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/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | apply PrefixAppend | case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β)
ih :
β m < n1β + n2β + 1,
β (e : PE V_N V_T),
I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case star_repetition
V_N V_T : Type
R : V_N β PE V_N V_T
eβ : PE V_N V_T
xs_1β xs_2β ysβ : List V_T
n1β n2β : β
aβΒΉ : Interpretation V_N V_T R (eβ, xs_1β ++ xs_2β ++ ysβ) (n1β, some xs_1β)
aβ : Interpretation V_N V_T R (eβ.star, xs_2β ++ ysβ) (n2β, some xs_2β... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n _ (R A) | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ ys.IsPrefix xs | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpreta... | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ ys.IsPrefix xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih : β m < n + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
β’ n < n + 1
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_1 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
A : V_N
n : β
ih_1 : Interpretation V_N V_T R (R A, xs) (n, some ys)
ih : Interpretation V_N V_T R (R A, xs) (n, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | specialize ih n2 _ e2 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ ys.IsPrefix xs | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ n2 < n1 + n2 + 1
V_N ... | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretat... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | omega | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
β’ n2 < n1 + n2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih : β m < n1 + n2 + 1, β (e : PE V_N V_T), Interpretation V_N V_T R (e, xs) (m, some ys) β ys.IsPrefix xs
ih_2 : Interpretat... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/PEG.lean | InterpretationPrefix | [264, 1] | [285, 20] | exact ih ih_2 | V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) β ys.IsPrefix xs
β’ ys.IsPrefix xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V_N V_T : Type
R : V_N β PE V_N V_T
xs ys : List V_T
e1 e2 : PE V_N V_T
n1 n2 : β
ih_1 : Interpretation V_N V_T R (e1, xs) (n1, none)
ih_2 : Interpretation V_N V_T R (e2, xs) (n2, some ys)
ih : Interpretation V_N V_T R (e2, xs) (n2, some ys) β ys.IsPrefix xs
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | induction F | F : Formula
Ο : PredName β PredName
h1 : F.predVarSet = β
β’ sub Ο F = F | case pred_const_
Ο : PredName β PredName
aβΒΉ : PredName
aβ : List VarName
h1 : (pred_const_ aβΒΉ aβ).predVarSet = β
β’ sub Ο (pred_const_ aβΒΉ aβ) = pred_const_ aβΒΉ aβ
case pred_var_
Ο : PredName β PredName
aβΒΉ : PredName
aβ : List VarName
h1 : (pred_var_ aβΒΉ aβ).predVarSet = β
β’ sub Ο (pred_var_ aβΒΉ aβ) = pred_var_ aβΒΉ ... | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β PredName
h1 : F.predVarSet = β
β’ sub Ο F = F
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case pred_const_ X xs =>
simp only [sub] | Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = β
β’ sub Ο (pred_const_ X xs) = pred_const_ X xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = β
β’ sub Ο (pred_const_ X xs) = pred_const_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case pred_var_ X xs =>
simp only [predVarSet] at h1
simp at h1 | Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = β
β’ sub Ο (pred_var_ X xs) = pred_var_ X xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = β
β’ sub Ο (pred_var_ X xs) = pred_var_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case eq_ x y =>
simp only [sub] | Ο : PredName β PredName
x y : VarName
h1 : (eq_ x y).predVarSet = β
β’ sub Ο (eq_ x y) = eq_ x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
x y : VarName
h1 : (eq_ x y).predVarSet = β
β’ sub Ο (eq_ x y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case true_ | false_ =>
simp only [sub] | Ο : PredName β PredName
h1 : false_.predVarSet = β
β’ sub Ο false_ = false_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
h1 : false_.predVarSet = β
β’ sub Ο false_ = false_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case not_ phi phi_ih =>
simp only [predVarSet] at h1
simp only [sub]
congr!
exact phi_ih h1 | Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.not_.predVarSet = β
β’ sub Ο phi.not_ = phi.not_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.not_.predVarSet = β
β’ sub Ο phi.not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [predVarSet] at h1
simp only [sub]
congr!
exact phi_ih h1 | Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : (exists_ x phi).predVarSet = β
β’ sub Ο (exists_ x phi) = exists_ x phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : (exists_ x phi).predVarSet = β
β’ sub Ο (exists_ x phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case def_ X xs =>
simp only [sub] | Ο : PredName β PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = β
β’ sub Ο (def_ X xs) = def_ X xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = β
β’ sub Ο (def_ X xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = β
β’ sub Ο (pred_const_ X xs) = pred_const_ X xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = β
β’ sub Ο (pred_const_ X xs) = pred_const_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = β
β’ sub Ο (pred_var_ X xs) = pred_var_ X xs | Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = β
β’ sub Ο (pred_var_ X xs) = pred_var_ X xs | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = β
β’ sub Ο (pred_var_ X xs) = pred_var_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp at h1 | Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = β
β’ sub Ο (pred_var_ X xs) = pred_var_ X xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = β
β’ sub Ο (pred_var_ X xs) = pred_var_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | Ο : PredName β PredName
x y : VarName
h1 : (eq_ x y).predVarSet = β
β’ sub Ο (eq_ x y) = eq_ x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
x y : VarName
h1 : (eq_ x y).predVarSet = β
β’ sub Ο (eq_ x y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | Ο : PredName β PredName
h1 : false_.predVarSet = β
β’ sub Ο false_ = false_ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
h1 : false_.predVarSet = β
β’ sub Ο false_ = false_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.not_.predVarSet = β
β’ sub Ο phi.not_ = phi.not_ | Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi.not_ = phi.not_ | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.not_.predVarSet = β
β’ sub Ο phi.not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi.not_ = phi.not_ | Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ (sub Ο phi).not_ = phi.not_ | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi.not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ (sub Ο phi).not_ = phi.not_ | case h.e'_1
Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi = phi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ (sub Ο phi).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1 | case h.e'_1
Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi = phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
Ο : PredName β PredName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : (phi.iff_ psi).predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : phi.predVarSet βͺ psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : (phi.iff_ psi).predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [Finset.union_eq_empty] at h1 | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : phi.predVarSet βͺ psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : phi.predVarSet = β
β§ psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : phi.predVarSet βͺ psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | cases h1 | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : phi.predVarSet = β
β§ psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi | case intro
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
leftβ : phi.predVarSet = β
rightβ : psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1 : phi.predVarSet = β
β§ psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ (sub Ο phi).iff_ (sub Ο psi) = phi.iff_ psi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ sub Ο (phi.iff_ psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ (sub Ο phi).iff_ (sub Ο psi) = phi.iff_ psi | case h.e'_1
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ sub Ο phi = phi
case h.e'_2
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = ... | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ (sub Ο phi).iff_ (sub Ο psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1_left | case h.e'_1
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ sub Ο phi = phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ sub Ο phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact psi_ih h1_right | case h.e'_2
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ sub Ο psi = psi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
Ο : PredName β PredName
phi psi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
psi_ih : psi.predVarSet = β
β sub Ο psi = psi
h1_left : phi.predVarSet = β
h1_right : psi.predVarSet = β
β’ sub Ο psi = psi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : (exists_ x phi).predVarSet = β
β’ sub Ο (exists_ x phi) = exists_ x phi | Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο (exists_ x phi) = exists_ x phi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : (exists_ x phi).predVarSet = β
β’ sub Ο (exists_ x phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο (exists_ x phi) = exists_ x phi | Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ exists_ x (sub Ο phi) = exists_ x phi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο (exists_ x phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ exists_ x (sub Ο phi) = exists_ x phi | case h.e'_2
Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi = phi | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ exists_ x (sub Ο phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1 | case h.e'_2
Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi = phi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
Ο : PredName β PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = β
β sub Ο phi = phi
h1 : phi.predVarSet = β
β’ sub Ο phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | Ο : PredName β PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = β
β’ sub Ο (def_ X xs) = def_ X xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ο : PredName β PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = β
β’ sub Ο (def_ X xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | induction E generalizing F V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β PredName
F : Formula
β’ Holds D I V E (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (Ο P) []) else I.pred_var_ P ds }
V E F | case nil
D : Type
I : Interpretation D
Ο : PredName β PredName
V : VarAssignment D
F : Formula
β’ Holds D I V [] (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (Ο P) []) else I.pred_var_ P ds }
V [] F
case con... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
Ο : PredName β PredName
F : Formula
β’ Holds D I V E (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case nil.def_ X xs =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I V [] (sub Ο (def_ X xs)) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (Ο P) []) else I.pred_var_ P ds }
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I V [] (sub Ο (def_ X xs)) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case cons.def_ hd tl ih X xs =>
simp only [Holds] at ih
simp at ih
simp only [sub]
simp only [Holds]
split_ifs
case _ c1 =>
specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q
simp only [sub_no_predVar hd.q Ο hd.h2] at ih
apply ih
case _ c1 =>
specialize ih V (def_ X xs)
... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (Ο P) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ :=... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | induction F generalizing V | case cons
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D... | case cons.pred_const_
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] ... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pred_const_ X xs =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pred_var_ X xs =>
simp only [sub]
split_ifs
case pos c1 =>
simp only [Holds]
simp
simp only [if_pos c1]
case neg c1 =>
simp only [Holds]
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case eq_ x y =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case true_ | false_ =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case not_ phi phi_ih =>
simp only [Holds] at phi_ih
simp only [sub]
simp only [Holds]
congr! 1
apply phi_ih | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [Holds] at phi_ih
simp only [sub]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intros d
apply phi_ih | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | split_ifs | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | case pos
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pos c1 =>
simp only [Holds]
simp
simp only [if_pos c1] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case neg c1 =>
simp only [Holds]
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [if_pos c1] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [if_neg c1] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | case a.h.e'_1.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at psi_ih | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | case a.h.e'_1.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply psi_ih | case a.h.e'_2.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then H... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | intros d | case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply forall_congr' | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply exists_congr | D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tailβ... | case h
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
headβ : Definition
tailβ : List Definition
tail_ihβ :
β (V : VarAssignment D) (F : Formula),
Holds D I V tailβ (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I V [] (sub Ο (def_ X xs)) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (Ο P) []) else I.pred_var_ P ds }
... | D : Type
I : Interpretation D
Ο : PredName β PredName
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I V [] (def_ X xs) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (Ο P) []) else I.pred_var_ P ds }
V [] ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I V [] (sub Ο (def_ X xs)) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Ho... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I V [] (def_ X xs) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (Ο P) []) else I.pred_var_ P ds }
V [] ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
X : DefName
xs : List VarName
V : VarAssignment D
β’ Holds D I V [] (def_ X xs) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (Ο P) ... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ :=... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp at ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | split_ifs | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | case pos
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := f... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q
simp only [sub_no_predVar hd.q Ο hd.h2] at ih
apply ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih V (def_ X xs)
simp only [sub] at ih
exact ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name β§ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub Ο hd.q) β
Holds D
{ nonempty := β―, pred_const_ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub_no_predVar hd.q Ο hd.h2] at ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name β§ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub Ο hd.q) β
Holds D
{ nonempty := β―, pred_const_ ... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name β§ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D
{ nonempty := β―, pred_const_ := I.pre... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name β§ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name β§ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q β
Holds D
{ nonempty := β―, pred_const_ := I.pre... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name β§ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl h... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih V (def_ X xs) | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
ih :
Holds D I V tl (sub Ο (def_ X xs)) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
β (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub Ο F) β
Holds D
{ nonempty := β―, pr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] at ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
ih :
Holds D I V tl (sub Ο (def_ X xs)) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := ... | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
ih :
Holds D I V tl (sub Ο (def_ X xs)) β
Holds D
{ nonem... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | exact ih | D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) β
Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun P ds... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
Ο : PredName β PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : Β¬(X = hd.name β§ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) β
Holds D
{ nonempty := β―... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] at h1 | F : Formula
Ο : PredName β PredName
h1 : F.IsValid
β’ (sub Ο F).IsValid | F : Formula
Ο : PredName β PredName
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub Ο F).IsValid | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β PredName
h1 : F.IsValid
β’ (sub Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] | F : Formula
Ο : PredName β PredName
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub Ο F).IsValid | F : Formula
Ο : PredName β PredName
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub Ο F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β PredName
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ (sub Ο F).IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | intro D I V E | F : Formula
Ο : PredName β PredName
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub Ο F) | F : Formula
Ο : PredName β PredName
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
β’ Holds D I V E (sub Ο F) | Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο : PredName β PredName
h1 : β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub Ο F)
TACTIC:
|
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