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/Parsing/DFT.lean
|
dft_iff
|
[884, 1]
|
[900, 50]
|
intro ⟨s, h1, h2⟩
|
case mpr
α : Type
inst✝ : DecidableEq α
g : Graph α
S : List α
s' : α
⊢ (∃ s ∈ S, Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s') → s' ∈ reachable_from_list g S
|
case mpr
α : Type
inst✝ : DecidableEq α
g : Graph α
S : List α
s' s : α
h1 : s ∈ S
h2 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'
⊢ s' ∈ reachable_from_list g S
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
dft_iff
|
[884, 1]
|
[900, 50]
|
induction h2 with
| refl => exact reachable_from_list.base _ h1
| tail _ h ih =>
have ⟨_, h3, h4⟩ := h
exact reachable_from_list.step _ _ h3 h4 ih
|
case mpr
α : Type
inst✝ : DecidableEq α
g : Graph α
S : List α
s' s : α
h1 : s ∈ S
h2 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s'
⊢ s' ∈ reachable_from_list g S
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
dft_iff
|
[884, 1]
|
[900, 50]
|
exact reachable_from_list.base _ h1
|
case mpr.refl
α : Type
inst✝ : DecidableEq α
g : Graph α
S : List α
s' s : α
h1 : s ∈ S
⊢ s ∈ reachable_from_list g S
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
dft_iff
|
[884, 1]
|
[900, 50]
|
have ⟨_, h3, h4⟩ := h
|
case mpr.tail
α : Type
inst✝ : DecidableEq α
g : Graph α
S : List α
s' s : α
h1 : s ∈ S
b✝ c✝ : α
a✝ : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s b✝
h : ∃ l, (b✝, l) ∈ g ∧ c✝ ∈ l
ih : b✝ ∈ reachable_from_list g S
⊢ c✝ ∈ reachable_from_list g S
|
case mpr.tail
α : Type
inst✝ : DecidableEq α
g : Graph α
S : List α
s' s : α
h1 : s ∈ S
b✝ c✝ : α
a✝ : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s b✝
h : ∃ l, (b✝, l) ∈ g ∧ c✝ ∈ l
ih : b✝ ∈ reachable_from_list g S
w✝ : List α
h3 : (b✝, w✝) ∈ g
h4 : c✝ ∈ w✝
⊢ c✝ ∈ reachable_from_list g S
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/DFT.lean
|
dft_iff
|
[884, 1]
|
[900, 50]
|
exact reachable_from_list.step _ _ h3 h4 ih
|
case mpr.tail
α : Type
inst✝ : DecidableEq α
g : Graph α
S : List α
s' s : α
h1 : s ∈ S
b✝ c✝ : α
a✝ : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s b✝
h : ∃ l, (b✝, l) ∈ g ∧ c✝ ∈ l
ih : b✝ ∈ reachable_from_list g S
w✝ : List α
h3 : (b✝, w✝) ∈ g
h4 : c✝ ∈ w✝
⊢ c✝ ∈ reachable_from_list g S
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
DA.memAccepts
|
[145, 1]
|
[151, 67]
|
rfl
|
α σ : Type
D : DA α σ
input : List α
⊢ D.accepts input ↔ D.evalFrom D.startingState input ∈ D.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NA.memAccepts
|
[154, 1]
|
[161, 38]
|
rfl
|
α σ : Type
N : NA α σ
input : List α
⊢ N.accepts input ↔ ∃ s ∈ N.evalFrom N.startingStates input, s ∈ N.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
ext cs
|
α σ : Type
N : NA α σ
⊢ N.toDA.accepts = N.accepts
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.accepts cs ↔ N.accepts cs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp only [DA.memAccepts]
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.accepts cs ↔ N.accepts cs
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔ N.accepts cs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp only [NA.memAccepts]
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔ N.accepts cs
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp only [NA.toDA]
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ { step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∈
{S | ∃ s ∈ S, s ∈ N.acceptingStates} ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ { step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∈
{S | ∃ s ∈ S, s ∈ N.acceptingStates} ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
constructor
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
case h.a.mp
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) →
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ (∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates) →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
all_goals
simp
intro s a1 a2
apply Exists.intro s
tauto
|
case h.a.mp
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) →
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ (∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates) →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ (∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates) →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ ∀ x ∈ N.evalFrom N.startingStates cs,
x ∈ N.acceptingStates →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
intro s a1 a2
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ ∀ x ∈ N.evalFrom N.startingStates cs,
x ∈ N.acceptingStates →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ ∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
apply Exists.intro s
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ ∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∧
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
tauto
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∧
s ∈ N.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_prime
|
[136, 1]
|
[146, 8]
|
induction F
|
F : Formula
V : VarBoolAssignment
h1 : F.IsPrime
⊢ evalPrime V F = (V F = true)
|
case pred_const_
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_const_ a✝¹ a✝).IsPrime
⊢ evalPrime V (pred_const_ a✝¹ a✝) = (V (pred_const_ a✝¹ a✝) = true)
case pred_var_
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_var_ a✝¹ a✝).IsPrime
⊢ evalPrime V (pred_var_ a✝¹ a✝) = (V (pred_var_ a✝¹ a✝) = true)
case eq_
V : VarBoolAssignment
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).IsPrime
⊢ evalPrime V (eq_ a✝¹ a✝) = (V (eq_ a✝¹ a✝) = true)
case true_
V : VarBoolAssignment
h1 : true_.IsPrime
⊢ evalPrime V true_ = (V true_ = true)
case false_
V : VarBoolAssignment
h1 : false_.IsPrime
⊢ evalPrime V false_ = (V false_ = true)
case not_
V : VarBoolAssignment
a✝ : Formula
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : a✝.not_.IsPrime
⊢ evalPrime V a✝.not_ = (V a✝.not_ = true)
case imp_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.imp_ a✝).IsPrime
⊢ evalPrime V (a✝¹.imp_ a✝) = (V (a✝¹.imp_ a✝) = true)
case and_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.and_ a✝).IsPrime
⊢ evalPrime V (a✝¹.and_ a✝) = (V (a✝¹.and_ a✝) = true)
case or_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.or_ a✝).IsPrime
⊢ evalPrime V (a✝¹.or_ a✝) = (V (a✝¹.or_ a✝) = true)
case iff_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrime V (a✝¹.iff_ a✝) = (V (a✝¹.iff_ a✝) = true)
case forall_
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (forall_ a✝¹ a✝).IsPrime
⊢ evalPrime V (forall_ a✝¹ a✝) = (V (forall_ a✝¹ a✝) = true)
case exists_
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (exists_ a✝¹ a✝).IsPrime
⊢ evalPrime V (exists_ a✝¹ a✝) = (V (exists_ a✝¹ a✝) = true)
case def_
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrime V (def_ a✝¹ a✝) = (V (def_ a✝¹ a✝) = true)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_prime
|
[136, 1]
|
[146, 8]
|
case true_ | false_ | not_ | imp_ | and_ | or_ | iff_ =>
simp only [Formula.IsPrime] at h1
|
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrime V (a✝¹.iff_ a✝) = (V (a✝¹.iff_ a✝) = true)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_prime
|
[136, 1]
|
[146, 8]
|
case pred_const_ | pred_var_ | eq_ | forall_ | exists_ | def_ =>
rfl
|
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrime V (def_ a✝¹ a✝) = (V (def_ a✝¹ a✝) = true)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_prime
|
[136, 1]
|
[146, 8]
|
simp only [Formula.IsPrime] at h1
|
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrime V (a✝¹.iff_ a✝) = (V (a✝¹.iff_ a✝) = true)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_prime
|
[136, 1]
|
[146, 8]
|
rfl
|
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrime V (def_ a✝¹ a✝) = (V (def_ a✝¹ a✝) = true)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
induction F
|
F : Formula
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ F) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) F
|
case pred_const_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (pred_const_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (pred_const_ a✝¹ a✝)
case pred_var_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (pred_var_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (pred_var_ a✝¹ a✝)
case eq_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : VarName
⊢ evalPrime V (substPrime σ (eq_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (eq_ a✝¹ a✝)
case true_
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ true_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) true_
case false_
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ false_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) false_
case not_
σ : Formula → Formula
V : VarBoolAssignment
a✝ : Formula
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ a✝.not_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝.not_
case imp_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.imp_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.imp_ a✝)
case and_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.and_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.and_ a✝)
case or_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.or_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.or_ a✝)
case iff_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.iff_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.iff_ a✝)
case forall_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (forall_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (forall_ a✝¹ a✝)
case exists_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (exists_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (exists_ a✝¹ a✝)
case def_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
case pred_const_ | pred_var_ | eq_ | forall_ | exists_ | def_ =>
simp only [Formula.substPrime]
simp only [Formula.evalPrime]
simp
|
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
case true_ | false_ =>
rfl
|
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ false_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) false_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
case not_ phi phi_ih =>
simp only [Formula.substPrime]
simp only [Formula.evalPrime]
congr! 1
|
σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi.not_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp only [Formula.substPrime]
simp only [Formula.evalPrime]
congr! 1
|
σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V (substPrime σ (phi.iff_ psi)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
simp only [Formula.substPrime]
|
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝)
|
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
simp only [Formula.evalPrime]
|
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝)
|
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ decide (evalPrime V (σ (def_ a✝¹ a✝))) = true
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
simp
|
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ decide (evalPrime V (σ (def_ a✝¹ a✝))) = true
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
rfl
|
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ false_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) false_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
simp only [Formula.substPrime]
|
σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi.not_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_
|
σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi).not_ ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
simp only [Formula.evalPrime]
|
σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi).not_ ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_
|
σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ ¬evalPrime V (substPrime σ phi) ↔ ¬evalPrime (fun H => decide (evalPrime V (σ H))) phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
congr! 1
|
σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ ¬evalPrime V (substPrime σ phi) ↔ ¬evalPrime (fun H => decide (evalPrime V (σ H))) phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
simp only [Formula.substPrime]
|
σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V (substPrime σ (phi.iff_ psi)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi)
|
σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V ((substPrime σ phi).iff_ (substPrime σ psi)) ↔
evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
simp only [Formula.evalPrime]
|
σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V ((substPrime σ phi).iff_ (substPrime σ psi)) ↔
evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi)
|
σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ (evalPrime V (substPrime σ phi) ↔ evalPrime V (substPrime σ psi)) ↔
(evalPrime (fun H => decide (evalPrime V (σ H))) phi ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime
|
[193, 1]
|
[218, 13]
|
congr! 1
|
σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ (evalPrime V (substPrime σ phi) ↔ evalPrime V (substPrime σ psi)) ↔
(evalPrime (fun H => decide (evalPrime V (σ H))) phi ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime
|
[221, 1]
|
[232, 11]
|
simp only [Formula.IsTautoPrime] at h1
|
P : Formula
h1 : P.IsTautoPrime
σ : Formula → Formula
⊢ (substPrime σ P).IsTautoPrime
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ (substPrime σ P).IsTautoPrime
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime
|
[221, 1]
|
[232, 11]
|
simp only [Formula.IsTautoPrime]
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ (substPrime σ P).IsTautoPrime
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V (substPrime σ P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime
|
[221, 1]
|
[232, 11]
|
intro V
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V (substPrime σ P)
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime
|
[221, 1]
|
[232, 11]
|
simp only [evalPrime_substPrime_eq_evalPrime_evalPrime P σ V]
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ P)
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime (fun H => decide (evalPrime V (σ H))) P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime
|
[221, 1]
|
[232, 11]
|
apply h1
|
P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime (fun H => decide (evalPrime V (σ H))) P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
simp only [IsProof]
|
P : Formula
⊢ IsProof (P.imp_ P)
|
P : Formula
⊢ IsDeduct ∅ (P.imp_ P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
apply IsDeduct.mp_ (P.imp_ (P.imp_ P))
|
P : Formula
⊢ IsDeduct ∅ (P.imp_ P)
|
case a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P))
case a
P : Formula
⊢ IsDeduct ∅ (P.imp_ (P.imp_ P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
apply IsDeduct.mp_ (P.imp_ ((P.imp_ P).imp_ P))
|
case a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P))
|
case a.a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P)))
case a.a
P : Formula
⊢ IsDeduct ∅ (P.imp_ ((P.imp_ P).imp_ P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
apply IsDeduct.axiom_
|
case a.a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P)))
|
case a.a.a
P : Formula
⊢ IsAxiom ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
exact IsAxiom.prop_2_ P (P.imp_ P) P
|
case a.a.a
P : Formula
⊢ IsAxiom ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P)))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
apply IsDeduct.axiom_
|
case a.a
P : Formula
⊢ IsDeduct ∅ (P.imp_ ((P.imp_ P).imp_ P))
|
case a.a.a
P : Formula
⊢ IsAxiom (P.imp_ ((P.imp_ P).imp_ P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
exact IsAxiom.prop_1_ P (P.imp_ P)
|
case a.a.a
P : Formula
⊢ IsAxiom (P.imp_ ((P.imp_ P).imp_ P))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
apply IsDeduct.axiom_
|
case a
P : Formula
⊢ IsDeduct ∅ (P.imp_ (P.imp_ P))
|
case a.a
P : Formula
⊢ IsAxiom (P.imp_ (P.imp_ P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_5
|
[253, 1]
|
[265, 30]
|
exact IsAxiom.prop_1_ P P
|
case a.a
P : Formula
⊢ IsAxiom (P.imp_ (P.imp_ P))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsDeduct.mp_ (P.not_.imp_ (Q.not_.imp_ P.not_))
|
P Q : Formula
⊢ IsProof (P.not_.imp_ (P.imp_ Q))
|
case a
P Q : Formula
⊢ IsDeduct ∅ ((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q)))
case a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ (Q.not_.imp_ P.not_))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsDeduct.mp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)))
|
case a
P Q : Formula
⊢ IsDeduct ∅ ((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q)))
|
case a.a
P Q : Formula
⊢ IsDeduct ∅
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q))))
case a.a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsDeduct.axiom_
|
case a.a
P Q : Formula
⊢ IsDeduct ∅
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q))))
|
case a.a.a
P Q : Formula
⊢ IsAxiom
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q))))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsAxiom.prop_2_
|
case a.a.a
P Q : Formula
⊢ IsAxiom
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q))))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsDeduct.mp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))
|
case a.a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)))
|
case a.a.a
P Q : Formula
⊢ IsDeduct ∅ (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))))
case a.a.a
P Q : Formula
⊢ IsDeduct ∅ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsDeduct.axiom_
|
case a.a.a
P Q : Formula
⊢ IsDeduct ∅ (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))))
|
case a.a.a.a
P Q : Formula
⊢ IsAxiom (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsAxiom.prop_1_
|
case a.a.a.a
P Q : Formula
⊢ IsAxiom (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsDeduct.axiom_
|
case a.a.a
P Q : Formula
⊢ IsDeduct ∅ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))
|
case a.a.a.a
P Q : Formula
⊢ IsAxiom ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsAxiom.prop_3_
|
case a.a.a.a
P Q : Formula
⊢ IsAxiom ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsDeduct.axiom_
|
case a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ (Q.not_.imp_ P.not_))
|
case a.a
P Q : Formula
⊢ IsAxiom (P.not_.imp_ (Q.not_.imp_ P.not_))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_13_6_no_deduct
|
[270, 1]
|
[284, 26]
|
apply IsAxiom.prop_1_
|
case a.a
P Q : Formula
⊢ IsAxiom (P.not_.imp_ (Q.not_.imp_ P.not_))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
intro Γ
|
F : Formula
Δ : Set Formula
h1 : IsDeduct Δ F
⊢ ∀ (Γ : Set Formula), IsDeduct (Δ ∪ Γ) F
|
F : Formula
Δ : Set Formula
h1 : IsDeduct Δ F
Γ : Set Formula
⊢ IsDeduct (Δ ∪ Γ) F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
induction h1
|
F : Formula
Δ : Set Formula
h1 : IsDeduct Δ F
Γ : Set Formula
⊢ IsDeduct (Δ ∪ Γ) F
|
case axiom_
F : Formula
Δ Γ : Set Formula
phi✝ : Formula
a✝ : IsAxiom phi✝
⊢ IsDeduct (Δ ∪ Γ) phi✝
case assume_
F : Formula
Δ Γ : Set Formula
phi✝ : Formula
a✝ : phi✝ ∈ Δ
⊢ IsDeduct (Δ ∪ Γ) phi✝
case mp_
F : Formula
Δ Γ : Set Formula
phi✝ psi✝ : Formula
a✝¹ : IsDeduct Δ (phi✝.imp_ psi✝)
a✝ : IsDeduct Δ phi✝
a_ih✝¹ : IsDeduct (Δ ∪ Γ) (phi✝.imp_ psi✝)
a_ih✝ : IsDeduct (Δ ∪ Γ) phi✝
⊢ IsDeduct (Δ ∪ Γ) psi✝
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
case axiom_ h1_phi h1_1 =>
apply IsDeduct.axiom_
exact h1_1
|
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
case assume_ h1_phi h1_1 =>
apply IsDeduct.assume_
simp
left
exact h1_1
|
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct (Δ ∪ Γ) h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
apply IsDeduct.axiom_
|
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
exact h1_1
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
apply IsDeduct.assume_
|
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct (Δ ∪ Γ) h1_phi
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∪ Γ
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
simp
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∪ Γ
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∨ h1_phi ∈ Γ
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
left
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∨ h1_phi ∈ Γ
|
case a.h
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
exact h1_1
|
case a.h
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
apply IsDeduct.mp_ h1_phi
|
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_psi
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
exact h1_ih_1
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10
|
[287, 1]
|
[306, 20]
|
exact h1_ih_2
|
case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10_comm
|
[309, 1]
|
[316, 23]
|
simp only [Set.union_comm]
|
Q : Formula
Δ : Set Formula
h1 : IsDeduct Δ Q
⊢ ∀ (Γ : Set Formula), IsDeduct (Γ ∪ Δ) Q
|
Q : Formula
Δ : Set Formula
h1 : IsDeduct Δ Q
⊢ ∀ (Γ : Set Formula), IsDeduct (Δ ∪ Γ) Q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_10_comm
|
[309, 1]
|
[316, 23]
|
exact T_14_10 Q Δ h1
|
Q : Formula
Δ : Set Formula
h1 : IsDeduct Δ Q
⊢ ∀ (Γ : Set Formula), IsDeduct (Δ ∪ Γ) Q
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.C_14_11
|
[319, 1]
|
[327, 11]
|
intro Δ
|
P : Formula
h1 : IsProof P
⊢ ∀ (Δ : Set Formula), IsDeduct Δ P
|
P : Formula
h1 : IsProof P
Δ : Set Formula
⊢ IsDeduct Δ P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.C_14_11
|
[319, 1]
|
[327, 11]
|
obtain s1 := T_14_10 P ∅ h1 Δ
|
P : Formula
h1 : IsProof P
Δ : Set Formula
⊢ IsDeduct Δ P
|
P : Formula
h1 : IsProof P
Δ : Set Formula
s1 : IsDeduct (∅ ∪ Δ) P
⊢ IsDeduct Δ P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.C_14_11
|
[319, 1]
|
[327, 11]
|
simp at s1
|
P : Formula
h1 : IsProof P
Δ : Set Formula
s1 : IsDeduct (∅ ∪ Δ) P
⊢ IsDeduct Δ P
|
P : Formula
h1 : IsProof P
Δ : Set Formula
s1 : IsDeduct Δ P
⊢ IsDeduct Δ P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.C_14_11
|
[319, 1]
|
[327, 11]
|
exact s1
|
P : Formula
h1 : IsProof P
Δ : Set Formula
s1 : IsDeduct Δ P
⊢ IsDeduct Δ P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
induction h1
|
P Q : Formula
Δ : Set Formula
h1 : IsDeduct (Δ ∪ {P}) Q
⊢ IsDeduct Δ (P.imp_ Q)
|
case axiom_
P Q : Formula
Δ : Set Formula
phi✝ : Formula
a✝ : IsAxiom phi✝
⊢ IsDeduct Δ (P.imp_ phi✝)
case assume_
P Q : Formula
Δ : Set Formula
phi✝ : Formula
a✝ : phi✝ ∈ Δ ∪ {P}
⊢ IsDeduct Δ (P.imp_ phi✝)
case mp_
P Q : Formula
Δ : Set Formula
phi✝ psi✝ : Formula
a✝¹ : IsDeduct (Δ ∪ {P}) (phi✝.imp_ psi✝)
a✝ : IsDeduct (Δ ∪ {P}) phi✝
a_ih✝¹ : IsDeduct Δ (P.imp_ (phi✝.imp_ psi✝))
a_ih✝ : IsDeduct Δ (P.imp_ phi✝)
⊢ IsDeduct Δ (P.imp_ psi✝)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply IsDeduct.mp_ h1_phi
|
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ (h1_phi.imp_ (P.imp_ h1_phi))
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply IsDeduct.axiom_
|
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ (h1_phi.imp_ (P.imp_ h1_phi))
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom (h1_phi.imp_ (P.imp_ h1_phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
exact IsAxiom.prop_1_ h1_phi P
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom (h1_phi.imp_ (P.imp_ h1_phi))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply IsDeduct.axiom_
|
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ h1_phi
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
exact h1_1
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
simp at h1_1
|
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ ∪ {P}
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi = P ∨ h1_phi ∈ Δ
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
cases h1_1
|
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi = P ∨ h1_phi ∈ Δ
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
case inl
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h✝ : h1_phi = P
⊢ IsDeduct Δ (P.imp_ h1_phi)
case inr
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h✝ : h1_phi ∈ Δ
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
case inl h1_1 =>
subst h1_1
apply proof_imp_deduct
exact prop_id h1_phi
|
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi = P
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
subst h1_1
|
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi = P
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
Q : Formula
Δ : Set Formula
h1_phi : Formula
⊢ IsDeduct Δ (h1_phi.imp_ h1_phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply proof_imp_deduct
|
Q : Formula
Δ : Set Formula
h1_phi : Formula
⊢ IsDeduct Δ (h1_phi.imp_ h1_phi)
|
case h1
Q : Formula
Δ : Set Formula
h1_phi : Formula
⊢ IsProof (h1_phi.imp_ h1_phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
exact prop_id h1_phi
|
case h1
Q : Formula
Δ : Set Formula
h1_phi : Formula
⊢ IsProof (h1_phi.imp_ h1_phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply IsDeduct.mp_ h1_phi
|
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ (h1_phi.imp_ (P.imp_ h1_phi))
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply IsDeduct.axiom_
|
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ (h1_phi.imp_ (P.imp_ h1_phi))
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsAxiom (h1_phi.imp_ (P.imp_ h1_phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
exact IsAxiom.prop_1_ h1_phi P
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsAxiom (h1_phi.imp_ (P.imp_ h1_phi))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply IsDeduct.assume_
|
case a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct Δ h1_phi
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
exact h1_1
|
case a.a
P Q : Formula
Δ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Prop.lean
|
FOL.NV.T_14_3
|
[333, 1]
|
[366, 20]
|
apply IsDeduct.mp_ (P.imp_ h1_phi)
|
P Q : Formula
Δ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct (Δ ∪ {P}) (h1_phi.imp_ h1_psi)
a✝ : IsDeduct (Δ ∪ {P}) h1_phi
h1_ih_1 : IsDeduct Δ (P.imp_ (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (P.imp_ h1_phi)
⊢ IsDeduct Δ (P.imp_ h1_psi)
|
case a
P Q : Formula
Δ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct (Δ ∪ {P}) (h1_phi.imp_ h1_psi)
a✝ : IsDeduct (Δ ∪ {P}) h1_phi
h1_ih_1 : IsDeduct Δ (P.imp_ (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (P.imp_ h1_phi)
⊢ IsDeduct Δ ((P.imp_ h1_phi).imp_ (P.imp_ h1_psi))
case a
P Q : Formula
Δ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct (Δ ∪ {P}) (h1_phi.imp_ h1_psi)
a✝ : IsDeduct (Δ ∪ {P}) h1_phi
h1_ih_1 : IsDeduct Δ (P.imp_ (h1_phi.imp_ h1_psi))
h1_ih_2 : IsDeduct Δ (P.imp_ h1_phi)
⊢ IsDeduct Δ (P.imp_ h1_phi)
|
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