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/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.mul | [42, 11] | [46, 63] | linear_combination c * hx + b * hy | n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a * c - b * d = n * (x * c + b * y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c d x : ℤ
hx : a - b = n * x
y : ℤ
hy : c - d = n * y
⊢ a * c - b * d = n * (x * c + b * y)
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.pow | [54, 11] | [57, 45] | induction k | n a b : ℤ
k : ℕ
h : a ≡ b [ZMOD n]
⊢ a ^ k ≡ b ^ k [ZMOD n] | case zero
n a b : ℤ
h : a ≡ b [ZMOD n]
⊢ a ^ Nat.zero ≡ b ^ Nat.zero [ZMOD n]
case succ
n a b : ℤ
h : a ≡ b [ZMOD n]
n✝ : ℕ
n_ih✝ : a ^ n✝ ≡ b ^ n✝ [ZMOD n]
⊢ a ^ Nat.succ n✝ ≡ b ^ Nat.succ n✝ [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
n a b : ℤ
k : ℕ
h : a ≡ b [ZMOD n]
⊢ a ^ k ≡ b ^ k [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.pow | [54, 11] | [57, 45] | case zero => exact Int.ModEq.refl _ | n a b : ℤ
h : a ≡ b [ZMOD n]
⊢ a ^ Nat.zero ≡ b ^ Nat.zero [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b : ℤ
h : a ≡ b [ZMOD n]
⊢ a ^ Nat.zero ≡ b ^ Nat.zero [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.pow | [54, 11] | [57, 45] | case succ k hk => exact Int.ModEq.mul hk h | n a b : ℤ
h : a ≡ b [ZMOD n]
k : ℕ
hk : a ^ k ≡ b ^ k [ZMOD n]
⊢ a ^ Nat.succ k ≡ b ^ Nat.succ k [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b : ℤ
h : a ≡ b [ZMOD n]
k : ℕ
hk : a ^ k ≡ b ^ k [ZMOD n]
⊢ a ^ Nat.succ k ≡ b ^ Nat.succ k [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.pow | [54, 11] | [57, 45] | exact Int.ModEq.refl _ | n a b : ℤ
h : a ≡ b [ZMOD n]
⊢ a ^ Nat.zero ≡ b ^ Nat.zero [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b : ℤ
h : a ≡ b [ZMOD n]
⊢ a ^ Nat.zero ≡ b ^ Nat.zero [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.pow | [54, 11] | [57, 45] | exact Int.ModEq.mul hk h | n a b : ℤ
h : a ≡ b [ZMOD n]
k : ℕ
hk : a ^ k ≡ b ^ k [ZMOD n]
⊢ a ^ Nat.succ k ≡ b ^ Nat.succ k [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b : ℤ
h : a ≡ b [ZMOD n]
k : ℕ
hk : a ^ k ≡ b ^ k [ZMOD n]
⊢ a ^ Nat.succ k ≡ b ^ Nat.succ k [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.symm | [59, 11] | [61, 41] | obtain ⟨x, hx⟩ := h | n a b : ℤ
h : a ≡ b [ZMOD n]
⊢ b ≡ a [ZMOD n] | case intro
n a b x : ℤ
hx : a - b = n * x
⊢ b ≡ a [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
n a b : ℤ
h : a ≡ b [ZMOD n]
⊢ b ≡ a [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.symm | [59, 11] | [61, 41] | exact ⟨-x, by linear_combination - hx⟩ | case intro
n a b x : ℤ
hx : a - b = n * x
⊢ b ≡ a [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n a b x : ℤ
hx : a - b = n * x
⊢ b ≡ a [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.symm | [59, 11] | [61, 41] | linear_combination - hx | n a b x : ℤ
hx : a - b = n * x
⊢ b - a = n * -x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b x : ℤ
hx : a - b = n * x
⊢ b - a = n * -x
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.trans | [64, 11] | [67, 47] | obtain ⟨x, hx⟩ := h1 | n a b c : ℤ
h1 : a ≡ b [ZMOD n]
h2 : b ≡ c [ZMOD n]
⊢ a ≡ c [ZMOD n] | case intro
n a b c : ℤ
h2 : b ≡ c [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a ≡ c [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c : ℤ
h1 : a ≡ b [ZMOD n]
h2 : b ≡ c [ZMOD n]
⊢ a ≡ c [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.trans | [64, 11] | [67, 47] | obtain ⟨y, hy⟩ := h2 | case intro
n a b c : ℤ
h2 : b ≡ c [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a ≡ c [ZMOD n] | case intro.intro
n a b c x : ℤ
hx : a - b = n * x
y : ℤ
hy : b - c = n * y
⊢ a ≡ c [ZMOD n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n a b c : ℤ
h2 : b ≡ c [ZMOD n]
x : ℤ
hx : a - b = n * x
⊢ a ≡ c [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.trans | [64, 11] | [67, 47] | exact ⟨x + y, by linear_combination hx + hy⟩ | case intro.intro
n a b c x : ℤ
hx : a - b = n * x
y : ℤ
hy : b - c = n * y
⊢ a ≡ c [ZMOD n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n a b c x : ℤ
hx : a - b = n * x
y : ℤ
hy : b - c = n * y
⊢ a ≡ c [ZMOD n]
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.ModEq.trans | [64, 11] | [67, 47] | linear_combination hx + hy | n a b c x : ℤ
hx : a - b = n * x
y : ℤ
hy : b - c = n * y
⊢ a - c = n * (x + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c x : ℤ
hx : a - b = n * x
y : ℤ
hy : b - c = n * y
⊢ a - c = n * (x + y)
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_fac_zero | [72, 1] | [72, 64] | ring | n t : ℤ
⊢ n * t - 0 = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t : ℤ
⊢ n * t - 0 = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_fac_zero' | [73, 1] | [73, 65] | ring | n t : ℤ
⊢ t * n - 0 = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t : ℤ
⊢ t * n - 0 = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_zero_fac | [74, 1] | [74, 65] | ring | n t : ℤ
⊢ 0 - n * t = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t : ℤ
⊢ 0 - n * t = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_zero_fac' | [75, 1] | [75, 66] | ring | n t : ℤ
⊢ 0 - t * n = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t : ℤ
⊢ 0 - t * n = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self | [76, 1] | [76, 72] | ring | n a t : ℤ
⊢ a + n * t - a = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a + n * t - a = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self' | [77, 1] | [77, 73] | ring | n t a : ℤ
⊢ n * t + a - a = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t a : ℤ
⊢ n * t + a - a = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self'' | [78, 1] | [78, 74] | ring | n a t : ℤ
⊢ a + t * n - a = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a + t * n - a = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self''' | [79, 1] | [79, 75] | ring | n t a : ℤ
⊢ t * n + a - a = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t a : ℤ
⊢ t * n + a - a = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self | [80, 1] | [80, 73] | ring | n a t : ℤ
⊢ a - n * t - a = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - n * t - a = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self' | [81, 1] | [81, 74] | ring | n t a : ℤ
⊢ n * t - a - -a = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t a : ℤ
⊢ n * t - a - -a = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self'' | [82, 1] | [82, 75] | ring | n a t : ℤ
⊢ a - t * n - a = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - t * n - a = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self''' | [83, 1] | [83, 76] | ring | n t a : ℤ
⊢ t * n - a - -a = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n t a : ℤ
⊢ t * n - a - -a = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self_symm | [84, 1] | [84, 78] | ring | n a t : ℤ
⊢ a - (a + n * t) = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - (a + n * t) = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self_symm' | [85, 1] | [85, 79] | ring | n a t : ℤ
⊢ a - (n * t + a) = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - (n * t + a) = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self_symm'' | [86, 1] | [86, 80] | ring | n a t : ℤ
⊢ a - (a + t * n) = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - (a + t * n) = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_add_fac_self_symm''' | [87, 1] | [87, 81] | ring | n a t : ℤ
⊢ a - (t * n + a) = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - (t * n + a) = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self_symm | [88, 1] | [88, 77] | ring | n a t : ℤ
⊢ a - (a - n * t) = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - (a - n * t) = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self_symm' | [89, 1] | [89, 80] | ring | n a t : ℤ
⊢ -a - (n * t - a) = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ -a - (n * t - a) = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self_symm'' | [90, 1] | [90, 79] | ring | n a t : ℤ
⊢ a - (a - t * n) = n * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ a - (a - t * n) = n * t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/ModEq/Lemmas.lean | BrownCs22.Int.modEq_sub_fac_self_symm''' | [91, 1] | [91, 82] | ring | n a t : ℤ
⊢ -a - (t * n - a) = n * -t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a t : ℤ
⊢ -a - (t * n - a) = n * -t
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/Defs.lean | BrownCs22.Nat.quotient_remainder | [10, 1] | [12, 38] | use a/b | a b c : ℕ
h : a % b = c
⊢ ∃ q, a = q * b + c | case h
a b c : ℕ
h : a % b = c
⊢ a = a / b * b + c | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℕ
h : a % b = c
⊢ ∃ q, a = q * b + c
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/Defs.lean | BrownCs22.Nat.quotient_remainder | [10, 1] | [12, 38] | rw [← h, mul_comm, Nat.div_add_mod] | case h
a b c : ℕ
h : a % b = c
⊢ a = a / b * b + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b c : ℕ
h : a % b = c
⊢ a = a / b * b + c
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/Defs.lean | BrownCs22.Set.inter_union_cancel_left | [19, 1] | [20, 29] | simp | α : Type u
s t : Set α
⊢ s ∩ t ∪ s = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
s t : Set α
⊢ s ∩ t ∪ s = s
TACTIC:
|
https://github.com/brown-cs22/CS22-Lean-2024.git | f1ad3fa764a40b2057b4a173dbdebf97394178bc | BrownCs22/Library/Defs.lean | BrownCs22.Set.inter_union_cancel_right | [22, 1] | [23, 29] | simp | α : Type u
s t : Set α
⊢ s ∩ t ∪ t = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u
s t : Set α
⊢ s ∩ t ∪ t = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole | [57, 1] | [66, 43] | rw [← compl_ne_univ] | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
⊢ Set.Nonempty (s ∩ t) | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
⊢ (s ∩ t)ᶜ ≠ univ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
⊢ Set.Nonempty (s ∩ t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole | [57, 1] | [66, 43] | intro h' | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
⊢ (s ∩ t)ᶜ ≠ univ | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
⊢ (s ∩ t)ᶜ ≠ univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole | [57, 1] | [66, 43] | apply not_le.mpr h | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ False | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard s + ncard t ≤ Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole | [57, 1] | [66, 43] | rw [← Set.ncard_union_add_ncard_inter] | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard s + ncard t ≤ Fintype.card α | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) + ncard (s ∩ t) ≤ Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard s + ncard t ≤ Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole | [57, 1] | [66, 43] | apply Nat.le_of_add_le_add_right | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) + ncard (s ∩ t) ≤ Fintype.card α | case a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) + ncard (s ∩ t) + ?b ≤ Fintype.card α + ?b
case b
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) + ncard (s ∩ t) ≤ Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole | [57, 1] | [66, 43] | rw [add_assoc, Set.ncard_add_ncard_compl, h', Set.ncard_univ, ← Nat.card_eq_fintype_card, add_le_add_iff_right, ← Set.ncard_univ] | case a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) + ncard (s ∩ t) + ?b ≤ Fintype.card α + ?b
case b
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ℕ | case a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) ≤ ncard univ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) + ncard (s ∩ t) + ?b ≤ Fintype.card α + ?b
case b
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ℕ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole | [57, 1] | [66, 43] | apply Set.ncard_le_ncard (subset_univ _) | case a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) ≤ ncard univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : Fintype.card α < ncard s + ncard t
h' : (s ∩ t)ᶜ = univ
⊢ ncard (s ∪ t) ≤ ncard univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl | [68, 1] | [77, 10] | apply Set.ncard_pigeonhole | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ Set.Nonempty (sᶜ ∩ tᶜ) | case h
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ Fintype.card α < ncard sᶜ + ncard tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ Set.Nonempty (sᶜ ∩ tᶜ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl | [68, 1] | [77, 10] | apply Nat.lt_of_add_lt_add_left | case h
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ Fintype.card α < ncard sᶜ + ncard tᶜ | case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ?h.n + Fintype.card α < ?h.n + (ncard sᶜ + ncard tᶜ)
case h.n
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ Fintype.card α < ncard sᶜ + ncard tᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl | [68, 1] | [77, 10] | rw [← add_assoc, Set.ncard_add_ncard_compl] | case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ?h.n + Fintype.card α < ?h.n + (ncard sᶜ + ncard tᶜ)
case h.n
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ℕ | case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s + Fintype.card α < Nat.card α + ncard tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ?h.n + Fintype.card α < ?h.n + (ncard sᶜ + ncard tᶜ)
case h.n
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ℕ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl | [68, 1] | [77, 10] | simp only [Nat.card_eq_fintype_card, add_comm, add_lt_add_iff_left] | case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s + Fintype.card α < Nat.card α + ncard tᶜ | case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s < ncard tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s + Fintype.card α < Nat.card α + ncard tᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl | [68, 1] | [77, 10] | apply Nat.lt_of_add_lt_add_left | case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s < ncard tᶜ | case h.a.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ?h.a.n + ncard s < ?h.a.n + ncard tᶜ
case h.a.n
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s < ncard tᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl | [68, 1] | [77, 10] | rw [Set.ncard_add_ncard_compl, Nat.card_eq_fintype_card, add_comm] | case h.a.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ?h.a.n + ncard s < ?h.a.n + ncard tᶜ
case h.a.n
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ℕ | case h.a.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s + ncard t < Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ?h.a.n + ncard s < ?h.a.n + ncard tᶜ
case h.a.n
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ℕ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl | [68, 1] | [77, 10] | exact h | case h.a.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s + ncard t < Fintype.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.a
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ ncard s + ncard t < Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl' | [79, 1] | [85, 27] | intro h' | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ s ∪ t ≠ ⊤ | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
⊢ s ∪ t ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl' | [79, 1] | [85, 27] | apply not_le.mpr h | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ False | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ Fintype.card α ≤ ncard s + ncard t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl' | [79, 1] | [85, 27] | rw [← Nat.card_eq_fintype_card, ← Set.ncard_univ, ← top_eq_univ, ← h'] | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ Fintype.card α ≤ ncard s + ncard t | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ ncard (s ∪ t) ≤ ncard s + ncard t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ Fintype.card α ≤ ncard s + ncard t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | Set.ncard_pigeonhole_compl' | [79, 1] | [85, 27] | exact ncard_union_le s t | α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ ncard (s ∪ t) ≤ ncard s + ncard t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Fintype α
s t : Set α
h : ncard s + ncard t < Fintype.card α
h' : s ∪ t = ⊤
⊢ ncard (s ∪ t) ≤ ncard s + ncard t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [Ne.def, Set.top_eq_univ, ← Set.compl_empty_iff, ← Ne.def, ← Set.nonempty_iff_ne_empty] at ha | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
ha : s ∪ g • s ≠ ⊤
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s)) | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
ha : Set.Nonempty (s ∪ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
ha : s ∪ g • s ≠ ⊤
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | obtain ⟨a, ha⟩ := ha | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
ha : Set.Nonempty (s ∪ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s)) | case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
ha : Set.Nonempty (s ∪ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | have ha' : a ∈ (s ∩ g • s)ᶜ := by
rw [Set.compl_inter]
apply Set.mem_union_left
rw [Set.compl_union] at ha
apply Set.mem_of_mem_inter_left ha | case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s)) | case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply (IsPretransitive.mk_base_iff (⟨a, ha'⟩ : SubMulAction.ofFixingSubgroup G (s ∩ g • s))).mpr | case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s)) | case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
⊢ ∀ (x : ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s))), ∃ g_1, g_1 • { val := a, property := ha' } = x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
⊢ IsPretransitive ↥(fixingSubgroup G (s ∩ g • s)) ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rintro ⟨x, hx⟩ | case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
⊢ ∀ (x : ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s))), ∃ g_1, g_1 • { val := a, property := ha' } = x | case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
⊢ ∀ (x : ↥(SubMulAction.ofFixingSubgroup G (s ∩ g • s))), ∃ g_1, g_1 • { val := a, property := ha' } = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [SubMulAction.mem_ofFixingSubgroup_iff, Set.mem_inter_iff, not_and_or] at hx | case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx } | case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s ∨ x ∉ g • s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | cases' hx with hx hx | case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s ∨ x ∉ g • s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | case intro.mk.inl
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ }
case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s ∨ x ∉ g • s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [Set.compl_inter] | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ (s ∩ g • s)ᶜ | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ sᶜ ∪ (g • s)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ (s ∩ g • s)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply Set.mem_union_left | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ sᶜ ∪ (g • s)ᶜ | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ sᶜ ∪ (g • s)ᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [Set.compl_union] at ha | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ sᶜ | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ sᶜ ∩ (g • s)ᶜ
⊢ a ∈ sᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
⊢ a ∈ sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply Set.mem_of_mem_inter_left ha | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ sᶜ ∩ (g • s)ᶜ
⊢ a ∈ sᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ sᶜ ∩ (g • s)ᶜ
⊢ a ∈ sᶜ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | obtain ⟨⟨k, hk⟩, hkax⟩ := hs.exists_smul_eq
⟨a, (by intro ha'; apply ha; apply Set.mem_union_left _ ha')⟩
⟨x, hx⟩ | case intro.mk.inl
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | case intro.mk.inl.intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.inl
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | use ⟨k, (by
rw [mem_fixingSubgroup_iff] at hk ⊢
intro y hy
apply hk
apply Set.mem_of_mem_inter_left hy)⟩ | case intro.mk.inl.intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ { val := k, property := ⋯ } • { val := a, property := ha' } = { val := x, property := hx✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.inl.intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | intro ha' | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
⊢ a ∈ SubMulAction.ofFixingSubgroup G s | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha'✝ : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
ha' : a ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
⊢ a ∈ SubMulAction.ofFixingSubgroup G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply ha | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha'✝ : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
ha' : a ∈ s
⊢ False | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha'✝ : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
ha' : a ∈ s
⊢ a ∈ s ∪ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha'✝ : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
ha' : a ∈ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply Set.mem_union_left _ ha' | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha'✝ : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
ha' : a ∈ s
⊢ a ∈ s ∪ g • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha'✝ : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
ha' : a ∈ s
⊢ a ∈ s ∪ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [mem_fixingSubgroup_iff] at hk ⊢ | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ k ∈ fixingSubgroup G (s ∩ g • s) | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ ∀ y ∈ s ∩ g • s, k • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ k ∈ fixingSubgroup G (s ∩ g • s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | intro y hy | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ ∀ y ∈ s ∩ g • s, k • y = y | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
y : α
hy : y ∈ s ∩ g • s
⊢ k • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ ∀ y ∈ s ∩ g • s, k • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply hk | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
y : α
hy : y ∈ s ∩ g • s
⊢ k • y = y | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
y : α
hy : y ∈ s ∩ g • s
⊢ y ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
y : α
hy : y ∈ s ∩ g • s
⊢ k • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply Set.mem_of_mem_inter_left hy | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
y : α
hy : y ∈ s ∩ g • s
⊢ y ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := a, property := ⋯ } = { val := x, property := hx }
y : α
hy : y ∈ s ∩ g • s
⊢ y ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | simp only [← SetLike.coe_eq_coe] at hkax ⊢ | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ { val := k, property := ⋯ } • { val := a, property := ha' } = { val := x, property := hx✝ } | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : ↑({ val := k, property := hk } • { val := a, property := ⋯ }) = x
⊢ ↑({ val := k, property := ⋯ } • { val := a, property := ha' }) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := a, property := ⋯ } = { val := x, property := hx }
⊢ { val := k, property := ⋯ } • { val := a, property := ha' } = { val := x, property := hx✝ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | exact hkax | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : ↑({ val := k, property := hk } • { val := a, property := ⋯ }) = x
⊢ ↑({ val := k, property := ⋯ } • { val := a, property := ha' }) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ s
k : G
hk : k ∈ fixingSubgroup G s
hkax : ↑({ val := k, property := hk } • { val := a, property := ⋯ }) = x
⊢ ↑({ val := k, property := ⋯ } • { val := a, property := ha' }) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [SubMulAction.mem_ofFixingSubgroup_iff] | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
⊢ g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
⊢ g⁻¹ • x ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
⊢ g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | intro h | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
⊢ g⁻¹ • x ∉ s | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
⊢ g⁻¹ • x ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply hx | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ False | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ x ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [Set.mem_smul_set_iff_inv_smul_mem] | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ x ∈ g • s | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ g⁻¹ • x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ x ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | exact h | case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ g⁻¹ • x ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.inr
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
h : g⁻¹ • x ∈ s
⊢ g⁻¹ • x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [SubMulAction.mem_ofFixingSubgroup_iff] | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ g⁻¹ • a ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [← Set.mem_smul_set_iff_inv_smul_mem] | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ g⁻¹ • a ∉ s | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ a ∉ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ g⁻¹ • a ∉ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | intro h | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ a ∉ g • s | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
h : a ∈ g • s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
⊢ a ∉ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply ha | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
h : a ∈ g • s
⊢ False | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
h : a ∈ g • s
⊢ a ∈ s ∪ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
h : a ∈ g • s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply Set.mem_union_right _ h | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
h : a ∈ g • s
⊢ a ∈ s ∪ g • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
h : a ∈ g • s
⊢ a ∈ s ∪ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | obtain ⟨⟨k, hk⟩, hkax⟩ := hs.exists_smul_eq ⟨g⁻¹ • a, hg'a⟩ ⟨g⁻¹ • x, hg'x⟩ | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | use ⟨g * k * g⁻¹, (by
rw [mem_fixingSubgroup_iff] at hk ⊢
intro y hy
simp [← smul_smul, smul_eq_iff_eq_inv_smul g]
apply hk
rw [← Set.mem_smul_set_iff_inv_smul_mem]
exact Set.mem_of_mem_inter_right hy)⟩ | case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ } | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ { val := g * k * g⁻¹, property := ⋯ } • { val := a, property := ha' } = { val := x, property := hx✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ ∃ g_1, g_1 • { val := a, property := ha' } = { val := x, property := hx✝ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [mem_fixingSubgroup_iff] at hk ⊢ | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ g * k * g⁻¹ ∈ fixingSubgroup G (s ∩ g • s) | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ ∀ y ∈ s ∩ g • s, (g * k * g⁻¹) • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ g * k * g⁻¹ ∈ fixingSubgroup G (s ∩ g • s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | intro y hy | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ ∀ y ∈ s ∩ g • s, (g * k * g⁻¹) • y = y | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ (g * k * g⁻¹) • y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ ∀ y ∈ s ∩ g • s, (g * k * g⁻¹) • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | simp [← smul_smul, smul_eq_iff_eq_inv_smul g] | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ (g * k * g⁻¹) • y = y | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ k • g⁻¹ • y = g⁻¹ • y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ (g * k * g⁻¹) • y = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | apply hk | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ k • g⁻¹ • y = g⁻¹ • y | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ g⁻¹ • y ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ k • g⁻¹ • y = g⁻¹ • y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [← Set.mem_smul_set_iff_inv_smul_mem] | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ g⁻¹ • y ∈ s | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ y ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ g⁻¹ • y ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | exact Set.mem_of_mem_inter_right hy | case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ y ∈ g • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk✝ : k ∈ fixingSubgroup G s
hk : ∀ y ∈ s, k • y = y
hkax : { val := k, property := hk✝ } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
y : α
hy : y ∈ s ∩ g • s
⊢ y ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | simp only [← SetLike.coe_eq_coe] at hkax ⊢ | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ { val := g * k * g⁻¹, property := ⋯ } • { val := a, property := ha' } = { val := x, property := hx✝ } | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : ↑({ val := k, property := hk } • { val := g⁻¹ • a, property := hg'a }) = g⁻¹ • x
⊢ ↑({ val := g * k * g⁻¹, property := ⋯ } • { val := a, property := ha' }) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • { val := g⁻¹ • a, property := hg'a } = { val := g⁻¹ • x, property := hg'x }
⊢ { val := g * k * g⁻¹, property := ⋯ } • { val := a, property := ha' } = { val := x, property := hx✝ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | simp only [SetLike.val_smul] at hkax ⊢ | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : ↑({ val := k, property := hk } • { val := g⁻¹ • a, property := hg'a }) = g⁻¹ • x
⊢ ↑({ val := g * k * g⁻¹, property := ⋯ } • { val := a, property := ha' }) = x | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • g⁻¹ • a = g⁻¹ • x
⊢ { val := g * k * g⁻¹, property := ⋯ } • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : ↑({ val := k, property := hk } • { val := g⁻¹ • a, property := hg'a }) = g⁻¹ • x
⊢ ↑({ val := g * k * g⁻¹, property := ⋯ } • { val := a, property := ha' }) = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | rw [← smul_eq_iff_eq_inv_smul] at hkax | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • g⁻¹ • a = g⁻¹ • x
⊢ { val := g * k * g⁻¹, property := ⋯ } • a = x | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : g • { val := k, property := hk } • g⁻¹ • a = x
⊢ { val := g * k * g⁻¹, property := ⋯ } • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : { val := k, property := hk } • g⁻¹ • a = g⁻¹ • x
⊢ { val := g * k * g⁻¹, property := ⋯ } • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | change (g * k * g⁻¹) • a = x | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : g • { val := k, property := hk } • g⁻¹ • a = x
⊢ { val := g * k * g⁻¹, property := ⋯ } • a = x | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : g • { val := k, property := hk } • g⁻¹ • a = x
⊢ (g * k * g⁻¹) • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : g • { val := k, property := hk } • g⁻¹ • a = x
⊢ { val := g * k * g⁻¹, property := ⋯ } • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Jordan.lean | isPretransitive_ofFixingSubgroup_inter | [98, 1] | [153, 12] | simp only [← smul_smul] | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : g • { val := k, property := hk } • g⁻¹ • a = x
⊢ (g * k * g⁻¹) • a = x | case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : g • { val := k, property := hk } • g⁻¹ • a = x
⊢ g • k • g⁻¹ • a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : IsPretransitive ↥(fixingSubgroup G s) ↥(SubMulAction.ofFixingSubgroup G s)
g : G
a : α
ha : a ∈ (s ∪ g • s)ᶜ
ha' : a ∈ (s ∩ g • s)ᶜ
x : α
hx✝ : x ∈ SubMulAction.ofFixingSubgroup G (s ∩ g • s)
hx : x ∉ g • s
hg'x : g⁻¹ • x ∈ SubMulAction.ofFixingSubgroup G s
hg'a : g⁻¹ • a ∈ SubMulAction.ofFixingSubgroup G s
k : G
hk : k ∈ fixingSubgroup G s
hkax : g • { val := k, property := hk } • g⁻¹ • a = x
⊢ (g * k * g⁻¹) • a = x
TACTIC:
|
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