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/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at psi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply psi_ih | case a.h.e'_2.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | intros d | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply forall_congr' | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply exists_congr | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝... | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
... | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (τ P) ... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ :=... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | split_ifs | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | case pos
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := f... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q
simp only [sub_no_predVar hd.q τ hd.h2] at ih
apply ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih V (def_ X xs)
simp only [sub] at ih
exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub τ hd.q) ↔
Holds D
{ nonempty := ⋯, pred_const_ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub_no_predVar hd.q τ hd.h2] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub τ hd.q) ↔
Holds D
{ nonempty := ⋯, pred_const_ ... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pre... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pre... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl h... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih V (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := ... | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (sub τ (def_ X xs)) ↔
Holds D
{ nonem... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] at h1 | F : Formula
τ : PredName → PredName
h1 : F.IsValid
⊢ (sub τ F).IsValid | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
τ : PredName → PredName
h1 : F.IsValid
⊢ (sub τ F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | intro D I V E | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F) | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [substitution_theorem D I V E τ F] | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F) | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ ... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | apply h1 | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | induction E generalizing F V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
u v : VarName
F : Formula
a : D
h1 : ¬isFreeIn v F
h2 : ¬isBoundIn v F
⊢ Holds D I (Function.updateITE V u a) E F ↔
Holds D I (Function.updateITE V v a) E (Sub.Var.One.Rec.fastReplaceFree u v F) | case nil
D : Type
I : Interpretation D
u v : VarName
a : D
V : VarAssignment D
F : Formula
h1 : ¬isFreeIn v F
h2 : ¬isBoundIn v F
⊢ Holds D I (Function.updateITE V u a) [] F ↔
Holds D I (Function.updateITE V v a) [] (Sub.Var.One.Rec.fastReplaceFree u v F)
case cons
D : Type
I : Interpretation D
u v : VarName
a : D... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
u v : VarName
F : Formula
a : D
h1 : ¬isFreeIn v F
h2 : ¬isBoundIn v F
⊢ Holds D I (Function.updateITE V u a) E F ↔
Holds D I (Function.updateITE V v a) E (Sub.Var.One.Rec.fastReplaceFree u v F)
TA... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case nil.def_ X xs =>
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) [] (def_ X xs) ↔
Holds D I (Function.updateITE V v a) [] (Sub.Var.One.Rec.fastReplaceFree u v (def_ X xs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) [] (def_ X xs) ↔
Holds D I (Function.updateITE V v a) [] ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | induction F generalizing V | case cons
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.... | case cons.pred_const_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case pred_const_ X xs | pred_var_ X xs =>
simp only [isFreeIn] at h1
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds]
congr! 1
simp
simp only [List.map_eq_map_iff]
intro x a1
simp
simp only [Function.updateITE]
simp only [eq_comm]
split_ifs
case _ c1 c2 =>
rfl
case _ c1 c2 =>
... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case true_ | false_ =>
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case not_ phi phi_ih =>
simp only [isFreeIn] at h1
simp only [isBoundIn] at h2
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds]
congr! 1
exact phi_ih V h1 h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | intro x a1 | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE] | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [eq_comm] | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case pos
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.R... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
subst c2
contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | subst c2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE] | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs <;> tauto | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE] | case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs <;> tauto | case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isBoundIn] at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Va... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact phi_ih V h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Va... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isBoundIn] at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case intro
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case intro
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Va... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact phi_ih V h1_left h2_left | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Va... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact psi_ih V h1_right h2_right | case a.h.e'_2.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Va... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isBoundIn] at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case intro
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | case pos
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.R... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case pos c1 =>
subst c1
apply Holds_coincide_Var
intro x a1
simp only [isFreeIn] at a1
cases a1
case h1.intro a1_left a1_right =>
simp only [Function.updateITE]
simp only [if_neg a1_left]
split_ifs
case pos c2 =>
subst c2
tauto
case neg c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | subst c1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastRe... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.