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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact h2 x c3 | case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply h3 x | case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | apply ih_1 x a2 c3 | case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp only [List.length_map] at c2 | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | contradiction | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | simp at c2 | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | contradiction | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux | [129, 1] | [273, 17] | exact ih | D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β binders', V v = V' (Ο' v)
h3 : β (v : VarName), Ο' v β binders' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : VarName β VarName
binders : Finset VarName
F F' : Formula
Ο' : VarName β VarName
binders' : Finset VarName
X' : DefName
xs' : List VarName
ih_1 : β v β xs', v β binders' β Ο' v β binders'
V V' : VarAssignment D
h2 : β v β bin... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | apply substitution_theorem_aux D I (V β Ο) V E Ο β
F F' h1 | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ Holds D I (V β Ο) E F β Holds D I V E F' | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, (V β Ο) v = V (Ο v)
case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β (v : VarName), Ο v β β
β (V β Ο) v = V (Ο ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ Holds D I (V β Ο) E F β Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | simp | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, (V β Ο) v = V (Ο v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, (V β Ο) v = V (Ο v)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | simp | case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β (v : VarName), Ο v β β
β (V β Ο) v = V (Ο v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h3
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β (v : VarName), Ο v β β
β (V β Ο) v = V (Ο v)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_theorem | [276, 1] | [289, 9] | simp | case h4
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, v = Ο v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h4
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
β’ β v β β
, v = Ο v
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | simp only [IsValid] at h2 | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : F.IsValid
β’ F'.IsValid | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : F.IsValid
β’ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | simp only [IsValid] | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F' | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | intro D I V E | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F' | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (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
β’ Holds D I V E F' | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | simp only [β substitution_theorem D I V E Ο F F' h1] | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (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
β’ Holds D I V E F' | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (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
β’ Holds D I (V β Ο) E F | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (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
β’ Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Ind/Sub.lean | FOL.NV.Sub.Var.All.Ind.substitution_is_valid | [292, 1] | [304, 11] | apply h2 | Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (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
β’ Holds D I (V β Ο) E F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : VarName β VarName
F F' : Formula
h1 : IsSub Ο F F'
h2 : β (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
β’ Holds D I (V β Ο) E F
TACTIC:
|
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] | induction F generalizing binders V | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
... | case pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ aβΒΉ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty :=... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : 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] | case pred_const_ X xs =>
simp only [replace]
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
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, ... |
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 eq_ x y =>
simp only [replace]
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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 true_ | false_ =>
simp only [replace]
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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―,... |
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 not_ phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
congr! 1
exact phi_ih V binders h1 h2 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
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_ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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)... |
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 forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih (Function.updateITE V x d) (binders βͺ {x}) h1
intro v a1
simp only [Function.updateITE]
simp at a1
push_neg at a... | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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 : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.... | 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
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.... | Please generate a tactic in lean4 to solve the state.
STATE:
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
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, ... |
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
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, ... |
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 : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_var_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pr... | 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
if xs.length = ((Ο X xs.length).get β―).1.... | Please generate a tactic in lean4 to solve the state.
STATE:
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
h1 : admitsAux Ο binders (pred_var_ X xs)
h2 : β x β binders, V'... |
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 | 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
if xs.length = ((Ο X xs.length).get β―).1.... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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
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 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 _ c1 c2 =>
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 _ c1 =>
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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] | let opt := Ο X xs.length | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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] | let val := Option.get opt c1 | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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] | let zs := val.fst | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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] | let H := val.snd | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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.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] | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.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 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.le... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.le... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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.le... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.le... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.le... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.le... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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.... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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).i... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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).isS... |
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_... | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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―,... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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―,... |
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_ ... | Please generate a tactic in lean4 to solve the state.
STATE:
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)... |
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_ ... | Please generate a tactic in lean4 to solve the state.
STATE:
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)... |
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_ ... | Please generate a tactic in lean4 to solve the state.
STATE:
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)... |
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 :=... | Please generate a tactic in lean4 to solve the state.
STATE:
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)... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 bind... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = ... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = ... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = ... |
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 := ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = ... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 = ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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―, ... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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―, ... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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―, ... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 b... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 b... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 b... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 b... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 b... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 b... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 b... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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―, ... | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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,... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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’ Hol... |
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_,
... | Please generate a tactic in lean4 to solve the state.
STATE:
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’ Hol... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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’ Hol... |
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 :=... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : ... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : ... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : ... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 : ... |
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