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stringlengths 1
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_3 | [889, 1] | [913, 13] | simp | U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
⊢ IsDeduct ∅ (List.foldr forall_ (U.iff_ V) []) | U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
⊢ IsDeduct ∅ (U.iff_ V) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_3 | [889, 1] | [913, 13] | exact h2 | U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
⊢ IsDeduct ∅ (U.iff_ V) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_3 | [889, 1] | [913, 13] | simp | U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
⊢ IsDeduct ∅ (List.foldr forall_ (U.iff_ V) (hd :: tl)) | U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
⊢ IsDeduct ∅ (forall_ hd (List.foldr forall_ (U.iff_ V) tl)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_3 | [889, 1] | [913, 13] | apply generalization | U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
⊢ IsDeduct ∅ (forall_ hd (List.foldr forall_ (U.iff_ V) tl)) | case h1
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
⊢ IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
case h2
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
⊢ ∀ H ∈ ∅, ¬isFreeIn hd H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_3 | [889, 1] | [913, 13] | exact ih | case h1
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
⊢ IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_3 | [889, 1] | [913, 13] | simp | case h2
U V P_U P_V : Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
hd : VarName
tl : List VarName
ih : IsDeduct ∅ (List.foldr forall_ (U.iff_ V) tl)
⊢ ∀ H ∈ ∅, ¬isFreeIn hd H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | apply IsDeduct.mp_ P_U | U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ P_V | case a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (P_U.imp_ P_V)
case a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ P_U |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | apply IsDeduct.mp_ (P_U.iff_ P_V) | case a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (P_U.imp_ P_V) | case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ ((P_U.iff_ P_V).imp_ (P_U.imp_ P_V))
case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (P_U.iff_ P_V) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | simp only [def_iff_] | case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ ((P_U.iff_ P_V).imp_ (P_U.imp_ P_V)) | case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (((P_U.imp_ P_V).and_ (P_V.imp_ P_U)).imp_ (P_U.imp_ P_V)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | simp only [def_and_] | case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (((P_U.imp_ P_V).and_ (P_V.imp_ P_U)).imp_ (P_U.imp_ P_V)) | case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (((P_U.imp_ P_V).imp_ (P_V.imp_ P_U).not_).not_.imp_ (P_U.imp_ P_V)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | SC | case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (((P_U.imp_ P_V).imp_ (P_V.imp_ P_U).not_).not_.imp_ (P_U.imp_ P_V)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | apply proof_imp_deduct | case a.a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ (P_U.iff_ P_V) | case a.a.h1
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsProof (P_U.iff_ P_V) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | exact C_18_3 U V P_U P_V h1 h2 | case a.a.h1
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsProof (P_U.iff_ P_V) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.C_18_4 | [917, 1] | [935, 13] | exact h3 | case a
U V P_U P_V : Formula
Δ : Set Formula
h1 : IsReplOfFormulaInFormula U V P_U P_V
h2 : IsProof (U.iff_ V)
h3 : IsDeduct Δ P_U
⊢ IsDeduct Δ P_U | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | simp only [def_exists_] | P : Formula
v : VarName
⊢ IsProof ((forall_ v P).iff_ (exists_ v P.not_).not_) | P : Formula
v : VarName
⊢ IsProof ((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply C_18_4 P P.not_.not_ ((forall_ v P).iff_ (forall_ v P).not_.not_) | P : Formula
v : VarName
⊢ IsProof ((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_) | case h1
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).iff_ (forall_ v P).not_.not_)
((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_)
case h2
P : Formula
v : VarName
⊢ IsProof (P.iff_ P.not_.not_)
case h3
P : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v P).iff_ (forall_ v P).not_.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | simp only [def_iff_] | case h1
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).iff_ (forall_ v P).not_.not_)
((forall_ v P).iff_ (forall_ v P.not_.not_).not_.not_) | case h1
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P)))
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).and_ ((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | simp only [def_and_] | case h1
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P)))
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).and_ ((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P))) | case h1
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.not_ | case h1
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_).not_ | case h1.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_)
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.imp_ | case h1.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_
(((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_)
(((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_).imp_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_) | case h1.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).imp_ (forall_ v P).not_.not_)
((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_)
case h1.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.imp_ | case h1.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).imp_ (forall_ v P).not_.not_)
((forall_ v P).imp_ (forall_ v P.not_.not_).not_.not_) | case h1.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P)
case h1.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.same_ | case h1.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P) | case h1.a.a.a.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_18_5 | [938, 1] | [980, 7] | rfl | case h1.a.a.a.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_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.not_ | case h1.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_ | case h1.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.not_ | case h1.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_ | case h1.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.forall_ | case h1.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_) | case h1.a.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.diff_ | case h1.a.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_ | case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P = P
case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P.not_.not_ = P.not_.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | rfl | case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P = P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | rfl | case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P.not_.not_ = P.not_.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.not_ | case h1.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)).not_ | case h1.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P))
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.imp_ | case h1.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ ((forall_ v P).not_.not_.imp_ (forall_ v P))
((forall_ v P.not_.not_).not_.not_.imp_ (forall_ v P)) | case h1.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_
case h1.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.not_ | case h1.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_.not_ (forall_ v P.not_.not_).not_.not_ | case h1.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.not_ | case h1.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P).not_ (forall_ v P.not_.not_).not_ | case h1.a.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.forall_ | case h1.a.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P.not_.not_) | case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.diff_ | case h1.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ P P.not_.not_ | case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P = P
case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P.not_.not_ = P.not_.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | rfl | case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P = P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | rfl | case h1.a.a.a.a.a.a.a.a
P : Formula
v : VarName
⊢ P.not_.not_ = P.not_.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | apply IsReplOfFormulaInFormula.same_ | case h1.a.a.a.a
P : Formula
v : VarName
⊢ IsReplOfFormulaInFormula P P.not_.not_ (forall_ v P) (forall_ v P) | case h1.a.a.a.a.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_18_5 | [938, 1] | [980, 7] | rfl | case h1.a.a.a.a.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_18_5 | [938, 1] | [980, 7] | simp only [def_iff_] | case h2
P : Formula
v : VarName
⊢ IsProof (P.iff_ P.not_.not_) | case h2
P : Formula
v : VarName
⊢ IsProof ((P.imp_ P.not_.not_).and_ (P.not_.not_.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | simp only [def_and_] | case h2
P : Formula
v : VarName
⊢ IsProof ((P.imp_ P.not_.not_).and_ (P.not_.not_.imp_ P)) | case h2
P : Formula
v : VarName
⊢ IsProof ((P.imp_ P.not_.not_).imp_ (P.not_.not_.imp_ P).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | SC | case h2
P : Formula
v : VarName
⊢ IsProof ((P.imp_ P.not_.not_).imp_ (P.not_.not_.imp_ P).not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | simp only [def_iff_] | case h3
P : Formula
v : VarName
⊢ IsDeduct ∅ ((forall_ v P).iff_ (forall_ v P).not_.not_) | case h3
P : Formula
v : VarName
⊢ IsDeduct ∅ (((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | simp only [def_and_] | case h3
P : Formula
v : VarName
⊢ IsDeduct ∅ (((forall_ v P).imp_ (forall_ v P).not_.not_).and_ ((forall_ v P).not_.not_.imp_ (forall_ v P))) | case h3
P : Formula
v : VarName
⊢ IsDeduct ∅ (((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_5 | [938, 1] | [980, 7] | SC | case h3
P : Formula
v : VarName
⊢ IsDeduct ∅ (((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp only [Similar] at h1 | P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | cases h1 | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | case intro
P_u P_v : Formula
u v : VarName
left✝ : ¬isFreeIn v P_u
right✝ :
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | cases h1_right | P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right :
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
left✝ : ¬isFreeIn u P_v
right✝ : fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | cases h1_right_right | P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right : fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
left✝ : fastAdmits u v P_u
right✝ : fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | cases h1_right_right_right | P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right : fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
left✝ : fastAdmits v u P_v
right✝ : P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | cases h1_right_right_right_right | P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right : P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
left✝ : P_v = fastReplaceFree u v P_u
right✝ : P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply IsDeduct.mp_ ((forall_ v P_v).imp_ (forall_ u P_u)) | P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ (((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v)))
case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ v P_v).imp_ (forall_ u P_u)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply IsDeduct.mp_ ((forall_ u P_u).imp_ (forall_ v P_v)) | case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ (((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v))) | case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v))))
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ u P_u).imp_ (forall_ v P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp only [def_iff_] | case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v)))) | case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).and_ ((forall_ v P_v).imp_ (forall_ u P_u))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp only [def_and_] | case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).and_ ((forall_ v P_v).imp_ (forall_ u P_u))))) | case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_ ((forall_ v P_v).imp_ (forall_ u P_u)).not_).not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | SC | case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_ ((forall_ v P_v).imp_ (forall_ u P_u)).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply deduction_theorem | case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ u P_u).imp_ (forall_ v P_v)) | case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ u P_u}) (forall_ v P_v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp | case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ u P_u}) (forall_ v P_v) | case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} (forall_ v P_v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply generalization | case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} (forall_ v P_v) | case a.a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} P_v
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ u P_u}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | subst h1_right_right_right_right_left | case a.a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} P_v | case a.a.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (fastReplaceFree u v P_u) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply spec | case a.a.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (fastReplaceFree u v P_u) | case a.a.h1.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (forall_ u P_u)
case a.a.h1.h1.h2
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ fastAdmits u v P_u |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply IsDeduct.assume_ | case a.a.h1.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (forall_ u P_u) | case a.a.h1.h1.h1.a
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ forall_ u P_u ∈ {forall_ u P_u} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp | case a.a.h1.h1.h1.a
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ forall_ u P_u ∈ {forall_ u P_u} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | exact h1_right_right_left | case a.a.h1.h1.h2
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ fastAdmits u v P_u | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | intro H a1 | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ u P_u}, ¬isFreeIn v H | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ u P_u}
⊢ ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp at a1 | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ u P_u}
⊢ ¬isFreeIn v H | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ u P_u
⊢ ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | subst a1 | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ u P_u
⊢ ¬isFreeIn v H | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v (forall_ u P_u) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp only [isFreeIn] | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v (forall_ u P_u) | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬v = u ∧ isFreeIn v P_u) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬v = u ∧ isFreeIn v P_u) | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬v = u → ¬isFreeIn v P_u |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | intro _ | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬v = u → ¬isFreeIn v P_u | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬v = u
⊢ ¬isFreeIn v P_u |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | exact h1_left | case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬v = u
⊢ ¬isFreeIn v P_u | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply deduction_theorem | case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ v P_v).imp_ (forall_ u P_u)) | case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ v P_v}) (forall_ u P_u) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp | case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ v P_v}) (forall_ u P_u) | case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} (forall_ u P_u) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply generalization | case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} (forall_ u P_u) | case a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} P_u
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ v P_v}, ¬isFreeIn u H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | subst h1_right_right_right_right_right | case a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} P_u | case a.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (fastReplaceFree v u P_v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply spec | case a.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (fastReplaceFree v u P_v) | case a.h1.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (forall_ v P_v)
case a.h1.h1.h2
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ fastAdmits v u P_v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | apply IsDeduct.assume_ | case a.h1.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (forall_ v P_v) | case a.h1.h1.h1.a
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ forall_ v P_v ∈ {forall_ v P_v} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp | case a.h1.h1.h1.a
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ forall_ v P_v ∈ {forall_ v P_v} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | exact h1_right_right_right_left | case a.h1.h1.h2
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ fastAdmits v u P_v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | intro H a1 | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ v P_v}, ¬isFreeIn u H | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ v P_v}
⊢ ¬isFreeIn u H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp at a1 | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ v P_v}
⊢ ¬isFreeIn u H | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ v P_v
⊢ ¬isFreeIn u H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | subst a1 | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ v P_v
⊢ ¬isFreeIn u H | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn u (forall_ v P_v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp only [isFreeIn] | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn u (forall_ v P_v) | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬u = v ∧ isFreeIn u P_v) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | simp | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬u = v ∧ isFreeIn u P_v) | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬u = v → ¬isFreeIn u P_v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | intro _ | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬u = v → ¬isFreeIn u P_v | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬u = v
⊢ ¬isFreeIn u P_v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_6 | [983, 1] | [1036, 36] | exact h1_right_left | case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬u = v
⊢ ¬isFreeIn u P_v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_7 | [1040, 1] | [1051, 13] | apply C_18_4 (forall_ u P_u) (forall_ v P_v) Q Q' Δ h2 | P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q' | case h2
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
case h3
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_7 | [1040, 1] | [1051, 13] | apply T_18_6 P_u P_v u v h3 | case h2
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_7 | [1040, 1] | [1051, 13] | exact h1 | case h3
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.similar_not | [1054, 1] | [1065, 8] | simp only [Similar] at * | P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u.not_ ∧
¬isFreeIn u P_v.not_ ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.similar_not | [1054, 1] | [1065, 8] | simp only [isFreeIn] at * | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u.not_ ∧
¬isFreeIn u P_v.not_ ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.similar_not | [1054, 1] | [1065, 8] | simp only [fastAdmits] at * | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u.not_ ∧
fastAdmitsAux v u ∅ P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.similar_not | [1054, 1] | [1065, 8] | simp only [fastAdmitsAux] at * | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u.not_ ∧
fastAdmitsAux v u ∅ P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.similar_not | [1054, 1] | [1065, 8] | simp only [fastReplaceFree] at * | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_ | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = (fastReplaceFree u v P_u).not_ ∧ P_u.not_ = (fastReplaceFree v u P_v).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.similar_not | [1054, 1] | [1065, 8] | tauto | P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = (fastReplaceFree u v P_u).not_ ∧ P_u.not_ = (fastReplaceFree v u P_v).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_8 | [1068, 1] | [1083, 37] | simp only [def_exists_] | P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((exists_ u P_u).iff_ (exists_ v P_v)) | P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_8 | [1068, 1] | [1083, 37] | apply IsDeduct.mp_ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)) | P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_) | case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
(((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)).imp_ ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_))
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_8 | [1068, 1] | [1083, 37] | simp only [def_iff_] | case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
(((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)).imp_ ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_)) | case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).and_ ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_))).imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).and_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_8 | [1068, 1] | [1083, 37] | simp only [def_and_] | case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).and_ ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_))).imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).and_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_))) | case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).imp_
((forall_ v P_v.not_).imp_ (forall_ u P_u.not_)).not_).not_.imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).imp_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_).not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_18_8 | [1068, 1] | [1083, 37] | SC | case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).imp_
((forall_ v P_v.not_).imp_ (forall_ u P_u.not_)).not_).not_.imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).imp_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_).not_).not_) | no goals |
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