Logiweb codex of pogave

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S, Neg, MP, Gen, Ded, S1, S2, S3, S4, S5, S6, S7, S8, S9, Prop 3.2a, Prop 3.2b, Prop 3.2c, Prop 3.2d, Prop 3.2e, Prop 3.2f, Prop 3.2g, Prop 3.2h, pogave, rule div, S10, Prop 3.2i, Prop 3.2j_{1}, Prop 3.2j_{2}, Prop 3.2j, Prop 3.2k_{1}, Prop 3.2k_{2}, Prop 3.2k, Prop 3.2l_{1}, Prop 3.2l_{2}, Prop 3.2l, Prop 3.2m_{1}, Prop 3.2m_{2}, Prop 3.2m, Prop 3.2n_{1}, Prop 3.2n_{2}, Prop 3.2n, Prop 3.2o, Prop 3.4a_{1}, Prop 3.4a_{2}, Prop 3.4a, Prop 3.4b, Prop 3.4c_{1}, Prop 3.4c_{2}, Prop 3.4c, Prop 3.4d_{1}, Prop 3.4d_{2}, Prop 3.4d, ∗∥∗,

S

[S →^{stmt} x]

Neg

[Neg →^{proof} Rule tactic]

[Neg →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ¬\underline{b} ⇒ ¬\underline{a} ⊢ ¬\underline{b} ⇒ \underline{a} ⊢ \underline{b}]

MP

[MP →^{proof} Rule tactic]

[MP →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ \underline{a} ⊢ \underline{b}]

Gen

[Gen →^{proof} Rule tactic]

[Gen →^{stmt} S ⊢ ∀\underline{x}: ∀\underline{a}: \underline{a} ⊢ ∀_{obj}\underline{x}: \underline{a}]

Ded

[Ded →^{proof} Rule tactic] [Ded →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: λx.Ded_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ⊢⊢ \underline{a} ⊢ \underline{b}]

S1

[S1 →^{proof} Rule tactic] [S1 →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{a} = \underline{c} ⊢ \underline{b} = \underline{c}]

S2

[S2 →^{proof} Rule tactic] [S2 →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} = \underline{b} ⊢ \underline{a}′ = \underline{b}′]

S3

[S3 →^{proof} Rule tactic] [S3 →^{stmt} S ⊢ ∀\underline{a}: ¬0 = \underline{a}′]

S4

[S4 →^{proof} Rule tactic] [S4 →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a}′ = \underline{b}′ ⊢ \underline{a} = \underline{b}]

S5

[S5 →^{proof} Rule tactic] [S5 →^{stmt} S ⊢ ∀\underline{a}: \underline{a} + 0 = \underline{a}]

S6

[S6 →^{proof} Rule tactic] [S6 →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} + \underline{b}′ = \underline{a} + \underline{b}′]

S7

[S7 →^{proof} Rule tactic] [S7 →^{stmt} S ⊢ ∀\underline{a}: \underline{a} · 0 = 0]

S8

[S8 →^{proof} Rule tactic] [S8 →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} · \underline{b}′ = \underline{a} · \underline{b} + \underline{a}]

S9

[S9 →^{proof} Rule tactic]

[S9 →^{stmt} S ⊢ ∀\underline{x}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ⟨⌈\underline{b}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{x}⌉:=⌈0⌉⟩ ⊢⊢ ⟨⌈\underline{c}⌉≡^{0}⌈\underline{a}⌉|⌈\underline{x}⌉:=⌈\underline{x}′⌉⟩ ⊢⊢ \underline{b} ⊢

\underline{a} ⇒ \underline{c} ⊢ \underline{a}]

Prop 3.2a

[Prop 3.2a →^{stmt} S ⊢ ∀\underline{a}: \underline{a} = \underline{a}]

Prop 3.2b

[Prop 3.2b →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} = \underline{b} ⊢ \underline{b} = \underline{a}]

Prop 3.2c

[Prop 3.2c →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{b} = \underline{c} ⊢ \underline{a} = \underline{c}]

Prop 3.2d

[Prop 3.2d →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{c} ⊢ \underline{b} = \underline{c} ⊢ \underline{a} = \underline{b}]

Prop 3.2e

[Prop 3.2e →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{a} + \underline{c} = \underline{b} + \underline{c}]

Prop 3.2f

[Prop 3.2f →^{stmt} S ⊢ ∀\underline{a}: \underline{a} = 0 + \underline{a}]

Prop 3.2g

[Prop 3.2g →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a}′ + \underline{b} = \underline{a} + \underline{b}′]

Prop 3.2h

[Prop 3.2h →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} + \underline{b} = \underline{b} + \underline{a}]

pogave

[pogave →^{prio} Preassociative [pogave], [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}(∗)], [rule div], [S10], [Prop 3.2i], [Prop 3.2j_{1}], [Prop 3.2j_{2}], [Prop 3.2j], [Prop 3.2k_{1}], [Prop 3.2k_{2}], [Prop 3.2k], [Prop 3.2l_{1}], [Prop 3.2l_{2}], [Prop 3.2l], [Prop 3.2m_{1}], [Prop 3.2m_{2}], [Prop 3.2m], [Prop 3.2n_{1}], [Prop 3.2n_{2}], [Prop 3.2n], [Prop 3.2o], [Prop 3.4a_{1}], [Prop 3.4a_{2}], [Prop 3.4a], [Prop 3.4b], [Prop 3.4c_{1}], [Prop 3.4c_{2}], [Prop 3.4c], [Prop 3.4d_{1}], [Prop 3.4d_{2}], [Prop 3.4d]; 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 [∗′]; Preassociative [∗ · ∗], [∗ ·_{0} ∗]; Preassociative [∗ + ∗], [∗ +_{0} ∗], [∗ +_{1} ∗], [∗ − ∗], [∗ −_{0} ∗], [∗ −_{1} ∗]; 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 [∗ ∨ ∗], [∗ ∥ ∗], [∗ ¨∨ ∗]; Preassociative [∃∗: ∗], [∀∗: ∗], [∀_{obj}∗: ∗]; Postassociative [∗ ¨⇒ ∗], [∗ ⇒ ∗], [∗ ⇔ ∗]; 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] ∗], [∗\\∗], [∗∥∗]; ]

[pogave →^{pyk} “pogave”]

rule div

[rule div →^{pyk} “rule div”]

S10

[S10 →^{tex} “

S10”]

[S10 →^{pyk} “axiom s ten”]

Prop 3.2i

[Prop 3.2i →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: Prop 3.2e ≫ \underline{a} = \underline{b} ⇒ \underline{a} + \underline{c} = \underline{b} + \underline{c}; Prop 3.2h ≫ \underline{a} + \underline{c} = \underline{c} + \underline{a}; Prop 3.2h ≫ \underline{b} + \underline{c} = \underline{c} + \underline{b}; \underline{a} = \underline{b} ⊢ MP ⊳ \underline{a} = \underline{b} ⇒ \underline{a} + \underline{c} = \underline{b} + \underline{c} ⊳ \underline{a} = \underline{b} ≫ \underline{a} + \underline{c} = \underline{b} + \underline{c}; S1 ≫ \underline{a} + \underline{c} = \underline{b} + \underline{c} ⇒ \underline{a} + \underline{c} = \underline{c} + \underline{a} ⇒ \underline{b} + \underline{c} = \underline{c} + \underline{a}; MP ⊳ \underline{a} + \underline{c} = \underline{b} + \underline{c} ⇒ \underline{a} + \underline{c} = \underline{c} + \underline{a} ⇒ \underline{b} + \underline{c} = \underline{c} + \underline{a} ⊳ \underline{a} + \underline{c} = \underline{b} + \underline{c} ≫ \underline{a} + \underline{c} = \underline{c} + \underline{a} ⇒ \underline{b} + \underline{c} = \underline{c} + \underline{a}; MP ⊳ \underline{a} + \underline{c} = \underline{c} + \underline{a} ⇒ \underline{b} + \underline{c} = \underline{c} + \underline{a} ⊳ \underline{a} + \underline{c} = \underline{c} + \underline{a} ≫ \underline{b} + \underline{c} = \underline{c} + \underline{a}; Prop 3.2b ⊳ \underline{b} + \underline{c} = \underline{c} + \underline{a} ≫ \underline{c} + \underline{a} = \underline{b} + \underline{c}; Prop 3.2e ≫ \underline{c} + \underline{a} = \underline{b} + \underline{c} ⇒ \underline{b} + \underline{c} = \underline{c} + \underline{b} ⇒ \underline{c} + \underline{a} = \underline{c} + \underline{b}; MP ⊳ \underline{c} + \underline{a} = \underline{b} + \underline{c} ⇒ \underline{b} + \underline{c} = \underline{c} + \underline{b} ⇒ \underline{c} + \underline{a} = \underline{c} + \underline{b} ⊳ \underline{c} + \underline{a} = \underline{b} + \underline{c} ≫ \underline{b} + \underline{c} = \underline{c} + \underline{b} ⇒ \underline{c} + \underline{a} = \underline{c} + \underline{b}; MP ⊳ \underline{b} + \underline{c} = \underline{c} + \underline{b} ⇒ \underline{c} + \underline{a} = \underline{c} + \underline{b} ⊳ \underline{b} + \underline{c} = \underline{c} + \underline{b} ≫ \underline{c} + \underline{a} = \underline{c} + \underline{b}; Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{c} + \underline{a} = \underline{c} + \underline{b} ≫ \underline{a} = \underline{b} ⇒ \underline{c} + \underline{a} = \underline{c} + \underline{b}⌉, p_{0}, c)]

[Prop 3.2i →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{c} + \underline{a} = \underline{c} + \underline{b}]

[Prop 3.2i →^{tex} “ Prop\ 3.2i”]

[Prop 3.2i →^{pyk} “prop three two i”]

Prop 3.2j_{1}

[Prop 3.2j_{1} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: S5 ≫ \underline{a} + \underline{b} + 0 = \underline{a} + \underline{b}; S5 ≫ \underline{b} + 0 = \underline{b}; Prop 3.2i ⊳ \underline{b} + 0 = \underline{b} ≫ \underline{a} + \underline{b} + 0 = \underline{a} + \underline{b}; Prop 3.2d ⊳ \underline{a} + \underline{b} + 0 = \underline{a} + \underline{b} ⊳ \underline{a} + \underline{b} + 0 = \underline{a} + \underline{b} ≫ \underline{a} + \underline{b} + 0 = \underline{a} + \underline{b} + 0⌉, p_{0}, c)]

[Prop 3.2j_{1} →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} + \underline{b} + 0 = \underline{a} + \underline{b} + 0]

[Prop 3.2j_{1} →^{tex} “ Prop\ 3.2j 1”]

[Prop 3.2j_{1} →^{pyk} “prop three two j one”]

Prop 3.2j_{2}

[Prop 3.2j_{2} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} + \underline{b} + \underline{c} = \underline{a} + \underline{b} + \underline{c} ⊢ S6 ≫ \underline{a} + \underline{b} + \underline{c}′ = \underline{a} + \underline{b} + \underline{c}′; S2 ⊳ \underline{a} + \underline{b} + \underline{c} = \underline{a} + \underline{b} + \underline{c} ≫ \underline{a} + \underline{b} + \underline{c}′ = \underline{a}+\underline{b}+\underline{c}′; Prop 3.2c⊳\underline{a}+\underline{b}+\underline{c}′ = \underline{a}+\underline{b}+\underline{c}′ ⊳\underline{a}+\underline{b}+\underline{c}′ = \underline{a}+\underline{b}+\underline{c}′ ≫ \underline{a}+\underline{b}+\underline{c}′ = \underline{a} + \underline{b} + \underline{c}′; S6 ≫ \underline{b} + \underline{c}′ = \underline{b} + \underline{c}′; Prop 3.2i ⊳ \underline{b} + \underline{c}′ = \underline{b} + \underline{c}′ ≫ \underline{a} + \underline{b} + \underline{c}′ = \underline{a}+\underline{b}+\underline{c}′; S6 ≫ \underline{a}+\underline{b}+\underline{c}′ = \underline{a}+\underline{b}+\underline{c}′; Prop 3.2c⊳\underline{a}+\underline{b}+\underline{c}′ = \underline{a}+\underline{b}+\underline{c}′⊳\underline{a}+\underline{b}+\underline{c}′ = \underline{a}+\underline{b}+\underline{c}′ ≫ \underline{a}+\underline{b}+\underline{c}′ = \underline{a}+\underline{b}+\underline{c}′; Prop 3.2d⊳\underline{a}+\underline{b}+\underline{c}′ = \underline{a}+\underline{b}+\underline{c}′ ⊳\underline{a}+\underline{b}+\underline{c}′ = \underline{a} + \underline{b} + \underline{c}′ ≫ \underline{a} + \underline{b} + \underline{c}′ = \underline{a} + \underline{b} + \underline{c}′; Ded ⊳ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} + \underline{b} + \underline{c} = \underline{a} + \underline{b} + \underline{c} ⊢ \underline{a} + \underline{b} + \underline{c}′ = \underline{a} + \underline{b} + \underline{c}′ ≫ \underline{a} + \underline{b} + \underline{c} = \underline{a} + \underline{b} + \underline{c} ⇒ \underline{a} + \underline{b} + \underline{c}′ = \underline{a} + \underline{b} + \underline{c}′⌉, p_{0}, c)]

[Prop 3.2j_{2} →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} + \underline{b} + \underline{c} = \underline{a} + \underline{b} + \underline{c} ⇒ \underline{a} + \underline{b} + \underline{c}′ = \underline{a} + \underline{b} + \underline{c}′]

[Prop 3.2j_{2} →^{tex} “ Prop\ 3.2j 2”]

[Prop 3.2j_{2} →^{pyk} “prop three two j two”]

Prop 3.2j

[Prop 3.2j →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: Prop 3.2j_{1} ≫ \overline{x} + \overline{y} + 0 = \overline{x} + \overline{y} + 0; Prop 3.2j_{2} ≫ \overline{x} + \overline{y} + \overline{z} = \overline{x} + \overline{y} + \overline{z} ⇒ \overline{x} + \overline{y} + \overline{z}′ = \overline{x}+\overline{y}+\overline{z}′; S9 @ \overline{z}⊳\overline{x}+\overline{y}+0 = \overline{x}+\overline{y}+0⊳\overline{x}+\overline{y}+\overline{z} = \overline{x}+\overline{y}+\overline{z} ⇒ \overline{x}+\overline{y}+\overline{z}′ = \overline{x}+\overline{y}+\overline{z}′ ≫ \overline{x} + \overline{y} + \overline{z} = \overline{x} + \overline{y} + \overline{z}; Ded ⊳ \overline{x} + \overline{y} + \overline{z} = \overline{x} + \overline{y} + \overline{z} ≫ \underline{a} + \underline{b} + \underline{c} = \underline{a} + \underline{b} + \underline{c}⌉, p_{0}, c)]

[Prop 3.2j →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} + \underline{b} + \underline{c} = \underline{a} + \underline{b} + \underline{c}]

[Prop 3.2j →^{tex} “ Prop\ 3.2j”]

[Prop 3.2j →^{pyk} “prop three two j”]

Prop 3.2k_{1}

[Prop 3.2k_{1} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: S7 ≫ \underline{a}·0 = 0; S7 ≫ \underline{b}·0 = 0; Prop 3.2a ≫ 0 = 0; Prop 3.2b⊳\underline{b}·0 = 0 ≫ 0 = \underline{b}·0; Prop 3.2c ≫ \underline{a}·0 = 0 ⇒ 0 = \underline{b}·0 ⇒ \underline{a}·0 = \underline{b} · 0; \underline{a} · 0 = 0 ⇒ 0 = \underline{b} · 0 ⇒ \underline{a} · 0 = \underline{b} · 0 ⊳ \underline{a} · 0 = 0 ⊳ 0 = \underline{b} · 0 ≫ \underline{a} · 0 = \underline{b} · 0⌉, p_{0}, c)]

[Prop 3.2k_{1} →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} = \underline{b} ⊢ \underline{a} · 0 = \underline{b} · 0]

[Prop 3.2k_{1} →^{tex} “ Prop\ 3.2k 1”]

[Prop 3.2k_{1} →^{pyk} “prop three two k one”]

Prop 3.2k_{2}

[Prop 3.2k_{2} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c} = \underline{b} · \underline{c} ⊢ \underline{a} = \underline{b} ⊢ \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c} = \underline{b} · \underline{c} ⊳ \underline{a} = \underline{b} ≫ \underline{a} · \underline{c} = \underline{b} · \underline{c}; S8 ≫ \underline{a} · \underline{c}′ = \underline{a} · \underline{c} + \underline{a}; S8 ≫ \underline{b} · \underline{c}′ = \underline{b} · \underline{c} + \underline{b}; Prop 3.2e ≫ \underline{a} · \underline{c} = \underline{b} · \underline{c} ⇒ \underline{a} · \underline{c} + \underline{a} = \underline{b} · \underline{c} + \underline{a}; \underline{a} · \underline{c} = \underline{b} · \underline{c} ⇒ \underline{a} · \underline{c} + \underline{a} = \underline{b}·\underline{c}+\underline{a}⊳\underline{a}·\underline{c} = \underline{b}·\underline{c} ≫ \underline{a}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{a}; Prop 3.2i ≫ \underline{a} = \underline{b} ⇒ \underline{b}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b}; \underline{a} = \underline{b} ⇒ \underline{b} · \underline{c} + \underline{a} = \underline{b} · \underline{c} + \underline{b} ⊳ \underline{a} · \underline{c} = \underline{b} · \underline{c} ≫ \underline{b} · \underline{c} + \underline{a} = \underline{b} · \underline{c} + \underline{b}; Prop 3.2c ≫ \underline{a} · \underline{c} + \underline{a} = \underline{b}·\underline{c}+\underline{a} ⇒ \underline{b}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b} ⇒ \underline{a}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b}; \underline{a}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{a} ⇒ \underline{b}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b} ⇒ \underline{a}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b}⊳\underline{a}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{a}⊳\underline{b}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b} ≫ \underline{a}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b}; \underline{a}·\underline{c}+\underline{a} = \underline{b}·\underline{c}+\underline{b}⊳\underline{a}·\underline{c}′ = \underline{a}·\underline{c}+\underline{a}⊳\underline{b}·\underline{c}′ = \underline{b}·\underline{c}+\underline{b} ≫ \underline{a}·\underline{c}′ = \underline{b}·\underline{c}′; Ded⊳∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⇒ \underline{a}·\underline{c} = \underline{b} · \underline{c} ⊢ \underline{a} = \underline{b} ⊢ \underline{a} · \underline{c}′ = \underline{b} · \underline{c}′ ≫ \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c} = \underline{b} · \underline{c} ⊢ \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c}′ = \underline{b} · \underline{c}′⌉, p_{0}, c)]

[Prop 3.2k_{2} →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c} = \underline{b} · \underline{c} ⊢ \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c}′ = \underline{b} · \underline{c}′]

[Prop 3.2k_{2} →^{tex} “ Prop\ 3.2k 2”]

[Prop 3.2k_{2} →^{pyk} “prop three two k two”]

Prop 3.2k

[Prop 3.2k →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: Prop 3.2k_{1} ≫ \overline{x} = \overline{y} ⇒ \overline{x} · 0 = \overline{y} · 0; Prop 3.2k_{2} ≫ \overline{x} = \overline{y} ⇒ \overline{x} · \overline{z} = \overline{y} · \overline{z} ⇒ \overline{x} = \overline{y} ⇒ \overline{x} · \overline{z}′ = \overline{y} · \overline{z}′; S9 @ \overline{z} ⊳ \overline{x} = \overline{y} ⇒ \overline{x} · 0 = \overline{y} · 0 ⊳ \overline{x} = \overline{y} ⇒ \overline{x} · \overline{z} = \overline{y} · \overline{z} ⇒ \overline{x} = \overline{y} ⇒ \overline{x} · \overline{z}′ = \overline{y} · \overline{z}′ ≫ \overline{x} = \overline{y} ⇒ \overline{x} · \overline{z} = \overline{y} · \overline{z}; Ded ⊳ \overline{x} = \overline{y} ⇒ \overline{x} · \overline{z} = \overline{y} · \overline{z} ≫ \underline{a} = \underline{b} ⊢ \underline{a} · \underline{c} = \underline{b} · \underline{c}⌉, p_{0}, c)]

[Prop 3.2k →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{a} · \underline{c} = \underline{b} · \underline{c}]

[Prop 3.2k →^{tex} “ Prop\ 3.2k”]

[Prop 3.2k →^{pyk} “prop three two k”]

Prop 3.2l_{1}

[Prop 3.2l_{1} →^{proof} λc.λx.P(⌈S ⊢ S7 ≫ 0 · 0 = 0⌉, p_{0}, c)]

[Prop 3.2l_{1} →^{stmt} S ⊢ ∀\underline{a}: 0 · 0 = 0]

[Prop 3.2l_{1} →^{tex} “ Prop\ 3.2l 1”]

[Prop 3.2l_{1} →^{pyk} “prop three two l one”]

Prop 3.2l_{2}

[Prop 3.2l_{2} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{a}: 0 · \underline{a} = 0 ⊢ S8 ≫ 0 · \underline{a}′ = 0 · \underline{a} + 0; S5 ≫ 0 · \underline{a} + 0 = 0 · \underline{a}; Prop 3.2c ⊳ 0 · \underline{a} + 0 = 0 · \underline{a} ⊳ 0 · \underline{a}′ = 0 · \underline{a} + 0 ≫ 0 · \underline{a}′ = 0 · \underline{a}; Prop 3.2c ⊳ 0 · \underline{a}′ = 0 · \underline{a} ⊳ 0 · \underline{a} = 0 ≫ 0 · \underline{a}′ = 0; Ded ⊳ ∀\underline{a}: 0 · \underline{a} = 0 ⊢ 0 · \underline{a}′ = 0 ≫ 0 · \underline{a} = 0 ⇒ 0 · \underline{a}′ = 0⌉, p_{0}, c)]

[Prop 3.2l_{2} →^{stmt} S ⊢ ∀\underline{a}: 0 · \underline{a} = 0 ⇒ 0 · \underline{a}′ = 0]

[Prop 3.2l_{2} →^{tex} “ Prop\ 3.2l 2”]

[Prop 3.2l_{2} →^{pyk} “prop three two l two”]

Prop 3.2l

[Prop 3.2l →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: Prop 3.2l_{1} ≫ 0 · 0 = 0; Prop 3.2l_{2} ≫ 0 · \overline{x} = 0 ⇒ 0 · \overline{x}′ = 0; S9 @ \overline{x} ⊳ 0 · 0 = 0 ⊳ 0 · \overline{x} = 0 ⇒ 0 · \overline{x}′ = 0 ≫ 0 · \overline{x} = 0; Ded ⊳ 0 · \overline{x} = 0 ≫ 0 · \underline{a} = 0⌉, p_{0}, c)]

[Prop 3.2l →^{stmt} S ⊢ ∀\underline{a}: 0 · \underline{a} = 0]

[Prop 3.2l →^{tex} “ Prop\ 3.2l”]

[Prop 3.2l →^{pyk} “prop three two l”]

Prop 3.2m_{1}

[Prop 3.2m_{1} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: S7 ≫ \underline{a}′ · 0 = 0; Prop 3.2f ≫ 0 = 0 + 0; S7 ≫ 0 = \underline{a} · 0; Prop 3.2e ≫ 0 = \underline{a} · 0 ⇒ 0 + 0 = \underline{a} · 0 + 0; 0 = \underline{a} · 0 ⇒ 0 + 0 = \underline{a} · 0 + 0 ⊳ 0 = \underline{a} · 0 ≫ 0 + 0 = \underline{a} · 0 + 0; Prop 3.2c ≫ 0 = 0 + 0 ⇒ 0 + 0 = \underline{a} · 0 + 0 ⇒ 0 = \underline{a} · 0 + 0; 0 = 0 + 0 ⇒ 0 + 0 = \underline{a} · 0 + 0 ⇒ 0 = \underline{a} · 0 + 0 ⊳ 0 = 0 + 0 ⊳ 0 + 0 = \underline{a} · 0 + 0 ≫ 0 = \underline{a} · 0 + 0; Prop 3.2c ≫ \underline{a}′ · 0 = 0 ⇒ 0 = \underline{a} · 0 + 0 ⇒ \underline{a}′ · 0 = \underline{a} · 0 + 0; \underline{a}′ · 0 = 0 ⇒ 0 = \underline{a} · 0 + 0 ⇒ \underline{a}′ · 0 = \underline{a} · 0 + 0 ⊳ \underline{a}′ · 0 = 0 ⊳ \underline{a}′ · 0 = 0 ⇒ 0 = \underline{a} · 0 + 0 ⇒ \underline{a}′ · 0 = \underline{a} · 0 + 0 ≫ \underline{a}′ · 0 = \underline{a} · 0 + 0⌉, p_{0}, c)]

[Prop 3.2m_{1} →^{stmt} S ⊢ ∀\underline{a}: \underline{a}′ · 0 = \underline{t} · 0 + 0]

[Prop 3.2m_{1} →^{tex} “ Prop\ 3.2m 1”]

[Prop 3.2m_{1} →^{pyk} “prop three two m one”]

Prop 3.2m_{2}

[Prop 3.2m_{2} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{a}: ∀\underline{b}: \underline{a}′ · \underline{b} = \underline{a} · \underline{b} + \underline{b} ⊢ S8 ≫ \underline{a}′ · \underline{b}′ = \underline{a}′ ·\underline{b}′ +\underline{a}′; Prop 3.2e ≫ \underline{a}′ ·\underline{b} = \underline{a}·\underline{b}+\underline{b} ⇒ \underline{a}′ ·\underline{b}+\underline{a}′ = \underline{a}·\underline{b}+\underline{b}+\underline{a}′; \underline{a}′ ·\underline{b} = \underline{a}·\underline{b}+\underline{b} ⇒ \underline{a}′ · \underline{b} + \underline{a}′ = \underline{a} · \underline{b} + \underline{b} + \underline{a}′ ⊳ \underline{a}′ · \underline{b} = \underline{a} · \underline{b} + \underline{b} ≫ \underline{a}′ · \underline{b} + \underline{a}′ = \underline{a} · \underline{b} + \underline{b} + \underline{a}′; S6 ≫ \underline{b} + \underline{a}′ = \underline{b} + \underline{a}′; Prop 3.2g ≫ \underline{b}′ + \underline{a} = \underline{b} + \underline{a}′; Prop 3.2d ⊳ \underline{b} + \underline{a}′ = \underline{b} + \underline{a}′ ⊳ \underline{b}′ + \underline{a} = \underline{b} + \underline{a}′ ≫ \underline{b} + \underline{a}′ = \underline{b}′ + \underline{a}; Prop 3.2h ≫ \underline{b}′ + \underline{a} = \underline{a} + \underline{b}′; Prop 3.2c ⊳ \underline{b} + \underline{a}′ = \underline{b}′ + \underline{a} ⊳ \underline{b}′ + \underline{a} = \underline{a} + \underline{b}′ ≫ \underline{b} + \underline{a}′ = \underline{a} + \underline{b}′; Prop 3.2i ⊳ \underline{b} + \underline{a}′ = \underline{a} + \underline{b}′ ≫ \underline{a} · \underline{b} + \underline{b} + \underline{a}′ = \underline{a} · \underline{b} + \underline{a} + \underline{b}′; S8 ≫ \underline{a} · \underline{b}′ = \underline{a} · \underline{b} + \underline{a}; Prop 3.2e ⊳ \underline{a} · \underline{b}′ = \underline{a} · \underline{b} + \underline{a} ≫ \underline{a} · \underline{b}′ + \underline{b}′ = \underline{a} · \underline{b} + \underline{a} + \underline{b}′; Prop 3.2d ⊳ \underline{a} · \underline{b}′ + \underline{b}′ = \underline{a} · \underline{b} + \underline{a} + \underline{b}′ ⊳ \underline{a} · \underline{b} + \underline{b} + \underline{a}′ = \underline{a} · \underline{b} + \underline{a} + \underline{b}′ ≫ \underline{a} · \underline{b} + \underline{b} + \underline{a}′ = \underline{a} · \underline{b}′ + \underline{b}′; Prop 3.2c ⊳ \underline{a}′ · \underline{b} + \underline{a}′ = \underline{a}·\underline{b}+\underline{b}+\underline{a}′ ⊳\underline{a}·\underline{b}+\underline{b}+\underline{a}′ = \underline{a}·\underline{b}′ +\underline{b}′ ≫ \underline{a}′ ·\underline{b}+\underline{a}′ = \underline{a}·\underline{b}′ +\underline{b}′; Prop 3.2c⊳\underline{a}′ ·\underline{b}′ = \underline{a}′ · \underline{b}′ + \underline{a}′ ⊳ \underline{a}′ · \underline{b} + \underline{a}′ = \underline{a} · \underline{b}′ + \underline{b}′ ≫ \underline{a}′ · \underline{b}′ = \underline{a} · \underline{b}′ + \underline{b}′; Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a}′ · \underline{b} = \underline{a} · \underline{b} + \underline{b} ⊢ \underline{a}′ · \underline{b}′ = \underline{a} · \underline{b}′ + \underline{b}′ ≫ \underline{a}′ · \underline{b} = \underline{a} · \underline{b} + \underline{b} ⇒ \underline{a}′ · \underline{b}′ = \underline{a} · \underline{b}′ + \underline{b}′⌉, p_{0}, c)]

[Prop 3.2m_{2} →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a}′ · \underline{b} = \underline{a} · \underline{b} + \underline{b} ⇒ \underline{a}′ · \underline{b}′ = \underline{a} · \underline{b}′ + \underline{b}′]

[Prop 3.2m_{2} →^{tex} “ Prop\ 3.2m 2”]

[Prop 3.2m_{2} →^{pyk} “prop three two m two”]

Prop 3.2m

[Prop 3.2m →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: Prop 3.2m_{1} ≫ \overline{x}′ · 0 = \overline{x} · 0 + 0; Prop 3.2m_{2} ≫ \overline{x}′ · \overline{y} = \overline{x} · \overline{y} + \overline{y} ⇒ \overline{x}′ · \overline{y}′ = \overline{x} · \overline{y}′ + \overline{y}; S9 @ \overline{y} ⊳ \overline{x}′ · 0 = \overline{x} · 0 + 0 ⊳ \overline{x}′ · \overline{y} = \overline{x} · \overline{y} + \overline{y} ⇒ \overline{x}′ · \overline{y}′ = \overline{x} · \overline{y}′ + \overline{y} ≫ \overline{x}′ · \overline{y} = \overline{x} · \overline{y} + \overline{y}; Ded ⊳ \overline{x}′ · \overline{y} = \overline{x} · \overline{y} + \overline{y} ≫ \underline{a}′ · \underline{b} = \underline{a} · \underline{b} + \underline{b}⌉, p_{0}, c)]

[Prop 3.2m →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a}′ · \underline{b} = \underline{a} · \underline{b} + \underline{b}]

[Prop 3.2m →^{tex} “ Prop\ 3.2m”]

[Prop 3.2m →^{pyk} “prop three two m”]

Prop 3.2n_{1}

[Prop 3.2n_{1} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: S7 ≫ \underline{a} · 0 = 0; Prop 3.2l ≫ 0 · \underline{a} = 0; Prop 3.2c ⊳ \underline{a} · 0 = 0 ⊳ 0 · \underline{a} = 0 ≫ \underline{a} · 0 = 0 · \underline{a}⌉, p_{0}, c)]

[Prop 3.2n_{1} →^{stmt} S ⊢ ∀\underline{a}: \underline{a} · 0 = 0 · \underline{a}]

[Prop 3.2n_{1} →^{tex} “ Prop\ 3.2n 1”] [Prop 3.2n_{1} →^{pyk} “prop three two n one”]

Prop 3.2n_{2}

[Prop 3.2n_{2} →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{a}: ∀\underline{b}: \underline{a} · \underline{b} = \underline{b} · \underline{a} ⊢ S8 ≫ \underline{a} · \underline{b}′ = \underline{a} · \underline{b} + \underline{a}; Prop 3.2e ⊳ \underline{a} · \underline{b} = \underline{b} · \underline{a} ≫ \underline{a} · \underline{b} + \underline{a} = \underline{b} · \underline{a} + \underline{a}; Prop 3.2b ⊳ Prop 3.2m ≫ \underline{b} · \underline{a} + \underline{a} = \underline{b}′ · \underline{a}; Prop 3.2c ⊳ L_{e} ⊳ \underline{b} · \underline{a} + \underline{a} = \underline{b}′ · \underline{a} ≫ \underline{a} · \underline{b} + \underline{a} = \underline{b}′ · \underline{a}; Prop 3.2c ⊳ \underline{a} · \underline{b}′ = \underline{a} · \underline{b} + \underline{a} ⊳ \underline{a} · \underline{b} + \underline{a} = \underline{b}′ · \underline{a} ≫ \underline{a} · \underline{b}′ = \underline{b}′ · \underline{a}; Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} · \underline{b} = \underline{b} · \underline{a} ⊢ \underline{a} · \underline{b}′ = \underline{b}′ · \underline{a} ≫ \underline{a} · \underline{b} = \underline{b} · \underline{a} ⇒ \underline{a} · \underline{b}′ = \underline{b}′ · \underline{s}⌉, p_{0}, c)]

[Prop 3.2n_{2} →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} · \underline{b} = \underline{b} · \underline{a} ⇒ \underline{a} · \underline{b}′ = \underline{b}′ · \underline{a}]

[Prop 3.2n_{2} →^{tex} “ Prop\ 3.2n 2”]

[Prop 3.2n_{2} →^{pyk} “prop three two n two”]

Prop 3.2n

[Prop 3.2n →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: Prop 3.2n_{1} ≫ \overline{x} · 0 = 0 · \overline{x}; Prop 3.2n_{2} ≫ \overline{x} · \overline{y} = \overline{y} · \overline{x} ⇒ \overline{x} · \overline{y}′ = \overline{y}′ · \overline{x}; S9 @ \overline{y} ⊳ \overline{x} · 0 = 0 · \overline{x} ⊳ \overline{x} · \overline{y} = \overline{y} · \overline{x} ⇒ \overline{x} · \overline{y}′ = \overline{y}′ · \overline{x} ≫ \overline{x} · \overline{y} = \overline{y} = \overline{x}; Ded ⊳ \overline{x} · \overline{y} = \overline{y} = \overline{x} ≫ \underline{a} · \underline{b} = \underline{b} · \underline{a}⌉, p_{0}, c)]

[Prop 3.2n →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} · \underline{b} = \underline{b} · \underline{a}]

[Prop 3.2n →^{tex} “ Prop\ 3.2n”]

[Prop 3.2n →^{pyk} “prop three two n”]

Prop 3.2o

[Prop 3.2o →^{proof} λc.λx.P(⌈S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ Prop 3.2k ≫ \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c} = \underline{b} · \underline{c}; \underline{a} = \underline{b} ⇒ \underline{a} · \underline{c} = \underline{b} · \underline{c} ⊳ \underline{a} = \underline{b} ≫ \underline{a} · \underline{c} = \underline{b} · \underline{c}; Prop 3.2n ≫ \underline{a} · \underline{c} = \underline{c}·\underline{a}; Prop 3.2n ≫ \underline{b}·\underline{c} = \underline{c}·\underline{b}; Prop 3.2c ≫ \underline{a}·\underline{c} = \underline{b}·\underline{c} ⇒ \underline{b}·\underline{c} = \underline{c}·\underline{b} ⇒ \underline{a}·\underline{c} = \underline{c}·\underline{b}; \underline{a}·\underline{c} = \underline{b}·\underline{c} ⇒ \underline{b}·\underline{c} = \underline{c}·\underline{b} ⇒ \underline{a}·\underline{c} = \underline{c}·\underline{b}⊳\underline{a}·\underline{c} = \underline{b}·\underline{c}⊳\underline{b}·\underline{c} = \underline{c}·\underline{b} ≫ \underline{a}·\underline{c} = \underline{c}·\underline{b}; S1 ≫ \underline{a}·\underline{c} = \underline{c}·\underline{b} ⇒ \underline{a}·\underline{c} = \underline{c}·\underline{a} ⇒ \underline{c}·\underline{a} = \underline{c}·\underline{b}; \underline{a}·\underline{c} = \underline{c}·\underline{b} ⇒ \underline{a}·\underline{c} = \underline{c}·\underline{a} ⇒ \underline{c}·\underline{a} = \underline{c}·\underline{b}⊳\underline{a}·\underline{c} = \underline{c}·\underline{b}⊳\underline{a}·\underline{c} = \underline{c}·\underline{a} ≫ \underline{c}·\underline{a} = \underline{c}·\underline{b}; Ded⊳∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{c}·\underline{a} = \underline{c}·\underline{b} ≫ \underline{a} = \underline{b} ⇒ \underline{c}·\underline{a} = \underline{c}·\underline{b}⌉, p_{0}, c)]

[Prop 3.2o →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} = \underline{b} ⊢ \underline{c} · \underline{a} = \underline{c} · \underline{b}]

[Prop 3.2o →^{tex} “ Prop\ 3.2o”] [Prop 3.2o →^{pyk} “prop three two o”]

Prop 3.4a_{1}

[Prop 3.4a_{1} →^{tex} “ Prop\ 3.4a 1”] [Prop 3.4a_{1} →^{pyk} “prop three four a one”]

Prop 3.4a_{2}

[Prop 3.4a_{2} →^{tex} “ Prop\ 3.4a 2”] [Prop 3.4a_{2} →^{pyk} “prop three four a two”]

Prop 3.4a

[Prop 3.4a →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} · \underline{b} + \underline{c} = \underline{a} · \underline{b} + \underline{a} · \underline{c}] [Prop 3.4a →^{tex} “ Prop\ 3.4a”] [Prop 3.4a →^{pyk} “prop three four a”]

Prop 3.4b

[Prop 3.4b →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{b} + \underline{c} · \underline{a} = \underline{b} · \underline{a} + \underline{c} · \underline{b}] [Prop 3.4b →^{tex} “ Prop\ 3.4b”] [Prop 3.4b →^{pyk} “prop three four b”]

Prop 3.4c_{1}

[Prop 3.4c_{1} →^{tex} “ Prop\ 3.4c 1”] [Prop 3.4c_{1} →^{pyk} “prop three four c one”]

Prop 3.4c_{2}

[Prop 3.4c_{2} →^{tex} “ Prop\ 3.4c 2”] [Prop 3.4c_{2} →^{pyk} “prop three four c two”]

Prop 3.4c

[Prop 3.4c →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} · \underline{b} · \underline{c} = \underline{a} · \underline{b} · \underline{c}] [Prop 3.4c →^{tex} “ Prop\ 3.4c”] [Prop 3.4c →^{pyk} “prop three four c”]

Prop 3.4d_{1}

[Prop 3.4d_{1} →^{tex} “ Prop\ 3.4d 1”] [Prop 3.4d_{1} →^{pyk} “prop three four d one”]

Prop 3.4d_{2}

[Prop 3.4d_{2} →^{tex} “ Prop\ 3.4d 2”] [Prop 3.4d_{2} →^{pyk} “prop three four d two”]

Prop 3.4d

[Prop 3.4d →^{stmt} S ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} + \underline{c} = \underline{b} + \underline{c} ⇒ \underline{a} = \underline{b}] [Prop 3.4d →^{tex} “ Prop\ 3.4d”] [Prop 3.4d →^{pyk} “prop three four d”]

∗∥∗

[x∥y →^{tex} “#1.

\| #2.”]

[∗∥∗ →^{pyk} “" divides "”]

The pyk compiler, version 0.grue.20060417+ by Klaus Grue GRD-2006-06-21.UTC:11:42:14.848103 = MJD-53907.TAI:11:42:47.848103 = LGT-4657606967848103e-6