Datasets:
internlm
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Lean-Github

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https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
simp
case h1 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ X ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds }.pred_const_ = I.pred_const_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
simp only [predVarOccursIn_iff_mem_predVarSet]
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q β†’ ({ nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ X ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds }.pred_var_ P ds ↔ I.pred_var_ P ds)
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet β†’ ((if h : (Ο„ P ds.length).isSome = true then if ds.length = ((Ο„ P ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ P ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ P ds.length).get β‹―).2 else I.pred_var_ P ds else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
simp only [hd.h2]
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet β†’ ((if h : (Ο„ P ds.length).isSome = true then if ds.length = ((Ο„ P ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ P ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ P ds.length).get β‹―).2 else I.pred_var_ P ds else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ βˆ… β†’ ((if h : (Ο„ P ds.length).isSome = true then if ds.length = ((Ο„ P ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ P ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ P ds.length).get β‹―).2 else I.pred_var_ P ds else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
simp
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ βˆ… β†’ ((if h : (Ο„ P ds.length).isSome = true then if ds.length = ((Ο„ P ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ P ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ P ds.length).get β‹―).2 else I.pred_var_ P ds else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
apply Holds_coincide_PredVar
D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ X ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds } V tl (def_ X xs) ↔ Holds D I V tl (def_ X xs)
case h1 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ X ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds }.pred_const_ = I.pred_const_ case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β†’ ({ nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ X ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds }.pred_var_ P ds ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
simp
case h1 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ X ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds }.pred_const_ = I.pred_const_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
simp only [predVarOccursIn]
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β†’ ({ nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if h : (Ο„ X ds.length).isSome = true then if ds.length = ((Ο„ X ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ X ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ X ds.length).get β‹―).2 else I.pred_var_ X ds else I.pred_var_ X ds }.pred_var_ P ds ↔ I.pred_var_ P ds)
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), False β†’ ((if h : (Ο„ P ds.length).isSome = true then if ds.length = ((Ο„ P ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ P ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ P ds.length).get β‹―).2 else I.pred_var_ P ds else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean
FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux
[90, 1]
[226, 15]
simp
case h2 D : Type I : Interpretation D V' : VarAssignment D c : Char Ο„ : PredName β†’ β„• β†’ Option (List VarName Γ— Formula) X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V' x = V x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), False β†’ ((if h : (Ο„ P ds.length).isSome = true then if ds.length = ((Ο„ P ds.length).get β‹―).1.length then Holds D I (Function.updateListITE V' ((Ο„ P ds.length).get β‹―).1 ds) (hd :: tl) ((Ο„ P ds.length).get β‹―).2 else I.pred_var_ P ds else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
induction F generalizing binders V
D : Type I : Interpretation D V V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula binders : Finset VarName F : Formula h1 : admitsAux Ο„ binders F h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E F ↔ Holds D I V E (replace Ο„ F)
case pred_const_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ : PredName a✝ : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (pred_const_ a✝¹ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_const_ a✝¹ a✝) ↔ Holds D I V E (replace Ο„ (pred_const_ a✝¹ a✝)) case pred_var_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ : PredName a✝ : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (pred_var_ a✝¹ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ a✝¹ a✝) ↔ Holds D I V E (replace Ο„ (pred_var_ a✝¹ a✝)) case eq_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ a✝ : VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (eq_ a✝¹ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (eq_ a✝¹ a✝) ↔ Holds D I V E (replace Ο„ (eq_ a✝¹ a✝)) case true_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders true_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E true_ ↔ Holds D I V E (replace Ο„ true_) case false_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders false_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E false_ ↔ Holds D I V E (replace Ο„ false_) case not_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝ ↔ Holds D I V E (replace Ο„ a✝)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders a✝.not_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝.not_ ↔ Holds D I V E (replace Ο„ a✝.not_) case imp_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝¹ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝¹ ↔ Holds D I V E (replace Ο„ a✝¹)) a_ih✝ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝ ↔ Holds D I V E (replace Ο„ a✝)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (a✝¹.imp_ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (a✝¹.imp_ a✝) ↔ Holds D I V E (replace Ο„ (a✝¹.imp_ a✝)) case and_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝¹ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝¹ ↔ Holds D I V E (replace Ο„ a✝¹)) a_ih✝ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝ ↔ Holds D I V E (replace Ο„ a✝)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (a✝¹.and_ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (a✝¹.and_ a✝) ↔ Holds D I V E (replace Ο„ (a✝¹.and_ a✝)) case or_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝¹ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝¹ ↔ Holds D I V E (replace Ο„ a✝¹)) a_ih✝ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝ ↔ Holds D I V E (replace Ο„ a✝)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (a✝¹.or_ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (a✝¹.or_ a✝) ↔ Holds D I V E (replace Ο„ (a✝¹.or_ a✝)) case iff_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝¹ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝¹ ↔ Holds D I V E (replace Ο„ a✝¹)) a_ih✝ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝ ↔ Holds D I V E (replace Ο„ a✝)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (a✝¹.iff_ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (a✝¹.iff_ a✝) ↔ Holds D I V E (replace Ο„ (a✝¹.iff_ a✝)) case forall_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝ ↔ Holds D I V E (replace Ο„ a✝)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (forall_ a✝¹ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (forall_ a✝¹ a✝) ↔ Holds D I V E (replace Ο„ (forall_ a✝¹ a✝)) case exists_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders a✝ β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E a✝ ↔ Holds D I V E (replace Ο„ a✝)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (exists_ a✝¹ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (exists_ a✝¹ a✝) ↔ Holds D I V E (replace Ο„ (exists_ a✝¹ a✝)) case def_ D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula a✝¹ : DefName a✝ : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ a✝¹ a✝) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (def_ a✝¹ a✝) ↔ Holds D I V E (replace Ο„ (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
case pred_const_ X xs => simp only [replace] simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (pred_const_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_const_ X xs) ↔ Holds D I V E (replace Ο„ (pred_const_ X xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
case eq_ x y => simp only [replace] simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x y : VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (eq_ x y) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (eq_ x y) ↔ Holds D I V E (replace Ο„ (eq_ x y))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
case true_ | false_ => simp only [replace] simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders false_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E false_ ↔ Holds D I V E (replace Ο„ false_)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
case not_ phi phi_ih => simp only [admitsAux] at h1 simp only [replace] simp only [Holds] congr! 1 exact phi_ih V binders h1 h2
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi.not_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi.not_ ↔ Holds D I V E (replace Ο„ phi.not_)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
case forall_ x phi phi_ih | exists_ x phi phi_ih => simp only [admitsAux] at h1 simp only [replace] simp only [Holds] first | apply forall_congr' | apply exists_congr intro d apply phi_ih (Function.updateITE V x d) (binders βˆͺ {x}) h1 intro v a1 simp only [Function.updateITE] simp at a1 push_neg at a1 cases a1 case h.intro a1_left a1_right => simp only [if_neg a1_right] exact h2 v a1_left
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (exists_ x phi) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (exists_ x phi) ↔ Holds D I V E (replace Ο„ (exists_ x phi))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (pred_const_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_const_ X xs) ↔ Holds D I V E (replace Ο„ (pred_const_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (pred_const_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_const_ X xs) ↔ Holds D I V E (pred_const_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (pred_const_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_const_ X xs) ↔ Holds D I V E (pred_const_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [admitsAux] at h1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (pred_var_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 ∧ (βˆ€ x ∈ binders, Β¬(isFreeIn x (Ο„ X xs.length).2 ∧ x βˆ‰ (Ο„ X xs.length).1)) ∧ xs.length = (Ο„ X xs.length).1.length h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp at h1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 ∧ (βˆ€ x ∈ binders, Β¬(isFreeIn x (Ο„ X xs.length).2 ∧ x βˆ‰ (Ο„ X xs.length).1)) ∧ xs.length = (Ο„ X xs.length).1.length h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1 : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 ∧ (βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1) ∧ xs.length = (Ο„ X xs.length).1.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
cases h1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1 : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 ∧ (βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1) ∧ xs.length = (Ο„ X xs.length).1.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
case intro D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x left✝ : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 right✝ : (βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1) ∧ xs.length = (Ο„ X xs.length).1.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
cases h1_right
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right : (βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1) ∧ xs.length = (Ο„ X xs.length).1.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
case intro D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 left✝ : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 right✝ : xs.length = (Ο„ X xs.length).1.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
obtain s1 := Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id (Ο„ X xs.length).fst xs) (Ο„ X xs.length).snd h1_left
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (V ∘ Function.updateListITE id (Ο„ X xs.length).1 xs) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Function.updateListITE_comp] at s1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (V ∘ Function.updateListITE id (Ο„ X xs.length).1 xs) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE (V ∘ id) (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp at s1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE (V ∘ id) (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [s2] at s1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) s2 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s2 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
clear s2
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s2 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (pred_var_ X xs) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ (if (List.map V xs).length = (Ο„ X (List.map V xs).length).1.length then Holds D I (Function.updateListITE V' (Ο„ X (List.map V xs).length).1 (List.map V xs)) E (Ο„ X (List.map V xs).length).2 else I.pred_var_ X (List.map V xs)) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ (if (List.map V xs).length = (Ο„ X (List.map V xs).length).1.length then Holds D I (Function.updateListITE V' (Ο„ X (List.map V xs).length).1 (List.map V xs)) E (Ο„ X (List.map V xs).length).2 else I.pred_var_ X (List.map V xs)) ↔ Holds D I V E (replace Ο„ (pred_var_ X xs))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ (if (List.map V xs).length = (Ο„ X (List.map V xs).length).1.length then Holds D I (Function.updateListITE V' (Ο„ X (List.map V xs).length).1 (List.map V xs)) E (Ο„ X (List.map V xs).length).2 else I.pred_var_ X (List.map V xs)) ↔ Holds D I V E (if xs.length = (Ο„ X xs.length).1.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 else pred_var_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ (if (List.map V xs).length = (Ο„ X (List.map V xs).length).1.length then Holds D I (Function.updateListITE V' (Ο„ X (List.map V xs).length).1 (List.map V xs)) E (Ο„ X (List.map V xs).length).2 else I.pred_var_ X (List.map V xs)) ↔ Holds D I V E (if xs.length = (Ο„ X xs.length).1.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 else pred_var_ X xs)
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ (if xs.length = (Ο„ X xs.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 else I.pred_var_ X (List.map V xs)) ↔ Holds D I V E (if xs.length = (Ο„ X xs.length).1.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 else pred_var_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [if_pos h1_right_right]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ (if xs.length = (Ο„ X xs.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 else I.pred_var_ X (List.map V xs)) ↔ Holds D I V E (if xs.length = (Ο„ X xs.length).1.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 else pred_var_ X xs)
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
exact s1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply Holds_coincide_Var
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I (Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2
case h1 D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ βˆ€ (v : VarName), isFreeIn v (Ο„ X xs.length).2 β†’ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
intro v a1
case h1 D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) ⊒ βˆ€ (v : VarName), isFreeIn v (Ο„ X xs.length).2 β†’ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
case h1 D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
by_cases c1 : v ∈ (Ο„ X xs.length).fst
case h1 D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v case neg D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply Function.updateListITE_mem_eq_len V V' v (Ο„ X xs.length).fst (List.map V xs) c1
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ (Ο„ X xs.length).1.length = (List.map V xs).length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ (Ο„ X xs.length).1.length = (List.map V xs).length
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ (Ο„ X xs.length).1.length = xs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
symm
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ (Ο„ X xs.length).1.length = xs.length
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ xs.length = (Ο„ X xs.length).1.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
exact h1_right_right
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v ∈ (Ο„ X xs.length).1 ⊒ xs.length = (Ο„ X xs.length).1.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
by_cases c2 : v ∈ binders
case neg D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v ∈ binders ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v case neg D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v βˆ‰ binders ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
specialize h1_right_left v c2 a1
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v ∈ binders ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v ∈ binders h1_right_left : v ∈ (Ο„ X xs.length).1 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
contradiction
case pos D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v ∈ binders h1_right_left : v ∈ (Ο„ X xs.length).1 ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
specialize h2 v c2
case neg D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v βˆ‰ binders ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
case neg D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v βˆ‰ binders h2 : V v = V' v ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply Function.updateListITE_mem'
case neg D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v βˆ‰ binders h2 : V v = V' v ⊒ Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs) v = Function.updateListITE V' (Ο„ X xs.length).1 (List.map V xs) v
case neg.h1 D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v βˆ‰ binders h2 : V v = V' v ⊒ V v = V' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
exact h2
case neg.h1 D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : PredName xs : List VarName V : VarAssignment D binders : Finset VarName h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2 h1_right_left : βˆ€ x ∈ binders, isFreeIn x (Ο„ X xs.length).2 β†’ x ∈ (Ο„ X xs.length).1 h1_right_right : xs.length = (Ο„ X xs.length).1.length s1 : Holds D I (Function.updateListITE V (Ο„ X xs.length).1 (List.map V xs)) E (Ο„ X xs.length).2 ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id (Ο„ X xs.length).1 xs) (Ο„ X xs.length).2) v : VarName a1 : isFreeIn v (Ο„ X xs.length).2 c1 : v βˆ‰ (Ο„ X xs.length).1 c2 : v βˆ‰ binders h2 : V v = V' v ⊒ V v = V' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x y : VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (eq_ x y) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (eq_ x y) ↔ Holds D I V E (replace Ο„ (eq_ x y))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x y : VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (eq_ x y) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (eq_ x y) ↔ Holds D I V E (eq_ x y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x y : VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (eq_ x y) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (eq_ x y) ↔ Holds D I V E (eq_ x y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders false_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E false_ ↔ Holds D I V E (replace Ο„ false_)
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders false_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E false_ ↔ Holds D I V E false_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders false_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E false_ ↔ Holds D I V E false_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [admitsAux] at h1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi.not_ h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi.not_ ↔ Holds D I V E (replace Ο„ phi.not_)
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi.not_ ↔ Holds D I V E (replace Ο„ phi.not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi.not_ ↔ Holds D I V E (replace Ο„ phi.not_)
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi.not_ ↔ Holds D I V E (replace Ο„ phi).not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi.not_ ↔ Holds D I V E (replace Ο„ phi).not_
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Β¬Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Β¬Holds D I V E (replace Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
congr! 1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Β¬Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Β¬Holds D I V E (replace Ο„ phi)
case a.h.e'_1.a D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
exact phi_ih V binders h1 h2
case a.h.e'_1.a D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [admitsAux] at h1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (phi.iff_ psi) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (phi.iff_ psi) ↔ Holds D I V E (replace Ο„ (phi.iff_ psi))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi ∧ admitsAux Ο„ binders psi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (phi.iff_ psi) ↔ Holds D I V E (replace Ο„ (phi.iff_ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi ∧ admitsAux Ο„ binders psi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (phi.iff_ psi) ↔ Holds D I V E (replace Ο„ (phi.iff_ psi))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi ∧ admitsAux Ο„ binders psi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (phi.iff_ psi) ↔ Holds D I V E ((replace Ο„ phi).iff_ (replace Ο„ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi ∧ admitsAux Ο„ binders psi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (phi.iff_ psi) ↔ Holds D I V E ((replace Ο„ phi).iff_ (replace Ο„ psi))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi ∧ admitsAux Ο„ binders psi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi) ↔ (Holds D I V E (replace Ο„ phi) ↔ Holds D I V E (replace Ο„ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
cases h1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders phi ∧ admitsAux Ο„ binders psi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi) ↔ (Holds D I V E (replace Ο„ phi) ↔ Holds D I V E (replace Ο„ psi))
case intro D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x left✝ : admitsAux Ο„ binders phi right✝ : admitsAux Ο„ binders psi ⊒ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi) ↔ (Holds D I V E (replace Ο„ phi) ↔ Holds D I V E (replace Ο„ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
congr! 1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : admitsAux Ο„ binders phi h1_right : admitsAux Ο„ binders psi ⊒ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi) ↔ (Holds D I V E (replace Ο„ phi) ↔ Holds D I V E (replace Ο„ psi))
case a.h.e'_1.a D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : admitsAux Ο„ binders phi h1_right : admitsAux Ο„ binders psi ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi) case a.h.e'_2.a D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : admitsAux Ο„ binders phi h1_right : admitsAux Ο„ binders psi ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
exact phi_ih V binders h1_left h2
case a.h.e'_1.a D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : admitsAux Ο„ binders phi h1_right : admitsAux Ο„ binders psi ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
exact psi_ih V binders h1_right h2
case a.h.e'_2.a D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula phi psi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) psi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders psi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)) V : VarAssignment D binders : Finset VarName h2 : βˆ€ x βˆ‰ binders, V x = V' x h1_left : admitsAux Ο„ binders phi h1_right : admitsAux Ο„ binders psi ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E psi ↔ Holds D I V E (replace Ο„ psi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [admitsAux] at h1
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (exists_ x phi) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (exists_ x phi) ↔ Holds D I V E (replace Ο„ (exists_ x phi))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (exists_ x phi) ↔ Holds D I V E (replace Ο„ (exists_ x phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (exists_ x phi) ↔ Holds D I V E (replace Ο„ (exists_ x phi))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (exists_ x phi) ↔ Holds D I V E (exists_ x (replace Ο„ phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (exists_ x phi) ↔ Holds D I V E (exists_ x (replace Ο„ phi))
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ (βˆƒ d, Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V x d) E (replace Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
first | apply forall_congr' | apply exists_congr
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ (βˆƒ d, Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V x d) E (replace Ο„ phi)
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V x a) E (replace Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
intro d
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V x a) E (replace Ο„ phi)
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x d) E phi ↔ Holds D I (Function.updateITE V x d) E (replace Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply phi_ih (Function.updateITE V x d) (binders βˆͺ {x}) h1
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x d) E phi ↔ Holds D I (Function.updateITE V x d) E (replace Ο„ phi)
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D ⊒ βˆ€ x_1 βˆ‰ binders βˆͺ {x}, Function.updateITE V x d x_1 = V' x_1
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
intro v a1
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D ⊒ βˆ€ x_1 βˆ‰ binders βˆͺ {x}, Function.updateITE V x d x_1 = V' x_1
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders βˆͺ {x} ⊒ Function.updateITE V x d v = V' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Function.updateITE]
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders βˆͺ {x} ⊒ Function.updateITE V x d v = V' v
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders βˆͺ {x} ⊒ (if v = x then d else V v) = V' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp at a1
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders βˆͺ {x} ⊒ (if v = x then d else V v) = V' v
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders ∧ Β¬v = x ⊒ (if v = x then d else V v) = V' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
push_neg at a1
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders ∧ Β¬v = x ⊒ (if v = x then d else V v) = V' v
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders ∧ v β‰  x ⊒ (if v = x then d else V v) = V' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
cases a1
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1 : v βˆ‰ binders ∧ v β‰  x ⊒ (if v = x then d else V v) = V' v
case h.intro D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName left✝ : v βˆ‰ binders right✝ : v β‰  x ⊒ (if v = x then d else V v) = V' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
case h.intro a1_left a1_right => simp only [if_neg a1_right] exact h2 v a1_left
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1_left : v βˆ‰ binders a1_right : v β‰  x ⊒ (if v = x then d else V v) = V' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply forall_congr'
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ (βˆ€ (d : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x d) E phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V x d) E (replace Ο„ phi)
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V x a) E (replace Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply exists_congr
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ (βˆƒ d, Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V x d) E (replace Ο„ phi)
case h D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ βˆ€ (a : D), Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V x a) E (replace Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [if_neg a1_right]
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1_left : v βˆ‰ binders a1_right : v β‰  x ⊒ (if v = x then d else V v) = V' v
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1_left : v βˆ‰ binders a1_right : v β‰  x ⊒ V v = V' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
exact h2 v a1_left
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula x : VarName phi : Formula phi_ih : βˆ€ (V : VarAssignment D) (binders : Finset VarName), admitsAux Ο„ binders phi β†’ (βˆ€ x βˆ‰ binders, V x = V' x) β†’ (Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E phi ↔ Holds D I V E (replace Ο„ phi)) V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ (binders βˆͺ {x}) phi h2 : βˆ€ x βˆ‰ binders, V x = V' x d : D v : VarName a1_left : v βˆ‰ binders a1_right : v β‰  x ⊒ V v = V' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
cases E
D : Type I : Interpretation D V' : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) E (Ο„ X ds.length).2 else I.pred_var_ X ds } V E (def_ X xs) ↔ Holds D I V E (replace Ο„ (def_ X xs))
case nil D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) [] (Ο„ X ds.length).2 else I.pred_var_ X ds } V [] (def_ X xs) ↔ Holds D I V [] (replace Ο„ (def_ X xs)) case cons D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x head✝ : Definition tail✝ : List Definition ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (head✝ :: tail✝) (Ο„ X ds.length).2 else I.pred_var_ X ds } V (head✝ :: tail✝) (def_ X xs) ↔ Holds D I V (head✝ :: tail✝) (replace Ο„ (def_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
case nil => simp only [replace] simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) [] (Ο„ X ds.length).2 else I.pred_var_ X ds } V [] (def_ X xs) ↔ Holds D I V [] (replace Ο„ (def_ X xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) [] (Ο„ X ds.length).2 else I.pred_var_ X ds } V [] (def_ X xs) ↔ Holds D I V [] (replace Ο„ (def_ X xs))
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) [] (Ο„ X ds.length).2 else I.pred_var_ X ds } V [] (def_ X xs) ↔ Holds D I V [] (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) [] (Ο„ X ds.length).2 else I.pred_var_ X ds } V [] (def_ X xs) ↔ Holds D I V [] (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [replace]
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } V (hd :: tl) (def_ X xs) ↔ Holds D I V (hd :: tl) (replace Ο„ (def_ X xs))
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } V (hd :: tl) (def_ X xs) ↔ Holds D I V (hd :: tl) (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [Holds]
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } V (hd :: tl) (def_ X xs) ↔ Holds D I V (hd :: tl) (def_ X xs)
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition ⊒ (if X = hd.name ∧ xs.length = hd.args.length then Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateListITE V hd.args (List.map V xs)) tl hd.q else Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } V tl (def_ X xs)) ↔ 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)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
split_ifs
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition ⊒ (if X = hd.name ∧ xs.length = hd.args.length then Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateListITE V hd.args (List.map V xs)) tl hd.q else Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } V tl (def_ X xs)) ↔ 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)
case pos D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition h✝ : X = hd.name ∧ xs.length = hd.args.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q case neg D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition h✝ : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } V tl (def_ X xs) ↔ Holds D I V tl (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply Holds_coincide_PredVar
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
case h1 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_const_ = I.pred_const_ case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q β†’ ({ nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_var_ P ds ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp
case h1 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_const_ = I.pred_const_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [predVarOccursIn_iff_mem_predVarSet]
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q β†’ ({ nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_var_ P ds ↔ I.pred_var_ P ds)
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet β†’ ((if ds.length = (Ο„ P ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ P ds.length).1 ds) (hd :: tl) (Ο„ P ds.length).2 else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [hd.h2]
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet β†’ ((if ds.length = (Ο„ P ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ P ds.length).1 ds) (hd :: tl) (Ο„ P ds.length).2 else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ βˆ… β†’ ((if ds.length = (Ο„ P ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ P ds.length).1 ds) (hd :: tl) (Ο„ P ds.length).2 else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊒ βˆ€ (P : PredName) (ds : List D), (P, ds.length) ∈ βˆ… β†’ ((if ds.length = (Ο„ P ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ P ds.length).1 ds) (hd :: tl) (Ο„ P ds.length).2 else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
apply Holds_coincide_PredVar
D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds } V tl (def_ X xs) ↔ Holds D I V tl (def_ X xs)
case h1 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_const_ = I.pred_const_ case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β†’ ({ nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_var_ P ds ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp
case h1 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_const_ = I.pred_const_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp only [predVarOccursIn]
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) β†’ ({ nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (Ο„ X ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ X ds.length).1 ds) (hd :: tl) (Ο„ X ds.length).2 else I.pred_var_ X ds }.pred_var_ P ds ↔ I.pred_var_ P ds)
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), False β†’ ((if ds.length = (Ο„ P ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ P ds.length).1 ds) (hd :: tl) (Ο„ P ds.length).2 else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux
[109, 1]
[238, 15]
simp
case h2 D : Type I : Interpretation D V' : VarAssignment D Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux Ο„ binders (def_ X xs) h2 : βˆ€ x βˆ‰ binders, V x = V' x hd : Definition tl : List Definition c1 : Β¬(X = hd.name ∧ xs.length = hd.args.length) ⊒ βˆ€ (P : PredName) (ds : List D), False β†’ ((if ds.length = (Ο„ P ds.length).1.length then Holds D I (Function.updateListITE V' (Ο„ P ds.length).1 ds) (hd :: tl) (Ο„ P ds.length).2 else I.pred_var_ P ds) ↔ I.pred_var_ P ds)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem
[241, 1]
[266, 8]
apply substitution_theorem_aux D I V V E Ο„ βˆ… F
D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admits Ο„ F ⊒ Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (Ο„ X ds.length).1; let H := (Ο„ X ds.length).2; if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ X ds } V E F ↔ Holds D I V E (replace Ο„ F)
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admits Ο„ F ⊒ admitsAux Ο„ βˆ… F case h2 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admits Ο„ F ⊒ βˆ€ x βˆ‰ βˆ…, V x = V x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem
[241, 1]
[266, 8]
simp only [admits] at h1
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admits Ο„ F ⊒ admitsAux Ο„ βˆ… F
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admitsAux Ο„ βˆ… F ⊒ admitsAux Ο„ βˆ… F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem
[241, 1]
[266, 8]
exact h1
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admitsAux Ο„ βˆ… F ⊒ admitsAux Ο„ βˆ… F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem
[241, 1]
[266, 8]
intro X _
case h2 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admits Ο„ F ⊒ βˆ€ x βˆ‰ βˆ…, V x = V x
case h2 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admits Ο„ F X : VarName a✝ : X βˆ‰ βˆ… ⊒ V X = V X
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_theorem
[241, 1]
[266, 8]
rfl
case h2 D : Type I : Interpretation D V : VarAssignment D E : Env Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula F : Formula h1 : admits Ο„ F X : VarName a✝ : X βˆ‰ βˆ… ⊒ V X = V X
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_is_valid
[269, 1]
[282, 11]
simp only [IsValid] at h2
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : F.IsValid ⊒ (replace Ο„ F).IsValid
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ (replace Ο„ F).IsValid
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_is_valid
[269, 1]
[282, 11]
simp only [IsValid]
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ (replace Ο„ F).IsValid
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replace Ο„ F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_is_valid
[269, 1]
[282, 11]
intro D I V E
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊒ βˆ€ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replace Ο„ F)
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (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 (replace Ο„ F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/All/Rec/Sub.lean
FOL.NV.Sub.Pred.All.Rec.substitution_is_valid
[269, 1]
[282, 11]
obtain s1 := substitution_theorem D I V E Ο„ F h1
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (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 (replace Ο„ F)
F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : βˆ€ (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 s1 : Holds D { nonempty := β‹―, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (Ο„ X ds.length).1; let H := (Ο„ X ds.length).2; if ds.length = zs.length then Holds D I (Function.updateListITE V zs ds) E H else I.pred_var_ X ds } V E F ↔ Holds D I V E (replace Ο„ F) ⊒ Holds D I V E (replace Ο„ F)