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stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
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stringlengths 1
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | apply Exists.intro [] | case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ ∃ s_1 ∈ RegExp.languageOf α s, a :: w ++ s_1 = w | case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ [] ∈ RegExp.languageOf α s ∧ a :: w ++ [] = w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | sorry | case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ [] ∈ RegExp.languageOf α s ∧ a :: w ++ [] = w | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp only [RegExp.languageOf] | α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ w ∈ RegExp.languageOf α (r.delta.concat (RegExp.derivative a s)) ↔ a :: w ∈ RegExp.languageOf α (r.concat s) | α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = x} ↔
a :: w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp | α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = x} ↔
a :: w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x} | α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w) ↔
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | constructor | α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w) ↔
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | intro a1 | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | apply Exists.elim a1 | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α r.delta ∧ ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), a_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | intro xs a2 | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α r.delta ∧ ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), a_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w
xs : List α
a2 : xs ∈ RegExp.languageOf α r.delta ∧ ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | clear a1 | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w
xs : List α
a2 : xs ∈ RegExp.languageOf α r.delta ∧ ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α r.delta ∧ ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | sorry | case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α r.delta ∧ ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | sorry | case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : ¬r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α (RegExp.derivative a s), r_1 ++ s_1 = w | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp only [RegExp.derivative] | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a e.closure) ↔ a :: w ∈ RegExp.languageOf α e.closure | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a e).concat e.closure) ↔ a :: w ∈ RegExp.languageOf α e.closure |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp only [RegExp.languageOf] | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a e).concat e.closure) ↔ a :: w ∈ RegExp.languageOf α e.closure | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ w ∈
{x |
∃ r ∈ RegExp.languageOf α (RegExp.derivative a e),
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x} ↔
a :: w ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ w ∈
{x |
∃ r ∈ RegExp.languageOf α (RegExp.derivative a e),
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x} ↔
a :: w ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l} | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ (∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w) ↔
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | constructor | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ (∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w) ↔
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ (∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w) →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w
case mpr
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ (∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w) →
∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | sorry | case mpr
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ (∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w) →
∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | intro a1 | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
⊢ (∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w) →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
a1 : ∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | apply Exists.elim a1 | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
a1 : ∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
a1 : ∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α (RegExp.derivative a e) ∧
∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ a_1 ++ a.join = w) →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | intro xs a2 | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
a1 : ∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α (RegExp.derivative a e) ∧
∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ a_1 ++ a.join = w) →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
a1 : ∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a e) ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | clear a1 | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w : List α
a1 : ∃ r ∈ RegExp.languageOf α (RegExp.derivative a e), ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ r ++ a.join = w
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a e) ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a e) ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | cases a2 | case mp
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a e) ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case mp.intro
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
left✝ : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
right✝ : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | case _ a2_left a2_right =>
apply Exists.elim a2_right
intro ys a3
clear a2_right
cases a3
case _ a3_left a3_right =>
apply Exists.intro [(a :: w)]
simp
simp only [← ih]
sorry | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | apply Exists.elim a2_right | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∀ (a_1 : List (List α)),
(∀ r ∈ a_1, r ∈ RegExp.languageOf α e) ∧ xs ++ a_1.join = w →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | intro ys a3 | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
⊢ ∀ (a_1 : List (List α)),
(∀ r ∈ a_1, r ∈ RegExp.languageOf α e) ∧ xs ++ a_1.join = w →
∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | clear a2_right | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α e) ∧ xs ++ a.join = w
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | cases a3 | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α e) ∧ xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | case intro
α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
left✝ : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
right✝ : xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | case _ a3_left a3_right =>
apply Exists.intro [(a :: w)]
simp
simp only [← ih]
sorry | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | apply Exists.intro [(a :: w)] | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ (∀ r ∈ [a :: w], r ∈ RegExp.languageOf α e) ∧ [a :: w].join = a :: w |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ (∀ r ∈ [a :: w], r ∈ RegExp.languageOf α e) ∧ [a :: w].join = a :: w | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ a :: w ∈ RegExp.languageOf α e |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | simp only [← ih] | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ a :: w ∈ RegExp.languageOf α e | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a e) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/Brzozowski.lean | derivative_def | [295, 1] | [407, 12] | sorry | α : Type
inst✝ : DecidableEq α
a : α
e : RegExp α
ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
w xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a e)
ys : List (List α)
a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α e
a3_right : xs ++ ys.join = w
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a e) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders F
⊢ fastAdmitsAux v u binders F | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders true_
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders false_
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝.not_
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | all_goals
simp only [admitsAux] at h2
simp only [fastAdmitsAux] | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders true_
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders false_
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝.not_
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {a✝¹}) a✝
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {a✝¹}) a✝
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | simp only [admitsAux] at h2 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u binders (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | simp only [fastAdmitsAux] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | by_cases c1 : v = x | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | left | case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case pos.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | exact c1 | case pos.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | right | case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi | case neg.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ fastAdmitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | apply phi_ih | case neg.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ fastAdmitsAux v u (binders ∪ {x}) phi | case neg.h.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
case neg.h.h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ admitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | simp | case neg.h.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x} | case neg.h.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | tauto | case neg.h.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | exact h2 | case neg.h.h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
h2 : admitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ admitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux | [217, 1] | [240, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
⊢ v ∈ a✝ → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders F | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | all_goals
simp only [admitsAux] | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
apply phi_ih
simp
left
exact h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | simp only [admitsAux] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | apply phi_ih | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ admitsAux v u (binders ∪ {x}) phi | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∪ {x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | simp | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∪ {x} | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∨ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | left | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∨ v = x | case h1.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | exact h1 | case h1.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux | [243, 1] | [259, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
⊢ admitsAux v u binders F | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders true_
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders false_
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝.not_
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | all_goals
simp only [fastAdmitsAux] at h1
simp only [admitsAux] | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
⊢ admitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders true_
⊢ admitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders false_
⊢ admitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝.not_
⊢ admitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
⊢ admitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
⊢ admitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
⊢ admitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
⊢ admitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
⊢ admitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
⊢ admitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
cases h1
case inl h1 =>
apply mem_binders_imp_admitsAux
subst h1
simp
case inr h1 =>
apply phi_ih
exact h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | simp only [fastAdmitsAux] at h1 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | simp only [admitsAux] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | cases h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi | case inl
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h✝ : v = x
⊢ admitsAux v u (binders ∪ {x}) phi
case inr
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h✝ : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | case inl h1 =>
apply mem_binders_imp_admitsAux
subst h1
simp | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | case inr h1 =>
apply phi_ih
exact h1 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | apply mem_binders_imp_admitsAux | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ v ∈ binders ∪ {x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | subst h1 | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ v ∈ binders ∪ {x} | case h1
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
⊢ v ∈ binders ∪ {v} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | simp | case h1
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
⊢ v ∈ binders ∪ {v} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | apply phi_ih | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | exact h1 | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux | [262, 1] | [284, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits | [287, 1] | [297, 46] | simp only [admits] | F : Formula
v u : VarName
⊢ admits v u F ↔ fastAdmits v u F | F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmits v u F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits | [287, 1] | [297, 46] | simp only [fastAdmits] | F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmits v u F | F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmitsAux v u ∅ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits | [287, 1] | [297, 46] | constructor | F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmitsAux v u ∅ F | case mp
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F → fastAdmitsAux v u ∅ F
case mpr
F : Formula
v u : VarName
⊢ fastAdmitsAux v u ∅ F → admitsAux v u ∅ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits | [287, 1] | [297, 46] | apply admitsAux_imp_fastAdmitsAux | case mp
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F → fastAdmitsAux v u ∅ F | case mp.h1
F : Formula
v u : VarName
⊢ v ∉ ∅ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits | [287, 1] | [297, 46] | simp | case mp.h1
F : Formula
v u : VarName
⊢ v ∉ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits | [287, 1] | [297, 46] | exact fastAdmitsAux_imp_admitsAux F v u ∅ | case mpr
F : Formula
v u : VarName
⊢ fastAdmitsAux v u ∅ F → admitsAux v u ∅ F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | induction F generalizing binders | F : Formula
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders F | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (eq_ a✝¹ a✝)
case true_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders true_
case false_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | all_goals
simp only [fastAdmitsAux] | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_const_ a✝¹ a✝)
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_var_ a✝¹ a✝)
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (eq_ a✝¹ a✝)
case true_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders true_
case false_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders false_
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝.not_
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.imp_ a✝)
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.and_ a✝)
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.or_ a✝)
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (a✝¹.iff_ a✝)
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (forall_ a✝¹ a✝)
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (exists_ a✝¹ a✝)
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝) | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → v ∉ binders
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ fastAdmitsAux v v (binders ∪ {a✝¹}) a✝
case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ fastAdmitsAux v v (binders ∪ {a✝¹}) a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | all_goals
tauto | case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v = a✝¹ ∨ v = a✝ → v ∉ binders
case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝
case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | simp only [fastAdmitsAux] | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝) | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | by_cases c1 : v = x | v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi | case pos
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
case neg
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | left | case pos
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi | case pos.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | exact c1 | case pos.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | right | case neg
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi | case neg.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ fastAdmitsAux v v (binders ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | apply phi_ih | case neg.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ fastAdmitsAux v v (binders ∪ {x}) phi | case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | simp | case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x} | case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | tauto | case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self | [302, 1] | [321, 10] | tauto | case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmits_self | [324, 1] | [331, 7] | simp only [fastAdmits] | F : Formula
v : VarName
⊢ fastAdmits v v F | F : Formula
v : VarName
⊢ fastAdmitsAux v v ∅ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmits_self | [324, 1] | [331, 7] | apply fastAdmitsAux_self | F : Formula
v : VarName
⊢ fastAdmitsAux v v ∅ F | case h1
F : Formula
v : VarName
⊢ v ∉ ∅ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.fastAdmits_self | [324, 1] | [331, 7] | simp | case h1
F : Formula
v : VarName
⊢ v ∉ ∅ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux | [335, 1] | [348, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v F
⊢ fastAdmitsAux v u binders F | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v true_
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v false_
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝.not_
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux | [335, 1] | [348, 10] | all_goals
simp only [isFreeIn] at h1
simp only [fastAdmitsAux] | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v true_
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v false_
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝.not_
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(v = a✝¹ ∨ v = a✝)
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux | [335, 1] | [348, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(v = a✝¹ ∨ v = a✝)
⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isFreeIn v a✝
⊢ fastAdmitsAux v u binders a✝
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝)
⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝)
⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux | [335, 1] | [348, 10] | simp only [isFreeIn] at h1 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux | [335, 1] | [348, 10] | simp only [fastAdmitsAux] | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux | [335, 1] | [348, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ a✝
⊢ v ∈ a✝ → u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmits | [351, 1] | [358, 50] | simp only [fastAdmits] | F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ fastAdmits v u F | F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ fastAdmitsAux v u ∅ F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmits | [351, 1] | [358, 50] | exact not_isFreeIn_imp_fastAdmitsAux F v u ∅ h1 | F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ fastAdmitsAux v u ∅ F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux | [362, 1] | [385, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u F
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders F | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)
case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders true_
case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders false_
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝.not_
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders a✝.not_
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.imp_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝)
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.and_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝)
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.or_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝)
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.iff_ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝)
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (forall_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝)
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (exists_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝)
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝) |
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