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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 | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : β(def_ X xs).primeSet β Ξ_U
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_U) (evalPrimeFfToNot V (def_ X xs)) | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : β(def_ X xs).primeSet β Ξ_U
F : Formula := def_ X xs
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_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 | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : β(def_ X xs).primeSet β Ξ_U
F : Formula := def_ X xs
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_U) (evalPrimeFfToNot V (def_ X xs)) | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : β{def_ X xs} β Ξ_U
F : Formula := def_ X xs
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_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 | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : β{def_ X xs} β Ξ_U
F : Formula := def_ X xs
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_U) (evalPrimeFfToNot V (def_ X xs)) | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_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] | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_U) (evalPrimeFfToNot V (def_ X xs)) | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Ξ_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] | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Ξ_U)
(if evalPrime V (def_ X xs) then def_ X xs else (def_ X xs).not_) | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Ξ_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_ | Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Ξ_U)
(if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) | case a
Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) β (fun a => if evalPrime V a then a else a.not_) '' Ξ_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
Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ (if V (def_ X xs) = true then def_ X xs else (def_ X xs).not_) β (fun a => if evalPrime V a then a else a.not_) '' Ξ_U | case a
Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ β x β Ξ_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
Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ β x β Ξ_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
Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ F β Ξ_U β§ (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
Ξ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
F : Formula := def_ X xs
h1 : def_ X xs β Ξ_U
β’ F β Ξ_U β§ (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 | Ξ_U : Set Formula
V : VarBoolAssignment
aβΒΉ : VarName
aβ : Formula
a_ihβ : βaβ.primeSet β Ξ_U β IsDeduct (evalPrimeFfToNot V '' Ξ_U) (evalPrimeFfToNot V aβ)
h1 : β(exists_ aβΒΉ aβ).primeSet β Ξ_U
β’ IsDeduct (evalPrimeFfToNot V '' Ξ_U) (evalPrimeFfToNot V (exists_ aβΒΉ aβ)) | 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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ P | case a
P U : Formula
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ ((U.not_.imp_ P).imp_ P)
case a
P U : Formula
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ (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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ ((U.not_.imp_ P).imp_ P) | case a.a
P U : Formula
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P))
case a.a
P U : Formula
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ (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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ ((U.imp_ P).imp_ ((U.not_.imp_ P).imp_ P)) | case a.a.h1
P U : Formula
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ 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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ 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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ (U.imp_ P) | case a.a.h1
P U : Formula
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct (Ξ βͺ {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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct (Ξ βͺ {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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct Ξ (U.not_.imp_ P) | case a.h1
P U : Formula
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct (Ξ βͺ {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
Ξ : Set Formula
h1 : IsDeduct (Ξ βͺ {U}) P
h2 : IsDeduct (Ξ βͺ {U.not_}) P
β’ IsDeduct (Ξ βͺ {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
β’ evalPrimeFfToNot (Function.updateITE V F' true) F = Function.updateITE (evalPrimeFfToNot V) F' F F | case pred_const_
F' : Formula
V : VarBoolAssignment
aβΒΉ : PredName
aβ : List VarName
h1 : (pred_const_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (pred_const_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ aβΒΉ aβ) (pred_const_ aβΒΉ aβ)
case pred_var_
F' : Formula
V : VarBoolAssignment
aβΒΉ : PredName
aβ : List VarName
h1 : (pred_var_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (pred_var_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ aβΒΉ aβ) (pred_var_ aβΒΉ aβ)
case eq_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : VarName
h1 : (eq_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (eq_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (eq_ aβΒΉ aβ) (eq_ aβΒΉ aβ)
case true_
F' : Formula
V : VarBoolAssignment
h1 : true_.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_ true_
case false_
F' : Formula
V : VarBoolAssignment
h1 : false_.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) false_ = Function.updateITE (evalPrimeFfToNot V) F' false_ false_
case not_
F' : Formula
V : VarBoolAssignment
aβ : Formula
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : aβ.not_.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) aβ.not_ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ.not_
case imp_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ aβΒΉ
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (aβΒΉ.imp_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (aβΒΉ.imp_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.imp_ aβ) (aβΒΉ.imp_ aβ)
case and_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ aβΒΉ
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (aβΒΉ.and_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (aβΒΉ.and_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.and_ aβ) (aβΒΉ.and_ aβ)
case or_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ aβΒΉ
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (aβΒΉ.or_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (aβΒΉ.or_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.or_ aβ) (aβΒΉ.or_ aβ)
case iff_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ aβΒΉ
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (aβΒΉ.iff_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (aβΒΉ.iff_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.iff_ aβ) (aβΒΉ.iff_ aβ)
case forall_
F' : Formula
V : VarBoolAssignment
aβΒΉ : VarName
aβ : Formula
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (forall_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (forall_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (forall_ aβΒΉ aβ) (forall_ aβΒΉ aβ)
case exists_
F' : Formula
V : VarBoolAssignment
aβΒΉ : VarName
aβ : Formula
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (exists_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (exists_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (exists_ aβΒΉ aβ) (exists_ aβΒΉ aβ)
case def_
F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (def_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ aβΒΉ aβ) (def_ aβΒΉ aβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_true | [936, 1] | [950, 38] | case pred_const_ | pred_var_ | eq_ | forall_ | exists_ | def_ =>
unfold Function.updateITE
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
split_ifs <;> tauto | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (def_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ aβΒΉ aβ) (def_ aβΒΉ aβ) | 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] | case true_ | false_ | not_ | imp_ | and_ | or_ | iff_ =>
simp only [Formula.IsPrime] at h1 | F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ aβΒΉ
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (aβΒΉ.iff_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (aβΒΉ.iff_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.iff_ aβ) (aβΒΉ.iff_ aβ) | 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] | unfold Function.updateITE | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (def_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ aβΒΉ aβ) (def_ aβΒΉ aβ) | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ aβΒΉ aβ) =
if def_ aβΒΉ aβ = F' then def_ aβΒΉ aβ else evalPrimeFfToNot V (def_ aβΒΉ aβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_true | [936, 1] | [950, 38] | simp only [evalPrimeFfToNot] | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ aβΒΉ aβ) =
if def_ aβΒΉ aβ = F' then def_ aβΒΉ aβ else evalPrimeFfToNot V (def_ aβΒΉ aβ) | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if evalPrime (fun c => if c = F' then true else V c) (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then def_ aβΒΉ aβ else if evalPrime V (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_true | [936, 1] | [950, 38] | simp only [Formula.evalPrime] | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if evalPrime (fun c => if c = F' then true else V c) (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then def_ aβΒΉ aβ else if evalPrime V (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if (if def_ aβΒΉ aβ = F' then true else V (def_ aβΒΉ aβ)) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then def_ aβΒΉ aβ else if V (def_ aβΒΉ aβ) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_true | [936, 1] | [950, 38] | split_ifs <;> tauto | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if (if def_ aβΒΉ aβ = F' then true else V (def_ aβΒΉ aβ)) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then def_ aβΒΉ aβ else if V (def_ aβΒΉ aβ) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ | 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] | simp only [Formula.IsPrime] at h1 | F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ aβΒΉ
a_ihβ :
aβ.IsPrime β evalPrimeFfToNot (Function.updateITE V F' true) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ aβ
h1 : (aβΒΉ.iff_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' true) (aβΒΉ.iff_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.iff_ aβ) (aβΒΉ.iff_ aβ) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | induction F | F F' : Formula
V : VarBoolAssignment
h1 : F.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) F = Function.updateITE (evalPrimeFfToNot V) F' F.not_ F | case pred_const_
F' : Formula
V : VarBoolAssignment
aβΒΉ : PredName
aβ : List VarName
h1 : (pred_const_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (pred_const_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ aβΒΉ aβ).not_ (pred_const_ aβΒΉ aβ)
case pred_var_
F' : Formula
V : VarBoolAssignment
aβΒΉ : PredName
aβ : List VarName
h1 : (pred_var_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (pred_var_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ aβΒΉ aβ).not_ (pred_var_ aβΒΉ aβ)
case eq_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : VarName
h1 : (eq_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (eq_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (eq_ aβΒΉ aβ).not_ (eq_ aβΒΉ aβ)
case true_
F' : Formula
V : VarBoolAssignment
h1 : true_.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_.not_ true_
case false_
F' : Formula
V : VarBoolAssignment
h1 : false_.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) false_ =
Function.updateITE (evalPrimeFfToNot V) F' false_.not_ false_
case not_
F' : Formula
V : VarBoolAssignment
aβ : Formula
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : aβ.not_.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) aβ.not_ =
Function.updateITE (evalPrimeFfToNot V) F' aβ.not_.not_ aβ.not_
case imp_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ.not_ aβΒΉ
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (aβΒΉ.imp_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (aβΒΉ.imp_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.imp_ aβ).not_ (aβΒΉ.imp_ aβ)
case and_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ.not_ aβΒΉ
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (aβΒΉ.and_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (aβΒΉ.and_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.and_ aβ).not_ (aβΒΉ.and_ aβ)
case or_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ.not_ aβΒΉ
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (aβΒΉ.or_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (aβΒΉ.or_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.or_ aβ).not_ (aβΒΉ.or_ aβ)
case iff_
F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ.not_ aβΒΉ
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (aβΒΉ.iff_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (aβΒΉ.iff_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.iff_ aβ).not_ (aβΒΉ.iff_ aβ)
case forall_
F' : Formula
V : VarBoolAssignment
aβΒΉ : VarName
aβ : Formula
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (forall_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (forall_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (forall_ aβΒΉ aβ).not_ (forall_ aβΒΉ aβ)
case exists_
F' : Formula
V : VarBoolAssignment
aβΒΉ : VarName
aβ : Formula
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (exists_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (exists_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (exists_ aβΒΉ aβ).not_ (exists_ aβΒΉ aβ)
case def_
F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (def_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ aβΒΉ aβ).not_ (def_ aβΒΉ aβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | case pred_const_ | pred_var_ | eq_ | forall_ | exists_ | def_ =>
unfold Function.updateITE
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
split_ifs <;> tauto | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (def_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ aβΒΉ aβ).not_ (def_ aβΒΉ aβ) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | case true_ | false_ | not_ | imp_ | and_ | or_ | iff_ =>
simp only [Formula.IsPrime] at h1 | F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ.not_ aβΒΉ
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (aβΒΉ.iff_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (aβΒΉ.iff_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.iff_ aβ).not_ (aβΒΉ.iff_ aβ) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | unfold Function.updateITE | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (def_ aβΒΉ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ aβΒΉ aβ).not_ (def_ aβΒΉ aβ) | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ aβΒΉ aβ) =
if def_ aβΒΉ aβ = F' then (def_ aβΒΉ aβ).not_ else evalPrimeFfToNot V (def_ aβΒΉ aβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | simp only [evalPrimeFfToNot] | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ aβΒΉ aβ) =
if def_ aβΒΉ aβ = F' then (def_ aβΒΉ aβ).not_ else evalPrimeFfToNot V (def_ aβΒΉ aβ) | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if evalPrime (fun c => if c = F' then false else V c) (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then (def_ aβΒΉ aβ).not_
else if evalPrime V (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | simp only [Formula.evalPrime] | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if evalPrime (fun c => if c = F' then false else V c) (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then (def_ aβΒΉ aβ).not_
else if evalPrime V (def_ aβΒΉ aβ) then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if (if def_ aβΒΉ aβ = F' then false else V (def_ aβΒΉ aβ)) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then (def_ aβΒΉ aβ).not_ else if V (def_ aβΒΉ aβ) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | split_ifs <;> tauto | F' : Formula
V : VarBoolAssignment
aβΒΉ : DefName
aβ : List VarName
h1 : (def_ aβΒΉ aβ).IsPrime
β’ (if (if def_ aβΒΉ aβ = F' then false else V (def_ aβΒΉ aβ)) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_) =
if def_ aβΒΉ aβ = F' then (def_ aβΒΉ aβ).not_ else if V (def_ aβΒΉ aβ) = true then def_ aβΒΉ aβ else (def_ aβΒΉ aβ).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrimeFfToNot_of_function_updateIte_false | [953, 1] | [967, 38] | simp only [Formula.IsPrime] at h1 | F' : Formula
V : VarBoolAssignment
aβΒΉ aβ : Formula
a_ihβΒΉ :
aβΒΉ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβΒΉ = Function.updateITE (evalPrimeFfToNot V) F' aβΒΉ.not_ aβΒΉ
a_ihβ :
aβ.IsPrime β
evalPrimeFfToNot (Function.updateITE V F' false) aβ = Function.updateITE (evalPrimeFfToNot V) F' aβ.not_ aβ
h1 : (aβΒΉ.iff_ aβ).IsPrime
β’ evalPrimeFfToNot (Function.updateITE V F' false) (aβΒΉ.iff_ aβ) =
Function.updateITE (evalPrimeFfToNot V) F' (aβΒΉ.iff_ aβ).not_ (aβΒΉ.iff_ aβ) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | apply Set.image_congr | U : Formula
Ξ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
β’ evalPrimeFfToNot (Function.updateITE V U b) '' Ξ = evalPrimeFfToNot V '' Ξ | case h
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
β’ β a β Ξ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | intro U' a1 | case h
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
β’ β a β Ξ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a | case h
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
β’ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | specialize h1_Ξ U' a1 | case h
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
β’ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' | case h
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
b : Bool
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | cases b | case h
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
b : Bool
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U' | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U'
case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | simp only [evalPrimeFfToNot_of_function_updateIte_false U' U V h1_Ξ] | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U' | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | simp only [Function.updateITE] | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U' | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | simp | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ U' = U β U'.not_ = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | intro a2 | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ U' = U β U'.not_ = evalPrimeFfToNot V U' | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
a2 : U' = U
β’ U'.not_ = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | subst a2 | case h.false
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
a2 : U' = U
β’ U'.not_ = evalPrimeFfToNot V U' | case h.false
Ξ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' β Ξ
h1_Ξ h1_U : U'.IsPrime
h2 : U' β Ξ
β’ U'.not_ = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | contradiction | case h.false
Ξ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' β Ξ
h1_Ξ h1_U : U'.IsPrime
h2 : U' β Ξ
β’ U'.not_ = evalPrimeFfToNot V U' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | simp only [evalPrimeFfToNot_of_function_updateIte_true U' U V h1_Ξ] | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U' | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | simp only [Function.updateITE] | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U' | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | simp | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U' | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ U' = U β U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | intro a2 | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
β’ U' = U β U' = evalPrimeFfToNot V U' | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
a2 : U' = U
β’ U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | subst a2 | case h.true
U : Formula
Ξ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U β Ξ
U' : Formula
a1 : U' β Ξ
h1_Ξ : U'.IsPrime
a2 : U' = U
β’ U' = evalPrimeFfToNot V U' | case h.true
Ξ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' β Ξ
h1_Ξ h1_U : U'.IsPrime
h2 : U' β Ξ
β’ U' = evalPrimeFfToNot V U' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.image_of_evalPrimeFfToNot_of_function_updateIte | [970, 1] | [996, 18] | contradiction | case h.true
Ξ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' β Ξ
h1_Ξ h1_U : U'.IsPrime
h2 : U' β Ξ
β’ U' = evalPrimeFfToNot V U' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | intro V | P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
h3 : β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot V U}) P
β’ β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ) P | P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
h3 : β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
β’ IsDeduct (evalPrimeFfToNot V '' Ξ) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | apply T_14_9_Deduct P U (Ξ.image (evalPrimeFfToNot V)) | P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
h3 : β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
β’ IsDeduct (evalPrimeFfToNot V '' Ξ) P | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
h3 : β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P
case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
h3 : β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | specialize h3 (Function.updateITE V U true) | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
h3 : β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [image_of_evalPrimeFfToNot_of_function_updateIte U Ξ V true h1_Ξ h1_U h2] at h3 | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [evalPrimeFfToNot_of_function_updateIte_true U U V h1_U] at h3 | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {Function.updateITE (evalPrimeFfToNot V) U U U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [Function.updateITE] at h3 | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {Function.updateITE (evalPrimeFfToNot V) U U U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {if True then U else evalPrimeFfToNot V U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [eq_self_iff_true, if_true] at h3 | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {if True then U else evalPrimeFfToNot V U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | exact h3 | case h1
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U}) P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | specialize h3 (Function.updateITE V U Bool.false) | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
h3 : β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U false) U})
P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [image_of_evalPrimeFfToNot_of_function_updateIte U Ξ V false h1_Ξ h1_U h2] at h3 | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U false) U})
P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U false) U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [evalPrimeFfToNot_of_function_updateIte_false U U V h1_U] at h3 | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {evalPrimeFfToNot (Function.updateITE V U false) U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [Function.updateITE] at h3 | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {if True then U.not_ else evalPrimeFfToNot V U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | simp only [eq_self_iff_true, if_true] at h3 | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {if True then U.not_ else evalPrimeFfToNot V U}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAuxAux | [999, 1] | [1021, 13] | exact h3 | case h2
P U : Formula
Ξ : Set Formula
h1_Ξ : β U' β Ξ, U'.IsPrime
h1_U : U.IsPrime
h2 : U β Ξ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P
β’ IsDeduct (evalPrimeFfToNot V '' Ξ βͺ {U.not_}) P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | induction Ξ_U using Finset.induction_on | P : Formula
Ξ_U : Finset Formula
h1 : Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P
β’ IsDeduct β
P | case empty
P : Formula
h1 : β
β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) β
)) P
β’ IsDeduct β
P
case insert
P aβΒ² : Formula
sβ : Finset Formula
aβΒΉ : aβΒ² β sβ
aβ : sβ β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) sβ)) P) β IsDeduct β
P
h1 : insert aβΒ² sβ β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert aβΒ² sβ))) P
β’ IsDeduct β
P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | case empty =>
simp at h2
exact h2 | P : Formula
h1 : β
β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) β
)) P
β’ IsDeduct β
P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | simp at h2 | P : Formula
h1 : β
β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) β
)) P
β’ IsDeduct β
P | P : Formula
h1 : β
β P.primeSet
h2 : IsDeduct β
P
β’ IsDeduct β
P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | exact h2 | P : Formula
h1 : β
β P.primeSet
h2 : IsDeduct β
P
β’ IsDeduct β
P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | apply Ξ_U_2 | P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h1 : insert U Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
β’ IsDeduct β
P | case h1
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h1 : insert U Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
β’ Ξ_U β P.primeSet
case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h1 : insert U Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | simp only [Finset.insert_subset_iff] at h1 | case h1
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h1 : insert U Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
β’ Ξ_U β P.primeSet | case h1
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
h1 : U β P.primeSet β§ Ξ_U β P.primeSet
β’ Ξ_U β P.primeSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | cases h1 | case h1
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
h1 : U β P.primeSet β§ Ξ_U β P.primeSet
β’ Ξ_U β P.primeSet | case h1.intro
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
leftβ : U β P.primeSet
rightβ : Ξ_U β P.primeSet
β’ Ξ_U β P.primeSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | case intro h1_left h1_right =>
exact h1_right | P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ Ξ_U β P.primeSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | exact h1_right | P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ Ξ_U β P.primeSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | simp only [Finset.insert_subset_iff] at h1 | case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h1 : insert U Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P | case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
h1 : U β P.primeSet β§ Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | simp at h2 | case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) (insert U Ξ_U))) P
h1 : U β P.primeSet β§ Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P | case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h1 : U β P.primeSet β§ Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | cases h1 | case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h1 : U β P.primeSet β§ Ξ_U β P.primeSet
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P | case h2.intro
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
leftβ : U β P.primeSet
rightβ : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | simp | P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P | P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' βΞ_U) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | apply propCompleteAuxAux P U Ξ_U | P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' βΞ_U) P | case h1_Ξ
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β U' β βΞ_U, U'.IsPrime
case h1_U
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ U.IsPrime
case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ U β βΞ_U
case h3
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' βΞ_U βͺ {evalPrimeFfToNot V U}) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | intro U' a1 | case h1_Ξ
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β U' β βΞ_U, U'.IsPrime | case h1_Ξ
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
U' : Formula
a1 : U' β βΞ_U
β’ U'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | apply mem_primeSet_isPrime P U' | case h1_Ξ
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
U' : Formula
a1 : U' β βΞ_U
β’ U'.IsPrime | case h1_Ξ
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
U' : Formula
a1 : U' β βΞ_U
β’ U' β P.primeSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | apply h1_right | case h1_Ξ
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
U' : Formula
a1 : U' β βΞ_U
β’ U' β P.primeSet | case h1_Ξ.a
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
U' : Formula
a1 : U' β βΞ_U
β’ U' β Ξ_U |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | exact a1 | case h1_Ξ.a
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
U' : Formula
a1 : U' β βΞ_U
β’ U' β Ξ_U | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | apply mem_primeSet_isPrime P U | case h1_U
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ U.IsPrime | case h1_U
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ U β P.primeSet |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | exact h1_left | case h1_U
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ U β P.primeSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | exact Ξ_U_1 | case h2
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ U β βΞ_U | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | simp | case h3
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' βΞ_U βͺ {evalPrimeFfToNot V U}) P | case h3
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.propCompleteAux | [1024, 1] | [1057, 19] | exact h2 | case h3
P U : Formula
Ξ_U : Finset Formula
Ξ_U_1 : U β Ξ_U
Ξ_U_2 :
Ξ_U β P.primeSet β (β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) Ξ_U)) P) β IsDeduct β
P
h2 : β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P
h1_left : U β P.primeSet
h1_right : Ξ_U β P.primeSet
β’ β (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' βΞ_U)) P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | simp only [IsProof] | P : Formula
h1 : P.IsTautoPrime
β’ IsProof P | P : Formula
h1 : P.IsTautoPrime
β’ IsDeduct β
P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | apply propCompleteAux P P.primeSet | P : Formula
h1 : P.IsTautoPrime
β’ IsDeduct β
P | case h1
P : Formula
h1 : P.IsTautoPrime
β’ P.primeSet β P.primeSet
case h2
P : Formula
h1 : P.IsTautoPrime
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) P.primeSet)) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | rfl | case h1
P : Formula
h1 : P.IsTautoPrime
β’ P.primeSet β P.primeSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | intro V | case h2
P : Formula
h1 : P.IsTautoPrime
β’ β (V : VarBoolAssignment), IsDeduct (β(Finset.image (evalPrimeFfToNot V) P.primeSet)) P | case h2
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ IsDeduct (β(Finset.image (evalPrimeFfToNot V) P.primeSet)) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | apply L_15_7 P P P.primeSet V (P.primeSet.image (evalPrimeFfToNot V)) | case h2
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ IsDeduct (β(Finset.image (evalPrimeFfToNot V) P.primeSet)) P | case h2.h1
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ βP.primeSet β βP.primeSet
case h2.h2
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ β(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' βP.primeSet
case h2.h3
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ P = evalPrimeFfToNot V P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | rfl | case h2.h1
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ βP.primeSet β βP.primeSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | simp only [Finset.coe_image] | case h2.h2
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ β(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' βP.primeSet | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | simp only [Formula.IsTautoPrime] at h1 | case h2.h3
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
β’ P = evalPrimeFfToNot V P | case h2.h3
P : Formula
h1 : β (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
β’ P = evalPrimeFfToNot V P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | simp only [evalPrimeFfToNot] | case h2.h3
P : Formula
h1 : β (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
β’ P = evalPrimeFfToNot V P | case h2.h3
P : Formula
h1 : β (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
β’ P = if evalPrime V P then P else P.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | specialize h1 V | case h2.h3
P : Formula
h1 : β (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
β’ P = if evalPrime V P then P else P.not_ | case h2.h3
P : Formula
V : VarBoolAssignment
h1 : evalPrime V P
β’ P = if evalPrime V P then P else P.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.prop_complete | [1063, 1] | [1078, 28] | simp only [if_pos h1] | case h2.h3
P : Formula
V : VarBoolAssignment
h1 : evalPrime V P
β’ P = if evalPrime V P then P else P.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | apply Finset.union_subset_iff.mpr | Ξ± : Type
instβ : DecidableEq Ξ±
A B C D : Finset Ξ±
h1 : A β C
h2 : B β D
β’ A βͺ B β C βͺ D | Ξ± : Type
instβ : DecidableEq Ξ±
A B C D : Finset Ξ±
h1 : A β C
h2 : B β D
β’ A β C βͺ D β§ B β C βͺ D |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Finset.lean | Finset.union_subset_left_right | [8, 1] | [23, 44] | constructor | Ξ± : Type
instβ : DecidableEq Ξ±
A B C D : Finset Ξ±
h1 : A β C
h2 : B β D
β’ A β C βͺ D β§ B β C βͺ D | case left
Ξ± : Type
instβ : DecidableEq Ξ±
A B C D : Finset Ξ±
h1 : A β C
h2 : B β D
β’ A β C βͺ D
case right
Ξ± : Type
instβ : DecidableEq Ξ±
A B C D : Finset Ξ±
h1 : A β C
h2 : B β D
β’ B β C βͺ D |
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