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/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case not_ phi phi_ih =>
simp only [Formula.primeSet] at h1
simp only [evalPrimeFfToNot] at phi_ih
simp only [evalPrimeFfToNot]
simp only [evalPrime]
simp
split_ifs
case _ c1 =>
simp only [c1] at phi_ih
simp at phi_ih
apply IsDeduct.mp_ phi
apply proof_imp_deduct
apply T_14_6
exact phi_ih h1
case _ c1 =>
simp only [c1] at phi_ih
simp at phi_ih
exact phi_ih h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.not_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.not_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case forall_ x phi phi_ih =>
let F := forall_ x phi
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
apply IsDeduct.assume_
simp
apply Exists.intro F
tauto
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case def_ X xs =>
let F := def_ X xs
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
apply IsDeduct.assume_
simp
apply Exists.intro F
tauto
|
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : ↑(def_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : ↑(def_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case and_ | or_ | iff_ | exists_ =>
sorry
|
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
let F := pred_const_ X xs
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.primeSet] at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_const_ X xs} ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_const_ X xs} ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_const_ X xs} ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot]
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_const_ X xs) then pred_const_ X xs else (pred_const_ X xs).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.evalPrime]
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_const_ X xs) then pred_const_ X xs else (pred_const_ X xs).not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_const_ X xs) then pred_const_ X xs else (pred_const_ X xs).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.assume_
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ (if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ (if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ (if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply Exists.intro F
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
tauto
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
let F := pred_var_ X xs
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.primeSet] at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_var_ X xs} ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_var_ X xs} ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_var_ X xs} ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot]
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_var_ X xs) then pred_var_ X xs else (pred_var_ X xs).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.evalPrime]
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_var_ X xs) then pred_var_ X xs else (pred_var_ X xs).not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_var_ X xs) then pred_var_ X xs else (pred_var_ X xs).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.assume_
|
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ (if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ (if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) = if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ (if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply Exists.intro F
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) = if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) = if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) = if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
tauto
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) = if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) = if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
let F := eq_ x y
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑(eq_ x y).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑(eq_ x y).primeSet ⊆ Δ_U
F : Formula := eq_ x y
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑(eq_ x y).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.primeSet] at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑(eq_ x y).primeSet ⊆ Δ_U
F : Formula := eq_ x y
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑{eq_ x y} ⊆ Δ_U
F : Formula := eq_ x y
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑(eq_ x y).primeSet ⊆ Δ_U
F : Formula := eq_ x y
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑{eq_ x y} ⊆ Δ_U
F : Formula := eq_ x y
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑{eq_ x y} ⊆ Δ_U
F : Formula := eq_ x y
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot]
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (eq_ x y) then eq_ x y else (eq_ x y).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.evalPrime]
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (eq_ x y) then eq_ x y else (eq_ x y).not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (eq_ x y) then eq_ x y else (eq_ x y).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.assume_
|
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ (if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ (if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ ∃ x_1 ∈ Δ_U, (if evalPrime V x_1 then x_1 else x_1.not_) = if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ (if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply Exists.intro F
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ ∃ x_1 ∈ Δ_U, (if evalPrime V x_1 then x_1 else x_1.not_) = if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ ∃ x_1 ∈ Δ_U, (if evalPrime V x_1 then x_1 else x_1.not_) = if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
tauto
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
F : Formula := eq_ x y
h1 : eq_ x y ∈ Δ_U
⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (eq_ x y) = true then eq_ x y else (eq_ x y).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.axiom_
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑true_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V true_)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑true_.primeSet ⊆ Δ_U
⊢ IsAxiom (evalPrimeFfToNot V true_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑true_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V true_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsAxiom.prop_true_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑true_.primeSet ⊆ Δ_U
⊢ IsAxiom (evalPrimeFfToNot V true_)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑true_.primeSet ⊆ Δ_U
⊢ IsAxiom (evalPrimeFfToNot V true_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.primeSet] at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑false_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑∅ ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑false_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑∅ ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑∅ ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot]
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V false_ then false_ else false_.not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.evalPrime]
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V false_ then false_ else false_.not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if False then false_ else false_.not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V false_ then false_ else false_.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if False then false_ else false_.not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) false_.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if False then false_ else false_.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
sorry
|
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) false_.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
h1 : True
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) false_.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.primeSet] at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.not_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.not_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot] at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot]
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V phi.not_ then phi.not_ else phi.not_.not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrime]
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V phi.not_ then phi.not_ else phi.not_.not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if ¬evalPrime V phi then phi.not_ else phi.not_.not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V phi.not_ then phi.not_ else phi.not_.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if ¬evalPrime V phi then phi.not_ else phi.not_.not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi.not_.not_ else phi.not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if ¬evalPrime V phi then phi.not_ else phi.not_.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
split_ifs
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi.not_.not_ else phi.not_)
|
case pos
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h✝ : evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
case neg
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h✝ : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi.not_.not_ else phi.not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case _ c1 =>
simp only [c1] at phi_ih
simp at phi_ih
apply IsDeduct.mp_ phi
apply proof_imp_deduct
apply T_14_6
exact phi_ih h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case _ c1 =>
simp only [c1] at phi_ih
simp at phi_ih
exact phi_ih h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [c1] at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if True then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if True then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if True then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.mp_ phi
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ phi.not_.not_)
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply proof_imp_deduct
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ phi.not_.not_)
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
case a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsProof (phi.imp_ phi.not_.not_)
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ phi.not_.not_)
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply T_14_6
|
case a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsProof (phi.imp_ phi.not_.not_)
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsProof (phi.imp_ phi.not_.not_)
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
exact phi_ih h1
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [c1] at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if False then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if False then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if False then phi else phi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
exact phi_ih h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.primeSet] at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑(phi.imp_ psi).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑(phi.primeSet ∪ psi.primeSet) ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑(phi.imp_ psi).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑(phi.primeSet ∪ psi.primeSet) ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑(phi.primeSet ∪ psi.primeSet) ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot] at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot] at psi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V psi)
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot]
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (phi.imp_ psi) then phi.imp_ psi else (phi.imp_ psi).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (phi.imp_ psi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
cases h1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (phi.imp_ psi) then phi.imp_ psi else (phi.imp_ psi).not_)
|
case intro
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
left✝ : ↑phi.primeSet ⊆ Δ_U
right✝ : ↑psi.primeSet ⊆ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (phi.imp_ psi) then phi.imp_ psi else (phi.imp_ psi).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1 : ↑phi.primeSet ⊆ Δ_U ∧ ↑psi.primeSet ⊆ Δ_U
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (phi.imp_ psi) then phi.imp_ psi else (phi.imp_ psi).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
split_ifs
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (phi.imp_ psi) then phi.imp_ psi else (phi.imp_ psi).not_)
|
case pos
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
h✝ : evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
case neg
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
h✝ : ¬evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (phi.imp_ psi) then phi.imp_ psi else (phi.imp_ psi).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case _ c1 =>
simp only [evalPrime] at c1
simp only [imp_iff_not_or] at c1
cases c1
case _ c1 =>
simp only [if_neg c1] at phi_ih
apply IsDeduct.mp_ (not_ phi)
apply proof_imp_deduct
apply T_13_6
apply phi_ih h1_left
case _ c1 =>
simp only [if_pos c1] at psi_ih
apply IsDeduct.mp_ psi
apply IsDeduct.axiom_
apply IsAxiom.prop_1_
apply psi_ih
exact h1_right
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrime] at c1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V phi → evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [imp_iff_not_or] at c1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V phi → evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi ∨ evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V phi → evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
cases c1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi ∨ evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
case inl
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
h✝ : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
case inr
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
h✝ : evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi ∨ evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case _ c1 =>
simp only [if_neg c1] at phi_ih
apply IsDeduct.mp_ (not_ phi)
apply proof_imp_deduct
apply T_13_6
apply phi_ih h1_left
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
case _ c1 =>
simp only [if_pos c1] at psi_ih
apply IsDeduct.mp_ psi
apply IsDeduct.axiom_
apply IsAxiom.prop_1_
apply psi_ih
exact h1_right
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [if_neg c1] at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.mp_ (not_ phi)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.not_.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply proof_imp_deduct
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.not_.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
case a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsProof (phi.not_.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.not_.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply T_13_6
|
case a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsProof (phi.not_.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsProof (phi.not_.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply phi_ih h1_left
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V phi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [if_pos c1] at psi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.mp_ psi
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (psi.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.axiom_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (psi.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
|
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsAxiom (psi.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (psi.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsAxiom.prop_1_
|
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsAxiom (psi.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsAxiom (psi.imp_ (phi.imp_ psi))
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply psi_ih
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ ↑psi.primeSet ⊆ Δ_U
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
exact h1_right
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ ↑psi.primeSet ⊆ Δ_U
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V psi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi
⊢ ↑psi.primeSet ⊆ Δ_U
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrime] at c1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬(evalPrime V phi → evalPrime V psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬evalPrime V (phi.imp_ psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at c1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬(evalPrime V phi → evalPrime V psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V phi ∧ ¬evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : ¬(evalPrime V phi → evalPrime V psi)
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
cases c1
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V phi ∧ ¬evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
case intro
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
left✝ : evalPrime V phi
right✝ : ¬evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1 : evalPrime V phi ∧ ¬evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [if_pos c1_left] at phi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih :
↑phi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V phi then phi else phi.not_)
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [if_neg c1_right] at psi_ih
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
psi_ih :
↑psi.primeSet ⊆ Δ_U →
IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (if evalPrime V psi then psi else psi.not_)
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.mp_ psi.not_
|
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (psi.not_.imp_ (phi.imp_ psi).not_)
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ psi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.mp_ phi
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (psi.not_.imp_ (phi.imp_ psi).not_)
|
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ (psi.not_.imp_ (phi.imp_ psi).not_))
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (psi.not_.imp_ (phi.imp_ psi).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply proof_imp_deduct
|
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ (psi.not_.imp_ (phi.imp_ psi).not_))
|
case a.a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsProof (phi.imp_ (psi.not_.imp_ (phi.imp_ psi).not_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) (phi.imp_ (psi.not_.imp_ (phi.imp_ psi).not_))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply T_14_8
|
case a.a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsProof (phi.imp_ (psi.not_.imp_ (phi.imp_ psi).not_))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.h1
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsProof (phi.imp_ (psi.not_.imp_ (phi.imp_ psi).not_))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
exact phi_ih h1_left
|
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
exact psi_ih h1_right
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
phi psi : Formula
h1_left : ↑phi.primeSet ⊆ Δ_U
h1_right : ↑psi.primeSet ⊆ Δ_U
c1_left : evalPrime V phi
c1_right : ¬evalPrime V psi
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) phi
psi_ih : ↑psi.primeSet ⊆ Δ_U → IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U) psi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
let F := forall_ x phi
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
F : Formula := forall_ x phi
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.primeSet] at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
F : Formula := forall_ x phi
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑{forall_ x phi} ⊆ Δ_U
F : Formula := forall_ x phi
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
F : Formula := forall_ x phi
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp at h1
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑{forall_ x phi} ⊆ Δ_U
F : Formula := forall_ x phi
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑{forall_ x phi} ⊆ Δ_U
F : Formula := forall_ x phi
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [evalPrimeFfToNot]
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (forall_ x phi) then forall_ x phi else (forall_ x phi).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp only [Formula.evalPrime]
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (forall_ x phi) then forall_ x phi else (forall_ x phi).not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (forall_ x phi) then forall_ x phi else (forall_ x phi).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply IsDeduct.assume_
|
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ (if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
Please generate a tactic in lean4 to solve the state.
STATE:
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
simp
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ (if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ ∃ x_1 ∈ Δ_U,
(if evalPrime V x_1 then x_1 else x_1.not_) =
if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ (if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
apply Exists.intro F
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ ∃ x_1 ∈ Δ_U,
(if evalPrime V x_1 then x_1 else x_1.not_) =
if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) = if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ ∃ x_1 ∈ Δ_U,
(if evalPrime V x_1 then x_1 else x_1.not_) =
if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
tauto
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) = if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
F : Formula := forall_ x phi
h1 : forall_ x phi ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) = if V (forall_ x phi) = true then forall_ x phi else (forall_ x phi).not_
TACTIC:
|
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