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.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
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✝)
|
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
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✝)
|
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
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✝)
|
F' : Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) =
if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝)
|
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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) =
if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else evalPrimeFfToNot V (def_ a✝¹ a✝)
|
F' : Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if evalPrime (fun c => if c = F' then true else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_
|
F' : Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if (if def_ a✝¹ a✝ = F' then true else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then def_ a✝¹ a✝ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).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
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✝)
|
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
⊢ evalPrimeFfToNot (Function.updateITE V F' false) F = Function.updateITE (evalPrimeFfToNot V) F' F.not_ 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' false) (pred_const_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_const_ a✝¹ a✝).not_ (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' false) (pred_var_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (pred_var_ a✝¹ a✝).not_ (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' false) (eq_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (eq_ a✝¹ a✝).not_ (eq_ a✝¹ a✝)
case true_
F' : Formula
V : VarBoolAssignment
h1 : true_.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) true_ = Function.updateITE (evalPrimeFfToNot V) F' true_.not_ true_
case false_
F' : Formula
V : VarBoolAssignment
h1 : false_.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) false_ =
Function.updateITE (evalPrimeFfToNot V) F' false_.not_ false_
case not_
F' : Formula
V : VarBoolAssignment
a✝ : Formula
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : a✝.not_.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) a✝.not_ =
Function.updateITE (evalPrimeFfToNot V) F' a✝.not_.not_ a✝.not_
case imp_
F' : Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ :
a✝¹.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (a✝¹.imp_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.imp_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.imp_ a✝).not_ (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' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (a✝¹.and_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.and_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.and_ a✝).not_ (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' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (a✝¹.or_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.or_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.or_ a✝).not_ (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' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝)
case forall_
F' : Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (forall_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (forall_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (forall_ a✝¹ a✝).not_ (forall_ a✝¹ a✝)
case exists_
F' : Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (exists_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (exists_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (exists_ a✝¹ a✝).not_ (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' false) (def_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)
|
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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)
|
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
a✝¹ a✝ : Formula
a_ih✝¹ :
a✝¹.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝)
|
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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (def_ a✝¹ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (def_ a✝¹ a✝).not_ (def_ a✝¹ a✝)
|
F' : Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) =
if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝)
|
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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrimeFfToNot (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) =
if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else evalPrimeFfToNot V (def_ a✝¹ a✝)
|
F' : Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_
else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if evalPrime (fun c => if c = F' then false else V c) (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_
else if evalPrime V (def_ a✝¹ a✝) then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_
|
F' : Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).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
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ (if (if def_ a✝¹ a✝ = F' then false else V (def_ a✝¹ a✝)) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).not_) =
if def_ a✝¹ a✝ = F' then (def_ a✝¹ a✝).not_ else if V (def_ a✝¹ a✝) = true then def_ a✝¹ a✝ else (def_ a✝¹ a✝).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
a✝¹ a✝ : Formula
a_ih✝¹ :
a✝¹.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝¹ = Function.updateITE (evalPrimeFfToNot V) F' a✝¹.not_ a✝¹
a_ih✝ :
a✝.IsPrime →
evalPrimeFfToNot (Function.updateITE V F' false) a✝ = Function.updateITE (evalPrimeFfToNot V) F' a✝.not_ a✝
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V F' false) (a✝¹.iff_ a✝) =
Function.updateITE (evalPrimeFfToNot V) F' (a✝¹.iff_ a✝).not_ (a✝¹.iff_ a✝)
|
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
Δ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
⊢ evalPrimeFfToNot (Function.updateITE V U b) '' Δ = evalPrimeFfToNot V '' Δ
|
case h
U : Formula
Δ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
⊢ ∀ a ∈ Δ, 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
Δ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
⊢ ∀ a ∈ Δ, evalPrimeFfToNot (Function.updateITE V U b) a = evalPrimeFfToNot V a
|
case h
U : Formula
Δ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
⊢ 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_Δ U' a1
|
case h
U : Formula
Δ : Set Formula
V : VarBoolAssignment
b : Bool
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'
|
case h
U : Formula
Δ : Set Formula
V : VarBoolAssignment
b : Bool
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ 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
Δ : Set Formula
V : VarBoolAssignment
b : Bool
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V U b) U' = evalPrimeFfToNot V U'
|
case h.false
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U'
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ 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_Δ]
|
case h.false
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V U false) U' = evalPrimeFfToNot V U'
|
case h.false
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ 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
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ Function.updateITE (evalPrimeFfToNot V) U U'.not_ U' = evalPrimeFfToNot V U'
|
case h.false
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ (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
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ (if U' = U then U'.not_ else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'
|
case h.false
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ U' = U → 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
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ U' = U → U'.not_ = evalPrimeFfToNot V U'
|
case h.false
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
a2 : U' = U
⊢ 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
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
a2 : U' = U
⊢ U'.not_ = evalPrimeFfToNot V U'
|
case h.false
Δ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' ∈ Δ
h1_Δ h1_U : U'.IsPrime
h2 : U' ∉ Δ
⊢ 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
Δ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' ∈ Δ
h1_Δ h1_U : U'.IsPrime
h2 : U' ∉ Δ
⊢ 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_Δ]
|
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ evalPrimeFfToNot (Function.updateITE V U true) U' = evalPrimeFfToNot V U'
|
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ 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
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ Function.updateITE (evalPrimeFfToNot V) U U' U' = evalPrimeFfToNot V U'
|
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ (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
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ (if U' = U then U' else evalPrimeFfToNot V U') = evalPrimeFfToNot V U'
|
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ 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]
|
intro a2
|
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
⊢ U' = U → U' = evalPrimeFfToNot V U'
|
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
a2 : 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]
|
subst a2
|
case h.true
U : Formula
Δ : Set Formula
V : VarBoolAssignment
h1_U : U.IsPrime
h2 : U ∉ Δ
U' : Formula
a1 : U' ∈ Δ
h1_Δ : U'.IsPrime
a2 : U' = U
⊢ U' = evalPrimeFfToNot V U'
|
case h.true
Δ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' ∈ Δ
h1_Δ h1_U : U'.IsPrime
h2 : U' ∉ Δ
⊢ 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
Δ : Set Formula
V : VarBoolAssignment
U' : Formula
a1 : U' ∈ Δ
h1_Δ h1_U : U'.IsPrime
h2 : U' ∉ Δ
⊢ 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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P
⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ) P
|
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
⊢ IsDeduct (evalPrimeFfToNot V '' Δ) 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 (Δ.image (evalPrimeFfToNot V))
|
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
⊢ IsDeduct (evalPrimeFfToNot V '' Δ) P
|
case h1
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
case h2
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
|
case h1
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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 Δ V true h1_Δ h1_U h2] at h3
|
case h1
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U true) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
|
case h1
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U true) U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
|
case h1
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
|
case h1
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U else evalPrimeFfToNot V U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
|
case h1
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
h3 : ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot V U}) P
V : VarBoolAssignment
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P
|
case h2
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U})
P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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 Δ V false h1_Δ h1_U h2] at h3
|
case h2
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 :
IsDeduct (evalPrimeFfToNot (Function.updateITE V U false) '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U})
P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P
|
case h2
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {evalPrimeFfToNot (Function.updateITE V U false) U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P
|
case h2
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {Function.updateITE (evalPrimeFfToNot V) U U.not_ U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P
|
case h2
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {if True then U.not_ else evalPrimeFfToNot V U}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P
|
case h2
P U : Formula
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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
Δ : Set Formula
h1_Δ : ∀ U' ∈ Δ, U'.IsPrime
h1_U : U.IsPrime
h2 : U ∉ Δ
V : VarBoolAssignment
h3 : IsDeduct (evalPrimeFfToNot V '' Δ ∪ {U.not_}) P
⊢ IsDeduct (evalPrimeFfToNot V '' Δ ∪ {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 Δ_U using Finset.induction_on
|
P : Formula
Δ_U : Finset Formula
h1 : Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P
⊢ IsDeduct ∅ P
|
case empty
P : Formula
h1 : ∅ ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P
⊢ IsDeduct ∅ P
case insert
P a✝² : Formula
s✝ : Finset Formula
a✝¹ : a✝² ∉ s✝
a✝ : s✝ ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) s✝)) P) → IsDeduct ∅ P
h1 : insert a✝² s✝ ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert a✝² s✝))) P
⊢ IsDeduct ∅ 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 : ∅ ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P
⊢ IsDeduct ∅ 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 : ∅ ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) ∅)) P
⊢ IsDeduct ∅ P
|
P : Formula
h1 : ∅ ⊆ P.primeSet
h2 : IsDeduct ∅ P
⊢ IsDeduct ∅ 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 : ∅ ⊆ P.primeSet
h2 : IsDeduct ∅ P
⊢ IsDeduct ∅ P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.propCompleteAux
|
[1024, 1]
|
[1057, 19]
|
apply Δ_U_2
|
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h1 : insert U Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
⊢ IsDeduct ∅ P
|
case h1
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h1 : insert U Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
⊢ Δ_U ⊆ P.primeSet
case h2
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h1 : insert U Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h1 : insert U Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
⊢ Δ_U ⊆ P.primeSet
|
case h1
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet
⊢ Δ_U ⊆ 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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet
⊢ Δ_U ⊆ P.primeSet
|
case h1.intro
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
left✝ : U ∈ P.primeSet
right✝ : Δ_U ⊆ P.primeSet
⊢ Δ_U ⊆ 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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ Δ_U ⊆ 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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ Δ_U ⊆ 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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h1 : insert U Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P
|
case h2
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) (insert U Δ_U))) P
h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P
|
case h2
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h1 : U ∈ P.primeSet ∧ Δ_U ⊆ P.primeSet
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P
|
case h2.intro
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
left✝ : U ∈ P.primeSet
right✝ : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P
|
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_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 Δ_U
|
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U) P
|
case h1_Δ
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime
case h1_U
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ U.IsPrime
case h2
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ U ∉ ↑Δ_U
case h3
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {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_Δ
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ U' ∈ ↑Δ_U, U'.IsPrime
|
case h1_Δ
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
U' : Formula
a1 : U' ∈ ↑Δ_U
⊢ 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_Δ
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
U' : Formula
a1 : U' ∈ ↑Δ_U
⊢ U'.IsPrime
|
case h1_Δ
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
U' : Formula
a1 : U' ∈ ↑Δ_U
⊢ U' ∈ 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_Δ
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
U' : Formula
a1 : U' ∈ ↑Δ_U
⊢ U' ∈ P.primeSet
|
case h1_Δ.a
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
U' : Formula
a1 : U' ∈ ↑Δ_U
⊢ U' ∈ Δ_U
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.propCompleteAux
|
[1024, 1]
|
[1057, 19]
|
exact a1
|
case h1_Δ.a
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
U' : Formula
a1 : U' ∈ ↑Δ_U
⊢ U' ∈ Δ_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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ U.IsPrime
|
case h1_U
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ U ∈ 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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ U ∈ P.primeSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.propCompleteAux
|
[1024, 1]
|
[1057, 19]
|
exact Δ_U_1
|
case h2
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ U ∉ ↑Δ_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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (evalPrimeFfToNot V '' ↑Δ_U ∪ {evalPrimeFfToNot V U}) P
|
case h3
P U : Formula
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_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
Δ_U : Finset Formula
Δ_U_1 : U ∉ Δ_U
Δ_U_2 :
Δ_U ⊆ P.primeSet → (∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) Δ_U)) P) → IsDeduct ∅ P
h2 : ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_U)) P
h1_left : U ∈ P.primeSet
h1_right : Δ_U ⊆ P.primeSet
⊢ ∀ (V : VarBoolAssignment), IsDeduct (insert (evalPrimeFfToNot V U) (evalPrimeFfToNot V '' ↑Δ_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
⊢ IsProof P
|
P : Formula
h1 : P.IsTautoPrime
⊢ IsDeduct ∅ 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
⊢ IsDeduct ∅ P
|
case h1
P : Formula
h1 : P.IsTautoPrime
⊢ P.primeSet ⊆ P.primeSet
case h2
P : Formula
h1 : P.IsTautoPrime
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(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
⊢ P.primeSet ⊆ 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
⊢ ∀ (V : VarBoolAssignment), IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P
|
case h2
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
⊢ IsDeduct (↑(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
⊢ IsDeduct (↑(Finset.image (evalPrimeFfToNot V) P.primeSet)) P
|
case h2.h1
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
⊢ ↑P.primeSet ⊆ ↑P.primeSet
case h2.h2
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.primeSet
case h2.h3
P : Formula
h1 : P.IsTautoPrime
V : VarBoolAssignment
⊢ 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
⊢ ↑P.primeSet ⊆ ↑P.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
⊢ ↑(Finset.image (evalPrimeFfToNot V) P.primeSet) = evalPrimeFfToNot V '' ↑P.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
⊢ P = evalPrimeFfToNot V P
|
case h2.h3
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
⊢ 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 : ∀ (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
⊢ P = evalPrimeFfToNot V P
|
case h2.h3
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
⊢ 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 : ∀ (V : VarBoolAssignment), evalPrime V P
V : VarBoolAssignment
⊢ P = if evalPrime V P then P else P.not_
|
case h2.h3
P : Formula
V : VarBoolAssignment
h1 : evalPrime V P
⊢ 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
⊢ P = if evalPrime V P then P else P.not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
induction xs using Finset.induction_on
|
x : VarName
xs : Finset VarName
h1 : x ∈ xs
⊢ x.length ≤ finset_var_name_max_len xs
|
case empty
x : VarName
h1 : x ∈ ∅
⊢ x.length ≤ finset_var_name_max_len ∅
case insert
x a✝² : VarName
s✝ : Finset VarName
a✝¹ : a✝² ∉ s✝
a✝ : x ∈ s✝ → x.length ≤ finset_var_name_max_len s✝
h1 : x ∈ insert a✝² s✝
⊢ x.length ≤ finset_var_name_max_len (insert a✝² s✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
case empty =>
simp at h1
|
x : VarName
h1 : x ∈ ∅
⊢ x.length ≤ finset_var_name_max_len ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
case insert hd tl a1 ih =>
simp at h1
cases h1
case inl c1 =>
subst c1
simp only [finset_var_name_max_len]
simp
case inr c1 =>
simp only [finset_var_name_max_len] at ih
simp only [finset_var_name_max_len]
simp
right
exact ih c1
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
h1 : x ∈ insert hd tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
simp at h1
|
x : VarName
h1 : x ∈ ∅
⊢ x.length ≤ finset_var_name_max_len ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
simp at h1
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
h1 : x ∈ insert hd tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
h1 : x = hd ∨ x ∈ tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
cases h1
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
h1 : x = hd ∨ x ∈ tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
case inl
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
h✝ : x = hd
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
case inr
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
h✝ : x ∈ tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
case inl c1 =>
subst c1
simp only [finset_var_name_max_len]
simp
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
c1 : x = hd
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
case inr c1 =>
simp only [finset_var_name_max_len] at ih
simp only [finset_var_name_max_len]
simp
right
exact ih c1
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
c1 : x ∈ tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
subst c1
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
c1 : x = hd
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
x : VarName
tl : Finset VarName
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
a1 : x ∉ tl
⊢ x.length ≤ finset_var_name_max_len (insert x tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
simp only [finset_var_name_max_len]
|
x : VarName
tl : Finset VarName
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
a1 : x ∉ tl
⊢ x.length ≤ finset_var_name_max_len (insert x tl)
|
x : VarName
tl : Finset VarName
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
a1 : x ∉ tl
⊢ x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) (insert x tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
simp
|
x : VarName
tl : Finset VarName
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
a1 : x ∉ tl
⊢ x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) (insert x tl)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
simp only [finset_var_name_max_len] at ih
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ finset_var_name_max_len tl
c1 : x ∈ tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
simp only [finset_var_name_max_len]
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ finset_var_name_max_len (insert hd tl)
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) (insert hd tl)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
simp
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) (insert hd tl)
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ hd.length ∨ x.length ≤ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
right
|
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ hd.length ∨ x.length ≤ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl
|
case h
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.finset_var_name_max_len_mem
|
[23, 1]
|
[46, 18]
|
exact ih c1
|
case h
x hd : VarName
tl : Finset VarName
a1 : hd ∉ tl
ih : x ∈ tl → x.length ≤ Finset.fold (fun m n => max m n) 0 (String.length ∘ VarName.toString) tl
c1 : x ∈ tl
⊢ x.length ≤ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
obtain s1 := finset_var_name_max_len_mem x xs h
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
⊢ finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
s1 : x.length ≤ finset_var_name_max_len xs
⊢ finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
simp only [tsub_lt_tsub_iff_right s1]
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
s1 : x.length ≤ finset_var_name_max_len xs
⊢ finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
s1 : x.length ≤ finset_var_name_max_len xs
⊢ finset_var_name_max_len xs < finset_var_name_max_len xs + 1
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
simp
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
s1 : x.length ≤ finset_var_name_max_len xs
⊢ finset_var_name_max_len xs < finset_var_name_max_len xs + 1
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
unfold fresh
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
⊢ fresh x c xs ∉ xs
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
⊢ (if h : x ∈ xs then
let_fun this := ⋯;
fresh { toString := x.toString ++ c.toString } c xs
else x) ∉
xs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
simp
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
⊢ (if h : x ∈ xs then
let_fun this := ⋯;
fresh { toString := x.toString ++ c.toString } c xs
else x) ∉
xs
|
x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
⊢ (if x ∈ xs then fresh { toString := x.toString ++ c.toString } c xs else x) ∉ xs
|
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