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
|
|---|---|---|---|---|---|---|---|---|
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_
|
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_
|
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
|
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))
|
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))
|
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))
|
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_)
|
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_)
|
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
|
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_
|
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_
|
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
|
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_)
|
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
|
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_)
|
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_)
|
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_)
|
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_)
|
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_
|
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
|
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_)
|
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_)
|
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_)
|
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_)
|
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_)
|
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_
|
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
|
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
|
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_
|
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_
|
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
|
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
|
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
|
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
|
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_
|
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_
|
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
|
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))
|
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))
|
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))
|
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))
|
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_)
|
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_)
|
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_
|
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
|
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)
|
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)
|
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)
|
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
|
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
|
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)
|
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_
|
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_
|
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_
|
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
|
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)
|
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
|
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
|
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
|
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
|
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
|
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_
|
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_
|
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_
|
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_
|
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_
|
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_
|
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
|
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_))
|
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
|
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
|
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
|
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))
|
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))
|
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))
|
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_)
|
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_)
|
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
|
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_
|
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_
|
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
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.L_15_7
|
[773, 1]
|
[916, 10]
|
let F := def_ X xs
|
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))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : ↑(def_ X xs).primeSet ⊆ Δ_U
F : Formula := def_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
|
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 : DefName
xs : List VarName
h1 : ↑(def_ X xs).primeSet ⊆ Δ_U
F : Formula := def_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : ↑{def_ X xs} ⊆ Δ_U
F : Formula := def_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
|
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 : DefName
xs : List VarName
h1 : ↑{def_ X xs} ⊆ Δ_U
F : Formula := def_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
|
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 : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs))
|
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_)
|
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 : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_)
|
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_)
|
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 : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_)
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) ∈ (fun a => if evalPrime V a then a else a.not_) '' Δ_U
|
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 : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ (if V (def_ X xs) = true then def_ X xs else (def_ 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 : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_
|
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 : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U, (if evalPrime V x then x else x.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_
|
case a
Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_
|
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 : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧ (if evalPrime V F then F else F.not_) = if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_
|
no goals
|
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
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
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
apply IsDeduct.mp_ (U.not_.imp_ P)
|
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ P
|
case a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P)
case a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ (U.not_.imp_ P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
apply IsDeduct.mp_ (U.imp_ P)
|
case a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ ((U.not_.imp_ P).imp_ P)
|
case a.a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))
case a.a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ (U.imp_ P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
apply proof_imp_deduct
|
case a.a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))
|
case a.a.h1
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
apply T_14_9
|
case a.a.h1
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsProof ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
apply deduction_theorem
|
case a.a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ (U.imp_ P)
|
case a.a.h1
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct (Δ ∪ {U}) P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
exact h1
|
case a.a.h1
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct (Δ ∪ {U}) P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
apply deduction_theorem
|
case a
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct Δ (U.not_.imp_ P)
|
case a.h1
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct (Δ ∪ {U.not_}) P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_9_Deduct
|
[919, 1]
|
[933, 13]
|
exact h2
|
case a.h1
P U : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {U}) P
h2 : IsDeduct (Δ ∪ {U.not_}) P
⊢ IsDeduct (Δ ∪ {U.not_}) P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrimeFfToNot_of_function_updateIte_true
|
[936, 1]
|
[950, 38]
|
induction F
|
F F' : Formula
V : VarBoolAssignment
h1 : F.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) F = Function.updateITE (evalPrimeFfToNot V) F' F F
|
case pred_const_
F' : Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_const_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (pred_const_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ a✝¹ a✝) (pred_const_ a✝¹ a✝)
case pred_var_
F' : Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_var_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (pred_var_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ a✝¹ a✝) (pred_var_ a✝¹ a✝)
case eq_
F' : Formula
V : VarBoolAssignment
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (eq_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (eq_ a✝¹ a✝) (eq_ a✝¹ a✝)
case true_
F' : Formula
V : VarBoolAssignment
h1 : true_.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_ true_
case false_
F' : Formula
V : VarBoolAssignment
h1 : false_.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) false_ = Function.updateITE (evalPrimeFfToNot V) F' false_ false_
case not_
F' : Formula
V : VarBoolAssignment
a✝ : Formula
a_ih✝ :
a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝
h1 : a✝.not_.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) a✝.not_ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝.not_
case imp_
F' : Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ :
a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹
a_ih✝ :
a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝
h1 : (a✝¹.imp_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.imp_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.imp_ a✝) (a✝¹.imp_ a✝)
case and_
F' : Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ :
a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹
a_ih✝ :
a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝
h1 : (a✝¹.and_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.and_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.and_ a✝) (a✝¹.and_ a✝)
case or_
F' : Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ :
a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹
a_ih✝ :
a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝
h1 : (a✝¹.or_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.or_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.or_ a✝) (a✝¹.or_ a✝)
case iff_
F' : Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ :
a✝¹.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹ a✝¹
a_ih✝ :
a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (a✝¹.iff_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝) (a✝¹.iff_ a✝)
case forall_
F' : Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝
h1 : (forall_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (forall_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (forall_ a✝¹ a✝) (forall_ a✝¹ a✝)
case exists_
F' : Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
a✝.IsPrime → evalPrimeFfToNot (Function.updateITE V F' true) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝ a✝
h1 : (exists_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (exists_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (exists_ a✝¹ a✝) (exists_ a✝¹ a✝)
case def_
F' : Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' true) (def_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝) (def_ a✝¹ a✝)
|
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