Datasets:
pkuAI4M
/
lean_github

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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_ :=...
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 ...
Please generate a tactic in lean4 to solve the state. STATE: 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 : Li...
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 ...
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...
Please generate a tactic in lean4 to solve the state. STATE: 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 : Li...
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 ⊒...
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....
Please generate a tactic in lean4 to solve the state. STATE: 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 : Li...
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....
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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...
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....
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....
Please generate a tactic in lean4 to solve the state. STATE: 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...
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....
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....
Please generate a tactic in lean4 to solve the state. STATE: 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...
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....
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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...
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...
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.arg...
Please generate a tactic in lean4 to solve the state. STATE: 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 : Li...
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.arg...
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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...
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.arg...
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.arg...
Please generate a tactic in lean4 to solve the state. STATE: 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...
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.arg...
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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...
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.len...
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
Please generate a tactic in lean4 to solve the state. STATE: 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)...
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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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
Please generate a tactic in lean4 to solve the state. STATE: F : Formula Ο„ : PredName β†’ β„• β†’ List VarName Γ— Formula h1 : admits Ο„ F h2 : F.IsValid ⊒ (replace Ο„ F).IsValid TACTIC:
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)
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
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)
Please generate a tactic in lean4 to solve the state. STATE: 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]
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 => ...
Please generate a tactic in lean4 to solve the state. STATE: 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) TACTIC:
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 [← s1]
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 => ...
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 => ...
Please generate a tactic in lean4 to solve the state. STATE: 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‹―, pre...
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]
apply h2
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 => ...
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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‹―, pre...
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
induction F
v : VarName F : Formula ⊒ occursIn v F ↔ v ∈ F.varSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ occursIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).varSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ occursIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).varSet case eq_ v a✝¹ a✝ : VarName ⊒ occursIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_...
Please generate a tactic in lean4 to solve the state. STATE: v : VarName F : Formula ⊒ occursIn v F ↔ v ∈ F.varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
all_goals simp only [occursIn] simp only [Formula.varSet]
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ occursIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).varSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ occursIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).varSet case eq_ v a✝¹ a✝ : VarName ⊒ occursIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_...
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ a✝.toFinset case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ a✝.toFinset case eq_ v a✝¹ a✝ : VarName ⊒ v = a✝¹ ∨ v = a✝ ↔ v ∈ {a✝¹, a✝} case true_ v : VarName ⊒ False ↔ v ∈ βˆ… case false_ v : VarName ⊒ False ↔ v ∈ ...
Please generate a tactic in lean4 to solve the state. STATE: case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ occursIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).varSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ occursIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).varSet cas...
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case pred_const_ X xs | pred_var_ X xs | def_ X xs => simp
v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case eq_ x y => simp
v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case true_ | false_ => tauto
v : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case not_ phi phi_ih => tauto
v : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi ↔ v ∈ phi.varSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi ↔ v ∈ phi.varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case imp_ phi psi phi_ih psi_ih | and_ phi psi phi_ih psi_ih | or_ phi psi phi_ih psi_ih | iff_ phi psi phi_ih psi_ih => simp tauto
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet βˆͺ psi.varSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet βˆͺ psi.varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case forall_ x phi phi_ih | exists_ x phi phi_ih => simp tauto
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet βˆͺ {x}
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet βˆͺ {x} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp only [occursIn]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ occursIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).varSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).varSet
Please generate a tactic in lean4 to solve the state. STATE: case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ occursIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp only [Formula.varSet]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).varSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ a✝.toFinset
Please generate a tactic in lean4 to solve the state. STATE: case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi ↔ v ∈ phi.varSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi ↔ v ∈ phi.varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet βˆͺ psi.varSet
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet βˆͺ psi.varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet βˆͺ {x}
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∨ v = x
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet βˆͺ {x} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∨ v = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∨ v = x TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
induction F
v : VarName F : Formula ⊒ isBoundIn v F ↔ v ∈ F.boundVarSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isBoundIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).boundVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isBoundIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).boundVarSet case eq_ v a✝¹ a✝ : VarName ⊒ isBoundIn v (eq_ a✝¹ a...
Please generate a tactic in lean4 to solve the state. STATE: v : VarName F : Formula ⊒ isBoundIn v F ↔ v ∈ F.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
all_goals simp only [isBoundIn] simp only [Formula.boundVarSet]
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isBoundIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).boundVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isBoundIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).boundVarSet case eq_ v a✝¹ a✝ : VarName ⊒ isBoundIn v (eq_ a✝¹ a...
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ False ↔ v ∈ βˆ… case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ False ↔ v ∈ βˆ… case eq_ v a✝¹ a✝ : VarName ⊒ False ↔ v ∈ βˆ… case true_ v : VarName ⊒ False ↔ v ∈ βˆ… case false_ v : VarName ⊒ False ↔ v ∈ βˆ… case not_ v : VarName a✝ : Formula a_i...
Please generate a tactic in lean4 to solve the state. STATE: case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isBoundIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).boundVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isBoundIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).boun...
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case pred_const_ X xs | pred_var_ X xs | def_ X xs => simp
v : VarName X : DefName xs : List VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case eq_ x y => simp
v x y : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x y : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case true_ | false_ => tauto
v : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case not_ phi phi_ih => tauto
v : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ isBoundIn v phi ↔ v ∈ phi.boundVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ isBoundIn v phi ↔ v ∈ phi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case imp_ phi psi phi_ih psi_ih | and_ phi psi phi_ih psi_ih | or_ phi psi phi_ih psi_ih | iff_ phi psi phi_ih psi_ih => simp tauto
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊒ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet βˆͺ psi.boundVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊒ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet βˆͺ psi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case forall_ x phi phi_ih | exists_ x phi phi_ih => simp tauto
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet βˆͺ {x}
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet βˆͺ {x} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp only [isBoundIn]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ False ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
Please generate a tactic in lean4 to solve the state. STATE: case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp only [Formula.boundVarSet]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ False ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ False ↔ v ∈ βˆ…
Please generate a tactic in lean4 to solve the state. STATE: case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ False ↔ v ∈ (def_ a✝¹ a✝).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v : VarName X : DefName xs : List VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v x y : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x y : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ isBoundIn v phi ↔ v ∈ phi.boundVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ isBoundIn v phi ↔ v ∈ phi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊒ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet βˆͺ psi.boundVarSet
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊒ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊒ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet βˆͺ psi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊒ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊒ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet βˆͺ {x}
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet βˆͺ {x} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊒ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
induction F
v : VarName F : Formula ⊒ isFreeIn v F ↔ v ∈ F.freeVarSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isFreeIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isFreeIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).freeVarSet case eq_ v a✝¹ a✝ : VarName ⊒ isFreeIn v (eq_ a✝¹ a✝) ↔ ...
Please generate a tactic in lean4 to solve the state. STATE: v : VarName F : Formula ⊒ isFreeIn v F ↔ v ∈ F.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
all_goals simp only [isFreeIn] simp only [Formula.freeVarSet]
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isFreeIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isFreeIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).freeVarSet case eq_ v a✝¹ a✝ : VarName ⊒ isFreeIn v (eq_ a✝¹ a✝) ↔ ...
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ a✝.toFinset case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ a✝.toFinset case eq_ v a✝¹ a✝ : VarName ⊒ v = a✝¹ ∨ v = a✝ ↔ v ∈ {a✝¹, a✝} case true_ v : VarName ⊒ False ↔ v ∈ βˆ… case false_ v : VarName ⊒ False ↔ v ∈ ...
Please generate a tactic in lean4 to solve the state. STATE: case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isFreeIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊒ isFreeIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).freeVar...
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case pred_const_ X xs | pred_var_ X xs | def_ X xs => simp
v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case eq_ x y => simp
v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case true_ | false_ => tauto
v : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case not_ phi phi_ih => tauto
v : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ isFreeIn v phi ↔ v ∈ phi.freeVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ isFreeIn v phi ↔ v ∈ phi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case imp_ phi psi phi_ih psi_ih | and_ phi psi phi_ih psi_ih | or_ phi psi phi_ih psi_ih | iff_ phi psi phi_ih psi_ih => simp tauto
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊒ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet βˆͺ psi.freeVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊒ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet βˆͺ psi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case forall_ x phi phi_ih | exists_ x phi phi_ih => simp tauto
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ Β¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x}
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ Β¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp only [isFreeIn]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ isFreeIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
Please generate a tactic in lean4 to solve the state. STATE: case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ isFreeIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp only [Formula.freeVarSet]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ a✝.toFinset
Please generate a tactic in lean4 to solve the state. STATE: case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊒ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName ⊒ v ∈ xs ↔ v ∈ xs.toFinset TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x y : VarName ⊒ v = x ∨ v = y ↔ v ∈ {x, y} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v : VarName ⊒ False ↔ v ∈ βˆ…
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName ⊒ False ↔ v ∈ βˆ… TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ isFreeIn v phi ↔ v ∈ phi.freeVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ isFreeIn v phi ↔ v ∈ phi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊒ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet βˆͺ psi.freeVarSet
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊒ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊒ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet βˆͺ psi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊒ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊒ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ Β¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x}
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ Β¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet ∧ Β¬v = x
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ Β¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x} TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ Β¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet ∧ Β¬v = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊒ Β¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet ∧ Β¬v = x TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
induction F
v : VarName F : Formula h1 : isFreeIn v F ⊒ isFreeInInd v F
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_const_ a✝¹ a✝) ⊒ isFreeInInd v (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_var_ a✝¹ a✝) ⊒ isFreeInInd v (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName h1 : isFreeIn v (eq_ a✝¹...
Please generate a tactic in lean4 to solve the state. STATE: v : VarName F : Formula h1 : isFreeIn v F ⊒ isFreeInInd v F TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
any_goals simp only [isFreeIn] at h1
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_const_ a✝¹ a✝) ⊒ isFreeInInd v (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_var_ a✝¹ a✝) ⊒ isFreeInInd v (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName h1 : isFreeIn v (eq_ a✝¹...
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : v ∈ a✝ ⊒ isFreeInInd v (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : v ∈ a✝ ⊒ isFreeInInd v (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName h1 : v = a✝¹ ∨ v = a✝ ⊒ isFreeInInd v (eq_ a✝¹ a✝) case not_ v : VarNam...
Please generate a tactic in lean4 to solve the state. STATE: case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_const_ a✝¹ a✝) ⊒ isFreeInInd v (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_var_ a✝¹ a✝) ⊒ isFreeInInd v (pred_var_ ...
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case pred_const_ X xs | pred_var_ X xs | eq_ x y | def_ X xs => first | exact isFreeInInd.pred_const_ X xs h1 | exact isFreeInInd.pred_var_ X xs h1 | exact isFreeInInd.eq_ x y h1 | exact isFreeInInd.def_ X xs h1
v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (def_ X xs)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (def_ X xs) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case not_ phi phi_ih => apply isFreeInInd.not_ exact phi_ih h1
v : VarName phi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi h1 : isFreeIn v phi ⊒ isFreeInInd v phi.not_
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi h1 : isFreeIn v phi ⊒ isFreeInInd v phi.not_ TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case imp_ phi psi phi_ih psi_ih | and_ phi psi phi_ih psi_ih | or_ phi psi phi_ih psi_ih | iff_ phi psi phi_ih psi_ih => cases h1 case inl c1 => first | apply isFreeInInd.imp_left_ | apply isFreeInInd.and_left_ | apply isFreeInInd.or_left_ | apply isFreeInInd.iff_left_ exact phi_ih c1 case inr c...
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi h1 : isFreeIn v phi ∨ isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi h1 : isFreeIn v phi ∨ isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
simp only [isFreeIn] at h1
case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : isFreeIn v (def_ a✝¹ a✝) ⊒ isFreeInInd v (def_ a✝¹ a✝)
case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : v ∈ a✝ ⊒ isFreeInInd v (def_ a✝¹ a✝)
Please generate a tactic in lean4 to solve the state. STATE: case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : isFreeIn v (def_ a✝¹ a✝) ⊒ isFreeInInd v (def_ a✝¹ a✝) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
first | exact isFreeInInd.pred_const_ X xs h1 | exact isFreeInInd.pred_var_ X xs h1 | exact isFreeInInd.eq_ x y h1 | exact isFreeInInd.def_ X xs h1
v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (def_ X xs)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (def_ X xs) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.pred_const_ X xs h1
v : VarName X : PredName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (pred_const_ X xs)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : PredName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (pred_const_ X xs) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.pred_var_ X xs h1
v : VarName X : PredName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (pred_var_ X xs)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : PredName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (pred_var_ X xs) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.eq_ x y h1
v x y : VarName h1 : v = x ∨ v = y ⊒ isFreeInInd v (eq_ x y)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v x y : VarName h1 : v = x ∨ v = y ⊒ isFreeInInd v (eq_ x y) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.def_ X xs h1
v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (def_ X xs)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊒ isFreeInInd v (def_ X xs) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.not_
v : VarName phi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi h1 : isFreeIn v phi ⊒ isFreeInInd v phi.not_
case a v : VarName phi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi h1 : isFreeIn v phi ⊒ isFreeInInd v phi
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi h1 : isFreeIn v phi ⊒ isFreeInInd v phi.not_ TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact phi_ih h1
case a v : VarName phi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi h1 : isFreeIn v phi ⊒ isFreeInInd v phi
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a v : VarName phi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi h1 : isFreeIn v phi ⊒ isFreeInInd v phi TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
cases h1
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi h1 : isFreeIn v phi ∨ isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi)
case inl v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi h✝ : isFreeIn v phi ⊒ isFreeInInd v (phi.iff_ psi) case inr v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi h✝ : isFreeIn ...
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi h1 : isFreeIn v phi ∨ isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case inl c1 => first | apply isFreeInInd.imp_left_ | apply isFreeInInd.and_left_ | apply isFreeInInd.or_left_ | apply isFreeInInd.iff_left_ exact phi_ih c1
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.iff_ psi)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.iff_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case inr c1 => first | apply isFreeInInd.imp_right_ | apply isFreeInInd.and_right_ | apply isFreeInInd.or_right_ | apply isFreeInInd.iff_right_ exact psi_ih c1
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi)
no goals
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
first | apply isFreeInInd.imp_left_ | apply isFreeInInd.and_left_ | apply isFreeInInd.or_left_ | apply isFreeInInd.iff_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v phi
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.iff_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact phi_ih c1
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v phi
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v phi TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.imp_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.imp_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v phi
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.imp_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.and_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.and_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v phi
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.and_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.or_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.or_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v phi
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.or_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.iff_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v phi
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v phi ⊒ isFreeInInd v (phi.iff_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
first | apply isFreeInInd.imp_right_ | apply isFreeInInd.and_right_ | apply isFreeInInd.or_right_ | apply isFreeInInd.iff_right_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v psi ⊒ isFreeInInd v psi
Please generate a tactic in lean4 to solve the state. STATE: v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v psi ⊒ isFreeInInd v (phi.iff_ psi) TACTIC:
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact psi_ih c1
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v psi ⊒ isFreeInInd v psi
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi β†’ isFreeInInd v phi psi_ih : isFreeIn v psi β†’ isFreeInInd v psi c1 : isFreeIn v psi ⊒ isFreeInInd v psi TACTIC: