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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 [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/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	induction h1 generalizing V V'	D : Type I : Interpretation D V V' : VarAssignment D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula h1 : IsSubAux σ binders F F' h2 : ∀ v ∈ binders, V v = V' (σ v) h3 : ∀ (v : VarName), σ v ∉ binders → V v = V' (σ v) h4 : ∀ v ∈ binders, v = σ v ⊢ Holds D I V E F ↔ Holds D I V' E F'	case pred_const_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName X✝ : PredName xs✝ : List VarName a✝ : ∀ v ∈ xs✝, v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (pred_const_ X✝ xs✝) ↔ Holds D I V' E (pred_const_ X✝ (List.map σ✝ xs✝)) case pred_var_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName X✝ : PredName xs✝ : List VarName a✝ : ∀ v ∈ xs✝, v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (pred_var_ X✝ xs✝) ↔ Holds D I V' E (pred_var_ X✝ (List.map σ✝ xs✝)) case eq_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName x✝ y✝ : VarName a✝ : ∀ (v : VarName), v = x✝ ∨ v = y✝ → v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (eq_ x✝ y✝) ↔ Holds D I V' E (eq_ (σ✝ x✝) (σ✝ y✝)) case true_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E true_ ↔ Holds D I V' E true_ case false_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E false_ ↔ Holds D I V' E false_ case not_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ phi'✝ : Formula a✝ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E phi✝.not_ ↔ Holds D I V' E phi'✝.not_ case imp_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.imp_ psi✝) ↔ Holds D I V' E (phi'✝.imp_ psi'✝) case and_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.and_ psi✝) ↔ Holds D I V' E (phi'✝.and_ psi'✝) case or_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.or_ psi✝) ↔ Holds D I V' E (phi'✝.or_ psi'✝) case iff_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.iff_ psi✝) ↔ Holds D I V' E (phi'✝.iff_ psi'✝) case forall_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName x✝ : VarName phi✝ phi'✝ : Formula a✝ : IsSubAux (Function.updateITE σ✝ x✝ x✝) (binders✝ ∪ {x✝}) phi✝ phi'✝ a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝ ∪ {x✝}, V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ (v : VarName), Function.updateITE σ✝ x✝ x✝ v ∉ binders✝ ∪ {x✝} → V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ v ∈ binders✝ ∪ {x✝}, v = Function.updateITE σ✝ x✝ x✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (forall_ x✝ phi✝) ↔ Holds D I V' E (forall_ x✝ phi'✝) case exists_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName x✝ : VarName phi✝ phi'✝ : Formula a✝ : IsSubAux (Function.updateITE σ✝ x✝ x✝) (binders✝ ∪ {x✝}) phi✝ phi'✝ a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝ ∪ {x✝}, V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ (v : VarName), Function.updateITE σ✝ x✝ x✝ v ∉ binders✝ ∪ {x✝} → V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ v ∈ binders✝ ∪ {x✝}, v = Function.updateITE σ✝ x✝ x✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (exists_ x✝ phi✝) ↔ Holds D I V' E (exists_ x✝ phi'✝) case def_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName X✝ : DefName xs✝ : List VarName a✝ : ∀ v ∈ xs✝, v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (def_ X✝ xs✝) ↔ Holds D I V' E (def_ X✝ (List.map σ✝ xs✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	case true_ \| false_ => simp only [Holds]	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E false_ ↔ Holds D I V' E false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	case not_ σ' binders' phi phi' _ ih_2 => simp only [Holds] congr! 1 exact ih_2 V V' h2 h3 h4	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux σ' binders' phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E phi.not_ ↔ Holds D I V' E phi'.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [Holds]	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E (pred_var_ X' xs') ↔ Holds D I V' E (pred_var_ X' (List.map σ' xs'))	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map V' (List.map σ' xs'))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map V' (List.map σ' xs'))	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map (V' ∘ σ') xs')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	congr! 1	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map (V' ∘ σ') xs')	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ List.map V xs' = List.map (V' ∘ σ') xs'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [List.map_eq_map_iff]	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ List.map V xs' = List.map (V' ∘ σ') xs'	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ ∀ x ∈ xs', V x = (V' ∘ σ') x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	intro x a1	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ ∀ x ∈ xs', V x = (V' ∘ σ') x	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' ⊢ V x = (V' ∘ σ') x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' ⊢ V x = (V' ∘ σ') x	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' ⊢ V x = V' (σ' x)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	by_cases c1 : x ∈ binders'	case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' ⊢ V x = V' (σ' x)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' c1 : x ∈ binders' ⊢ V x = V' (σ' x) case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' c1 : x ∉ binders' ⊢ V x = V' (σ' x)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact h2 x c1	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' c1 : x ∈ binders' ⊢ V x = V' (σ' x)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply h3	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' c1 : x ∉ binders' ⊢ V x = V' (σ' x)	case neg.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' c1 : x ∉ binders' ⊢ σ' x ∉ binders'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact ih_1 x a1 c1	case neg.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' c1 : x ∉ binders' ⊢ σ' x ∉ binders'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [Holds]	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E (eq_ x y) ↔ Holds D I V' E (eq_ (σ' x) (σ' y))	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ V x = V y ↔ V' (σ' x) = V' (σ' y)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	congr! 1	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ V x = V y ↔ V' (σ' x) = V' (σ' y)	case a.h.e'_2 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ V x = V' (σ' x) case a.h.e'_3 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ V y = V' (σ' y)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	by_cases c1 : x ∈ binders'	case a.h.e'_2 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ V x = V' (σ' x)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∈ binders' ⊢ V x = V' (σ' x) case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ V x = V' (σ' x)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact h2 x c1	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∈ binders' ⊢ V x = V' (σ' x)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply h3	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ V x = V' (σ' x)	case neg.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ σ' x ∉ binders'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply ih_1	case neg.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ σ' x ∉ binders'	case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ x = x ∨ x = y case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ x ∉ binders'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp	case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ x = x ∨ x = y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact c1	case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : x ∉ binders' ⊢ x ∉ binders'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	by_cases c1 : y ∈ binders'	case a.h.e'_3 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ V y = V' (σ' y)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∈ binders' ⊢ V y = V' (σ' y) case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ V y = V' (σ' y)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact h2 y c1	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∈ binders' ⊢ V y = V' (σ' y)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply h3	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ V y = V' (σ' y)	case neg.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ σ' y ∉ binders'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply ih_1	case neg.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ σ' y ∉ binders'	case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ y = x ∨ y = y case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ y ∉ binders'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp	case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ y = x ∨ y = y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact c1	case neg.a.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x y : VarName ih_1 : ∀ (v : VarName), v = x ∨ v = y → v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v c1 : y ∉ binders' ⊢ y ∉ binders'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [Holds]	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E false_ ↔ Holds D I V' E false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [Holds]	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux σ' binders' phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E phi.not_ ↔ Holds D I V' E phi'.not_	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux σ' binders' phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ ¬Holds D I V E phi ↔ ¬Holds D I V' E phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	congr! 1	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux σ' binders' phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ ¬Holds D I V E phi ↔ ¬Holds D I V' E phi'	case a.h.e'_1.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux σ' binders' phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E phi ↔ Holds D I V' E phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact ih_2 V V' h2 h3 h4	case a.h.e'_1.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux σ' binders' phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E phi ↔ Holds D I V' E phi'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [Holds]	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux σ' binders' phi phi' a✝ : IsSubAux σ' binders' psi psi' phi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E (phi.iff_ psi) ↔ Holds D I V' E (phi'.iff_ psi')	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux σ' binders' phi phi' a✝ : IsSubAux σ' binders' psi psi' phi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ (Holds D I V E phi ↔ Holds D I V E psi) ↔ (Holds D I V' E phi' ↔ Holds D I V' E psi')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	congr! 1	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux σ' binders' phi phi' a✝ : IsSubAux σ' binders' psi psi' phi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ (Holds D I V E phi ↔ Holds D I V E psi) ↔ (Holds D I V' E phi' ↔ Holds D I V' E psi')	case a.h.e'_1.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux σ' binders' phi phi' a✝ : IsSubAux σ' binders' psi psi' phi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E phi ↔ Holds D I V' E phi' case a.h.e'_2.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux σ' binders' phi phi' a✝ : IsSubAux σ' binders' psi psi' phi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E psi ↔ Holds D I V' E psi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply phi_ih_2 V V' h2 h3 h4	case a.h.e'_1.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux σ' binders' phi phi' a✝ : IsSubAux σ' binders' psi psi' phi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E phi ↔ Holds D I V' E phi'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply psi_ih_2 V V' h2 h3 h4	case a.h.e'_2.a D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux σ' binders' phi phi' a✝ : IsSubAux σ' binders' psi psi' phi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E psi ↔ Holds D I V' E psi'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp at ih_2	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders' ∪ {x}, V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' ∪ {x} → V v = V' (Function.updateITE σ' x x v)) → (∀ v ∈ binders' ∪ {x}, v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	have s1 : ∀ (v : VarName), ¬ v = x → v ∈ binders' → ¬ σ' v = x	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') ⊢ ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	intro v a1 a2 contra	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') ⊢ ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : ¬v = x a2 : v ∈ binders' contra : σ' v = x ⊢ False D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply a1	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : ¬v = x a2 : v ∈ binders' contra : σ' v = x ⊢ False D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : ¬v = x a2 : v ∈ binders' contra : σ' v = x ⊢ v = x D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [← contra]	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : ¬v = x a2 : v ∈ binders' contra : σ' v = x ⊢ v = x D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : ¬v = x a2 : v ∈ binders' contra : σ' v = x ⊢ v = σ' v D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	exact h4 v a2	case s1 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : ¬v = x a2 : v ∈ binders' contra : σ' v = x ⊢ v = σ' v D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	simp only [Holds]	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ (∃ d, Holds D I (Function.updateITE V x d) E phi) ↔ ∃ d, Holds D I (Function.updateITE V' x d) E phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	first \| apply forall_congr'\| apply exists_congr	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ (∃ d, Holds D I (Function.updateITE V x d) E phi) ↔ ∃ d, Holds D I (Function.updateITE V' x d) E phi'	case h D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	intro d	case h D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'	case h D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ Holds D I (Function.updateITE V x d) E phi ↔ Holds D I (Function.updateITE V' x d) E phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply ih_2	case h D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ Holds D I (Function.updateITE V x d) E phi ↔ Holds D I (Function.updateITE V' x d) E phi'	case h.h2 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ ∀ (v : VarName), v ∈ binders' ∨ v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v) case h.h3 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ ∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v) case h.h4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ ∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply forall_congr'	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) E phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V' x d) E phi'	case h D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	apply exists_congr	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ (∃ d, Holds D I (Function.updateITE V x d) E phi) ↔ ∃ d, Holds D I (Function.updateITE V' x d) E phi'	case h D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux	[129, 1]	[273, 17]	intro v a1	case h.h2 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ ∀ (v : VarName), v ∈ binders' ∨ v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case h.h2 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊢ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)

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