url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | Nat.ge_mod | [34, 1] | [58, 28] | apply ih | x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ Nat.modCore (succ n✝) (succ n + 1) < succ n + 1 | case a
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ n✝ < succ x | Please generate a tactic in lean4 to solve the state.
STATE:
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ Nat.modCore (succ n✝) (succ n + 1) < succ n + 1
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | Nat.ge_mod | [34, 1] | [58, 28] | apply Nat.lt_of_succ_le | case a
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ n✝ < succ x | case a.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ succ n✝ ≤ succ x | Please generate a tactic in lean4 to solve the state.
STATE:
case a
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ n✝ < succ x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | Nat.ge_mod | [34, 1] | [58, 28] | apply Nat.le_succ_of_le | case a.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ succ n✝ ≤ succ x | case a.h.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ succ n✝ ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ succ n✝ ≤ succ x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | Nat.ge_mod | [34, 1] | [58, 28] | rw [←hm] | case a.h.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ succ n✝ ≤ x | case a.h.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ x - n ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ succ n✝ ≤ x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | Nat.ge_mod | [34, 1] | [58, 28] | apply sub_le | case a.h.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ x - n ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : succ n ≤ x + 1
n✝ : ℕ
hm : x - n = succ n✝
⊢ x - n ≤ x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | Nat.ge_mod | [34, 1] | [58, 28] | apply Nat.succ_lt_succ_iff.mpr | case neg
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : ¬succ n ≤ x + 1
⊢ succ (succ x) < succ n + 1 | case neg
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : ¬succ n ≤ x + 1
⊢ x + 1 < n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : ¬succ n ≤ x + 1
⊢ succ (succ x) < succ n + 1
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | Nat.ge_mod | [34, 1] | [58, 28] | apply not_le.mp h | case neg
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : ¬succ n ≤ x + 1
⊢ x + 1 < n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x n : ℕ
ih : ∀ m < succ x, Nat.modCore (succ m) (succ n + 1) < succ n + 1
h : ¬succ n ≤ x + 1
⊢ x + 1 < n + 1
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | not_le_mod | [60, 1] | [61, 33] | simp only [not_le, Nat.ge_mod] | x n : ℕ
⊢ ¬n + 1 ≤ x % (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x n : ℕ
⊢ ¬n + 1 ≤ x % (n + 1)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | sumOfSequence_eq | [65, 1] | [77, 6] | simp [Nat.mul_add] | n : ℕ
⊢ sumOfSequence n = n * (n + 1) / 2 | n : ℕ
⊢ sumOfSequence n = (n * n + n) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ sumOfSequence n = n * (n + 1) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | sumOfSequence_eq | [65, 1] | [77, 6] | induction n | n : ℕ
⊢ sumOfSequence n = (n * n + n) / 2 | case zero
⊢ sumOfSequence Nat.zero = (Nat.zero * Nat.zero + Nat.zero) / 2
case succ
n✝ : ℕ
n_ih✝ : sumOfSequence n✝ = (n✝ * n✝ + n✝) / 2
⊢ sumOfSequence (Nat.succ n✝) = (Nat.succ n✝ * Nat.succ n✝ + Nat.succ n✝) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ sumOfSequence n = (n * n + n) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | sumOfSequence_eq | [65, 1] | [77, 6] | next => rfl | case zero
⊢ sumOfSequence Nat.zero = (Nat.zero * Nat.zero + Nat.zero) / 2
case succ
n✝ : ℕ
n_ih✝ : sumOfSequence n✝ = (n✝ * n✝ + n✝) / 2
⊢ sumOfSequence (Nat.succ n✝) = (Nat.succ n✝ * Nat.succ n✝ + Nat.succ n✝) / 2 | case succ
n✝ : ℕ
n_ih✝ : sumOfSequence n✝ = (n✝ * n✝ + n✝) / 2
⊢ sumOfSequence (Nat.succ n✝) = (Nat.succ n✝ * Nat.succ n✝ + Nat.succ n✝) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ sumOfSequence Nat.zero = (Nat.zero * Nat.zero + Nat.zero) / 2
case succ
n✝ : ℕ
n_ih✝ : sumOfSequence n✝ = (n✝ * n✝ + n✝) / 2
⊢ sumOfSequence (Nat.succ n✝) = (Nat.succ n✝ * Nat.succ n✝ + Nat.succ n✝) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | sumOfSequence_eq | [65, 1] | [77, 6] | next n ih =>
simp only [sumOfSequence, ih, Nat.add_succ, add_zero, Nat.succ_mul, Nat.mul_succ, Nat.succ_add]
simp [
Nat.add_div (b:=2),
Nat.add_assoc (_+n) n n,
←Nat.mul_two n,
Nat.add_div (b := n*2)
] | case succ
n✝ : ℕ
n_ih✝ : sumOfSequence n✝ = (n✝ * n✝ + n✝) / 2
⊢ sumOfSequence (Nat.succ n✝) = (Nat.succ n✝ * Nat.succ n✝ + Nat.succ n✝) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
n_ih✝ : sumOfSequence n✝ = (n✝ * n✝ + n✝) / 2
⊢ sumOfSequence (Nat.succ n✝) = (Nat.succ n✝ * Nat.succ n✝ + Nat.succ n✝) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | sumOfSequence_eq | [65, 1] | [77, 6] | rfl | ⊢ sumOfSequence Nat.zero = (Nat.zero * Nat.zero + Nat.zero) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ sumOfSequence Nat.zero = (Nat.zero * Nat.zero + Nat.zero) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | sumOfSequence_eq | [65, 1] | [77, 6] | simp only [sumOfSequence, ih, Nat.add_succ, add_zero, Nat.succ_mul, Nat.mul_succ, Nat.succ_add] | n : ℕ
ih : sumOfSequence n = (n * n + n) / 2
⊢ sumOfSequence (Nat.succ n) = (Nat.succ n * Nat.succ n + Nat.succ n) / 2 | n : ℕ
ih : sumOfSequence n = (n * n + n) / 2
⊢ Nat.succ ((n * n + n) / 2 + n) = Nat.succ (Nat.succ (n * n + n + n + n)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : sumOfSequence n = (n * n + n) / 2
⊢ sumOfSequence (Nat.succ n) = (Nat.succ n * Nat.succ n + Nat.succ n) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02Solutions.lean | sumOfSequence_eq | [65, 1] | [77, 6] | simp [
Nat.add_div (b:=2),
Nat.add_assoc (_+n) n n,
←Nat.mul_two n,
Nat.add_div (b := n*2)
] | n : ℕ
ih : sumOfSequence n = (n * n + n) / 2
⊢ Nat.succ ((n * n + n) / 2 + n) = Nat.succ (Nat.succ (n * n + n + n + n)) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : sumOfSequence n = (n * n + n) / 2
⊢ Nat.succ ((n * n + n) / 2 + n) = Nat.succ (Nat.succ (n * n + n + n + n)) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Exercise02SumOfSequence.lean | sumOfSequence_eq | [22, 1] | [24, 8] | sorry | n : ℕ
⊢ sumOfSequence n = n * (n + 1) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ sumOfSequence n = n * (n + 1) / 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.zero_eq | [96, 1] | [98, 6] | rfl | ⊢ 0 = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 = zero
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.one_eq_succ_zero | [100, 1] | [102, 6] | rfl | ⊢ 1 = succ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 1 = succ 0
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.two_eq_succ_one | [104, 1] | [106, 6] | rfl | ⊢ 2 = succ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 2 = succ 1
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.three_eq_succ_two | [108, 1] | [110, 6] | rfl | ⊢ 3 = succ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 3 = succ 2
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.three_eq_succ_succ_succ_zero | [124, 1] | [129, 15] | rw [three_eq_succ_two] | ⊢ 3 = succ (succ (succ zero)) | ⊢ succ 2 = succ (succ (succ zero)) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 3 = succ (succ (succ zero))
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.three_eq_succ_succ_succ_zero | [124, 1] | [129, 15] | rw [two_eq_succ_one] | ⊢ succ 2 = succ (succ (succ zero)) | ⊢ succ (succ 1) = succ (succ (succ zero)) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ succ 2 = succ (succ (succ zero))
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.three_eq_succ_succ_succ_zero | [124, 1] | [129, 15] | rw [one_eq_succ_zero] | ⊢ succ (succ 1) = succ (succ (succ zero)) | ⊢ succ (succ (succ 0)) = succ (succ (succ zero)) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ succ (succ 1) = succ (succ (succ zero))
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.three_eq_succ_succ_succ_zero | [124, 1] | [129, 15] | rw [zero_eq] | ⊢ succ (succ (succ 0)) = succ (succ (succ zero)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ succ (succ (succ 0)) = succ (succ (succ zero))
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.three_eq_succ_succ_succ_zero' | [135, 1] | [137, 69] | rw [three_eq_succ_two, two_eq_succ_one, one_eq_succ_zero, zero_eq] | ⊢ 3 = succ (succ (succ zero)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 3 = succ (succ (succ zero))
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_zero | [185, 1] | [188, 6] | intro y | ⊢ ∀ (x : Nat), x + 0 = x | y : Nat
⊢ y + 0 = y | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (x : Nat), x + 0 = x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_zero | [185, 1] | [188, 6] | rfl | y : Nat
⊢ y + 0 = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y : Nat
⊢ y + 0 = y
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_succ | [191, 1] | [194, 6] | intro x y | ⊢ ∀ (x y : Nat), x + succ y = succ (x + y) | x y : Nat
⊢ x + succ y = succ (x + y) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (x y : Nat), x + succ y = succ (x + y)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_succ | [191, 1] | [194, 6] | rfl | x y : Nat
⊢ x + succ y = succ (x + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ x + succ y = succ (x + y)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.zero_add | [197, 1] | [256, 22] | intro x | ⊢ ∀ (x : Nat), 0 + x = x | x : Nat
⊢ 0 + x = x | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (x : Nat), 0 + x = x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.zero_add | [197, 1] | [256, 22] | induction x | x : Nat
⊢ 0 + x = x | case zero
⊢ 0 + zero = zero
case succ
n✝ : Nat
n_ih✝ : 0 + n✝ = n✝
⊢ 0 + succ n✝ = succ n✝ | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ 0 + x = x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.zero_add | [197, 1] | [256, 22] | case zero =>
rfl | ⊢ 0 + zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 + zero = zero
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.zero_add | [197, 1] | [256, 22] | case succ x' ih =>
rw [add_succ, ih] | x' : Nat
ih : 0 + x' = x'
⊢ 0 + succ x' = succ x' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x' : Nat
ih : 0 + x' = x'
⊢ 0 + succ x' = succ x'
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.zero_add | [197, 1] | [256, 22] | rfl | ⊢ 0 + zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0 + zero = zero
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.zero_add | [197, 1] | [256, 22] | rw [add_succ, ih] | x' : Nat
ih : 0 + x' = x'
⊢ 0 + succ x' = succ x' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x' : Nat
ih : 0 + x' = x'
⊢ 0 + succ x' = succ x'
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.succ_add | [267, 1] | [271, 48] | induction y | x y : Nat
⊢ succ x + y = succ (x + y) | case zero
x : Nat
⊢ succ x + zero = succ (x + zero)
case succ
x n✝ : Nat
n_ih✝ : succ x + n✝ = succ (x + n✝)
⊢ succ x + succ n✝ = succ (x + succ n✝) | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ succ x + y = succ (x + y)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.succ_add | [267, 1] | [271, 48] | case zero => rfl | x : Nat
⊢ succ x + zero = succ (x + zero) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ succ x + zero = succ (x + zero)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.succ_add | [267, 1] | [271, 48] | case succ y ih => rw [add_succ, add_succ, ih] | x y : Nat
ih : succ x + y = succ (x + y)
⊢ succ x + succ y = succ (x + succ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
ih : succ x + y = succ (x + y)
⊢ succ x + succ y = succ (x + succ y)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.succ_add | [267, 1] | [271, 48] | rfl | x : Nat
⊢ succ x + zero = succ (x + zero) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ succ x + zero = succ (x + zero)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.succ_add | [267, 1] | [271, 48] | rw [add_succ, add_succ, ih] | x y : Nat
ih : succ x + y = succ (x + y)
⊢ succ x + succ y = succ (x + succ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
ih : succ x + y = succ (x + y)
⊢ succ x + succ y = succ (x + succ y)
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_comm | [287, 1] | [293, 10] | induction y | x y : Nat
⊢ x + y = y + x | case zero
x : Nat
⊢ x + zero = zero + x
case succ
x n✝ : Nat
n_ih✝ : x + n✝ = n✝ + x
⊢ x + succ n✝ = succ n✝ + x | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ x + y = y + x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_comm | [287, 1] | [293, 10] | case zero =>
rw [←zero_eq, zero_add, add_zero] | x : Nat
⊢ x + zero = zero + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ x + zero = zero + x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_comm | [287, 1] | [293, 10] | case succ y ih =>
sorry | x y : Nat
ih : x + y = y + x
⊢ x + succ y = succ y + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
ih : x + y = y + x
⊢ x + succ y = succ y + x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_comm | [287, 1] | [293, 10] | rw [←zero_eq, zero_add, add_zero] | x : Nat
⊢ x + zero = zero + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ x + zero = zero + x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_comm | [287, 1] | [293, 10] | sorry | x y : Nat
ih : x + y = y + x
⊢ x + succ y = succ y + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
ih : x + y = y + x
⊢ x + succ y = succ y + x
TACTIC:
|
https://github.com/alexkeizer/LeanMeetup.git | 98bea524b3beaa38c37d61d1fa7592c24fa36c52 | LeanMeetup/Demo01.lean | LeanDemo.add_assoc | [298, 1] | [300, 8] | sorry | x y z : Nat
⊢ x + y + z = x + (y + z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x + y + z = x + (y + z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.nonempty_ps | [48, 1] | [49, 73] | simp only [Super.ps, mem_setOf, eq_self_iff_true, true_or_iff] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
⊢ 1 ∈ s.ps c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
⊢ 1 ∈ s.ps c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | have pc : Continuous (s.potential c) := (Continuous.potential s).along_snd | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
⊢ IsCompact (s.ps c) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
⊢ IsCompact (s.ps c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
⊢ IsCompact (s.ps c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | have c1 : IsCompact {(1 : ℝ)} := isCompact_singleton | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
⊢ IsCompact (s.ps c) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
⊢ IsCompact (s.ps c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
⊢ IsComp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | convert c1.union ((s.isClosed_critical_not_a.snd_preimage c).isCompact.image pc) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
⊢ IsCompact (s.ps c) | case h.e'_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
⊢ s.ps c = {1} ∪ s.potential c '' {b |... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | apply Set.ext | case h.e'_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
⊢ s.ps c = {1} ∪ s.potential c '' {b |... | case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
⊢ ∀ (x : ℝ), x ∈ s.ps c ↔ x ∈ {1} ∪ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | intro p | case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
⊢ ∀ (x : ℝ), x ∈ s.ps c ↔ x ∈ {1} ∪ ... | case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ p ∈ s.ps c ↔ p ∈ {1} ∪ s.pot... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potenti... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | simp only [mem_setOf, Super.ps, mem_singleton_iff, mem_union, mem_image, Ne, ←
s.potential_eq_zero_of_onePreimage c] | case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ p ∈ s.ps c ↔ p ∈ {1} ∪ s.pot... | case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (p = 1 ∨ ¬p = 0 ∧ ∃ z, s.pot... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potenti... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | apply or_congr_right | case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (p = 1 ∨ ¬p = 0 ∧ ∃ z, s.pot... | case h.e'_3.h.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (¬p = 0 ∧ ∃ z, s.potential... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potenti... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | constructor | case h.e'_3.h.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (¬p = 0 ∧ ∃ z, s.potential... | case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (¬p = 0 ∧ ∃ z, s.potent... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.poten... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | intro ⟨p0, z, e, c⟩ | case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (¬p = 0 ∧ ∃ z, s.potent... | case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
p0 : ¬p = 0
z : S
e : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.po... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | rw [← e] at p0 | case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
p0 : ¬p = 0
z : S
e : ... | case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
z : S
p0 : ¬s.potentia... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | exact ⟨z, ⟨c, p0⟩, e⟩ | case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
z : S
p0 : ¬s.potentia... | case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (∃ x, (Critical (f c) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | intro ⟨z, ⟨c, p0⟩, e⟩ | case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c)
c1 : IsCompact {1}
p : ℝ
⊢ (∃ x, (Critical (f c) ... | case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
z : S
c : Critical (f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | rw [e] at p0 | case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
z : S
c : Critical (f... | case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
z : S
c : Critical (f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.compact_ps | [52, 1] | [61, 59] | exact ⟨p0, z, e, c⟩ | case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s.potential c✝)
c1 : IsCompact {1}
p : ℝ
z : S
c : Critical (f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.h.mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
pc : Continuous (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | cases' m with m m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p ∈ s.ps c
⊢ 0 < p | case inl
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p = 1
⊢ 0 < p
case inr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Sp... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p ∈ s.ps c
⊢ 0 < p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | simp only [m, zero_lt_one] | case inl
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p = 1
⊢ 0 < p
case inr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Sp... | case inr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p ≠ 0 ∧ ∃ z, s.potential c z = p ∧ Critical (f c) z
⊢ 0 < p | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p = 1
⊢ 0 < p
case inr
S : Type
inst... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | rcases m with ⟨p0, z, e, c⟩ | case inr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p ≠ 0 ∧ ∃ z, s.potential c z = p ∧ Critical (f c) z
⊢ 0 < p | case inr.intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c✝ : ℂ
p : ℝ
p0 : p ≠ 0
z : S
e : s.potential c✝ z = p
c : Critical (f c✝) z
⊢ 0 < p | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
p : ℝ
m : p ≠ 0 ∧ ∃ z, s.potential c z = p ∧ Cr... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | rw [← e] at p0 ⊢ | case inr.intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c✝ : ℂ
p : ℝ
p0 : p ≠ 0
z : S
e : s.potential c✝ z = p
c : Critical (f c✝) z
⊢ 0 < p | case inr.intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c✝ : ℂ
p : ℝ
z : S
p0 : s.potential c✝ z ≠ 0
e : s.potential c✝ z = p
c : Critical (f c✝) z
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c✝ : ℂ
p : ℝ
p0 : p ≠ 0
z : S
e :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.ps_pos | [64, 1] | [66, 44] | exact p0.symm.lt_of_le s.potential_nonneg | case inr.intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c✝ : ℂ
p : ℝ
z : S
p0 : s.potential c✝ z ≠ 0
e : s.potential c✝ z = p
c : Critical (f c✝) z
⊢ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c✝ : ℂ
p : ℝ
z : S
p0 : s.potenti... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.mem_ps | [73, 1] | [74, 95] | rw [← s.compact_ps.isClosed.closure_eq] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
inst✝ : OnePreimage s
⊢ s.p c ∈ s.ps c | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
inst✝ : OnePreimage s
⊢ s.p c ∈ closure (s.ps c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
inst✝ : OnePreimage s
⊢ s.p c ∈ s.ps c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.mem_ps | [73, 1] | [74, 95] | exact csInf_mem_closure s.nonempty_ps s.bddBelow_ps | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
inst✝ : OnePreimage s
⊢ s.p c ∈ closure (s.ps c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
inst✝ : OnePreimage s
⊢ s.p c ∈ closure (s.ps c)
TACTIC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | intro c p h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
⊢ LowerSemicontinuous s.p | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : p < s.p c
⊢ ∀ᶠ (x' : ℂ) in 𝓝 c, p < s.p x' | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
⊢ LowerSemicontinuous s.p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | contrapose h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : p < s.p c
⊢ ∀ᶠ (x' : ℂ) in 𝓝 c, p < s.p x' | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ¬∀ᶠ (x' : ℂ) in 𝓝 c, p < s.p x'
⊢ ¬p < s.p c | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : p < s.p c
⊢ ∀ᶠ (x' : ℂ)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [not_lt, Filter.not_eventually] at h ⊢ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ¬∀ᶠ (x' : ℂ) in 𝓝 c, p < s.p x'
⊢ ¬p < s.p c | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
⊢ s.p c ≤ p | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ¬∀ᶠ (x' : ℂ) in 𝓝 c, p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | apply le_of_forall_lt' | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
⊢ s.p c ≤ p | case H
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
⊢ ∀ (c_1 : ℝ), p < c_1 → s.p c < c_1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | intro q' pq' | case H
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
⊢ ∀ (c_1 : ℝ), p < c_1 → s.p c < c_1 | case H
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q' : ℝ
pq' : p < q'
⊢ s.p c < q' | Please generate a tactic in lean4 to solve the state.
STATE:
case H
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases exists_between pq' with ⟨q, pq, qq⟩ | case H
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q' : ℝ
pq' : p < q'
⊢ s.p c < q' | case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q' : ℝ
pq' : p < q'
q : ℝ
pq : p < ... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine lt_of_le_of_lt ?_ qq | case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q' : ℝ
pq' : p < q'
q : ℝ
pq : p < ... | case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q' : ℝ
pq' : p < q'
q : ℝ
pq : p < ... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | clear qq pq' q' | case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q' : ℝ
pq' : p < q'
q : ℝ
pq : p < ... | case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
⊢ s.p c ≤ q | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | by_cases q1 : 1 ≤ q | case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
⊢ s.p c ≤ q | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : 1 ≤ q
⊢ s.p c ≤ q
case... | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | exact _root_.trans s.p_le_one q1 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : 1 ≤ q
⊢ s.p c ≤ q
case... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : ¬1 ≤ q
⊢ s.p c ≤ q | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [not_le] at q1 | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : ¬1 ≤ q
⊢ s.p c ≤ q | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
⊢ s.p c ≤ q | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | set t : Set (ℂ × S) := {x | s.potential x.1 x.2 ≤ q ∧ Critical (f x.1) x.2 ∧ x.2 ≠ a} | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
⊢ s.p c ≤ q | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | have ct : IsClosed t :=
(isClosed_le (Continuous.potential s) continuous_const).inter s.isClosed_critical_not_a | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | set u := Prod.fst '' t | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | have cu : IsClosed u := isClosedMap_fst_of_compactSpace _ ct | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | suffices m : c ∈ u by
rcases(mem_image _ _ _).mp m with ⟨⟨c', z⟩, ⟨zp, zc, za⟩, cc⟩
simp only at cc za zc zp; simp only [cc] at za zc zp; clear cc c'
simp only [Ne, ← s.potential_eq_zero_of_onePreimage c] at za
refine _root_.trans (csInf_le s.bddBelow_ps ?_) zp; right; use za, z, rfl, zc | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine Filter.Frequently.mem_of_closed ?_ cu | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine h.mp (eventually_of_forall fun e h ↦ ?_) | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) :... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases exists_lt_of_csInf_lt s.nonempty_ps (lt_of_le_of_lt h pq) with ⟨r, m, rq⟩ | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) ... | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | cases' m with m m | case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : ... | case neg.intro.intro.inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | linarith | case neg.intro.intro.inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
... | case neg.intro.intro.inr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases m with ⟨r0, z, zr, zc⟩ | case neg.intro.intro.inr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
... | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rw [← zr, Ne, s.potential_eq_zero_of_onePreimage] at r0 | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreima... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rw [mem_image] | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreima... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine ⟨(e, z), ⟨?_, zc, r0⟩, rfl⟩ | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreima... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [zr] | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreima... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | exact rq.le | case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h✝ : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.inr.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreima... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases(mem_image _ _ _).mp m with ⟨⟨c', z⟩, ⟨zp, zc, za⟩, cc⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 : q < 1
t : Set (ℂ × S) := {x | s.... | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only at cc za zc zp | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [cc] at za zc zp | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | clear cc c' | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [Ne, ← s.potential_eq_zero_of_onePreimage c] at za | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
... |
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