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/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [predVarOccursIn_iff_mem_predVarSet] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [hd.h2] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.lengt... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | 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 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [predVarOccursIn] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V' x = V... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | induction F generalizing binders V | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds.length = (... | case pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β 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 := β―, pred_const_ :=... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | case pred_const_ X xs =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : 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
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | case eq_ x y =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : 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
Ο : PredName β β β List VarName Γ Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : 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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | case true_ | false_ =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | case not_ phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
congr! 1
exact phi_ih V binders h1 h2 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
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_ := I.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
Ο : PredName β β β 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 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih (Function.updateITE V x d) (binders 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
Ο : PredName β β β 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―, pred_const_ := I.p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : 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
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : 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
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : 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
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_var_ X xs)
h2 : 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
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2 β§
(β x β binders, Β¬(isFreeIn x (Ο X xs.length)... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2 β§
(β x β binders, Β¬(isFreeIn x (Ο X xs.length)... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1 :
Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2 β§
(β x β binder... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1 :
Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2 β§
(β x β binder... | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
leftβ : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
rightβ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1 :
Var.All.Rec.admits (Function.updateListITE i... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | cases h1_right | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right : (β x ... | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
leftβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | obtain s1 :=
Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id (Ο X xs.length).fst xs)
(Ο X xs.length).snd h1_left | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Function.updateListITE_comp] at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [s2] at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | clear s2 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [if_pos h1_right_right] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | apply Holds_coincide_Var | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : ... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListIT... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | intro v a1 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upda... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | by_cases c1 : v β (Ο X xs.length).fst | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upda... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | apply Function.updateListITE_mem_eq_len V V' v (Ο X xs.length).fst (List.map V xs) c1 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | symm | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | exact h1_right_right | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | 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
Ο : PredName β β β 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_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | by_cases c2 : v β binders | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | specialize h1_right_left v c2 a1 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | contradiction | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | 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
Ο : PredName β β β 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_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | specialize h2 v c2 | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_righ... | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : β x β binders, isFreeI... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V x = V' x
h1_left : Var.All.Rec.admits (Function.upd... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | apply Function.updateListITE_mem' | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : β x β binders, isFreeI... | case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : β x β binders, isFr... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | exact h2 | case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length).1 xs) (Ο X xs.length).2
h1_right_left : β x β binders, isFr... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1_left : Var.All.Rec.admits (Function.updateListITE id (Ο X xs.length... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : 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
Ο : PredName β β β List VarName Γ Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : 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
Ο : PredName β β β List VarName Γ Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : 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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : 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
Ο : PredName β β β List VarName Γ Formula
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : 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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if ds... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
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_ := I.pred_const_,
... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β 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_ := I.pred_const_,
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
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_ := I.pred_const_,
... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β 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_ := I.pred_const_,
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
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_ := I.pred_const_,
... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β 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_ := I.pred_const_,
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
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_ := I.pred_const_,
... | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
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
Ο : PredName β β β 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 ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | exact phi_ih V binders h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
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:
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V x = V' x) β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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_ := I.pred_cons... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β 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_const_ := I.pred_cons... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
(Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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_ := I.pred_cons... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β 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_const_ := I.pred_cons... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
(Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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_ := I.pred_cons... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β 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_const_ := I.pred_cons... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
(Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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_ := I.pred_cons... | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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
Ο : PredName β β β 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
(Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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_ := I.pred_cons... | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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
Ο : PredName β β β 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
(Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | exact phi_ih V binders h1_left h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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:
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V x = V' ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | exact psi_ih V binders h1_right h2 | case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
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:
case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
phi psi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V x = V' ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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
Ο : PredName β β β 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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | intro d | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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 h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 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
Ο : PredName β β β 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―, pred_const_... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | intro v a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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 h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Function.updateITE] | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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 h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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 h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | push_neg at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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 h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | cases a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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 h.intro
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | case h.intro a1_left a1_right =>
simp only [if_neg a1_right]
exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | apply forall_congr' | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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
Ο : PredName β β β 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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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
Ο : PredName β β β 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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [if_neg a1_right] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
x : VarName
phi : Formula
phi_ih :
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_ := I.p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
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
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | cases E | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_... | case nil
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : 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
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | case nil =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonem... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := ... | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonem... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
β’ Holds D
{ nonem... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
β’ Holds D
{ nonempty := β―, pred_const_ :=... | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
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
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : Li... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
β’ Holds D
{ nonempty := β―, pred_const_ :=... | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
β’ (if X = hd.name β§ xs.length = hd.args.length ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : Li... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
β’ (if X = hd.name β§ xs.length = hd.args.length ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
hβ : X = hd.name β§ xs.length = hd.args... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : Li... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.length
β’... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : Li... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [predVarOccursIn_iff_mem_predVarSet] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.... | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [hd.h2] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.... | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : X = hd.name β§ xs.length = hd.args.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.length = hd.arg... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : Li... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.length = hd.arg... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp only [predVarOccursIn] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.length = hd.arg... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.length = hd.arg... | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem_aux | [109, 1] | [238, 15] | simp | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition
tl : List Definition
c1 : Β¬(X = hd.name β§ xs.length = hd.arg... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
Ο : PredName β β β List VarName Γ Formula
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ X xs)
h2 : β x β binders, V x = V' x
hd : Definition... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem | [241, 1] | [266, 8] | apply substitution_theorem_aux D I V V E Ο β
F | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let zs := (Ο X ds.length).1;
let H := (Ο X ds.length).2;
if ds.len... | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ admitsAux Ο β
F
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ β x β β
, V x = V x | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let zs := (Ο X ds.length)... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem | [241, 1] | [266, 8] | simp only [admits] at h1 | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ admitsAux Ο β
F | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admitsAux Ο β
F
β’ admitsAux Ο β
F | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ admitsAux Ο β
F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem | [241, 1] | [266, 8] | exact h1 | case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admitsAux Ο β
F
β’ admitsAux Ο β
F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admitsAux Ο β
F
β’ admitsAux Ο β
F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem | [241, 1] | [266, 8] | intro X _ | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ β x β β
, V x = V x | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
X : VarName
aβ : X β β
β’ V X = V X | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
β’ β x β β
, V x = V x
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_theorem | [241, 1] | [266, 8] | rfl | case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
X : VarName
aβ : X β β
β’ V X = V X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β β β List VarName Γ Formula
F : Formula
h1 : admits Ο F
X : VarName
aβ : X β β
β’ V X = V X
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_is_valid | [269, 1] | [282, 11] | simp only [IsValid] at h2 | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : F.IsValid
β’ (replace Ο F).IsValid | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replace Ο F).IsValid | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : F.IsValid
β’ (replace Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_is_valid | [269, 1] | [282, 11] | simp only [IsValid] | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replace Ο F).IsValid | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο 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 (replace Ο F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (replace Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_is_valid | [269, 1] | [282, 11] | intro D I V E | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : 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 (replace Ο F) | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο 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 (replace Ο F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : 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 (replace Ο F)
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Sub.lean | FOL.NV.Sub.Pred.All.Rec.substitution_is_valid | [269, 1] | [282, 11] | obtain s1 := substitution_theorem D I V E Ο F h1 | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : 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 (replace Ο F) | F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο 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
s1 :
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β β β List VarName Γ Formula
h1 : admits Ο F
h2 : 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 (replace Ο F)
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.