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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 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isProofAltImpIsDeduct | [402, 1] | [446, 10] | sorry | 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.isDeductImpIsProofAlt | [449, 1] | [486, 55] | induction h1 | F : Formula
h1 : IsDeduct ∅ F
⊢ IsProofAlt F | case axiom_
F phi✝ : Formula
a✝ : IsAxiom phi✝
⊢ IsProofAlt phi✝
case assume_
F phi✝ : Formula
a✝ : phi✝ ∈ ∅
⊢ IsProofAlt phi✝
case mp_
F phi✝ psi✝ : Formula
a✝¹ : IsDeduct ∅ (phi✝.imp_ psi✝)
a✝ : IsDeduct ∅ phi✝
a_ih✝¹ : IsProofAlt (phi✝.imp_ psi✝)
a_ih✝ : IsProofAlt phi✝
⊢ IsProofAlt psi✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case axiom_ h1_phi h1_1 =>
induction h1_1
case prop_true_ =>
apply IsProofAlt.prop_true_
case prop_1_ h1_1_phi h1_1_psi =>
apply IsProofAlt.prop_1_
case prop_2_ h1_1_phi h1_1_psi h1_1_chi =>
apply IsProofAlt.prop_2_
case prop_3_ h1_1_phi h1_1_psi =>
apply IsProofAlt.prop_3_
case pred_1_ h1_1_v h1_1_phi h1_1_psi =>
apply IsProofAlt.pred_1_
case pred_2_ h1_1_v h1_1_t h1_1_phi h1_1_1 h1_1_ih_1 h1_1_ih_2 =>
apply IsProofAlt.pred_2_ h1_1_v h1_1_t h1_1_phi h1_1_1 h1_1_ih_1 h1_1_ih_2
case pred_3_ h1_1_v h1_1_phi h1_1_1 =>
apply IsProofAlt.pred_3_
exact h1_1_1
case eq_1_ h1_1 =>
apply IsProofAlt.eq_1_
case eq_2_pred_const_ h1_1_name h1_1_n h1_1_xs h1_1_ys =>
apply IsProofAlt.eq_2_pred_const_
case eq_2_eq_ h1_1_x_0 h1_1_y_0 h1_1_x_1 h1_1_y_1 =>
apply IsProofAlt.eq_2_eq_
case gen_ h1_1_v h1_1_phi h1_1_1 h1_1_ih =>
apply IsProofAlt.gen_
exact h1_1_ih
all_goals
sorry | F h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsProofAlt h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case assume_ h1_phi h1_1 =>
simp at h1_1 | F h1_phi : Formula
h1_1 : h1_phi ∈ ∅
⊢ IsProofAlt h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case mp_ h1_phi h1_psi h1_1 h1_2 h1_ih_1 h1_ih_2 =>
exact IsProofAlt.mp_ h1_phi h1_psi h1_ih_1 h1_ih_2 | F h1_phi h1_psi : Formula
h1_1 : IsDeduct ∅ (h1_phi.imp_ h1_psi)
h1_2 : IsDeduct ∅ h1_phi
h1_ih_1 : IsProofAlt (h1_phi.imp_ h1_psi)
h1_ih_2 : IsProofAlt h1_phi
⊢ IsProofAlt h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | induction h1_1 | F h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsProofAlt h1_phi | case prop_true_
F h1_phi : Formula
⊢ IsProofAlt true_
case prop_1_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt (phi✝.imp_ (psi✝.imp_ phi✝))
case prop_2_
F h1_phi phi✝ psi✝ chi✝ : Formula
⊢ IsProofAlt ((phi✝.imp_ (psi✝.imp_ chi✝)).imp_ ((phi✝.imp_ psi✝).imp_ (phi✝.imp_ chi✝)))
case prop_3_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt ((phi✝.not_.imp_ psi✝.not_).imp_ (psi✝.imp_ phi✝))
case pred_1_
F h1_phi : Formula
v✝ : VarName
phi✝ psi✝ : Formula
⊢ IsProofAlt ((forall_ v✝ (phi✝.imp_ psi✝)).imp_ ((forall_ v✝ phi✝).imp_ (forall_ v✝ psi✝)))
case pred_2_
F h1_phi : Formula
v✝ t✝ : VarName
phi✝ phi'✝ : Formula
a✝¹ : fastAdmits v✝ t✝ phi✝
a✝ : fastReplaceFree v✝ t✝ phi✝ = phi'✝
⊢ IsProofAlt ((forall_ v✝ phi✝).imp_ phi'✝)
case pred_3_
F h1_phi : Formula
v✝ : VarName
phi✝ : Formula
a✝ : ¬isFreeIn v✝ phi✝
⊢ IsProofAlt (phi✝.imp_ (forall_ v✝ phi✝))
case eq_1_
F h1_phi : Formula
v✝ : VarName
⊢ IsProofAlt (forall_ v✝ (eq_ v✝ v✝))
case eq_2_pred_const_
F h1_phi : Formula
name✝ : PredName
n✝ : ℕ
xs✝ ys✝ : Fin n✝ → VarName
⊢ IsProofAlt
(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 h1_phi : Formula
x_0✝ x_1✝ y_0✝ y_1✝ : VarName
⊢ IsProofAlt
(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 h1_phi : Formula
v✝ : VarName
phi✝ : Formula
a✝ : IsAxiom phi✝
a_ih✝ : IsProofAlt phi✝
⊢ IsProofAlt (forall_ v✝ phi✝)
case def_false_
F h1_phi : Formula
⊢ IsProofAlt (false_.iff_ true_.not_)
case def_and_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt ((phi✝.and_ psi✝).iff_ (phi✝.imp_ psi✝.not_).not_)
case def_or_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt ((phi✝.or_ psi✝).iff_ (phi✝.not_.imp_ psi✝))
case def_iff_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt
(((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 h1_phi : Formula
v✝ : VarName
phi✝ : Formula
⊢ IsProofAlt ((exists_ v✝ phi✝).iff_ (forall_ v✝ phi✝.not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case prop_true_ =>
apply IsProofAlt.prop_true_ | F h1_phi : Formula
⊢ IsProofAlt true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case prop_1_ h1_1_phi h1_1_psi =>
apply IsProofAlt.prop_1_ | F h1_phi h1_1_phi h1_1_psi : Formula
⊢ IsProofAlt (h1_1_phi.imp_ (h1_1_psi.imp_ h1_1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case prop_2_ h1_1_phi h1_1_psi h1_1_chi =>
apply IsProofAlt.prop_2_ | F h1_phi h1_1_phi h1_1_psi h1_1_chi : Formula
⊢ IsProofAlt ((h1_1_phi.imp_ (h1_1_psi.imp_ h1_1_chi)).imp_ ((h1_1_phi.imp_ h1_1_psi).imp_ (h1_1_phi.imp_ h1_1_chi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case prop_3_ h1_1_phi h1_1_psi =>
apply IsProofAlt.prop_3_ | F h1_phi h1_1_phi h1_1_psi : Formula
⊢ IsProofAlt ((h1_1_phi.not_.imp_ h1_1_psi.not_).imp_ (h1_1_psi.imp_ h1_1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case pred_1_ h1_1_v h1_1_phi h1_1_psi =>
apply IsProofAlt.pred_1_ | F h1_phi : Formula
h1_1_v : VarName
h1_1_phi h1_1_psi : Formula
⊢ IsProofAlt ((forall_ h1_1_v (h1_1_phi.imp_ h1_1_psi)).imp_ ((forall_ h1_1_v h1_1_phi).imp_ (forall_ h1_1_v h1_1_psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case pred_2_ h1_1_v h1_1_t h1_1_phi h1_1_1 h1_1_ih_1 h1_1_ih_2 =>
apply IsProofAlt.pred_2_ h1_1_v h1_1_t h1_1_phi h1_1_1 h1_1_ih_1 h1_1_ih_2 | F h1_phi : Formula
h1_1_v h1_1_t : VarName
h1_1_phi h1_1_1 : Formula
h1_1_ih_1 : fastAdmits h1_1_v h1_1_t h1_1_phi
h1_1_ih_2 : fastReplaceFree h1_1_v h1_1_t h1_1_phi = h1_1_1
⊢ IsProofAlt ((forall_ h1_1_v h1_1_phi).imp_ h1_1_1) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case pred_3_ h1_1_v h1_1_phi h1_1_1 =>
apply IsProofAlt.pred_3_
exact h1_1_1 | F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : ¬isFreeIn h1_1_v h1_1_phi
⊢ IsProofAlt (h1_1_phi.imp_ (forall_ h1_1_v h1_1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case eq_1_ h1_1 =>
apply IsProofAlt.eq_1_ | F h1_phi : Formula
h1_1 : VarName
⊢ IsProofAlt (forall_ h1_1 (eq_ h1_1 h1_1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case eq_2_pred_const_ h1_1_name h1_1_n h1_1_xs h1_1_ys =>
apply IsProofAlt.eq_2_pred_const_ | F h1_phi : Formula
h1_1_name : PredName
h1_1_n : ℕ
h1_1_xs h1_1_ys : Fin h1_1_n → VarName
⊢ IsProofAlt
(Forall_ (List.ofFn h1_1_xs)
(Forall_ (List.ofFn h1_1_ys)
((And_ (List.ofFn fun i => eq_ (h1_1_xs i) (h1_1_ys i))).imp_
((pred_const_ h1_1_name (List.ofFn h1_1_xs)).iff_ (pred_const_ h1_1_name (List.ofFn h1_1_ys)))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case eq_2_eq_ h1_1_x_0 h1_1_y_0 h1_1_x_1 h1_1_y_1 =>
apply IsProofAlt.eq_2_eq_ | F h1_phi : Formula
h1_1_x_0 h1_1_y_0 h1_1_x_1 h1_1_y_1 : VarName
⊢ IsProofAlt
(forall_ h1_1_x_0
(forall_ h1_1_y_0
(forall_ h1_1_x_1
(forall_ h1_1_y_1
(((eq_ h1_1_x_0 h1_1_x_1).and_ (eq_ h1_1_y_0 h1_1_y_1)).imp_
((eq_ h1_1_x_0 h1_1_y_0).iff_ (eq_ h1_1_x_1 h1_1_y_1))))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | case gen_ h1_1_v h1_1_phi h1_1_1 h1_1_ih =>
apply IsProofAlt.gen_
exact h1_1_ih | F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : IsAxiom h1_1_phi
h1_1_ih : IsProofAlt h1_1_phi
⊢ IsProofAlt (forall_ h1_1_v h1_1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | all_goals
sorry | case def_false_
F h1_phi : Formula
⊢ IsProofAlt (false_.iff_ true_.not_)
case def_and_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt ((phi✝.and_ psi✝).iff_ (phi✝.imp_ psi✝.not_).not_)
case def_or_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt ((phi✝.or_ psi✝).iff_ (phi✝.not_.imp_ psi✝))
case def_iff_
F h1_phi phi✝ psi✝ : Formula
⊢ IsProofAlt
(((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 h1_phi : Formula
v✝ : VarName
phi✝ : Formula
⊢ IsProofAlt ((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.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.prop_true_ | F h1_phi : Formula
⊢ IsProofAlt true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.prop_1_ | F h1_phi h1_1_phi h1_1_psi : Formula
⊢ IsProofAlt (h1_1_phi.imp_ (h1_1_psi.imp_ h1_1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.prop_2_ | F h1_phi h1_1_phi h1_1_psi h1_1_chi : Formula
⊢ IsProofAlt ((h1_1_phi.imp_ (h1_1_psi.imp_ h1_1_chi)).imp_ ((h1_1_phi.imp_ h1_1_psi).imp_ (h1_1_phi.imp_ h1_1_chi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.prop_3_ | F h1_phi h1_1_phi h1_1_psi : Formula
⊢ IsProofAlt ((h1_1_phi.not_.imp_ h1_1_psi.not_).imp_ (h1_1_psi.imp_ h1_1_phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.pred_1_ | F h1_phi : Formula
h1_1_v : VarName
h1_1_phi h1_1_psi : Formula
⊢ IsProofAlt ((forall_ h1_1_v (h1_1_phi.imp_ h1_1_psi)).imp_ ((forall_ h1_1_v h1_1_phi).imp_ (forall_ h1_1_v h1_1_psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.pred_2_ h1_1_v h1_1_t h1_1_phi h1_1_1 h1_1_ih_1 h1_1_ih_2 | F h1_phi : Formula
h1_1_v h1_1_t : VarName
h1_1_phi h1_1_1 : Formula
h1_1_ih_1 : fastAdmits h1_1_v h1_1_t h1_1_phi
h1_1_ih_2 : fastReplaceFree h1_1_v h1_1_t h1_1_phi = h1_1_1
⊢ IsProofAlt ((forall_ h1_1_v h1_1_phi).imp_ h1_1_1) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.pred_3_ | F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : ¬isFreeIn h1_1_v h1_1_phi
⊢ IsProofAlt (h1_1_phi.imp_ (forall_ h1_1_v h1_1_phi)) | case a
F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : ¬isFreeIn h1_1_v h1_1_phi
⊢ ¬isFreeIn h1_1_v h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | exact h1_1_1 | case a
F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : ¬isFreeIn h1_1_v h1_1_phi
⊢ ¬isFreeIn h1_1_v h1_1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.eq_1_ | F h1_phi : Formula
h1_1 : VarName
⊢ IsProofAlt (forall_ h1_1 (eq_ h1_1 h1_1)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.eq_2_pred_const_ | F h1_phi : Formula
h1_1_name : PredName
h1_1_n : ℕ
h1_1_xs h1_1_ys : Fin h1_1_n → VarName
⊢ IsProofAlt
(Forall_ (List.ofFn h1_1_xs)
(Forall_ (List.ofFn h1_1_ys)
((And_ (List.ofFn fun i => eq_ (h1_1_xs i) (h1_1_ys i))).imp_
((pred_const_ h1_1_name (List.ofFn h1_1_xs)).iff_ (pred_const_ h1_1_name (List.ofFn h1_1_ys)))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.eq_2_eq_ | F h1_phi : Formula
h1_1_x_0 h1_1_y_0 h1_1_x_1 h1_1_y_1 : VarName
⊢ IsProofAlt
(forall_ h1_1_x_0
(forall_ h1_1_y_0
(forall_ h1_1_x_1
(forall_ h1_1_y_1
(((eq_ h1_1_x_0 h1_1_x_1).and_ (eq_ h1_1_y_0 h1_1_y_1)).imp_
((eq_ h1_1_x_0 h1_1_y_0).iff_ (eq_ h1_1_x_1 h1_1_y_1))))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | apply IsProofAlt.gen_ | F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : IsAxiom h1_1_phi
h1_1_ih : IsProofAlt h1_1_phi
⊢ IsProofAlt (forall_ h1_1_v h1_1_phi) | case a
F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : IsAxiom h1_1_phi
h1_1_ih : IsProofAlt h1_1_phi
⊢ IsProofAlt h1_1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | exact h1_1_ih | case a
F h1_phi : Formula
h1_1_v : VarName
h1_1_phi : Formula
h1_1_1 : IsAxiom h1_1_phi
h1_1_ih : IsProofAlt h1_1_phi
⊢ IsProofAlt h1_1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | sorry | case def_exists_
F h1_phi : Formula
v✝ : VarName
phi✝ : Formula
⊢ IsProofAlt ((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.isDeductImpIsProofAlt | [449, 1] | [486, 55] | simp at h1_1 | F h1_phi : Formula
h1_1 : h1_phi ∈ ∅
⊢ IsProofAlt h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.isDeductImpIsProofAlt | [449, 1] | [486, 55] | exact IsProofAlt.mp_ h1_phi h1_psi h1_ih_1 h1_ih_2 | F h1_phi h1_psi : Formula
h1_1 : IsDeduct ∅ (h1_phi.imp_ h1_psi)
h1_2 : IsDeduct ∅ h1_phi
h1_ih_1 : IsProofAlt (h1_phi.imp_ h1_psi)
h1_ih_2 : IsProofAlt h1_phi
⊢ IsProofAlt h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | apply deduction_theorem | P : Formula
u v : VarName
⊢ IsProof ((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))) | case h1
P : Formula
u v : VarName
⊢ IsDeduct (∅ ∪ {forall_ u (forall_ v P)}) (forall_ v (forall_ u P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | simp | case h1
P : Formula
u v : VarName
⊢ IsDeduct (∅ ∪ {forall_ u (forall_ v P)}) (forall_ v (forall_ u P)) | case h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ v (forall_ u P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | apply generalization | case h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ v (forall_ u P)) | case h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ u P)
case h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {forall_ u (forall_ v P)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | apply generalization | case h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ u P) | case h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} P
case h1.h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {forall_ u (forall_ v P)}, ¬isFreeIn u H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | apply specId v P | case h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} P | case h1.h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | apply specId u (forall_ v P) | case h1.h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ v P) | case h1.h1.h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ u (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | apply IsDeduct.assume_ | case h1.h1.h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {forall_ u (forall_ v P)} (forall_ u (forall_ v P)) | case h1.h1.h1.h1.h1.a
P : Formula
u v : VarName
⊢ forall_ u (forall_ v P) ∈ {forall_ u (forall_ v P)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | simp | case h1.h1.h1.h1.h1.a
P : Formula
u v : VarName
⊢ forall_ u (forall_ v P) ∈ {forall_ u (forall_ v P)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | simp | case h1.h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {forall_ u (forall_ v P)}, ¬isFreeIn u H | case h1.h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | simp only [isFreeIn] | case h1.h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u (forall_ v P)) | case h1.h1.h2
P : Formula
u v : VarName
⊢ ¬(¬True ∧ ¬u = v ∧ isFreeIn u P) |
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