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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
94
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stringlengths 6
10
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stringlengths 1
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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | by_cases n0 : n = 0 | case a
N : Finset ℕ
n : ℕ
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a
N : Finset ℕ
n : ℕ
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | simp_rw [or_iff_right n0] | case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | constructor | case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n ∈ N | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) → n ∈ N
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ n ∈ N → ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rw [n0] | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (0 = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = 0) ↔ 0 = 0 ∨ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | simp | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (0 = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = 0) ↔ 0 = 0 ∨ 0 ∈ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (0 = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = 0) ↔ 0 = 0 ∨ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | intro h | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) → n ∈ N | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
⊢ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) → n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rcases h with ⟨x, ⟨x0, xN⟩, xn⟩ | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
⊢ n ∈ N | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x - 1 + 1 = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
⊢ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rw [Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.mpr x0)] at xn | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x - 1 + 1 = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x - 1 + 1 = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rwa [←xn] | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | intro h | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ n ∈ N → ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ n ∈ N → ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | exists n | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ (¬n = 0 ∧ n ∈ N) ∧ n - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | use ⟨n0,h⟩ | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ (¬n = 0 ∧ n ∈ N) ∧ n - 1 + 1 = n | case right
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ n - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ (¬n = 0 ∧ n ∈ N) ∧ n - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | exact Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.mpr n0) | case right
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ n - 1 + 1 = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ n - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | simp | A B : Finset ℕ
⊢ push A ≤ push B ↔ A ≤ B | A B : Finset ℕ
⊢ push A ⊆ push B ↔ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ push A ≤ push B ↔ A ≤ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | rw [push] | A B : Finset ℕ
⊢ push A ⊆ push B ↔ A ⊆ B | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ push B ↔ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ push A ⊆ push B ↔ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | rw [push] | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ push B ↔ A ⊆ B | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) ↔ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ push B ↔ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | constructor | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) ↔ A ⊆ B | case mp
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) → A ⊆ B
case mpr
A B : Finset ℕ
⊢ A ⊆ B → insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) ↔ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | intro AB | case mp
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) → A ⊆ B | case mp
A B : Finset ℕ
AB : insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
⊢ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) → A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | rw [Finset.subset_iff] at AB ⊢ | case mp
A B : Finset ℕ
AB : insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
⊢ A ⊆ B | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ ∀ ⦃x : ℕ⦄, x ∈ A → x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
⊢ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | intro x xA | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ ∀ ⦃x : ℕ⦄, x ∈ A → x ∈ B | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ ∀ ⦃x : ℕ⦄, x ∈ A → x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | have h : x + 1 ∈ insert 0 (Finset.image (fun n : ℕ ↦ n + 1) A) := by simpa | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x ∈ B | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | specialize AB h | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
⊢ x ∈ B | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | simp at AB | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ x ∈ B | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x ∈ B
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | assumption | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x ∈ B
⊢ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x ∈ B
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | simpa | A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | intro AB | case mpr
A B : Finset ℕ
⊢ A ⊆ B → insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | case mpr
A B : Finset ℕ
AB : A ⊆ B
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A B : Finset ℕ
⊢ A ⊆ B → insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | apply Finset.insert_subset_insert | case mpr
A B : Finset ℕ
AB : A ⊆ B
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | case mpr.h
A B : Finset ℕ
AB : A ⊆ B
⊢ Finset.image (fun n => n + 1) A ⊆ Finset.image (fun n => n + 1) B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A B : Finset ℕ
AB : A ⊆ B
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | apply Finset.image_mono | case mpr.h
A B : Finset ℕ
AB : A ⊆ B
⊢ Finset.image (fun n => n + 1) A ⊆ Finset.image (fun n => n + 1) B | case mpr.h.a
A B : Finset ℕ
AB : A ⊆ B
⊢ A ≤ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.h
A B : Finset ℕ
AB : A ⊆ B
⊢ Finset.image (fun n => n + 1) A ⊆ Finset.image (fun n => n + 1) B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | assumption | case mpr.h.a
A B : Finset ℕ
AB : A ⊆ B
⊢ A ≤ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.h.a
A B : Finset ℕ
AB : A ⊆ B
⊢ A ≤ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_sum | [47, 1] | [49, 23] | rw [push] | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (push N).sum (cons a f) | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (insert 0 (Finset.image (fun n => n + 1) N)).sum (cons a f) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (push N).sum (cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_sum | [47, 1] | [49, 23] | simp | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (insert 0 (Finset.image (fun n => n + 1) N)).sum (cons a f) | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = cons a f 0 + N.sum fun x => cons a f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (insert 0 (Finset.image (fun n => n + 1) N)).sum (cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_sum | [47, 1] | [49, 23] | rfl | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = cons a f 0 + N.sum fun x => cons a f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = cons a f 0 + N.sum fun x => cons a f (x + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_prod | [52, 1] | [53, 23] | rw [push] | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (cons a f) | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (insert 0 (Finset.image (fun n => n + 1) N)).prod (cons a f) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_prod | [52, 1] | [53, 23] | simp | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (insert 0 (Finset.image (fun n => n + 1) N)).prod (cons a f) | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = cons a f 0 * N.prod fun x => cons a f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (insert 0 (Finset.image (fun n => n + 1) N)).prod (cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_prod | [52, 1] | [53, 23] | rfl | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = cons a f 0 * N.prod fun x => cons a f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = cons a f 0 * N.prod fun x => cons a f (x + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [Set.range] | ⊢ Set.range push = {N | 0 ∈ N} | ⊢ {x | ∃ y, push y = x} = {N | 0 ∈ N} | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Set.range push = {N | 0 ∈ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | apply Set.ext | ⊢ {x | ∃ y, push y = x} = {N | 0 ∈ N} | case h
⊢ ∀ (x : Finset ℕ), x ∈ {x | ∃ y, push y = x} ↔ x ∈ {N | 0 ∈ N} | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ {x | ∃ y, push y = x} = {N | 0 ∈ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | simp | case h
⊢ ∀ (x : Finset ℕ), x ∈ {x | ∃ y, push y = x} ↔ x ∈ {N | 0 ∈ N} | case h
⊢ ∀ (x : Finset ℕ), (∃ y, push y = x) ↔ 0 ∈ x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
⊢ ∀ (x : Finset ℕ), x ∈ {x | ∃ y, push y = x} ↔ x ∈ {N | 0 ∈ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | intro N | case h
⊢ ∀ (x : Finset ℕ), (∃ y, push y = x) ↔ 0 ∈ x | case h
N : Finset ℕ
⊢ (∃ y, push y = N) ↔ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h
⊢ ∀ (x : Finset ℕ), (∃ y, push y = x) ↔ 0 ∈ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | constructor | case h
N : Finset ℕ
⊢ (∃ y, push y = N) ↔ 0 ∈ N | case h.mp
N : Finset ℕ
⊢ (∃ y, push y = N) → 0 ∈ N
case h.mpr
N : Finset ℕ
⊢ 0 ∈ N → ∃ y, push y = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : Finset ℕ
⊢ (∃ y, push y = N) ↔ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | intro h | case h.mp
N : Finset ℕ
⊢ (∃ y, push y = N) → 0 ∈ N | case h.mp
N : Finset ℕ
h : ∃ y, push y = N
⊢ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : Finset ℕ
⊢ (∃ y, push y = N) → 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rcases h with ⟨M, H⟩ | case h.mp
N : Finset ℕ
h : ∃ y, push y = N
⊢ 0 ∈ N | case h.mp.intro
N M : Finset ℕ
H : push M = N
⊢ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : Finset ℕ
h : ∃ y, push y = N
⊢ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [push] at H | case h.mp.intro
N M : Finset ℕ
H : push M = N
⊢ 0 ∈ N | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
N M : Finset ℕ
H : push M = N
⊢ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [← H] | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ N | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ insert 0 (Finset.image (fun n => n + 1) M) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | exact Finset.mem_insert_self 0 _ | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ insert 0 (Finset.image (fun n => n + 1) M) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ insert 0 (Finset.image (fun n => n + 1) M)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | intro N0 | case h.mpr
N : Finset ℕ
⊢ 0 ∈ N → ∃ y, push y = N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ ∃ y, push y = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
⊢ 0 ∈ N → ∃ y, push y = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | exists pop N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ ∃ y, push y = N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ push (pop N) = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ ∃ y, push y = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [push_pop] | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ push (pop N) = N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ insert 0 N = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ push (pop N) = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | exact Finset.insert_eq_of_mem N0 | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ insert 0 N = N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ insert 0 N = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | apply Filter.comap_embedding_atTop | ⊢ Filter.comap push atTop = atTop | case hm
⊢ ∀ (b₁ b₂ : Finset ℕ), push b₁ ≤ push b₂ ↔ b₁ ≤ b₂
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Filter.comap push atTop = atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | exact @push_le_push | case hm
⊢ ∀ (b₁ b₂ : Finset ℕ), push b₁ ≤ push b₂ ↔ b₁ ≤ b₂
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | Please generate a tactic in lean4 to solve the state.
STATE:
case hm
⊢ ∀ (b₁ b₂ : Finset ℕ), push b₁ ≤ push b₂ ↔ b₁ ≤ b₂
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | intro N | case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | case hu
N : Finset ℕ
⊢ ∃ b, N ≤ push b | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | exists pop N | case hu
N : Finset ℕ
⊢ ∃ b, N ≤ push b | case hu
N : Finset ℕ
⊢ N ≤ push (pop N) | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
N : Finset ℕ
⊢ ∃ b, N ≤ push b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | rw [push_pop] | case hu
N : Finset ℕ
⊢ N ≤ push (pop N) | case hu
N : Finset ℕ
⊢ N ≤ insert 0 N | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
N : Finset ℕ
⊢ N ≤ push (pop N)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | simp | case hu
N : Finset ℕ
⊢ N ≤ insert 0 N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
N : Finset ℕ
⊢ N ≤ insert 0 N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | nth_rw 1 [← push_comap_atTop] | A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) atTop l ↔ Filter.Tendsto f atTop l | A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) (Filter.comap push atTop) l ↔ Filter.Tendsto f atTop l | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) atTop l ↔ Filter.Tendsto f atTop l
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | apply Filter.tendsto_comap'_iff | A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) (Filter.comap push atTop) l ↔ Filter.Tendsto f atTop l | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Set.range push ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) (Filter.comap push atTop) l ↔ Filter.Tendsto f atTop l
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | rw [push_range] | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Set.range push ∈ atTop | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Set.range push ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | have h : {N : Finset ℕ | 0 ∈ N} = {N : Finset ℕ | {0} ≤ N} := by simp | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} ∈ atTop | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | 0 ∈ N} ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | rw [h] | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | 0 ∈ N} ∈ atTop | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | {0} ≤ N} ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | 0 ∈ N} ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | exact Filter.mem_atTop _ | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | {0} ≤ N} ∈ atTop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | {0} ≤ N} ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | simp | A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} = {N | {0} ≤ N} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} = {N | {0} ≤ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | induction' N using Finset.induction with n N Nn h | N : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n) | case empty
f : ℕ → ℂ
⊢ Complex.abs (∅.sum fun n => f n) ≤ ∅.sum fun n => Complex.abs (f n)
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs ((insert n N).sum fun n => f n) ≤ (insert n N).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
N : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | simp | case empty
f : ℕ → ℂ
⊢ Complex.abs (∅.sum fun n => f n) ≤ ∅.sum fun n => Complex.abs (f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
f : ℕ → ℂ
⊢ Complex.abs (∅.sum fun n => f n) ≤ ∅.sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | rw [Finset.sum_insert Nn] | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs ((insert n N).sum fun n => f n) ≤ (insert n N).sum fun n => Complex.abs (f n) | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ (insert n N).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs ((insert n N).sum fun n => f n) ≤ (insert n N).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | rw [Finset.sum_insert Nn] | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ (insert n N).sum fun n => Complex.abs (f n) | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ (insert n N).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | trans abs (f n) + abs (N.sum fun n ↦ f n) | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + Complex.abs (N.sum fun n => f n)
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | exact Complex.abs.add_le _ _ | f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + Complex.abs (N.sum fun n => f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | apply add_le_add_left | f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | case bc
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun x => Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | assumption | case bc
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun x => Complex.abs (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun x => Complex.abs (f x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | rw [Finset.subset_iff] | A B : Finset ℕ
⊢ B ⊆ A ∪ B \ A | A B : Finset ℕ
⊢ ∀ ⦃x : ℕ⦄, x ∈ B → x ∈ A ∪ B \ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ B ⊆ A ∪ B \ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | intro x Bx | A B : Finset ℕ
⊢ ∀ ⦃x : ℕ⦄, x ∈ B → x ∈ A ∪ B \ A | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∪ B \ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ ∀ ⦃x : ℕ⦄, x ∈ B → x ∈ A ∪ B \ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | rw [Finset.mem_union, Finset.mem_sdiff] | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∪ B \ A | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∪ B \ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | by_cases Ax : x ∈ A | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | case pos
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
case neg
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | left | case pos
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | case pos.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | exact Ax | case pos.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | right | case neg
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | case neg.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ B ∧ x ∉ A | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | exact ⟨Bx, Ax⟩ | case neg.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ B ∧ x ∉ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ B ∧ x ∉ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | mandelbrot_eq_multibrot | [27, 1] | [31, 6] | ext c | ⊢ mandelbrot = multibrot 2 | case h
c : ℂ
⊢ c ∈ mandelbrot ↔ c ∈ multibrot 2 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ mandelbrot = multibrot 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | mandelbrot_eq_multibrot | [27, 1] | [31, 6] | simp only [mandelbrot, mem_setOf_eq, multibrot, f_f'_iter, tendsto_inf_iff_tendsto_atInf,
tendsto_atInf_iff_norm_tendsto_atTop, Complex.norm_eq_abs] | case h
c : ℂ
⊢ c ∈ mandelbrot ↔ c ∈ multibrot 2 | case h
c : ℂ
⊢ ¬Tendsto (fun n => Complex.abs ((fun z => z ^ 2 + c)^[n] c)) atTop atTop ↔
¬Tendsto (fun x => Complex.abs ((f' 2 c)^[x] c)) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
⊢ c ∈ mandelbrot ↔ c ∈ multibrot 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | mandelbrot_eq_multibrot | [27, 1] | [31, 6] | rfl | case h
c : ℂ
⊢ ¬Tendsto (fun n => Complex.abs ((fun z => z ^ 2 + c)^[n] c)) atTop atTop ↔
¬Tendsto (fun x => Complex.abs ((f' 2 c)^[x] c)) atTop atTop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
⊢ ¬Tendsto (fun n => Complex.abs ((fun z => z ^ 2 + c)^[n] c)) atTop atTop ↔
¬Tendsto (fun x => Complex.abs ((f' 2 c)^[x] c)) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_mandelbrot | [34, 1] | [35, 62] | rw [mandelbrot_eq_multibrot] | ⊢ IsConnected mandelbrot | ⊢ IsConnected (multibrot 2) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected mandelbrot
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_mandelbrot | [34, 1] | [35, 62] | exact isConnected_multibrot 2 | ⊢ IsConnected (multibrot 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected (multibrot 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_compl_mandelbrot | [38, 1] | [39, 68] | rw [mandelbrot_eq_multibrot] | ⊢ IsConnected mandelbrotᶜ | ⊢ IsConnected (multibrot 2)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected mandelbrotᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Mandelbrot.lean | isConnected_compl_mandelbrot | [38, 1] | [39, 68] | exact isConnected_compl_multibrot 2 | ⊢ IsConnected (multibrot 2)ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ IsConnected (multibrot 2)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | closure_inter_subset_compl | [25, 1] | [28, 82] | rw [← vo.isClosed_compl.closure_eq] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ vᶜ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ closure vᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ vᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | closure_inter_subset_compl | [25, 1] | [28, 82] | apply closure_mono | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ closure vᶜ | case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ s ∩ u ⊆ vᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ closure vᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | closure_inter_subset_compl | [25, 1] | [28, 82] | exact _root_.trans (inter_subset_right _ _) (Disjoint.subset_compl_left d.symm) | case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ s ∩ u ⊆ vᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ s ∩ u ⊆ vᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | rw [←closure_subset_iff_isClosed, ←diff_eq_empty] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ IsClosed (s ∩ u) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ closure (s ∩ u) \ (s ∩ u) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ IsClosed (s ∩ u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | by_contra h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ closure (s ∩ u) \ (s ∩ u) = ∅ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : ¬closure (s ∩ u) \ (s ∩ u) = ∅
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ closure (s ∩ u) \ (s ∩ u) = ∅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | simp only [← ne_eq, ← nonempty_iff_ne_empty] at h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : ¬closure (s ∩ u) \ (s ∩ u) = ∅
⊢ False | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : (closure (s ∩ u) \ (s ∩ u)).Nonempty
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : ¬closure (s ∩ u) \ (s ∩ u) = ∅
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | rcases h with ⟨x, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : (closure (s ∩ u) \ (s ∩ u)).Nonempty
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) \ (s ∩ u)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : (closure (s ∩ u) \ (s ∩ u)).Nonempty
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | simp only [mem_diff, mem_inter_iff, not_and] at h | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) \ (s ∩ u)
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) \ (s ∩ u)
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | have sus : closure (s ∩ u) ⊆ s := by
nth_rw 2 [← sc.closure_eq]; apply closure_mono; apply inter_subset_left | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | have xs := sus h.1 | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | have m := not_or.mpr ⟨h.2 xs, not_mem_of_mem_compl (closure_inter_subset_compl vo d h.1)⟩ | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : ¬(x ∈ u ∨ x ∈ v)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | rw [← mem_union _ _ _] at m | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : ¬(x ∈ u ∨ x ∈ v)
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : x ∉ u ∪ v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : ¬(x ∈ u ∨ x ∈ v)
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | exact not_mem_subset suv m xs | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : x ∉ u ∪ v
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : x ∉ u ∪ v
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | nth_rw 2 [← sc.closure_eq] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ s | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ closure s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | apply closure_mono | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ closure s | case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ s ∩ u ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ closure s
TACTIC:
|
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