url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.specId | [259, 1] | [271, 13] | exact fastReplaceFree_self P v | case a.a.a
v : VarName
P : Formula
Δ : Set Formula
h1 : IsDeduct Δ (forall_ v P)
⊢ fastReplaceFree v v P = P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.specId | [259, 1] | [271, 13] | exact h1 | case a
v : VarName
P : Formula
Δ : Set Formula
h1 : IsDeduct Δ (forall_ v P)
⊢ IsDeduct Δ (forall_ v P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | simp only [fastAdmits] at h1 | P : Formula
v t : VarName
h1 : fastAdmits v t P
⊢ IsProof ((fastReplaceFree v t P).imp_ (exists_ v P)) | P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsProof ((fastReplaceFree v t P).imp_ (exists_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | simp only [def_exists_] | P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsProof ((fastReplaceFree v t P).imp_ (exists_ v P)) | P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsProof ((fastReplaceFree v t P).imp_ (forall_ v P.not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | simp only [IsProof] | P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsProof ((fastReplaceFree v t P).imp_ (forall_ v P.not_).not_) | P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsDeduct ∅ ((fastReplaceFree v t P).imp_ (forall_ v P.not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | apply IsDeduct.mp_ ((forall_ v P.not_).imp_ (fastReplaceFree v t P).not_) | P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsDeduct ∅ ((fastReplaceFree v t P).imp_ (forall_ v P.not_).not_) | case a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsDeduct ∅
(((forall_ v P.not_).imp_ (fastReplaceFree v t P).not_).imp_ ((fastReplaceFree v t P).imp_ (forall_ v P.not_).not_))
case a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsDeduct ∅ ((forall_ v P.not_).imp_ (fastReplaceFree v t P).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | SC | case a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsDeduct ∅
(((forall_ v P.not_).imp_ (fastReplaceFree v t P).not_).imp_ ((fastReplaceFree v t P).imp_ (forall_ v P.not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | apply IsDeduct.axiom_ | case a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsDeduct ∅ ((forall_ v P.not_).imp_ (fastReplaceFree v t P).not_) | case a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsAxiom ((forall_ v P.not_).imp_ (fastReplaceFree v t P).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | apply IsAxiom.pred_2_ v t | case a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ IsAxiom ((forall_ v P.not_).imp_ (fastReplaceFree v t P).not_) | case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastAdmits v t P.not_
case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastReplaceFree v t P.not_ = (fastReplaceFree v t P).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | simp only [fastAdmits] | case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastAdmits v t P.not_ | case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastAdmitsAux v t ∅ P.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | simp only [fastAdmitsAux] | case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastAdmitsAux v t ∅ P.not_ | case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastAdmitsAux v t ∅ P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | exact h1 | case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastAdmitsAux v t ∅ P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_3 | [276, 1] | [295, 10] | rfl | case a.a.a
P : Formula
v t : VarName
h1 : fastAdmitsAux v t ∅ P
⊢ fastReplaceFree v t P.not_ = (fastReplaceFree v t P).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_4 | [299, 1] | [311, 13] | apply IsDeduct.mp_ (fastReplaceFree v t P) | P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ IsDeduct Δ (exists_ v P) | case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ IsDeduct Δ ((fastReplaceFree v t P).imp_ (exists_ v P))
case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ IsDeduct Δ (fastReplaceFree v t P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_4 | [299, 1] | [311, 13] | apply proof_imp_deduct | case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ IsDeduct Δ ((fastReplaceFree v t P).imp_ (exists_ v P)) | case a.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ IsProof ((fastReplaceFree v t P).imp_ (exists_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_4 | [299, 1] | [311, 13] | apply T_17_3 | case a.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ IsProof ((fastReplaceFree v t P).imp_ (exists_ v P)) | case a.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ fastAdmits v t P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_4 | [299, 1] | [311, 13] | exact h1 | case a.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ fastAdmits v t P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_4 | [299, 1] | [311, 13] | exact h2 | case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : fastAdmits v t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
⊢ IsDeduct Δ (fastReplaceFree v t P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsIntroId | [316, 1] | [326, 13] | apply T_17_4 P v v Δ | P : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ P
⊢ IsDeduct Δ (exists_ v P) | case h1
P : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ P
⊢ fastAdmits v v P
case h2
P : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ P
⊢ IsDeduct Δ (fastReplaceFree v v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsIntroId | [316, 1] | [326, 13] | exact fastAdmits_self P v | case h1
P : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ P
⊢ fastAdmits v v P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsIntroId | [316, 1] | [326, 13] | simp only [fastReplaceFree_self] | case h2
P : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ P
⊢ IsDeduct Δ (fastReplaceFree v v P) | case h2
P : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ P
⊢ IsDeduct Δ P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsIntroId | [316, 1] | [326, 13] | exact h1 | case h2
P : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ P
⊢ IsDeduct Δ P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_6 | [329, 1] | [339, 7] | apply deduction_theorem | P : Formula
v : VarName
⊢ IsProof ((forall_ v P).imp_ (exists_ v P)) | case h1
P : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v P}) (exists_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_6 | [329, 1] | [339, 7] | simp | case h1
P : Formula
v : VarName
⊢ IsDeduct (∅ ∪ {forall_ v P}) (exists_ v P) | case h1
P : Formula
v : VarName
⊢ IsDeduct {forall_ v P} (exists_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_6 | [329, 1] | [339, 7] | apply existsIntroId | case h1
P : Formula
v : VarName
⊢ IsDeduct {forall_ v P} (exists_ v P) | case h1.h1
P : Formula
v : VarName
⊢ IsDeduct {forall_ v P} P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_6 | [329, 1] | [339, 7] | apply specId v | case h1.h1
P : Formula
v : VarName
⊢ IsDeduct {forall_ v P} P | case h1.h1.h1
P : Formula
v : VarName
⊢ IsDeduct {forall_ v P} (forall_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_6 | [329, 1] | [339, 7] | apply IsDeduct.assume_ | case h1.h1.h1
P : Formula
v : VarName
⊢ IsDeduct {forall_ v P} (forall_ v P) | case h1.h1.h1.a
P : Formula
v : VarName
⊢ forall_ v P ∈ {forall_ v P} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_6 | [329, 1] | [339, 7] | simp | case h1.h1.h1.a
P : Formula
v : VarName
⊢ forall_ v P ∈ {forall_ v P} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | induction h1 | F : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ F
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
⊢ IsDeduct Δ (forall_ v F) | case axiom_
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
phi✝ : Formula
a✝ : IsAxiom phi✝
⊢ IsDeduct Δ (forall_ v phi✝)
case assume_
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
phi✝ : Formula
a✝ : phi✝ ∈ Δ
⊢ IsDeduct Δ (forall_ v phi✝)
case mp_
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
phi✝ psi✝ : Formula
a✝¹ : IsDeduct Δ (phi✝.imp_ psi✝)
a✝ : IsDeduct Δ phi✝
a_ih✝¹ : IsDeduct Δ (forall_ v (phi✝.imp_ psi✝))
a_ih✝ : IsDeduct Δ (forall_ v phi✝)
⊢ IsDeduct Δ (forall_ v psi✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | case axiom_ h1_phi h1_1 =>
apply IsDeduct.axiom_
apply IsAxiom.gen_
exact h1_1 | F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ (forall_ v h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsDeduct.axiom_ | F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ (forall_ v h1_phi) | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom (forall_ v h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsAxiom.gen_ | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom (forall_ v h1_phi) | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | exact h1_1 | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsDeduct.mp_ h1_phi | F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ (forall_ v h1_phi) | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ (h1_phi.imp_ (forall_ v h1_phi))
case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsDeduct.axiom_ | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ (h1_phi.imp_ (forall_ v h1_phi)) | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsAxiom (h1_phi.imp_ (forall_ v h1_phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsAxiom.pred_3_ | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsAxiom (h1_phi.imp_ (forall_ v h1_phi)) | case a.a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ ¬isFreeIn v h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | exact h2 h1_phi h1_1 | case a.a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ ¬isFreeIn v h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsDeduct.assume_ | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ h1_phi | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | exact h1_1 | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsDeduct.mp_ (forall_ v h1_phi) | F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ (forall_ v h1_psi) | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ ((forall_ v h1_phi).imp_ (forall_ v h1_psi))
case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ (forall_ v h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsDeduct.mp_ (forall_ v (h1_phi.imp_ h1_psi)) | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ ((forall_ v h1_phi).imp_ (forall_ v h1_psi)) | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ ((forall_ v (h1_phi.imp_ h1_psi)).imp_ ((forall_ v h1_phi).imp_ (forall_ v h1_psi)))
case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsDeduct.axiom_ | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ ((forall_ v (h1_phi.imp_ h1_psi)).imp_ ((forall_ v h1_phi).imp_ (forall_ v h1_psi))) | case a.a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsAxiom ((forall_ v (h1_phi.imp_ h1_psi)).imp_ ((forall_ v h1_phi).imp_ (forall_ v h1_psi))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | apply IsAxiom.pred_1_ | case a.a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsAxiom ((forall_ v (h1_phi.imp_ h1_psi)).imp_ ((forall_ v h1_phi).imp_ (forall_ v h1_psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | exact h1_ih_1 | case a.a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_7 | [342, 1] | [368, 20] | exact h1_ih_2 | case a
F : Formula
v : VarName
Δ : Set Formula
h2 : ∀ H ∈ Δ, ¬isFreeIn v H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct Δ (forall_ v (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (forall_ v h1_phi)
⊢ IsDeduct Δ (forall_ v h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | rw [← fastReplaceFree_inverse P v t h1] | P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct Δ (forall_ v P) | P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct Δ (forall_ v (fastReplaceFree t v (fastReplaceFree v t P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | apply IsDeduct.mp_ (forall_ t (fastReplaceFree v t P)) | P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct Δ (forall_ v (fastReplaceFree t v (fastReplaceFree v t P))) | case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct Δ ((forall_ t (fastReplaceFree v t P)).imp_ (forall_ v (fastReplaceFree t v (fastReplaceFree v t P))))
case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct Δ (forall_ t (fastReplaceFree v t P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | apply proof_imp_deduct | case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct Δ ((forall_ t (fastReplaceFree v t P)).imp_ (forall_ v (fastReplaceFree t v (fastReplaceFree v t P)))) | case a.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsProof ((forall_ t (fastReplaceFree v t P)).imp_ (forall_ v (fastReplaceFree t v (fastReplaceFree v t P)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | apply deduction_theorem | case a.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsProof ((forall_ t (fastReplaceFree v t P)).imp_ (forall_ v (fastReplaceFree t v (fastReplaceFree v t P)))) | case a.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct (∅ ∪ {forall_ t (fastReplaceFree v t P)}) (forall_ v (fastReplaceFree t v (fastReplaceFree v t P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | simp | case a.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct (∅ ∪ {forall_ t (fastReplaceFree v t P)}) (forall_ v (fastReplaceFree t v (fastReplaceFree v t P))) | case a.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct {forall_ t (fastReplaceFree v t P)} (forall_ v (fastReplaceFree t v (fastReplaceFree v t P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | apply generalization | case a.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct {forall_ t (fastReplaceFree v t P)} (forall_ v (fastReplaceFree t v (fastReplaceFree v t P))) | case a.h1.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct {forall_ t (fastReplaceFree v t P)} (fastReplaceFree t v (fastReplaceFree v t P))
case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ∀ H ∈ {forall_ t (fastReplaceFree v t P)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | apply spec | case a.h1.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct {forall_ t (fastReplaceFree v t P)} (fastReplaceFree t v (fastReplaceFree v t P)) | case a.h1.h1.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct {forall_ t (fastReplaceFree v t P)} (forall_ t (fastReplaceFree v t P))
case a.h1.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ fastAdmits t v (fastReplaceFree v t P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | apply IsDeduct.assume_ | case a.h1.h1.h1.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct {forall_ t (fastReplaceFree v t P)} (forall_ t (fastReplaceFree v t P)) | case a.h1.h1.h1.h1.a
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ forall_ t (fastReplaceFree v t P) ∈ {forall_ t (fastReplaceFree v t P)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | simp | case a.h1.h1.h1.h1.a
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ forall_ t (fastReplaceFree v t P) ∈ {forall_ t (fastReplaceFree v t P)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | apply fastReplaceFree_fastAdmits | case a.h1.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ fastAdmits t v (fastReplaceFree v t P) | case a.h1.h1.h1.h2.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬occursIn t P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | exact h1 | case a.h1.h1.h1.h2.h1
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬occursIn t P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | simp | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ∀ H ∈ {forall_ t (fastReplaceFree v t P)}, ¬isFreeIn v H | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬isFreeIn v (forall_ t (fastReplaceFree v t P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | simp only [isFreeIn] | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬isFreeIn v (forall_ t (fastReplaceFree v t P)) | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬(¬v = t ∧ isFreeIn v (fastReplaceFree v t P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | simp | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬(¬v = t ∧ isFreeIn v (fastReplaceFree v t P)) | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬v = t → ¬isFreeIn v (fastReplaceFree v t P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | intro a1 contra | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ ¬v = t → ¬isFreeIn v (fastReplaceFree v t P) | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
a1 : ¬v = t
contra : isFreeIn v (fastReplaceFree v t P)
⊢ False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | exact not_isFreeIn_fastReplaceFree P v t a1 contra | case a.h1.h1.h2
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
a1 : ¬v = t
contra : isFreeIn v (fastReplaceFree v t P)
⊢ False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.univIntro | [374, 1] | [399, 59] | exact generalization (fastReplaceFree v t P) t Δ h2 h3 | case a
P : Formula
v t : VarName
Δ : Set Formula
h1 : ¬occursIn t P
h2 : IsDeduct Δ (fastReplaceFree v t P)
h3 : ∀ H ∈ Δ, ¬isFreeIn t H
⊢ IsDeduct Δ (forall_ t (fastReplaceFree v t P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | induction h1 | F : Formula
h1 : IsProofAlt F
⊢ IsDeduct ∅ F | case prop_true_
F : Formula
⊢ IsDeduct ∅ true_
case prop_1_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅ (phi✝.imp_ (psi✝.imp_ phi✝))
case prop_2_
F phi✝ psi✝ chi✝ : Formula
⊢ IsDeduct ∅ ((phi✝.imp_ (psi✝.imp_ chi✝)).imp_ ((phi✝.imp_ psi✝).imp_ (phi✝.imp_ chi✝)))
case prop_3_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅ ((phi✝.not_.imp_ psi✝.not_).imp_ (psi✝.imp_ phi✝))
case pred_1_
F : Formula
v✝ : VarName
phi✝ psi✝ : Formula
⊢ IsDeduct ∅ ((forall_ v✝ (phi✝.imp_ psi✝)).imp_ ((forall_ v✝ phi✝).imp_ (forall_ v✝ psi✝)))
case pred_2_
F : Formula
v✝ t✝ : VarName
phi✝ phi'✝ : Formula
a✝¹ : fastAdmits v✝ t✝ phi✝
a✝ : fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ IsDeduct ∅ ((forall_ v✝ phi✝).imp_ phi'✝)
case pred_3_
F : Formula
v✝ : VarName
phi✝ : Formula
a✝ : ¬isFreeIn v✝ phi✝
⊢ IsDeduct ∅ (phi✝.imp_ (forall_ v✝ phi✝))
case eq_1_
F : Formula
v✝ : VarName
⊢ IsDeduct ∅ (forall_ v✝ (eq_ v✝ v✝))
case eq_2_pred_const_
F : Formula
name✝ : PredName
n✝ : ℕ
xs✝ ys✝ : Fin n✝ → VarName
⊢ IsDeduct ∅
(Forall_ (List.ofFn xs✝)
(Forall_ (List.ofFn ys✝)
((And_ (List.ofFn fun i => eq_ (xs✝ i) (ys✝ i))).imp_
((pred_const_ name✝ (List.ofFn xs✝)).iff_ (pred_const_ name✝ (List.ofFn ys✝))))))
case eq_2_eq_
F : Formula
x_0✝ x_1✝ y_0✝ y_1✝ : VarName
⊢ IsDeduct ∅
(forall_ x_0✝
(forall_ x_1✝
(forall_ y_0✝
(forall_ y_1✝ (((eq_ x_0✝ y_0✝).and_ (eq_ x_1✝ y_1✝)).imp_ ((eq_ x_0✝ x_1✝).iff_ (eq_ y_0✝ y_1✝)))))))
case gen_
F : Formula
v✝ : VarName
phi✝ : Formula
a✝ : IsProofAlt phi✝
a_ih✝ : IsDeduct ∅ phi✝
⊢ IsDeduct ∅ (forall_ v✝ phi✝)
case mp_
F phi✝ psi✝ : Formula
a✝¹ : IsProofAlt (phi✝.imp_ psi✝)
a✝ : IsProofAlt phi✝
a_ih✝¹ : IsDeduct ∅ (phi✝.imp_ psi✝)
a_ih✝ : IsDeduct ∅ phi✝
⊢ IsDeduct ∅ psi✝
case def_false_
F : Formula
⊢ IsDeduct ∅ (false_.iff_ true_.not_)
case def_and_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅ ((phi✝.and_ psi✝).iff_ (phi✝.imp_ psi✝.not_).not_)
case def_or_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅ ((phi✝.or_ psi✝).iff_ (phi✝.not_.imp_ psi✝))
case def_iff_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅
(((phi✝.iff_ psi✝).imp_ ((phi✝.imp_ psi✝).imp_ (psi✝.imp_ phi✝).not_).not_).imp_
(((phi✝.imp_ psi✝).imp_ (psi✝.imp_ phi✝).not_).not_.imp_ (phi✝.iff_ psi✝)).not_).not_
case def_exists_
F : Formula
v✝ : VarName
phi✝ : Formula
⊢ IsDeduct ∅ ((exists_ v✝ phi✝).iff_ (forall_ v✝ phi✝.not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case prop_true_ =>
apply IsDeduct.axiom_
apply IsAxiom.prop_true_ | F : Formula
⊢ IsDeduct ∅ true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case prop_1_ h1_phi h1_psi =>
apply IsDeduct.axiom_
apply IsAxiom.prop_1_ | F h1_phi h1_psi : Formula
⊢ IsDeduct ∅ (h1_phi.imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case prop_2_ h1_phi h1_psi h1_chi =>
apply IsDeduct.axiom_
apply IsAxiom.prop_2_ | F h1_phi h1_psi h1_chi : Formula
⊢ IsDeduct ∅ ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case prop_3_ h1_phi h1_psi =>
apply IsDeduct.axiom_
apply IsAxiom.prop_3_ | F h1_phi h1_psi : Formula
⊢ IsDeduct ∅ ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case pred_1_ h1_v h1_phi h1_psi =>
apply IsDeduct.axiom_
apply IsAxiom.pred_1_ | F : Formula
h1_v : VarName
h1_phi h1_psi : Formula
⊢ IsDeduct ∅ ((forall_ h1_v (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_v h1_phi).imp_ (forall_ h1_v h1_psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case pred_2_ h1_v h1_t h1_phi h1_phi' h1_1 h1_ih_1 =>
apply IsDeduct.axiom_
exact IsAxiom.pred_2_ h1_v h1_t h1_phi h1_phi' h1_1 h1_ih_1 | F : Formula
h1_v h1_t : VarName
h1_phi h1_phi' : Formula
h1_1 : fastAdmits h1_v h1_t h1_phi
h1_ih_1 : fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ IsDeduct ∅ ((forall_ h1_v h1_phi).imp_ h1_phi') | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case pred_3_ h1_v h1_phi h1_1 =>
apply IsDeduct.axiom_
apply IsAxiom.pred_3_
exact h1_1 | F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : ¬isFreeIn h1_v h1_phi
⊢ IsDeduct ∅ (h1_phi.imp_ (forall_ h1_v h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case eq_1_ h1 =>
apply IsDeduct.axiom_
apply IsAxiom.eq_1_ | F : Formula
h1 : VarName
⊢ IsDeduct ∅ (forall_ h1 (eq_ h1 h1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case eq_2_pred_const_ h1_name h1_n h1_xs h1_ys =>
apply IsDeduct.axiom_
apply IsAxiom.eq_2_pred_const_ | F : Formula
h1_name : PredName
h1_n : ℕ
h1_xs h1_ys : Fin h1_n → VarName
⊢ IsDeduct ∅
(Forall_ (List.ofFn h1_xs)
(Forall_ (List.ofFn h1_ys)
((And_ (List.ofFn fun i => eq_ (h1_xs i) (h1_ys i))).imp_
((pred_const_ h1_name (List.ofFn h1_xs)).iff_ (pred_const_ h1_name (List.ofFn h1_ys)))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case eq_2_eq_ h1_x_0 h1_y_0 h1_x_1 h1_y_1 =>
apply IsDeduct.axiom_
apply IsAxiom.eq_2_eq_ | F : Formula
h1_x_0 h1_y_0 h1_x_1 h1_y_1 : VarName
⊢ IsDeduct ∅
(forall_ h1_x_0
(forall_ h1_y_0
(forall_ h1_x_1
(forall_ h1_y_1
(((eq_ h1_x_0 h1_x_1).and_ (eq_ h1_y_0 h1_y_1)).imp_ ((eq_ h1_x_0 h1_y_0).iff_ (eq_ h1_x_1 h1_y_1))))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | case mp_ h1_phi h1_psi h1_1 h1_2 h1_ih_1 h1_ih_2 =>
exact IsDeduct.mp_ h1_phi h1_psi h1_ih_1 h1_ih_2 | F h1_phi h1_psi : Formula
h1_1 : IsProofAlt (h1_phi.imp_ h1_psi)
h1_2 : IsProofAlt h1_phi
h1_ih_1 : IsDeduct ∅ (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct ∅ h1_phi
⊢ IsDeduct ∅ h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | all_goals
sorry | case def_false_
F : Formula
⊢ IsDeduct ∅ (false_.iff_ true_.not_)
case def_and_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅ ((phi✝.and_ psi✝).iff_ (phi✝.imp_ psi✝.not_).not_)
case def_or_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅ ((phi✝.or_ psi✝).iff_ (phi✝.not_.imp_ psi✝))
case def_iff_
F phi✝ psi✝ : Formula
⊢ IsDeduct ∅
(((phi✝.iff_ psi✝).imp_ ((phi✝.imp_ psi✝).imp_ (psi✝.imp_ phi✝).not_).not_).imp_
(((phi✝.imp_ psi✝).imp_ (psi✝.imp_ phi✝).not_).not_.imp_ (phi✝.iff_ psi✝)).not_).not_
case def_exists_
F : Formula
v✝ : VarName
phi✝ : Formula
⊢ IsDeduct ∅ ((exists_ v✝ phi✝).iff_ (forall_ v✝ phi✝.not_).not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F : Formula
⊢ IsDeduct ∅ true_ | case a
F : Formula
⊢ IsAxiom true_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.prop_true_ | case a
F : Formula
⊢ IsAxiom true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F h1_phi h1_psi : Formula
⊢ IsDeduct ∅ (h1_phi.imp_ (h1_psi.imp_ h1_phi)) | case a
F h1_phi h1_psi : Formula
⊢ IsAxiom (h1_phi.imp_ (h1_psi.imp_ h1_phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.prop_1_ | case a
F h1_phi h1_psi : Formula
⊢ IsAxiom (h1_phi.imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F h1_phi h1_psi h1_chi : Formula
⊢ IsDeduct ∅ ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi))) | case a
F h1_phi h1_psi h1_chi : Formula
⊢ IsAxiom ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.prop_2_ | case a
F h1_phi h1_psi h1_chi : Formula
⊢ IsAxiom ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F h1_phi h1_psi : Formula
⊢ IsDeduct ∅ ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi)) | case a
F h1_phi h1_psi : Formula
⊢ IsAxiom ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.prop_3_ | case a
F h1_phi h1_psi : Formula
⊢ IsAxiom ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F : Formula
h1_v : VarName
h1_phi h1_psi : Formula
⊢ IsDeduct ∅ ((forall_ h1_v (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_v h1_phi).imp_ (forall_ h1_v h1_psi))) | case a
F : Formula
h1_v : VarName
h1_phi h1_psi : Formula
⊢ IsAxiom ((forall_ h1_v (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_v h1_phi).imp_ (forall_ h1_v h1_psi))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.pred_1_ | case a
F : Formula
h1_v : VarName
h1_phi h1_psi : Formula
⊢ IsAxiom ((forall_ h1_v (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_v h1_phi).imp_ (forall_ h1_v h1_psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F : Formula
h1_v h1_t : VarName
h1_phi h1_phi' : Formula
h1_1 : fastAdmits h1_v h1_t h1_phi
h1_ih_1 : fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ IsDeduct ∅ ((forall_ h1_v h1_phi).imp_ h1_phi') | case a
F : Formula
h1_v h1_t : VarName
h1_phi h1_phi' : Formula
h1_1 : fastAdmits h1_v h1_t h1_phi
h1_ih_1 : fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ IsAxiom ((forall_ h1_v h1_phi).imp_ h1_phi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | exact IsAxiom.pred_2_ h1_v h1_t h1_phi h1_phi' h1_1 h1_ih_1 | case a
F : Formula
h1_v h1_t : VarName
h1_phi h1_phi' : Formula
h1_1 : fastAdmits h1_v h1_t h1_phi
h1_ih_1 : fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ IsAxiom ((forall_ h1_v h1_phi).imp_ h1_phi') | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : ¬isFreeIn h1_v h1_phi
⊢ IsDeduct ∅ (h1_phi.imp_ (forall_ h1_v h1_phi)) | case a
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : ¬isFreeIn h1_v h1_phi
⊢ IsAxiom (h1_phi.imp_ (forall_ h1_v h1_phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.pred_3_ | case a
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : ¬isFreeIn h1_v h1_phi
⊢ IsAxiom (h1_phi.imp_ (forall_ h1_v h1_phi)) | case a.a
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : ¬isFreeIn h1_v h1_phi
⊢ ¬isFreeIn h1_v h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | exact h1_1 | case a.a
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : ¬isFreeIn h1_v h1_phi
⊢ ¬isFreeIn h1_v h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F : Formula
h1 : VarName
⊢ IsDeduct ∅ (forall_ h1 (eq_ h1 h1)) | case a
F : Formula
h1 : VarName
⊢ IsAxiom (forall_ h1 (eq_ h1 h1)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.eq_1_ | case a
F : Formula
h1 : VarName
⊢ IsAxiom (forall_ h1 (eq_ h1 h1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F : Formula
h1_name : PredName
h1_n : ℕ
h1_xs h1_ys : Fin h1_n → VarName
⊢ IsDeduct ∅
(Forall_ (List.ofFn h1_xs)
(Forall_ (List.ofFn h1_ys)
((And_ (List.ofFn fun i => eq_ (h1_xs i) (h1_ys i))).imp_
((pred_const_ h1_name (List.ofFn h1_xs)).iff_ (pred_const_ h1_name (List.ofFn h1_ys)))))) | case a
F : Formula
h1_name : PredName
h1_n : ℕ
h1_xs h1_ys : Fin h1_n → VarName
⊢ IsAxiom
(Forall_ (List.ofFn h1_xs)
(Forall_ (List.ofFn h1_ys)
((And_ (List.ofFn fun i => eq_ (h1_xs i) (h1_ys i))).imp_
((pred_const_ h1_name (List.ofFn h1_xs)).iff_ (pred_const_ h1_name (List.ofFn h1_ys)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.eq_2_pred_const_ | case a
F : Formula
h1_name : PredName
h1_n : ℕ
h1_xs h1_ys : Fin h1_n → VarName
⊢ IsAxiom
(Forall_ (List.ofFn h1_xs)
(Forall_ (List.ofFn h1_ys)
((And_ (List.ofFn fun i => eq_ (h1_xs i) (h1_ys i))).imp_
((pred_const_ h1_name (List.ofFn h1_xs)).iff_ (pred_const_ h1_name (List.ofFn h1_ys)))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsDeduct.axiom_ | F : Formula
h1_x_0 h1_y_0 h1_x_1 h1_y_1 : VarName
⊢ IsDeduct ∅
(forall_ h1_x_0
(forall_ h1_y_0
(forall_ h1_x_1
(forall_ h1_y_1
(((eq_ h1_x_0 h1_x_1).and_ (eq_ h1_y_0 h1_y_1)).imp_ ((eq_ h1_x_0 h1_y_0).iff_ (eq_ h1_x_1 h1_y_1))))))) | case a
F : Formula
h1_x_0 h1_y_0 h1_x_1 h1_y_1 : VarName
⊢ IsAxiom
(forall_ h1_x_0
(forall_ h1_y_0
(forall_ h1_x_1
(forall_ h1_y_1
(((eq_ h1_x_0 h1_x_1).and_ (eq_ h1_y_0 h1_y_1)).imp_ ((eq_ h1_x_0 h1_y_0).iff_ (eq_ h1_x_1 h1_y_1))))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply IsAxiom.eq_2_eq_ | case a
F : Formula
h1_x_0 h1_y_0 h1_x_1 h1_y_1 : VarName
⊢ IsAxiom
(forall_ h1_x_0
(forall_ h1_y_0
(forall_ h1_x_1
(forall_ h1_y_1
(((eq_ h1_x_0 h1_x_1).and_ (eq_ h1_y_0 h1_y_1)).imp_ ((eq_ h1_x_0 h1_y_0).iff_ (eq_ h1_x_1 h1_y_1))))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | apply generalization | F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : IsProofAlt h1_phi
h1_ih : IsDeduct ∅ h1_phi
⊢ IsDeduct ∅ (forall_ h1_v h1_phi) | case h1
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : IsProofAlt h1_phi
h1_ih : IsDeduct ∅ h1_phi
⊢ IsDeduct ∅ h1_phi
case h2
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : IsProofAlt h1_phi
h1_ih : IsDeduct ∅ h1_phi
⊢ ∀ H ∈ ∅, ¬isFreeIn h1_v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | exact h1_ih | case h1
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : IsProofAlt h1_phi
h1_ih : IsDeduct ∅ h1_phi
⊢ IsDeduct ∅ h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | simp | case h2
F : Formula
h1_v : VarName
h1_phi : Formula
h1_1 : IsProofAlt h1_phi
h1_ih : IsDeduct ∅ h1_phi
⊢ ∀ H ∈ ∅, ¬isFreeIn h1_v H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | exact IsDeduct.mp_ h1_phi h1_psi h1_ih_1 h1_ih_2 | F h1_phi h1_psi : Formula
h1_1 : IsProofAlt (h1_phi.imp_ h1_psi)
h1_2 : IsProofAlt h1_phi
h1_ih_1 : IsDeduct ∅ (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct ∅ h1_phi
⊢ IsDeduct ∅ h1_psi | no goals |
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