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Split (1)

train · 219k rows

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/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [s1] at h2	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply h2	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ h1_X = P ∧ (List.map V h1_ts).length = zs.length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ h1_X = P ∧ (List.map V h1_ts).length = zs.length	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ h1_X = P ∧ h1_ts.length = zs.length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	exact h1_1	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length → (Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ J.pred_var_ P (List.map V h1_ts)) ⊢ h1_X = P ∧ h1_ts.length = zs.length	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x h1_y : VarName V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E (eq_ h1_x h1_y) ↔ Holds D J V E (eq_ h1_x h1_y)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E false_ ↔ Holds D J V E false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E h1_phi'.not_ ↔ Holds D J V E h1_phi.not_	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ ¬Holds D I V E h1_phi' ↔ ¬Holds D J V E h1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	congr! 1	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ ¬Holds D I V E h1_phi' ↔ ¬Holds D J V E h1_phi	case a.h.e'_1.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E h1_phi' ↔ Holds D J V E h1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	exact h1_ih V h2	case a.h.e'_1.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E h1_phi' ↔ Holds D J V E h1_phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_psi h1_phi' h1_psi' : Formula a✝¹ : IsSub P zs H h1_phi h1_phi' a✝ : IsSub P zs H h1_psi h1_psi' h1_ih_1 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) h1_ih_2 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E (h1_phi'.iff_ h1_psi') ↔ Holds D J V E (h1_phi.iff_ h1_psi)	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_psi h1_phi' h1_psi' : Formula a✝¹ : IsSub P zs H h1_phi h1_phi' a✝ : IsSub P zs H h1_psi h1_psi' h1_ih_1 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) h1_ih_2 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ (Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') ↔ (Holds D J V E h1_phi ↔ Holds D J V E h1_psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	congr! 1	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_psi h1_phi' h1_psi' : Formula a✝¹ : IsSub P zs H h1_phi h1_phi' a✝ : IsSub P zs H h1_psi h1_psi' h1_ih_1 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) h1_ih_2 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ (Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') ↔ (Holds D J V E h1_phi ↔ Holds D J V E h1_psi)	case a.h.e'_1.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_psi h1_phi' h1_psi' : Formula a✝¹ : IsSub P zs H h1_phi h1_phi' a✝ : IsSub P zs H h1_psi h1_psi' h1_ih_1 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) h1_ih_2 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E h1_phi' ↔ Holds D J V E h1_phi case a.h.e'_2.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_psi h1_phi' h1_psi' : Formula a✝¹ : IsSub P zs H h1_phi h1_phi' a✝ : IsSub P zs H h1_psi h1_psi' h1_ih_1 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) h1_ih_2 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E h1_psi' ↔ Holds D J V E h1_psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	exact h1_ih_1 V h2	case a.h.e'_1.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_psi h1_phi' h1_psi' : Formula a✝¹ : IsSub P zs H h1_phi h1_phi' a✝ : IsSub P zs H h1_psi h1_psi' h1_ih_1 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) h1_ih_2 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E h1_phi' ↔ Holds D J V E h1_phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	exact h1_ih_2 V h2	case a.h.e'_2.a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_psi h1_phi' h1_psi' : Formula a✝¹ : IsSub P zs H h1_phi h1_phi' a✝ : IsSub P zs H h1_psi h1_psi' h1_ih_1 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) h1_ih_2 : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E h1_psi' ↔ Holds D J V E h1_psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E (exists_ h1_x h1_phi') ↔ Holds D J V E (exists_ h1_x h1_phi)	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi') ↔ ∃ d, Holds D J (Function.updateITE V h1_x d) E h1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	first \| apply forall_congr' \| apply exists_congr	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi') ↔ ∃ d, Holds D J (Function.updateITE V h1_x d) E h1_phi	case h D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' ↔ Holds D J (Function.updateITE V h1_x a) E h1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	intro d	case h D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' ↔ Holds D J (Function.updateITE V h1_x a) E h1_phi	case h D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) d : D ⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi' ↔ Holds D J (Function.updateITE V h1_x d) E h1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply h1_ih	case h D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) d : D ⊢ Holds D I (Function.updateITE V h1_x d) E h1_phi' ↔ Holds D J (Function.updateITE V h1_x d) E h1_phi	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) d : D ⊢ ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	intro Q ds a1	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) d : D ⊢ ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds)	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	specialize h2 Q ds a1	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	have s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ Holds D I (Function.updateListITE V zs ds) E H := by apply Holds_coincide_Var intro v a1 apply Function.updateListITE_updateIte intro contra subst contra contradiction	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ Holds D I (Function.updateListITE V zs ds) E H ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [h2] at s1	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ Holds D I (Function.updateListITE V zs ds) E H ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	exact s1	case h.h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ J.pred_var_ P ds	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply forall_congr'	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ (∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi') ↔ ∀ (d : D), Holds D J (Function.updateITE V h1_x d) E h1_phi	case h D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' ↔ Holds D J (Function.updateITE V h1_x a) E h1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply exists_congr	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ (∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi') ↔ ∃ d, Holds D J (Function.updateITE V h1_x d) E h1_phi	case h D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ ∀ (a : D), Holds D I (Function.updateITE V h1_x a) E h1_phi' ↔ Holds D J (Function.updateITE V h1_x a) E h1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply Holds_coincide_Var	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds ⊢ Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ Holds D I (Function.updateListITE V zs ds) E H	case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds ⊢ ∀ (v : VarName), isFreeIn v H → Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	intro v a1	case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds ⊢ ∀ (v : VarName), isFreeIn v H → Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v	case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H ⊢ Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply Function.updateListITE_updateIte	case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H ⊢ Function.updateListITE (Function.updateITE V h1_x d) zs ds v = Function.updateListITE V zs ds v	case h1.h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H ⊢ ¬v = h1_x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	intro contra	case h1.h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H ⊢ ¬v = h1_x	case h1.h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H contra : v = h1_x ⊢ False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	subst contra	case h1.h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : ¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H contra : v = h1_x ⊢ False	case h1.h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H h1_1 : ¬isFreeIn v H ⊢ False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	contradiction	case h1.h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : ∀ (V : VarAssignment D), (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D d : D Q : PredName ds : List D a1✝ : Q = P ∧ ds.length = zs.length h2 : Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds v : VarName a1 : isFreeIn v H h1_1 : ¬isFreeIn v H ⊢ False	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	cases E	D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊢ Holds D I V E (def_ X xs) ↔ Holds D J V E (def_ X xs)	case nil D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) [] H ↔ J.pred_var_ P ds) ⊢ Holds D I V [] (def_ X xs) ↔ Holds D J V [] (def_ X xs) case cons D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D head✝ : Definition tail✝ : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (head✝ :: tail✝) H ↔ J.pred_var_ P ds) ⊢ Holds D I V (head✝ :: tail✝) (def_ X xs) ↔ Holds D J V (head✝ :: tail✝) (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	case nil => simp only [Holds]	D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) [] H ↔ J.pred_var_ P ds) ⊢ Holds D I V [] (def_ X xs) ↔ Holds D J V [] (def_ X xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) [] H ↔ J.pred_var_ P ds) ⊢ Holds D I V [] (def_ X xs) ↔ Holds D J V [] (def_ X xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [Holds]	D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) ⊢ Holds D I V (hd :: tl) (def_ X xs) ↔ Holds D J V (hd :: tl) (def_ X xs)	D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) ⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q else Holds D I V tl (def_ X xs)) ↔ if X = hd.name ∧ xs.length = hd.args.length then Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q else Holds D J V tl (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	split_ifs	D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) ⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q else Holds D I V tl (def_ X xs)) ↔ if X = hd.name ∧ xs.length = hd.args.length then Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q else Holds D J V tl (def_ X xs)	case pos D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) h✝ : X = hd.name ∧ xs.length = hd.args.length ⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q case neg D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) h✝ : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ Holds D I V tl (def_ X xs) ↔ Holds D J V tl (def_ X xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply Holds_coincide_PredVar	D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D J (Function.updateListITE V hd.args (List.map V xs)) tl hd.q	case h1 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ I.pred_const_ = J.pred_const_ case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → (I.pred_var_ P ds ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	exact h3_const	case h1 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ I.pred_const_ = J.pred_const_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [predVarOccursIn_iff_mem_predVarSet]	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → (I.pred_var_ P ds ↔ J.pred_var_ P ds)	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet → (I.pred_var_ P ds ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [hd.h2]	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet → (I.pred_var_ P ds ↔ J.pred_var_ P ds)	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ ∅ → (I.pred_var_ P ds ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ ∅ → (I.pred_var_ P ds ↔ J.pred_var_ P ds)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	apply Holds_coincide_PredVar	D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ Holds D I V tl (def_ X xs) ↔ Holds D J V tl (def_ X xs)	case h1 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ I.pred_const_ = J.pred_const_ case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → (I.pred_var_ P ds ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	exact h3_const	case h1 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ I.pred_const_ = J.pred_const_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp only [predVarOccursIn]	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → (I.pred_var_ P ds ↔ J.pred_var_ P ds)	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_theorem	[131, 1]	[245, 15]	simp	case h2 D : Type I J : Interpretation D A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X : DefName xs : List VarName V : VarAssignment D hd : Definition tl : List Definition h2 : ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) (hd :: tl) H ↔ J.pred_var_ P ds) c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ J.pred_var_ P ds)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp only [IsValid] at h2	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : F.IsValid ⊢ F'.IsValid	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ F'.IsValid
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp only [IsValid]	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ F'.IsValid	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	intro D I V E	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I V E F'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	let J : Interpretation D := { nonempty := I.nonempty pred_const_ := I.pred_const_ pred_var_ := fun (Q : PredName) (ds : List D) => if (Q = P ∧ ds.length = zs.length) then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds }	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I V E F'	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } ⊢ Holds D I V E F'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	obtain s1 := substitution_theorem D I J V E F P zs H F' h1	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } ⊢ Holds D I V E F'	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → I.pred_const_ = J.pred_const_ → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ Holds D I V E F'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp only [Interpretation.pred_var_] at s1	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) → I.pred_const_ = J.pred_const_ → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ Holds D I V E F'	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ Holds D I V E F'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp only [s2]	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) s2 : Holds D I V E F' ↔ Holds D J V E F ⊢ Holds D I V E F'	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) s2 : Holds D I V E F' ↔ Holds D J V E F ⊢ Holds D J V E F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	apply h2	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) s2 : Holds D I V E F' ↔ Holds D J V E F ⊢ Holds D J V E F	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	apply s1	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ Holds D I V E F' ↔ Holds D J V E F	case h2 F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds) case h3_const F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ True case h3_var F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	intro Q ds a1	case h2 F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ ∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)	case h2 F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length ⊢ Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	cases a1	case h2 F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1 : Q = P ∧ ds.length = zs.length ⊢ Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds	case h2.intro F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D left✝ : Q = P right✝ : ds.length = zs.length ⊢ Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	case h2.intro a1_left a1_right => simp simp only [if_pos a1_right]	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1_left : Q = P a1_right : ds.length = zs.length ⊢ Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1_left : Q = P a1_right : ds.length = zs.length ⊢ Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1_left : Q = P a1_right : ds.length = zs.length ⊢ Holds D I (Function.updateListITE V zs ds) E H ↔ if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp only [if_pos a1_right]	F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1_left : Q = P a1_right : ds.length = zs.length ⊢ Holds D I (Function.updateListITE V zs ds) E H ↔ if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp	case h3_const F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ True	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	intro Q ds a1	case h3_var F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) ⊢ ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)	case h3_var F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1 : ¬(Q = P ∧ ds.length = zs.length) ⊢ I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Pred/One/Ind/Sub.lean	FOL.NV.Sub.Pred.One.Ind.substitution_is_valid	[248, 1]	[282, 11]	simp only [if_neg a1]	case h3_var F F' : Formula P : PredName zs : List VarName H : Formula h1 : IsSub P zs H F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env J : Interpretation D := { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun Q ds => if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds } s1 : (∀ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ if True ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ P ds)) → True → (∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds)) → (Holds D I V E F' ↔ Holds D J V E F) Q : PredName ds : List D a1 : ¬(Q = P ∧ ds.length = zs.length) ⊢ I.pred_var_ Q ds ↔ if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ Q ds	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	induction s	α : Type s : Str α ⊢ ∃ n, List.reverse s ∈ exp α n	case nil α : Type ⊢ ∃ n, [].reverse ∈ exp α n case cons α : Type head✝ : α tail✝ : List α tail_ih✝ : ∃ n, tail✝.reverse ∈ exp α n ⊢ ∃ n, (head✝ :: tail✝).reverse ∈ exp α n
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	case nil => apply Exists.intro 0 exact exp.zero	α : Type ⊢ ∃ n, [].reverse ∈ exp α n	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	case cons hd tl ih => apply Exists.elim ih intro n a1 apply Exists.intro (n + 1) simp exact exp.succ n hd tl.reverse a1	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n ⊢ ∃ n, (hd :: tl).reverse ∈ exp α n	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	apply Exists.intro 0	α : Type ⊢ ∃ n, [].reverse ∈ exp α n	α : Type ⊢ [].reverse ∈ exp α 0
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	exact exp.zero	α : Type ⊢ [].reverse ∈ exp α 0	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	apply Exists.elim ih	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n ⊢ ∃ n, (hd :: tl).reverse ∈ exp α n	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n ⊢ ∀ (a : ℕ), tl.reverse ∈ exp α a → ∃ n, (hd :: tl).reverse ∈ exp α n
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	intro n a1	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n ⊢ ∀ (a : ℕ), tl.reverse ∈ exp α a → ∃ n, (hd :: tl).reverse ∈ exp α n	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n n : ℕ a1 : tl.reverse ∈ exp α n ⊢ ∃ n, (hd :: tl).reverse ∈ exp α n
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	apply Exists.intro (n + 1)	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n n : ℕ a1 : tl.reverse ∈ exp α n ⊢ ∃ n, (hd :: tl).reverse ∈ exp α n	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n n : ℕ a1 : tl.reverse ∈ exp α n ⊢ (hd :: tl).reverse ∈ exp α (n + 1)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	simp	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n n : ℕ a1 : tl.reverse ∈ exp α n ⊢ (hd :: tl).reverse ∈ exp α (n + 1)	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n n : ℕ a1 : tl.reverse ∈ exp α n ⊢ tl.reverse ++ [hd] ∈ exp α (n + 1)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp	[53, 1]	[67, 40]	exact exp.succ n hd tl.reverse a1	α : Type hd : α tl : List α ih : ∃ n, tl.reverse ∈ exp α n n : ℕ a1 : tl.reverse ∈ exp α n ⊢ tl.reverse ++ [hd] ∈ exp α (n + 1)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.str_mem_exp	[70, 1]	[77, 13]	obtain s1 := rev_str_mem_exp s.reverse	α : Type s : Str α ⊢ ∃ n, s ∈ exp α n	α : Type s : Str α s1 : ∃ n, (List.reverse s).reverse ∈ exp α n ⊢ ∃ n, s ∈ exp α n
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.str_mem_exp	[70, 1]	[77, 13]	simp only [List.reverse_reverse] at s1	α : Type s : Str α s1 : ∃ n, (List.reverse s).reverse ∈ exp α n ⊢ ∃ n, s ∈ exp α n	α : Type s : Str α s1 : ∃ n, s ∈ exp α n ⊢ ∃ n, s ∈ exp α n
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.str_mem_exp	[70, 1]	[77, 13]	exact s1	α : Type s : Str α s1 : ∃ n, s ∈ exp α n ⊢ ∃ n, s ∈ exp α n	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp_str_len	[80, 1]	[91, 48]	induction s	α : Type s : Str α ⊢ List.reverse s ∈ exp α (List.length s)	case nil α : Type ⊢ [].reverse ∈ exp α [].length case cons α : Type head✝ : α tail✝ : List α tail_ih✝ : tail✝.reverse ∈ exp α tail✝.length ⊢ (head✝ :: tail✝).reverse ∈ exp α (head✝ :: tail✝).length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp_str_len	[80, 1]	[91, 48]	case nil => simp exact exp.zero	α : Type ⊢ [].reverse ∈ exp α [].length	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp_str_len	[80, 1]	[91, 48]	case cons hd tl ih => simp exact exp.succ tl.length hd tl.reverse ih	α : Type hd : α tl : List α ih : tl.reverse ∈ exp α tl.length ⊢ (hd :: tl).reverse ∈ exp α (hd :: tl).length	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp_str_len	[80, 1]	[91, 48]	simp	α : Type ⊢ [].reverse ∈ exp α [].length	α : Type ⊢ [] ∈ exp α 0
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp_str_len	[80, 1]	[91, 48]	exact exp.zero	α : Type ⊢ [] ∈ exp α 0	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp_str_len	[80, 1]	[91, 48]	simp	α : Type hd : α tl : List α ih : tl.reverse ∈ exp α tl.length ⊢ (hd :: tl).reverse ∈ exp α (hd :: tl).length	α : Type hd : α tl : List α ih : tl.reverse ∈ exp α tl.length ⊢ tl.reverse ++ [hd] ∈ exp α (tl.length + 1)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.rev_str_mem_exp_str_len	[80, 1]	[91, 48]	exact exp.succ tl.length hd tl.reverse ih	α : Type hd : α tl : List α ih : tl.reverse ∈ exp α tl.length ⊢ tl.reverse ++ [hd] ∈ exp α (tl.length + 1)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.str_mem_exp_str_len	[94, 1]	[101, 13]	obtain s1 := rev_str_mem_exp_str_len s.reverse	α : Type s : Str α ⊢ s ∈ exp α (List.length s)	α : Type s : Str α s1 : (List.reverse s).reverse ∈ exp α (List.reverse s).length ⊢ s ∈ exp α (List.length s)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.str_mem_exp_str_len	[94, 1]	[101, 13]	simp at s1	α : Type s : Str α s1 : (List.reverse s).reverse ∈ exp α (List.reverse s).length ⊢ s ∈ exp α (List.length s)	α : Type s : Str α s1 : s ∈ exp α (List.length s) ⊢ s ∈ exp α (List.length s)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.str_mem_exp_str_len	[94, 1]	[101, 13]	exact s1	α : Type s : Str α s1 : s ∈ exp α (List.length s) ⊢ s ∈ exp α (List.length s)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.mem_exp_imp_str_len_eq	[104, 1]	[116, 17]	induction h1	α : Type s : Str α n : ℕ h1 : s ∈ exp α n ⊢ List.length s = n	case zero α : Type s : Str α n : ℕ ⊢ [].length = 0 case succ α : Type s : Str α n n✝ : ℕ a✝¹ : α s✝ : Str α a✝ : s✝ ∈ exp α n✝ a_ih✝ : List.length s✝ = n✝ ⊢ List.length (s✝ ++ [a✝¹]) = n✝ + 1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.mem_exp_imp_str_len_eq	[104, 1]	[116, 17]	case zero => simp	α : Type s : Str α n : ℕ ⊢ [].length = 0	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.mem_exp_imp_str_len_eq	[104, 1]	[116, 17]	case succ m a s ih_1 ih_2 => simp exact ih_2	α : Type s✝ : Str α n m : ℕ a : α s : Str α ih_1 : s ∈ exp α m ih_2 : List.length s = m ⊢ List.length (s ++ [a]) = m + 1	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.mem_exp_imp_str_len_eq	[104, 1]	[116, 17]	simp	α : Type s : Str α n : ℕ ⊢ [].length = 0	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.mem_exp_imp_str_len_eq	[104, 1]	[116, 17]	simp	α : Type s✝ : Str α n m : ℕ a : α s : Str α ih_1 : s ∈ exp α m ih_2 : List.length s = m ⊢ List.length (s ++ [a]) = m + 1	α : Type s✝ : Str α n m : ℕ a : α s : Str α ih_1 : s ∈ exp α m ih_2 : List.length s = m ⊢ List.length s = m
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.mem_exp_imp_str_len_eq	[104, 1]	[116, 17]	exact ih_2	α : Type s✝ : Str α n m : ℕ a : α s : Str α ih_1 : s ∈ exp α m ih_2 : List.length s = m ⊢ List.length s = m	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.all_str_mem_kleene_closure	[153, 1]	[160, 24]	simp only [kleene_closure]	α : Type s : Str α ⊢ s ∈ kleene_closure α	α : Type s : Str α ⊢ s ∈ ⋃ n, exp α n
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.all_str_mem_kleene_closure	[153, 1]	[160, 24]	simp	α : Type s : Str α ⊢ s ∈ ⋃ n, exp α n	α : Type s : Str α ⊢ ∃ i, s ∈ exp α i
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.all_str_mem_kleene_closure	[153, 1]	[160, 24]	exact str_mem_exp s	α : Type s : Str α ⊢ ∃ i, s ∈ exp α i	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.thm_2	[192, 1]	[198, 36]	symm	α : Type s t u : Str α ⊢ s ++ (t ++ u) = s ++ t ++ u	α : Type s t u : Str α ⊢ s ++ t ++ u = s ++ (t ++ u)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Strings.thm_2	[192, 1]	[198, 36]	exact (List.append_assoc s t u)	α : Type s t u : Str α ⊢ s ++ t ++ u = s ++ (t ++ u)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_3_a	[237, 1]	[243, 9]	simp only [concat]	α : Type L : Language α ⊢ concat L ∅ = ∅	α : Type L : Language α ⊢ {x \| ∃ s ∈ L, ∃ t ∈ ∅, s ++ t = x} = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_3_a	[237, 1]	[243, 9]	simp	α : Type L : Language α ⊢ {x \| ∃ s ∈ L, ∃ t ∈ ∅, s ++ t = x} = ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_3_b	[246, 1]	[252, 9]	simp only [concat]	α : Type L : Language α ⊢ concat ∅ L = ∅	α : Type L : Language α ⊢ {x \| ∃ s ∈ ∅, ∃ t ∈ L, s ++ t = x} = ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_3_b	[246, 1]	[252, 9]	simp	α : Type L : Language α ⊢ {x \| ∃ s ∈ ∅, ∃ t ∈ L, s ++ t = x} = ∅	no goals

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