url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | obtain s1 := Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateListITE id zs xs) c H | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Function.updateListITE_comp] at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [s1] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply Holds_coincide_Var | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | intro v a1 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get ... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | by_cases c3 : v β zs | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply Function.updateListITE_mem_eq_len V' V v zs (List.map V xs) c3 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [β c2] | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Function.updateListITE_not_mem V v zs (List.map V xs) c3] | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Function.updateListITE_not_mem V' v zs (List.map V xs) c3] | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply h2 | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).g... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | intro contra | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).g... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).g... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).g... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).g... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Finset.eq_empty_iff_forall_not_mem] at h1 | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).g... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [c1] at h1 | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [β c2] at h1 | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp at h1 | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | specialize h1 v contra a1 | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | contradiction | case neg.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.lengt... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.len... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds ... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ ... | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty :=... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | exact phi_ih V binders h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty :=... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_con... | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempt... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | exact phi_ih V binders h1_left h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempt... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | exact psi_ih V binders h1_right h2 | case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempt... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | intro d | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply phi_ih (Function.updateITE V x d) (binders βͺ {x}) h1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | intro v a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Function.updateITE] | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | push_neg at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | cases a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... | case h.intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ no... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case h.intro a1_left a1_right =>
simp only [if_neg a1_right]
exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply forall_congr' | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [if_neg a1_right] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, ... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | cases E | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_con... | case nil
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_co... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case nil =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
β’ Holds D
{ nonempty :=... | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
β’ Holds D
{ nonempty :=... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
β’ Holds D
{ nonempty :=... | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
β’ (if X = hd.name β§ xs.length... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
β’ (if X = hd.name β§ xs.length... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
hβ : X = hd.name β§ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length ... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [predVarOccursIn_iff_mem_predVarSet] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [hd.h2] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.lengt... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [predVarOccursIn] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | ext cs | Ξ± : Type
R : RegExp Ξ±
β’ RegExp.languageOf Ξ± R.closure = RegExp.languageOf Ξ± (RegExp.epsilon.union (R.concat R.closure)) | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ cs β RegExp.languageOf Ξ± R.closure β cs β RegExp.languageOf Ξ± (RegExp.epsilon.union (R.concat R.closure)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | simp only [RegExp.languageOf] | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ cs β RegExp.languageOf Ξ± R.closure β cs β RegExp.languageOf Ξ± (RegExp.epsilon.union (R.concat R.closure)) | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ cs β {l | β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = l} β
cs β
{[]} βͺ
{x |
β r β RegExp.languageOf Ξ± R,
β s β {l | β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = l}, r ++ s = x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | simp | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ cs β {l | β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = l} β
cs β
{[]} βͺ
{x |
β r β RegExp.languageOf Ξ± R,
β s β {l | β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = l}, r ++ s = x} | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ (β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs) β
cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | constructor | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ (β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs) β
cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ (β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs) β
cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs
case h.mpr
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ (cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | intro a1 | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ (β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs) β
cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | apply Exists.elim a1 | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs
β’ β (a : List (List Ξ±)),
(β r β a, r β RegExp.languageOf Ξ± R) β§ a.join = cs β
cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | intro xs a2 | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs
β’ β (a : List (List Ξ±)),
(β r β a, r β RegExp.languageOf Ξ± R) β§ a.join = cs β
cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs
xs : List (List Ξ±)
a2 : (β r β xs, r β RegExp.languageOf Ξ± R) β§ xs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | clear a1 | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs
xs : List (List Ξ±)
a2 : (β r β xs, r β RegExp.languageOf Ξ± R) β§ xs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
xs : List (List Ξ±)
a2 : (β r β xs, r β RegExp.languageOf Ξ± R) β§ xs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | cases a2 | case h.mp
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
xs : List (List Ξ±)
a2 : (β r β xs, r β RegExp.languageOf Ξ± R) β§ xs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | case h.mp.intro
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
xs : List (List Ξ±)
leftβ : β r β xs, r β RegExp.languageOf Ξ± R
rightβ : xs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | case _ a2_left a2_right =>
simp only [β a2_right]
cases xs
case nil =>
left
simp
case cons hd tl =>
right
simp at a2_left
cases a2_left
case _ a2_left_left a2_left_right =>
apply Exists.intro hd
tauto | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
xs : List (List Ξ±)
a2_left : β r β xs, r β RegExp.languageOf Ξ± R
a2_right : xs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | simp only [β a2_right] | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
xs : List (List Ξ±)
a2_left : β r β xs, r β RegExp.languageOf Ξ± R
a2_right : xs.join = cs
β’ cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
xs : List (List Ξ±)
a2_left : β r β xs, r β RegExp.languageOf Ξ± R
a2_right : xs.join = cs
β’ xs.join = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = xs.join |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | cases xs | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
xs : List (List Ξ±)
a2_left : β r β xs, r β RegExp.languageOf Ξ± R
a2_right : xs.join = cs
β’ xs.join = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = xs.join | case nil
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a2_left : β r β [], r β RegExp.languageOf Ξ± R
a2_right : [].join = cs
β’ [].join = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = [].join
case cons
Ξ± : Type
R : RegExp Ξ±
cs headβ : List Ξ±
tailβ : List (List Ξ±)
a2_left : β r β headβ ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | case nil =>
left
simp | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a2_left : β r β [], r β RegExp.languageOf Ξ± R
a2_right : [].join = cs
β’ [].join = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = [].join | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | case cons hd tl =>
right
simp at a2_left
cases a2_left
case _ a2_left_left a2_left_right =>
apply Exists.intro hd
tauto | Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_left : β r β hd :: tl, r β RegExp.languageOf Ξ± R
a2_right : (hd :: tl).join = cs
β’ (hd :: tl).join = [] β¨
β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | left | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a2_left : β r β [], r β RegExp.languageOf Ξ± R
a2_right : [].join = cs
β’ [].join = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = [].join | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a2_left : β r β [], r β RegExp.languageOf Ξ± R
a2_right : [].join = cs
β’ [].join = [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | simp | case h
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a2_left : β r β [], r β RegExp.languageOf Ξ± R
a2_right : [].join = cs
β’ [].join = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | right | Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_left : β r β hd :: tl, r β RegExp.languageOf Ξ± R
a2_right : (hd :: tl).join = cs
β’ (hd :: tl).join = [] β¨
β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join | case h
Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_left : β r β hd :: tl, r β RegExp.languageOf Ξ± R
a2_right : (hd :: tl).join = cs
β’ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | simp at a2_left | case h
Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_left : β r β hd :: tl, r β RegExp.languageOf Ξ± R
a2_right : (hd :: tl).join = cs
β’ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join | case h
Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_right : (hd :: tl).join = cs
a2_left : hd β RegExp.languageOf Ξ± R β§ β a β tl, a β RegExp.languageOf Ξ± R
β’ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | cases a2_left | case h
Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_right : (hd :: tl).join = cs
a2_left : hd β RegExp.languageOf Ξ± R β§ β a β tl, a β RegExp.languageOf Ξ± R
β’ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join | case h.intro
Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_right : (hd :: tl).join = cs
leftβ : hd β RegExp.languageOf Ξ± R
rightβ : β a β tl, a β RegExp.languageOf Ξ± R
β’ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | case _ a2_left_left a2_left_right =>
apply Exists.intro hd
tauto | Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_right : (hd :: tl).join = cs
a2_left_left : hd β RegExp.languageOf Ξ± R
a2_left_right : β a β tl, a β RegExp.languageOf Ξ± R
β’ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | apply Exists.intro hd | Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_right : (hd :: tl).join = cs
a2_left_left : hd β RegExp.languageOf Ξ± R
a2_left_right : β a β tl, a β RegExp.languageOf Ξ± R
β’ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = (hd :: tl).join | Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_right : (hd :: tl).join = cs
a2_left_left : hd β RegExp.languageOf Ξ± R
a2_left_right : β a β tl, a β RegExp.languageOf Ξ± R
β’ hd β RegExp.languageOf Ξ± R β§ β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ hd ++ a.join = (hd :: tl).join |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | tauto | Ξ± : Type
R : RegExp Ξ±
cs hd : List Ξ±
tl : List (List Ξ±)
a2_right : (hd :: tl).join = cs
a2_left_left : hd β RegExp.languageOf Ξ± R
a2_left_right : β a β tl, a β RegExp.languageOf Ξ± R
β’ hd β RegExp.languageOf Ξ± R β§ β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ hd ++ a.join = (hd :: tl).join | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | intro a1 | case h.mpr
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
β’ (cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs) β
β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs | case h.mpr
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs
β’ β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | cases a1 | case h.mpr
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
a1 : cs = [] β¨ β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs
β’ β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs | case h.mpr.inl
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
hβ : cs = []
β’ β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs
case h.mpr.inr
Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
hβ : β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs
β’ β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.j... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | case _ left =>
apply Exists.intro []
simp
simp only [left] | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
left : cs = []
β’ β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | case _ right =>
apply Exists.elim right
intro xs a2
clear right
cases a2
case _ a2_left a2_right =>
apply Exists.elim a2_right
intro ys a3
clear a2_right
cases a3
case _ a3_left a3_right =>
apply Exists.intro ([xs] ++ ys)
simp
tauto | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
right : β r β RegExp.languageOf Ξ± R, β a, (β r β a, r β RegExp.languageOf Ξ± R) β§ r ++ a.join = cs
β’ β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | apply Exists.intro [] | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
left : cs = []
β’ β rs, (β r β rs, r β RegExp.languageOf Ξ± R) β§ rs.join = cs | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
left : cs = []
β’ (β r β [], r β RegExp.languageOf Ξ± R) β§ [].join = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | simp | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
left : cs = []
β’ (β r β [], r β RegExp.languageOf Ξ± R) β§ [].join = cs | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
left : cs = []
β’ [] = cs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExp.lean | equiv_language_of_closure | [36, 1] | [83, 18] | simp only [left] | Ξ± : Type
R : RegExp Ξ±
cs : List Ξ±
left : cs = []
β’ [] = cs | no goals |
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