url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact phi_ih h1 | case h.e'_2
P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ replace P zs H (def_ X xs) = def_ X xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P : PredName
zs : List VarName
H : Formula
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ replace P zs H (def_ X xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | set E_ref := E | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E P zs H) V E F ↔ Holds D I V E (replace P zs H F) | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := E
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E P zs H) V E F ↔ Holds D I V E (r... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | induction E generalizing F binders V | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := E
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F) | case nil
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
V : VarAssignment D
F : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := []
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
F : Formula
P : PredName
zs : List VarName
H : Formula
binders : Finset VarName
h1 : admitsAux P zs H binders F
h2 : ∀ x ∉ binders, V x = V' x
E_ref : Env := E
⊢ Holds D (I' D I V' E_ref P zs H) V E... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case nil.def_ X xs =>
simp only [replace]
simp only [E_ref]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Hol... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | induction F generalizing binders V | case cons
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail... | case cons.pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pred_const_ X xs =>
simp only [replace]
simp only [Holds]
simp only [I']
simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case eq_ x y =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case true_ | false_ =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case not_ phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
congr! 1
exact phi_ih V binders h1 h2 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1
intro v a1
simp only [Function.updateITE]
simp at a1
push_neg at a... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [I'] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [I'] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Interpretation.usingPred] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case neg c1 =>
split_ifs
case pos c2 =>
contradiction
case neg c2 =>
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Sub.Var.All.Rec.admits] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tai... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | have s1 :
Holds D I (V ∘ Function.updateListITE id zs xs) E_ref H ↔
Holds D I V E_ref (Sub.Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H) :=
by
exact Sub.Var.All.Rec.substitution_theorem D I V E_ref (Function.updateListITE id zs xs) H h1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Function.updateListITE_comp] at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [s2] at s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pos c2 =>
exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case neg _ =>
exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact Sub.Var.All.Rec.substitution_theorem D I V E_ref (Function.updateListITE id zs xs) H h1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Holds_coincide_Var | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | intro v a1 | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | by_cases c2 : v ∈ zs | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Function.updateListITE_mem_eq_len V V' v zs (List.map V xs) c2 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases c1 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | case pos.intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref :=... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pos.intro c1_left c1_right =>
simp
symm
exact c1_right | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | symm | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact c1_right | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | by_cases c3 : v ∈ binders | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | specialize h1_right v c3 a1 | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | contradiction | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Function.updateListITE_mem' | case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := ta... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact h2 v c3 | case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := ta... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact s1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case pos c2 =>
contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case neg c2 =>
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref :... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact phi_ih V binders h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binder... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tai... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref :... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact phi_ih V binders h1_left h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binder... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact psi_ih V binders h1_right h2 | case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binder... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | intro d | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | intro v a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Function.updateITE] | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | push_neg at a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | cases a1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h.intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := t... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | case h.intro a1_left a1_right =>
simp only [if_neg a1_right]
exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply forall_congr' | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [if_neg a1_right] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tail✝;
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Hol... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Hol... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [E_ref] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (def_ X xs) ↔ Hol... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' [] P zs H) V [] (def_ X xs) ↔ Holds D I... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' [] P zs H) V [] (def_ X xs) ↔ Holds D I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
E_ref : Env := []
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [E_ref] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, V x = V' x) →
let E_ref := tl;
Holds D (I' D ... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
admitsAux P zs H binders F →
(∀ x ∉ binders, ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [replace_no_predVar P zs H hd.q hd.h2] at ih | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ... | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | apply Holds_coincide_PredVar | D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = hd.name ... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [I'] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = ... | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [Interpretation.usingPred] | case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux | [188, 1] | [334, 13] | simp only [predVarOccursIn_iff_mem_predVarSet] | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = ... | case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (def_ X xs)
h2 : ∀ x ∉ binders, V x = V' x
c1 : X = ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
hd : Definition
tl : List Definition
E_ref : Env := hd :: tl
X : DefName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H... |
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