internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Margaris/Fol.lean	FOL.NV.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)

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)