline{x} ∗ −1 + 1 = \underline{x} ∗ −1 + \underline{x} ∗ 1 ≫ \underline{x} ∗ −1 + \underline{x} ∗ 1 = \underline{x}∗−1+1; eqTransitivity⊳\underline{x}∗−1+\underline{x}∗1 = \underline{x}∗−1+1⊳\underline{x}∗−1+1 = 0 ≫ \underline{x}∗−1+\underline{x}∗1 = 0; PositiveToRight(Eq) ⊳ \underline{x} ∗ −1 + \underline{x} ∗ 1 = 0 ≫ \underline{x} ∗ −1 = 0 + −\underline{x} ∗ 1; plus0Left ≫ 0 + −\underline{x} ∗ 1 = −\underline{x} ∗ 1; eqTransitivity ⊳ \underline{x} ∗ −1 = 0 + −\underline{x} ∗ 1 ⊳ 0 + −\underline{x} ∗ 1 = −\underline{x} ∗ 1 ≫ \underline{x} ∗ −1 = −\underline{x} ∗ 1; times1 ≫ \underline{x} ∗ 1 = \underline{x}; EqNegated ⊳ \underline{x} ∗ 1 = \underline{x} ≫ −\underline{x} ∗ 1 = −\underline{x}; eqTransitivity ⊳ \underline{x} ∗ −1 = −\underline{x} ∗ 1 ⊳ −\underline{x} ∗ 1 = −\underline{x} ≫ \underline{x} ∗ −1 = −\underline{x}⌉, p_{0}, c)]

[Times(−1) →^{stmt} ZFsub ⊢ ∀\underline{x}: \underline{x} ∗ −1 = −\underline{x}]

[Times(−1) →^{tex} “Times(-1)”]

[Times(−1) →^{pyk} “lemma times(-1)”]

Times(−1)Left

[Times(−1)Left →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: Times(−1) ≫ \underline{x} ∗ −1 = −\underline{x}; timesCommutativity ≫ −1 ∗ \underline{x} = \underline{x} ∗ −1; eqTransitivity ⊳ −1 ∗ \underline{x} = \underline{x} ∗ −1 ⊳ \underline{x} ∗ −1 = −\underline{x} ≫ −1 ∗ \underline{x} = −\underline{x}⌉, p_{0}, c)]

[Times(−1)Left →^{stmt} ZFsub ⊢ ∀\underline{x}: −1 ∗ \underline{x} = −\underline{x}]

[Times(−1)Left →^{tex} “Times(-1)Left”]

[Times(−1)Left →^{pyk} “lemma times(-1)Left”]

−0 = 0

[−0 = 0 →^{proof} λc.λx.P(⌈ZFsub ⊢ Negative ≫ 0 + −0 = 0; plus0 ≫ 0 + 0 = 0; UniqueNegative ⊳ 0 + −0 = 0 ⊳ 0 + 0 = 0 ≫ −0 = 0⌉, p_{0}, c)]

[−0 = 0 →^{stmt} ZFsub ⊢ −0 = 0]

[−0 = 0 →^{tex} “-0=0”]

[−0 = 0 →^{pyk} “lemma -0=0”]

SFsymmetry

[SFsymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ FromSF ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; NumericalDifference ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fy)}[\underline{m}]+−\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}]+−\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}]+−\underline{(fx)}[\underline{m}]); SubLessLeft⊳if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)}; SF(\underline{(fx)}, \underline{(fy)}) ⊢ MP ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)} ⊳ SF(\underline{(fx)}, \underline{(fy)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fx)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fy)}, \underline{(fx)})⌉, p_{0}, c)]

[SFsymmetry →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fx)})]

[SFsymmetry →^{tex} “SFsymmetry”]

[SFsymmetry →^{pyk} “lemma sameFsymmetry”]

SFtransitivity

[SFtransitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ PositiveHalved ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ ˙¬ 0 <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{(ϵ)}; FromSF ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ ˙¬ 0 <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; FromSF ⊳ SF(\underline{(fy)}, \underline{(fz)}) ⊳ ˙¬ 0 <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = rec1 + 1 ∗ \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)}; AddEquations(Less) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ⊳ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)}; TwoHalves ≫ rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} = \underline{(ϵ)}; SubLessRight ⊳ rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} = \underline{(ϵ)} ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= rec1 + 1 ∗ \underline{(ϵ)} + rec1 + 1 ∗ \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}]+−\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}]+−\underline{(fz)}[\underline{m}]) = rec1+1∗\underline{(ϵ)}+rec1+1∗\underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}])+if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; insertMiddleTerm(Numerical) ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]); leqLessTransitivity ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) + if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}])+if(0 <= \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fy)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fy)}) ⇒ SF(\underline{(fy)}, \underline{(fz)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ MP2 ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⇒ SF(\underline{(fy)}, \underline{(fz)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ SF(\underline{(fy)}, \underline{(fz)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fz)})⌉, p_{0}, c)]

[SFtransitivity →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)}, \underline{(fy)}) ⊢ SF(\underline{(fy)}, \underline{(fz)}) ⊢ SF(\underline{(fx)}, \underline{(fz)})]

[SFtransitivity →^{tex} “SFtransitivity”]

[SFtransitivity →^{pyk} “lemma sameFtransitivity”]

= fToSameF

[= fToSameF →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ From = f ⊳ \underline{(fx)} =_{f} \underline{(fy)} ≫ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]; PositiveToLeft(Eq)(1term) ⊳ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = 0; SameNumerical ⊳ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = 0 ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= 0, 0, −0); |0| = 0 ≫ if(0 <= 0, 0, −0) = 0; eqTransitivity ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= 0, 0, −0) ⊳ if(0 <= 0, 0, −0) = 0 ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = 0; eqSymmetry ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = 0 ≫ 0 = if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]); SubLessLeft ⊳ 0 = if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; Weakening ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ 0 <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; ExistIntro @ c_{Ex} @ 0 ⊳ 0 <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ \underline{(fx)} =_{f} \underline{(fy)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; \underline{(fx)} =_{f} \underline{(fy)} ⊢ MP ⊳ \underline{(fx)} =_{f} \underline{(fy)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊳ \underline{(fx)} =_{f} \underline{(fy)} ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)}, \underline{(fy)})⌉, p_{0}, c)]

[= fToSameF →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} =_{f} \underline{(fy)} ⊢ SF(\underline{(fx)}, \underline{(fy)})]

[= fToSameF →^{tex} “=fToSameF ”]

[= fToSameF →^{pyk} “lemma =f to sameF”]

PlusF(Sym)

[PlusF(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}]⌉, p_{0}, c)]

[PlusF(Sym) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}]]

[PlusF(Sym) →^{tex} “PlusF(Sym)”]

[PlusF(Sym) →^{pyk} “lemma plusF(Sym)”]

TimesF(Sym)

[TimesF(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}]⌉, p_{0}, c)]

[TimesF(Sym) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}]] [TimesF(Sym) →^{tex} “TimesF(Sym)”]

[TimesF(Sym) →^{pyk} “lemma timesF(Sym)”]

f2R(Plus)

[f2R(Plus) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⊢ \underline{var fv} ∈ R(\underline{(fz)}) ⊢ FromInR ⊳ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ≫ SF(\underline{(fv)}, \underline{(fx)} +_{f} \underline{(fy)}); FromInR ⊳ \underline{var fv} ∈ R(\underline{(fz)}) ≫ SF(\underline{var fv}, \underline{(fz)}); SFsymmetry ⊳ SF(\underline{var fv}, \underline{(fz)}) ≫ SF(\underline{(fz)}, \underline{var fv}); SFtransitivity ⊳ SF(\underline{(fv)}, \underline{(fx)} +_{f} \underline{(fy)}) ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ≫ SF(\underline{(fv)}, \underline{(fz)}); SFtransitivity ⊳ SF(\underline{(fv)}, \underline{(fz)}) ⊳ SF(\underline{(fz)}, \underline{var fv}) ≫ SF(\underline{(fv)}, \underline{var fv}); ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⊢ \underline{var fv} ∈ R(\underline{(fz)}) ⊢ SF(\underline{(fv)}, \underline{var fv}) ≫ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⇒ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}); SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ MP ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⇒ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}) ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ≫ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}); To == XX ⊳ \underline{(fv)} ∈ R(\underline{(fx)} +_{f} \underline{(fy)}) ⇒ \underline{var fv} ∈ R(\underline{(fz)}) ⇒ SF(\underline{(fv)}, \underline{var fv}) ≫ R(\underline{(fx)} +_{f} \underline{(fy)}) == R(\underline{(fz)}); PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}); ==Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)}+_{f} \underline{(fy)})⊳R(\underline{(fx)}+_{f} \underline{(fy)}) == R(\underline{(fz)}) ≫ R((fx)+_{f} (fy)) == R(\underline{(fz)})⌉, p_{0}, c)]

[f2R(Plus) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fv)}: ∀\underline{var fv}: SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fz)}) ⊢ R((fx) +_{f} (fy)) == R(\underline{(fz)})]

[f2R(Plus) →^{tex} “f2R(Plus)”]

[f2R(Plus) →^{pyk} “lemma f2R(Plus)”]

f2R(Times)

[f2R(Times) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fz)}) ⊢ To == ⊳SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fz)}) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fz)}); TimesR ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}); ==Transitivity ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ⊳ R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fz)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fz)})⌉, p_{0}, c)]

[f2R(Times) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fz)}) ⊢ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fz)})]

[f2R(Times) →^{tex} “f2R(Times)”]

[f2R(Times) →^{pyk} “lemma f2R(Times)”]

PlusR(Sym)

[PlusR(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}); ==Symmetry ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)}) ≫ R(\underline{(fx)} +_{f} \underline{(fy)}) == R((fx) +_{f} (fy))⌉, p_{0}, c)]

[PlusR(Sym) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)} +_{f} \underline{(fy)}) == R((fx) +_{f} (fy))]

[PlusR(Sym) →^{tex} “PlusR(Sym)”]

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

TimesR(Sym)

[TimesR(Sym) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: TimesR ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}); ==Symmetry ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fx)}) ∗ ∗R(\underline{(fy)})⌉, p_{0}, c)]

[TimesR(Sym) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)} ∗_{f} \underline{(fy)}) == R(\underline{(fx)}) ∗ ∗R(\underline{(fy)})]

[TimesR(Sym) →^{tex} “TimesR(Sym)”]

[TimesR(Sym) →^{pyk} “lemma timesR(Sym)”]

LessLeq(R)

[LessLeq(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ WeakenOr2 ⊳ \underline{(rx)} << \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}⌉, p_{0}, c)]

[LessLeq(R) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} << \underline{(ry)} ⊢ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}]

[LessLeq(R) →^{tex} “LessLeq(R)”]

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

eqLeq(R)

[eqLeq(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ WeakenOr1 ⊳ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}⌉, p_{0}, c)] [eqLeq(R) →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}]

[eqLeq(R) →^{tex} “eqLeq(R)”]

[eqLeq(R) →^{pyk} “lemma eqLeq(R)”]

SubLessRight(R)

[SubLessRight(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ SENC2 ⊳ \underline{(rx)} == \underline{(ry)} ≫ \underline{(fy)} ∈ \underline{(rx)}; From << XX ⊳ \underline{(rz)} << \underline{(rx)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ \underline{(fy)} ∈ \underline{(rx)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rz)} << \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ MP2 ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rz)} << \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(rz)} << \underline{(rx)} ≫ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; to << XX ⊳ \underline{(fz)} ∈ \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rz)} << \underline{(ry)}⌉, p_{0}, c)]

[SubLessRight(R) →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rz)} << \underline{(rx)} ⊢ \underline{(rz)} << \underline{(ry)}]

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

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

SubLessLeft(R)

[SubLessLeft(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ SENC2 ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(fy)} ∈ \underline{(ry)} ≫ \underline{(fy)} ∈ \underline{(rx)}; From << XX ⊳ \underline{(rx)} << \underline{(rz)} ⊳ \underline{(fy)} ∈ \underline{(rx)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: Ded ⊳ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ \underline{(fy)} ∈ \underline{(ry)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ MP2 ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(rx)} << \underline{(rz)} ≫ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; to << XX ⊳ \underline{(fy)} ∈ \underline{(ry)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(ry)} << \underline{(rz)}⌉, p_{0}, c)]

[SubLessLeft(R) →^{stmt} ZFsub ⊢ ∀\underline{(ϵ)}: ∀\underline{m}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(rx)} << \underline{(rz)} ⊢ \underline{(ry)} << \underline{(rz)}]

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

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

<< TransitivityHelper(Q)

[<< TransitivityHelper(Q) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} + −\underline{u} ⊢ \underline{y} <= \underline{z} + −\underline{u} ⊢ ˙¬ 0 <= \underline{u} ⇒ ˙¬ ˙¬ 0 = \underline{u} ⊢ PositiveNegated ⊳ ˙¬ 0 <= \underline{u} ⇒ ˙¬ ˙¬ 0 = \underline{u} ≫ ˙¬ − \underline{u} <= 0 ⇒ ˙¬ ˙¬ − \underline{u} = 0; LessAdditionLeft ⊳ ˙¬ − \underline{u} <= 0 ⇒ ˙¬ ˙¬ − \underline{u} = 0 ≫ ˙¬ \underline{y} + −\underline{u} <= \underline{y} + 0 ⇒ ˙¬ ˙¬ \underline{y} + −\underline{u} = \underline{y} + 0; plus0 ≫ \underline{y} + 0 = \underline{y}; SubLessRight ⊳ \underline{y} + 0 = \underline{y} ⊳ ˙¬ \underline{y} + −\underline{u} <= \underline{y} + 0 ⇒ ˙¬ ˙¬ \underline{y} + −\underline{u} = \underline{y} + 0 ≫ ˙¬ \underline{y}+−\underline{u} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{y}+−\underline{u} = \underline{y}; leqLessTransitivity⊳\underline{x} <= \underline{y}+−\underline{u}⊳ ˙¬ \underline{y}+−\underline{u} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{y} + −\underline{u} = \underline{y} ≫ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y}; LessLeqTransitivity ⊳ ˙¬ \underline{x} <= \underline{y} ⇒ ˙¬ ˙¬ \underline{x} = \underline{y} ⊳ \underline{y} <= \underline{z} + −\underline{u} ≫ ˙¬ \underline{x} <= \underline{z} + −\underline{u} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} + −\underline{u}; LessLeq ⊳ ˙¬ \underline{x} <= \underline{z} + −\underline{u} ⇒ ˙¬ ˙¬ \underline{x} = \underline{z} + −\underline{u} ≫ \underline{x} <= \underline{z} + −\underline{u}⌉, p_{0}, c)]

[<< TransitivityHelper(Q) →^{stmt} ZFsub ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} + −\underline{u} ⊢ \underline{y} <= \underline{z} + −\underline{u} ⊢ ˙¬ 0 <= \underline{u} ⇒ ˙¬ ˙¬ 0 = \underline{u} ⊢ \underline{x} <= \underline{z} + −\underline{u}]

[<< TransitivityHelper(Q) →^{tex} “<<TransitivityHelper(Q)”]

[<< TransitivityHelper(Q) →^{pyk} “lemma <<TransitivityHelper(Q)”]

<< Transitivity

[<< Transitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From <<

(2) ⊳ \underline{(rx)} << \underline{(ry)} ≫ t_{Ex} ∈ \underline{(ry)}; ThirdGeqSeries ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(ry)} << \underline{(rz)} ⊳ \underline{(fx)} ∈ \underline{(rx)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ \underline{(fz)} ∈ \underline{(rz)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; FirstConjunct ⊳ ˙¬ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)}; SecondConjunct ⊳ ˙¬ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⇒ ˙¬ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; << TransitivityHelper(Q) ⊳ \underline{(fx)}[\underline{m}] <= t_{Ex}[\underline{m}] + −\underline{(ϵ)} ⊳ t_{Ex}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ \underline{(fx)} ∈ \underline{(rx)} ⊢ \underline{(fz)} ∈ \underline{(rz)} ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ MP2 ⊳ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(ry)} << \underline{(rz)} ≫ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ExistIntro @ a_{Ex} @ c_{Ex} ⊳ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; to << XX ⊳ \underline{(fx)} ∈ \underline{(rx)} ⇒ \underline{(fz)} ∈ \underline{(rz)} ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ a_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(rx)} << \underline{(rz)}⌉, p_{0}, c)]

[<< Transitivity →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀\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”]

<<== Reflexivity

[<<== Reflexivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ==Reflexivity ≫ \underline{(rx)} == \underline{(rx)}; eqLeq(R) ⊳ \underline{(rx)} == \underline{(rx)} ≫ ˙¬ \underline{(rx)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(rx)}⌉, p_{0}, c)]

[<<== Reflexivity →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ˙¬ \underline{(rx)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(rx)}]

[<<== Reflexivity →^{tex} “<<==Reflexivity”]

[<<== Reflexivity →^{pyk} “lemma <<==Reflexivity”]

<<== AntisymmetryHelper(Q)

[<<== AntisymmetryHelper(Q) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ \underline{x} <= \underline{y} + −\underline{z} ⊢ \underline{y} <= \underline{x} + −\underline{z} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} + −\underline{z} ≫ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z}; plusAssociativity ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + −\underline{z} + \underline{z}; plusCommutativity ≫ −\underline{z} + \underline{z} = \underline{z} + −\underline{z}; lemma eqAdditionLeft ⊳ −\underline{z} + \underline{z} = \underline{z} + −\underline{z} ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y} + \underline{z} + −\underline{z}; x = x + (y − y) ≫ \underline{y} = \underline{y} + \underline{z} + −\underline{z}; eqSymmetry ⊳ \underline{y} = \underline{y} + \underline{z} + −\underline{z} ≫ \underline{y} + \underline{z} + −\underline{z} = \underline{y}; eqTransitivity4⊳\underline{y}+−\underline{z}+\underline{z} = \underline{y}+−\underline{z}+\underline{z}⊳\underline{y}+−\underline{z}+\underline{z} = \underline{y}+\underline{z}+−\underline{z}⊳\underline{y}+\underline{z}+−\underline{z} = \underline{y} ≫ \underline{y} + −\underline{z} + \underline{z} = \underline{y}; subLeqRight ⊳ \underline{y} + −\underline{z} + \underline{z} = \underline{y} ⊳ \underline{x} + \underline{z} <= \underline{y} + −\underline{z} + \underline{z} ≫ \underline{x} + \underline{z} <= \underline{y}; leqTransitivity ⊳ \underline{x} + \underline{z} <= \underline{y} ⊳ \underline{y} <= \underline{x} + −\underline{z} ≫ \underline{x} + \underline{z} <= \underline{x} + −\underline{z}; leqSubtractionLeft ⊳ \underline{x} + \underline{z} <= \underline{x} + −\underline{z} ≫ \underline{z} <= −\underline{z}; toNotLess ⊳ \underline{z} <= −\underline{z} ≫ ˙¬ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z}; NegativeLessPositive ⊳ ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ≫ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z}; FromContradiction ⊳ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z} ⊳ ˙¬ ˙¬ − \underline{z} <= \underline{z} ⇒ ˙¬ ˙¬ − \underline{z} = \underline{z} ≫ \underline{a}⌉, p_{0}, c)]

[<<== AntisymmetryHelper(Q) →^{stmt} ZFsub ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ 0 <= \underline{z} ⇒ ˙¬ ˙¬ 0 = \underline{z} ⊢ \underline{x} <= \underline{y} + −\underline{z} ⊢ \underline{y} <= \underline{x} + −\underline{z} ⊢ \underline{a}]

[<<== AntisymmetryHelper(Q) →^{tex} “<<==AntisymmetryHelper(Q)”]

[<<== AntisymmetryHelper(Q) →^{pyk} “lemma <<==AntisymmetryHelper(Q)”]

<<== Antisymmetry

[<<== Antisymmetry →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ⊢ Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ≫ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)}; ExpandDisjuncts ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊳ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ≫ ˙¬ \underline{(rx)} == \underline{(ry)} ⇒ ˙¬ \underline{(ry)} == \underline{(rx)} ⇒ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)}; AutoImply ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(ry)} == \underline{(rx)} ⊢==Symmetry ⊳ \underline{(ry)} == \underline{(rx)} ≫ \underline{(rx)} == \underline{(ry)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: \underline{(ry)} == \underline{(rx)} ⊢ \underline{(rx)} == \underline{(ry)} ≫ \underline{(ry)} == \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⊢ FirstConjunct ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ≫ \underline{(rx)} << \underline{(ry)}; From << (1) ⊳ \underline{(rx)} << \underline{(ry)} ≫ j_{Ex} ∈ \underline{(rx)}; From << (2) ⊳ \underline{(rx)} << \underline{(ry)} ≫ t_{Ex} ∈ \underline{(ry)}; 0 < 1 ≫ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1; From << XX ⊳ \underline{(rx)} << \underline{(ry)} ⊳ j_{Ex} ∈ \underline{(rx)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ a_{Ex} <= c_{Ex} ⇒ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1; SecondConjunct ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ≫ \underline{(ry)} << \underline{(rx)}; From << XX ⊳ \underline{(ry)} << \underline{(rx)} ⊳ t_{Ex} ∈ \underline{(ry)} ⊳ j_{Ex} ∈ \underline{(rx)} ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ≫ a_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1; ExistIntro @ b_{Ex} @ a_{Ex} ⊳ a_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1 ≫ b_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1; thirdGeq ≫ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex}; FirstConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ a_{Ex} <= c_{Ex}; MP ⊳ a_{Ex} <= c_{Ex} ⇒ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1 ⊳ a_{Ex} <= c_{Ex} ≫ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1; SecondConjunct ⊳ ˙¬ a_{Ex} <= c_{Ex} ⇒ ˙¬ b_{Ex} <= c_{Ex} ≫ b_{Ex} <= c_{Ex}; MP ⊳ b_{Ex} <= c_{Ex} ⇒ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1 ⊳ b_{Ex} <= c_{Ex} ≫ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1; <<== AntisymmetryHelper(Q) ⊳ ˙¬ 0 <= 1 ⇒ ˙¬ ˙¬ 0 = 1 ⊳ j_{Ex}[c_{Ex}] <= t_{Ex}[c_{Ex}] + −1 ⊳ t_{Ex}[c_{Ex}] <= j_{Ex}[c_{Ex}] + −1 ≫ \underline{(rx)} == \underline{(ry)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⊢ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)}; From3Disjuncts ⊳ ˙¬ \underline{(rx)} == \underline{(ry)} ⇒ ˙¬ \underline{(ry)} == \underline{(rx)} ⇒ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(ry)} == \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)}⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ \underline{(rx)} == \underline{(ry)}⌉, p_{0}, c)]

[<<== Antisymmetry →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rx)} ⇒ \underline{(ry)} == \underline{(rx)} ⊢ \underline{(rx)} == \underline{(ry)}]

[<<== Antisymmetry →^{tex} “<<==Antisymmetry”]

[<<== Antisymmetry →^{pyk} “lemma <<==Antisymmetry”]

<<== Transitivity

[<<== Transitivity →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢<< Transitivity ⊳ \underline{(rx)} << \underline{(ry)} ⊳ \underline{(ry)} << \underline{(rz)} ≫ \underline{(rx)} << \underline{(rz)}; LessLeq(R) ⊳ \underline{(rx)} << \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ SubLessRight(R) ⊳ \underline{(ry)} == \underline{(rz)} ⊳ \underline{(rx)} << \underline{(ry)} ≫ \underline{(rx)} << \underline{(rz)}; LessLeq(R) ⊳ \underline{(rx)} << \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢==Symmetry ⊳ \underline{(rx)} == \underline{(ry)} ≫ \underline{(ry)} == \underline{(rx)}; SubLessLeft(R) ⊳ \underline{(ry)} == \underline{(rx)} ⊳ \underline{(ry)} << \underline{(rz)} ≫ \underline{(rx)} << \underline{(rz)}; LessLeq(R) ⊳ \underline{(rx)} << \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ == Transitivity ⊳ \underline{(rx)} == \underline{(ry)} ⊳ \underline{(ry)} == \underline{(rz)} ≫ \underline{(rx)} == \underline{(rz)}; eqLeq(R) ⊳ \underline{(rx)} == \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ⊢ Repetition ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ≫ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)}; Repetition ⊳ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ≫ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} << \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} << \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; Ded ⊳ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: \underline{(rx)} == \underline{(ry)} ⊢ \underline{(ry)} == \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}; From2 ∗ 2Disjuncts ⊳ ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊳ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ⊳ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ⊳ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} << \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ⊳ \underline{(rx)} == \underline{(ry)} ⇒ \underline{(ry)} == \underline{(rz)} ⇒ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)} ≫ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}⌉, p_{0}, c)]

[<<== Transitivity →^{stmt} ZFsub ⊢ ∀\underline{(rx)}: ∀\underline{(ry)}: ∀\underline{(rz)}: ˙¬ \underline{(rx)} << \underline{(ry)} ⇒ \underline{(rx)} == \underline{(ry)} ⊢ ˙¬ \underline{(ry)} << \underline{(rz)} ⇒ \underline{(ry)} == \underline{(rz)} ⊢ ˙¬ \underline{(rx)} << \underline{(rz)} ⇒ \underline{(rx)} == \underline{(rz)}]

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

Plus0f

[Plus0f →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: PlusF ≫ \underline{(fx)} +_{f} 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0f[\underline{m}]; 0f ≫ 0f[\underline{m}] = 0; lemma eqAdditionLeft ⊳ 0f[\underline{m}] = 0 ≫ \underline{(fx)}[\underline{m}] + 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0; plus0 ≫ \underline{(fx)}[\underline{m}] + 0 = \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} +_{f} 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0f[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + 0f[\underline{m}] = \underline{(fx)}[\underline{m}] + 0 ⊳ \underline{(fx)}[\underline{m}] + 0 = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}+_{f}0f[\underline{m}] = \underline{(fx)}[\underline{m}]; To = f⊳\underline{(fx)}+_{f}0f[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}+_{f}0f =_{f} \underline{(fx)}⌉, p_{0}, c)]

[Plus0f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: \underline{(fx)} +_{f} 0f =_{f} \underline{(fx)}]

[Plus0f →^{tex} “Plus0f”]

[Plus0f →^{pyk} “lemma plus0f”]

Plus00

[Plus00 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: Plus0f ≫ \underline{(fx)} +_{f} 0f =_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} +_{f} 0f =_{f} \underline{(fx)} ≫ SF(\underline{(fx)} +_{f} 0f, \underline{(fx)}); f2R(Plus) ⊳ SF(\underline{(fx)} +_{f} 0f, \underline{(fx)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)}); Repetition ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)})⌉, p_{0}, c)]

[Plus00 →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: R((fx) +_{f} (fy)) == R(\underline{(fx)})]

[Plus00 →^{tex} “Plus00”]

[Plus00 →^{pyk} “lemma plus00”]

== Addition

[== Addition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From == ⊳R(\underline{(fx)}) == R(\underline{(fy)}) ≫ SF(\underline{(fx)}, \underline{(fy)}); FromSF ⊳ SF(\underline{(fx)}, \underline{(fy)}) ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ⊳ c_{Ex} <= \underline{m} ≫ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)}; insertMiddleTerm(Difference) ≫ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; EqNegated ⊳ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; AddEquations ⊳ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; eqTransitivity ⊳ \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}]; SameNumerical⊳\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}] ≫ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]); SubLessLeft ⊳ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) = if(0 <= \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}], \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) ⊳ ˙¬ if(0 <= \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}]+−\underline{(fy)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], \underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}], −\underline{(fx)}[\underline{m}] + −\underline{(fy)}[\underline{m}]) = \underline{(ϵ)} ≫ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ ˙¬ if(0 <= \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}], \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(fy)}+_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]+ −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ MP ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)}; ToSF ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ if(0 <= \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}], −\underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]) = \underline{(ϵ)} ≫ SF(\underline{(fx)} +_{f} \underline{(fz)}, \underline{(fy)} +_{f} \underline{(fz)}); To == ⊳SF(\underline{(fx)} +_{f} \underline{(fz)}, \underline{(fy)} +_{f} \underline{(fz)}) ≫ R(\underline{(fx)} +_{f} \underline{(fz)}) == R(\underline{(fy)} +_{f} \underline{(fz)}); PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fz)}); PlusR ≫ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}); ==Symmetry ⊳ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); ==Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fz)}) ⊳ R(\underline{(fx)} +_{f} \underline{(fz)}) == R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}); ==Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fz)}) ⊳ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))⌉, p_{0}, c)]

[== Addition →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))]

[== Addition →^{tex} “==Addition”]

[== Addition →^{pyk} “lemma ==Addition”]

== AdditionLeft

[== AdditionLeft →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢== Addition ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); PlusCommutativity(R) ≫ R((fx) +_{f} (fy)) == R((fx)+_{f} (fy)); PlusCommutativity(R) ≫ R((fx)+_{f} (fy)) == R((fx)+_{f} (fy)); == Transitivity ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); == Transitivity ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))⌉, p_{0}, c)]

[== AdditionLeft →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))]

[== AdditionLeft →^{tex} “==AdditionLeft”]

[== AdditionLeft →^{pyk} “lemma ==AdditionLeft”]

<< Addition

[<< Addition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ From << ⊳R(\underline{(fx)}) << R(\underline{(fy)}) ≫ \underline{(fx)} <_{f} \underline{(fy)}; From < f ⊳ \underline{(fx)} <_{f} \underline{(fy)} ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ≫ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)}; MP ⊳ c_{Ex} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}]+−\underline{(ϵ)}⊳c_{Ex} <= \underline{m} ≫ \underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}]+−\underline{(ϵ)}; leqAddition⊳\underline{(fx)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}]; subLeqLeft ⊳ \underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}]⊳\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}]+−\underline{(ϵ)}+\underline{(fz)}[\underline{m}] ≫ \underline{(fx)}+_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}]; Three2threeTerms ≫ \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; PlusF(Sym) ≫ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; eqAddition ⊳ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]+−\underline{(ϵ)} = \underline{(fy)}+_{f} \underline{(fz)}[\underline{m}]+−\underline{(ϵ)}; eqTransitivity ⊳\underline{(fy)}[\underline{m}]+−\underline{(ϵ)}+\underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; subLeqRight ⊳ \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)}[\underline{m}] + −\underline{(ϵ)} + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⊢ c_{Ex} <= \underline{m} ⊢ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ MP ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ≫ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)}; To < f ⊳ ˙¬ 0 <= \underline{(ϵ)} ⇒ ˙¬ ˙¬ 0 = \underline{(ϵ)} ⇒ c_{Ex} <= \underline{m} ⇒ \underline{(fx)} +_{f} \underline{(fz)}[\underline{m}] <= \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] + −\underline{(ϵ)} ≫ \underline{(fx)}+_{f} \underline{(fz)} <_{f} \underline{(fy)}+_{f} \underline{(fz)}; To << ⊳\underline{(fx)}+_{f} \underline{(fz)} <_{f} \underline{(fy)}+_{f} \underline{(fz)} ≫ R(\underline{(fx)}+_{f} \underline{(fz)}) << R(\underline{(fy)}+_{f} \underline{(fz)}); PlusR(Sym) ≫ R(\underline{(fx)}+_{f} \underline{(fz)}) == R((fx)+_{f} (fy)); PlusR(Sym) ≫ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); SubLessLeft(R) ⊳ R(\underline{(fx)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ⊳ R(\underline{(fx)} +_{f} \underline{(fz)}) << R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) << R(\underline{(fy)} +_{f} \underline{(fz)}); SubLessRight(R) ⊳ R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) << R(\underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy))⌉, p_{0}, c)]

[<< Addition →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy))]

[<< Addition →^{tex} “<<Addition”]

[<< Addition →^{pyk} “lemma <<Addition”]

<<== Addition

[<<== Addition →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢<< Addition ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ≫ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)); LessLeq(R) ⊳ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ≫ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢== Addition ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); eqLeq(R) ⊳ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) << R(\underline{(fy)}) ⊢ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)); ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ Repetition ⊳ ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ≫ ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}); FromDisjuncts ⊳ ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ⊳ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ⊳ R(\underline{(fx)}) == R(\underline{(fy)}) ⇒ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy)) ≫ ˙¬ R((fx)+_{f} (fy)) << R((fx)+_{f} (fy)) ⇒ R((fx)+_{f} (fy)) == R((fx)+_{f} (fy))⌉, p_{0}, c)]

[<<== Addition →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ R(\underline{(fx)}) << R(\underline{(fy)}) ⇒ R(\underline{(fx)}) == R(\underline{(fy)}) ⊢ ˙¬ R((fx) +_{f} (fy)) << R((fx) +_{f} (fy)) ⇒ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))]

[<<== Addition →^{tex} “<<==Addition”]

[<<== Addition →^{pyk} “lemma <<==Addition”]

PlusAssociativity(F)

[PlusAssociativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]; eqAddition ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; plusAssociativity ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; lemma eqAdditionLeft ⊳ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; eqTransitivity6 ⊳ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}]⊳\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; To = f ⊳ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}⌉, p_{0}, c)]

[PlusAssociativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}]

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

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

PlusAssociativity(R)

[PlusAssociativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: PlusAssociativity(F) ≫ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}; = fToSameF ⊳ \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)} ≫ SF(\underline{(fx)}+_{f} \underline{(fy)}+_{f} \underline{(fz)}, \underline{(fx)}+_{f} \underline{(fy)}+_{f} \underline{(fz)}); f2R(Plus)⊳SF(\underline{(fx)}+_{f} \underline{(fy)}+_{f} \underline{(fz)}, \underline{(fx)}+_{f} \underline{(fy)} +_{f} \underline{(fz)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}); PlusR(Sym) ≫ R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); == Transitivity ⊳ R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ⊳ R((fx) +_{f} (fy)) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) ≫ R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)})⌉, p_{0}, c)]

[PlusAssociativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)}) == R(\underline{(fx)} +_{f} \underline{(fy)} +_{f} \underline{(fz)})]

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

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

Negative(R)

[Negative(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: PlusF ≫ \underline{(fx)} +_{f} −_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −_{f} \underline{(fx)}[\underline{m}]; MinusF ≫ −_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}]; lemma eqAdditionLeft⊳−_{f} \underline{(fx)}[\underline{m}] = −\underline{(fx)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}]+−_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}]; Negative ≫ \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}] = 0; 0f ≫ 0f[\underline{m}] = 0; eqSymmetry ⊳ 0f[\underline{m}] = 0 ≫ 0 = 0f[\underline{m}]; eqTransitivity5 ⊳ \underline{(fx)} +_{f} −_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −_{f} \underline{(fx)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + −_{f} \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + −\underline{(fx)}[\underline{m}] = 0⊳0 = 0f[\underline{m}] ≫ \underline{(fx)}+_{f} −_{f} \underline{(fx)}[\underline{m}] = 0f[\underline{m}]; To = f⊳\underline{(fx)}+_{f} −_{f} \underline{(fx)}[\underline{m}] = 0f[\underline{m}] ≫ \underline{(fx)} +_{f} −_{f} \underline{(fx)} =_{f} 0f; = fToSameF ⊳ \underline{(fx)} +_{f} −_{f} \underline{(fx)} =_{f} 0f ≫ SF(\underline{(fx)} +_{f} −_{f} \underline{(fx)}, 0f); f2R(Plus) ⊳ SF(\underline{(fx)} +_{f} −_{f} \underline{(fx)}, 0f) ≫ R((fx) +_{f} (fy)) == R(0f); Repetition⊳R((fx)+_{f} (fy)) == R(0f) ≫ R((fx)+_{f} (fy)) == R(0f)⌉, p_{0}, c)]

[Negative(R) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: R((fx) +_{f} (fy)) == R(0f)]

[Negative(R) →^{tex} “Negative(R)”]

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

PlusCommutativity(F)

[PlusCommutativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: PlusF ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]; plusCommutativity ≫ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ⊳ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}]; To = f ⊳ \underline{(fx)} +_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} +_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} +_{f} \underline{(fy)} =_{f} \underline{(fy)} +_{f} \underline{(fx)}⌉, p_{0}, c)]

[PlusCommutativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}+_{f} \underline{(fy)} =_{f} \underline{(fy)}+_{f} \underline{(fx)}]

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

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

PlusCommutativity(R)

[PlusCommutativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: PlusCommutativity(F) ≫ \underline{(fx)} +_{f} \underline{(fy)} =_{f} \underline{(fy)} +_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} +_{f} \underline{(fy)} =_{f} \underline{(fy)} +_{f} \underline{(fx)} ≫ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fy)} +_{f} \underline{(fx)}); f2R(Plus) ⊳ SF(\underline{(fx)} +_{f} \underline{(fy)}, \underline{(fy)} +_{f} \underline{(fx)}) ≫ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fx)}); PlusR(Sym) ≫ R(\underline{(fy)} +_{f} \underline{(fx)}) == R((fx) +_{f} (fy)); == Transitivity ⊳ R((fx) +_{f} (fy)) == R(\underline{(fy)} +_{f} \underline{(fx)}) ⊳ R(\underline{(fy)} +_{f} \underline{(fx)}) == R((fx) +_{f} (fy)) ≫ R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))⌉, p_{0}, c)]

[PlusCommutativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R((fx) +_{f} (fy)) == R((fx) +_{f} (fy))]

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

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

TimesAssociativity(F)

[TimesAssociativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]; eqMultiplication ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; timesAssociativity ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; EqMultiplicationLeft ⊳ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; eqTransitivity6 ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}]; To = f ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}⌉, p_{0}, c)]

[TimesAssociativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}]

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

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

TimesAssociativity(R)

[TimesAssociativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: TimesAssociativity(F) ≫ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}; = fToSameF ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)} ≫ SF(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)}∗_{f} \underline{(fz)}) ≫ R(\underline{(fx)}∗_{f} \underline{(fy)})∗∗R(\underline{(fz)}) == R(\underline{(fx)}∗_{f} \underline{(fy)}∗_{f} \underline{(fz)}); TimesR(Sym) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ∗ ∗R(\underline{(fz)}); ==Transitivity ⊳ R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)}) ∗ ∗R(\underline{(fz)}) ⊳ R(\underline{(fx)} ∗_{f} \underline{(fy)}) ∗ ∗R(\underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)})⌉, p_{0}, c)]

[TimesAssociativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} ∗_{f} \underline{(fz)})]

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

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

Times1f

[Times1f →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: TimesF ≫ \underline{(fx)} ∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}]; 1f ≫ 1f[\underline{m}] = 1; EqMultiplicationLeft ⊳ 1f[\underline{m}] = 1 ≫ \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1; times1 ≫ \underline{(fx)}[\underline{m}] ∗ 1 = \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} ∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ 1 ⊳ \underline{(fx)}[\underline{m}] ∗ 1 = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}]; To = f ⊳\underline{(fx)}∗_{f} 1f[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}∗_{f} 1f =_{f} \underline{(fx)}⌉, p_{0}, c)]

[Times1f →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: \underline{(fx)} ∗_{f} 1f =_{f} \underline{(fx)}]

[Times1f →^{tex} “Times1f”]

[Times1f →^{pyk} “lemma times1f”]

Times01

[Times01 →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: Times1f ≫ \underline{(fx)} ∗_{f} 1f =_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} ∗_{f} 1f =_{f} \underline{(fx)} ≫ SF(\underline{(fx)} ∗_{f} 1f, \underline{(fx)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} 1f, \underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)}); Repetition ⊳ R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)})⌉, p_{0}, c)]

[Times01 →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: R(\underline{(fx)}) ∗ ∗R(1f) == R(\underline{(fx)})]

[Times01 →^{tex} “Times01”]

[Times01 →^{pyk} “lemma times01”]

TimesCommutativity(F)

[TimesCommutativity(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]; timesCommutativity ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}]; eqTransitivity4 ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}] ⊳ \underline{(fy)}[\underline{m}] ∗ \underline{(fx)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}]; To = f ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] = \underline{(fy)} ∗_{f} \underline{(fx)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} =_{f} \underline{(fy)} ∗_{f} \underline{(fx)}⌉, p_{0}, c)]

[TimesCommutativity(F) →^{stmt} ZFsub ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)}∗_{f} \underline{(fy)} =_{f} \underline{(fy)}∗_{f} \underline{(fx)}]

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

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

TimesCommutativity(R)

[TimesCommutativity(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: TimesCommutativity(F) ≫ \underline{(fx)} ∗_{f} \underline{(fy)} =_{f} \underline{(fy)} ∗_{f} \underline{(fx)}; = fToSameF ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} =_{f} \underline{(fy)} ∗_{f} \underline{(fx)} ≫ SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fy)} ∗_{f} \underline{(fx)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} \underline{(fy)}, \underline{(fy)} ∗_{f} \underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)} ∗_{f} \underline{(fx)}); TimesR(Sym) ≫ R(\underline{(fy)} ∗_{f} \underline{(fx)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)}); == Transitivity ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)} ∗_{f} \underline{(fx)}) ⊳ R(\underline{(fy)} ∗_{f} \underline{(fx)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)})⌉, p_{0}, c)]

[TimesCommutativity(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}) == R(\underline{(fy)}) ∗ ∗R(\underline{(fx)})]

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

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

Distribution(F)

[Distribution(F) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: TimesF ≫ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}]; PlusF ≫ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; EqMultiplicationLeft ⊳ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}]; Distribution ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}]; TimesF(Sym) ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)}∗_{f} \underline{(fz)}[\underline{m}]; AddEquations⊳\underline{(fx)}[\underline{m}]∗\underline{(fy)}[\underline{m}] = \underline{(fx)}∗_{f} \underline{(fy)}[\underline{m}]⊳\underline{(fx)}[\underline{m}]∗\underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}]; PlusF(Sym) ≫ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}]; eqTransitivity6 ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}] = \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] ⊳ \underline{(fx)} ∗_{f} \underline{(fy)}[\underline{m}] + \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}[\underline{m}]; To = f ⊳ \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}[\underline{m}] = \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}[\underline{m}] ≫ \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fz)} =_{f} \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}⌉, p_{0}, c)]

[Distribution(F) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}]

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

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

Distribution(R)

[Distribution(R) →^{proof} λc.λx.P(⌈ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Distribution(F) ≫ \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fz)} =_{f} \underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}; = fToSameF⊳\underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fz)} =_{f} \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)} ≫ SF(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}); f2R(Times) ⊳ SF(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fz)}, \underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)} +_{f} \underline{(fz)}) == R(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}); PlusR(Sym) ≫ R(\underline{(fx)} ∗_{f} \underline{(fy)} +_{f} \underline{(fx)} ∗_{f} \underline{(fz)}) == R((fx) +_{f} (fy)); ==Transitivity ⊳ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)}+_{f} \underline{(fz)}) == R(\underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)})⊳R(\underline{(fx)}∗_{f} \underline{(fy)}+_{f} \underline{(fx)}∗_{f} \underline{(fz)}) == R((fx) +_{f} (fy)) ≫ R(\underline{(fx)}) ∗ ∗R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy))⌉, p_{0}, c)]

[Distribution(R) →^{stmt} ZFsub ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: R(\underline{(fx)}) ∗ ∗R(\underline{(fy)} +_{f} \underline{(fz)}) == R((fx) +_{f} (fy))]

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

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

R(∗)

[R(x) →^{tex} “R(#1.

)”]

[R(∗) →^{pyk} “R( " )”]

− − R(∗)

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

[− − 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”]

{∗, ∗}

[{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”]

−∗

[−x →^{tex} “-#1.”]

[−∗ →^{pyk} “- "”]

−_{f}∗

[−_{f}x →^{tex} “- {f}#1.”]

[−_{f}∗ →^{pyk} “-f "”]

∗ ∈ ∗

[x ∈ y →^{tex} “#1.

\mathrel{\in} #2.”]

[∗ ∈ ∗ →^{pyk} “" in0 "”]

∗(∗, ∗)

[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) →^{tex} “#1.

∗ {f}#2.”]

[∗ ∗_{f} ∗ →^{pyk} “" ∗f "”]

∗ ∗ ∗∗

[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 + (−y)]⌉)]

[x − y →^{tex} “#1.

-#2.”]

[∗ − ∗ →^{pyk} “" - "”]

∗ +_{f} ∗

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

[∗ +_{f} ∗ →^{pyk} “" +f "”]

∗ −_{f} ∗

[(fx) −_{f} (fy) →^{tex} “#1.

- {f}#2.”]

[∗ −_{f} ∗ →^{pyk} “" -f "”]

∗ + +∗

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

[x + +y →^{tex} “#1.

++#2.”]

[∗ + +∗ →^{pyk} “" ++ "”]

R(∗) − −R(∗)

[R((fx)) − −R((fy)) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[R((fx)) − −R((fy)) ¨= R((fx)) + +R(−_{f}(fy))]⌉)]

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

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

| ∗ |

[|x| →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[|x| ¨= if(0 <= x, x, −x)]⌉)]

[|x| →^{tex} “|#1.|”]

[| ∗ | →^{pyk} “| " |”]

if(∗, ∗, ∗)

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

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

∗ = ∗

[x = y →^{tex} “#1.

= #2.”]

[∗ = ∗ →^{pyk} “" = "”]

∗ ≠ ∗

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

[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} ∗

[x =_{f} y →^{tex} “#1.

= {f}#2.”]

[∗ =_{f} ∗ →^{pyk} “" =f "”]

∗ <_{f} ∗

[x <_{f} y →^{tex} “#1.

< {f}#2.”]

[∗ <_{f} ∗ →^{pyk} “" <f "”]

SF(∗, ∗)

[SF(x, y) →^{tex} “SF(#1. , #2. )”]

[SF(∗, ∗) →^{pyk} “" sameF "”]

∗ == ∗

[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 << 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 ¨= (s ∈ x ⇒ s ∈ y)]⌉)]

[x ⊆ y →^{tex} “#1.

\mathrel{\subseteq} #2.”] [∗ ⊆ ∗ →^{pyk} “" is subset of "”]

˙¬ ∗

[ ˙¬ x →^{tex} “\dot{\neg}\, #1.”]

[ ˙¬ ∗ →^{pyk} “not0 "”]

∗ ∉ ∗

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

[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]⌉)]

[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)]⌉)]

[x ˙∧ y →^{tex} “#1.

\mathrel{\dot{\wedge}} #2.”]

[∗ ˙∧ ∗ →^{pyk} “" and0 "”]

∗ ˙∨ ∗

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

[x ˙∨ y →^{tex} “#1.

\mathrel{\dot{\vee}} #2.”]

[∗ ˙∨ ∗ →^{pyk} “" or0 "”]

∗ ˙⇔ ∗

[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-2006-09-15.UTC:09:33:20.992497 = MJD-53993.TAI:09:33:53.992497 = LGT-4665029633992497e-6