Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
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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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition ⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition ⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition h✝ : X = hd.name ∧ xs.length = hd.args.length ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition h✝ : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → ({ nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → ({ nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet → ((if ds.length = (τ P ds.length).1.length then Holds D I (Function.updateListITE V' (τ P ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ hd.q.predVarSet → ((if ds.length = (τ P ds.length).1.length then Holds D I (Function.updateListITE V' (τ P ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ ∅ → ((if ds.length = (τ P ds.length).1.length then Holds D I (Function.updateListITE V' (τ P ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : X = hd.name ∧ xs.length = hd.args.length ⊢ ∀ (P : PredName) (ds : List D), (P, ds.length) ∈ ∅ → ((if ds.length = (τ P ds.length).1.length then Holds D I (Function.updateListITE V' (τ P ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → ({ nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → ({ nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V' (τ X ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), False → ((if ds.length = (τ P ds.length).1.length then Holds D I (Function.updateListITE V' (τ P ds.length).1 ds) (hd :: tl) (τ 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 τ : PredName → ℕ → List VarName × Formula X : DefName xs : List VarName V : VarAssignment D binders : Finset VarName h1 : admitsAux τ binders (def_ X xs) h2 : ∀ x ∉ binders, V x = V' x hd : Definition tl : List Definition c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) ⊢ ∀ (P : PredName) (ds : List D), False → ((if ds.length = (τ P ds.length).1.length then Holds D I (Function.updateListITE V' (τ P ds.length).1 ds) (hd :: tl) (τ 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 τ ∅ F
D : Type I : Interpretation D V : VarAssignment D E : Env τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admits τ F ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (τ X ds.length).1; let H := (τ 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 τ F)
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admits τ F ⊢ admitsAux τ ∅ F case h2 D : Type I : Interpretation D V : VarAssignment D E : Env τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admits τ F ⊢ ∀ x ∉ ∅, 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 τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admits τ F ⊢ admitsAux τ ∅ F
case h1 D : Type I : Interpretation D V : VarAssignment D E : Env τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admitsAux τ ∅ F ⊢ admitsAux τ ∅ 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 τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admitsAux τ ∅ F ⊢ admitsAux τ ∅ 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 τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admits τ F ⊢ ∀ x ∉ ∅, V x = V x
case h2 D : Type I : Interpretation D V : VarAssignment D E : Env τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admits τ F X : VarName a✝ : X ∉ ∅ ⊢ 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 τ : PredName → ℕ → List VarName × Formula F : Formula h1 : admits τ F X : VarName a✝ : X ∉ ∅ ⊢ 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 τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : F.IsValid ⊢ (replace τ F).IsValid
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ (replace τ 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 τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ (replace τ F).IsValid
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replace τ 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 τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (replace τ F)
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I V E (replace τ 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 τ F h1
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I V E (replace τ F)
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env s1 : Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (τ X ds.length).1; let H := (τ 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 τ F) ⊢ Holds D I V E (replace τ 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]
simp only [← s1]
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env s1 : Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (τ X ds.length).1; let H := (τ 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 τ F) ⊢ Holds D I V E (replace τ F)
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env s1 : Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (τ X ds.length).1; let H := (τ 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 τ F) ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V (τ X ds.length).1 ds) E (τ X ds.length).2 else I.pred_var_ X ds } V E 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]
apply h2
F : Formula τ : PredName → ℕ → List VarName × Formula h1 : admits τ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env s1 : Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => let zs := (τ X ds.length).1; let H := (τ 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 τ F) ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun X ds => if ds.length = (τ X ds.length).1.length then Holds D I (Function.updateListITE V (τ X ds.length).1 ds) E (τ X ds.length).2 else I.pred_var_ X ds } V E F
no goals
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_ a✝¹ a✝).varSet case true_ v : VarName ⊢ occursIn v true_ ↔ v ∈ true_.varSet case false_ v : VarName ⊢ occursIn v false_ ↔ v ∈ false_.varSet case not_ v : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).varSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).varSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).varSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).varSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ occursIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).varSet
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_ a✝¹ a✝).varSet case true_ v : VarName ⊢ occursIn v true_ ↔ v ∈ true_.varSet case false_ v : VarName ⊢ occursIn v false_ ↔ v ∈ false_.varSet case not_ v : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).varSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).varSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).varSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).varSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ occursIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).varSet
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 ∈ ∅ case false_ v : VarName ⊢ False ↔ v ∈ ∅ case not_ v : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝ ↔ v ∈ a✝.varSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ v = a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝.varSet ∪ {a✝¹} case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ v = a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝.varSet ∪ {a✝¹} case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset
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
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
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 ∈ ∅
no goals
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
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 ∪ psi.varSet
no goals
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 ∪ {x}
no goals
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
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
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
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
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 ∈ ∅
no goals
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
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 ∪ 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
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
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 ∪ {x}
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∨ v = x
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
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✝) ↔ v ∈ (eq_ a✝¹ a✝).boundVarSet case true_ v : VarName ⊢ isBoundIn v true_ ↔ v ∈ true_.boundVarSet case false_ v : VarName ⊢ isBoundIn v false_ ↔ v ∈ false_.boundVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝.not_ ↔ v ∈ a✝.not_.boundVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).boundVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).boundVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).boundVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).boundVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
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✝) ↔ v ∈ (eq_ a✝¹ a✝).boundVarSet case true_ v : VarName ⊢ isBoundIn v true_ ↔ v ∈ true_.boundVarSet case false_ v : VarName ⊢ isBoundIn v false_ ↔ v ∈ false_.boundVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝.not_ ↔ v ∈ a✝.not_.boundVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).boundVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).boundVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).boundVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).boundVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ False ↔ v ∈ ∅ case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ False ↔ v ∈ ∅ case eq_ v a✝¹ a✝ : VarName ⊢ False ↔ v ∈ ∅ case true_ v : VarName ⊢ False ↔ v ∈ ∅ case false_ v : VarName ⊢ False ↔ v ∈ ∅ case not_ v : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ v = a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ∪ {a✝¹} case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ v = a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ∪ {a✝¹} case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ False ↔ v ∈ ∅
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 ∈ ∅
no goals
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 ∈ ∅
no goals
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 ∈ ∅
no goals
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
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 ∪ psi.boundVarSet
no goals
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 ∪ {x}
no goals
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
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 ∈ ∅
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 ∈ ∅
no goals
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 ∈ ∅
no goals
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 ∈ ∅
no goals
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
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 ∪ 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
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
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 ∪ {x}
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x
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
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✝) ↔ v ∈ (eq_ a✝¹ a✝).freeVarSet case true_ v : VarName ⊢ isFreeIn v true_ ↔ v ∈ true_.freeVarSet case false_ v : VarName ⊢ isFreeIn v false_ ↔ v ∈ false_.freeVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝.not_ ↔ v ∈ a✝.not_.freeVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).freeVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).freeVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).freeVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).freeVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).freeVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).freeVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isFreeIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
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✝) ↔ v ∈ (eq_ a✝¹ a✝).freeVarSet case true_ v : VarName ⊢ isFreeIn v true_ ↔ v ∈ true_.freeVarSet case false_ v : VarName ⊢ isFreeIn v false_ ↔ v ∈ false_.freeVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝.not_ ↔ v ∈ a✝.not_.freeVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).freeVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).freeVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).freeVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).freeVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).freeVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).freeVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isFreeIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
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 ∈ ∅ case false_ v : VarName ⊢ False ↔ v ∈ ∅ case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ ¬v = a✝¹ ∧ isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet \ {a✝¹} case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ ¬v = a✝¹ ∧ isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet \ {a✝¹} case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset
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
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
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 ∈ ∅
no goals
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
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 ∪ psi.freeVarSet
no goals
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 ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x}
no goals
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
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
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
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
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 ∈ ∅
no goals
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
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 ∪ 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
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
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 ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x}
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet ∧ ¬v = x
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 ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet ∧ ¬v = x
no goals
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 a✝) ⊢ isFreeInInd v (eq_ a✝¹ a✝) case true_ v : VarName h1 : isFreeIn v true_ ⊢ isFreeInInd v true_ case false_ v : VarName h1 : isFreeIn v false_ ⊢ isFreeInInd v false_ case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝.not_ ⊢ isFreeInInd v a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.imp_ a✝) ⊢ isFreeInInd v (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.and_ a✝) ⊢ isFreeInInd v (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.or_ a✝) ⊢ isFreeInInd v (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.iff_ a✝) ⊢ isFreeInInd v (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (forall_ a✝¹ a✝) ⊢ isFreeInInd v (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (exists_ a✝¹ a✝) ⊢ isFreeInInd v (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : isFreeIn v (def_ a✝¹ a✝) ⊢ isFreeInInd v (def_ a✝¹ a✝)
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 a✝) ⊢ isFreeInInd v (eq_ a✝¹ a✝) case true_ v : VarName h1 : isFreeIn v true_ ⊢ isFreeInInd v true_ case false_ v : VarName h1 : isFreeIn v false_ ⊢ isFreeInInd v false_ case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝.not_ ⊢ isFreeInInd v a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.imp_ a✝) ⊢ isFreeInInd v (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.and_ a✝) ⊢ isFreeInInd v (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.or_ a✝) ⊢ isFreeInInd v (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.iff_ a✝) ⊢ isFreeInInd v (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (forall_ a✝¹ a✝) ⊢ isFreeInInd v (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (exists_ a✝¹ a✝) ⊢ isFreeInInd v (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : isFreeIn v (def_ a✝¹ a✝) ⊢ isFreeInInd v (def_ a✝¹ a✝)
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 : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝ ⊢ isFreeInInd v a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : ¬v = a✝¹ ∧ isFreeIn v a✝ ⊢ isFreeInInd v (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : ¬v = a✝¹ ∧ isFreeIn v a✝ ⊢ isFreeInInd v (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : v ∈ a✝ ⊢ isFreeInInd v (def_ a✝¹ a✝)
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
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 → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi.not_
no goals
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 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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi h1 : isFreeIn v phi ∨ isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
no goals
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✝)
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
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
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
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
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
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 → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi.not_
case a v : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi
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 → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi
no goals
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → 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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi h✝ : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi) case inr v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi h✝ : isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi)
no goals
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
no goals
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
no goals
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.imp_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.and_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.or_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
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 → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
no goals