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/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | have h1 := h0 x (f x + f x) | R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (t x : R), 0 ≤ (f t - x) * f x + (f x - t) * f t
x : R
⊢ x * f x ≤ 0 | R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (t x : R), 0 ≤ (f t - x) * f x + (f x - t) * f t
x : R
h1 : 0 ≤ (f x - (f x + f x)) * f (f x + f x) + (f (f x + f x) - x) * f x
⊢ x * f x ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (t x : R), 0 ≤ (f t - x) * f x + (f x - t) * f t
x : R
⊢ x * f x ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | rwa [sub_add_cancel_left, sub_mul, neg_mul, mul_comm,
← add_sub_assoc, neg_add_self, zero_sub, neg_nonneg] at h1 | R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (t x : R), 0 ≤ (f t - x) * f x + (f x - t) * f t
x : R
h1 : 0 ≤ (f x - (f x + f x)) * f (f x + f x) + (f (f x + f x) - x) * f x
⊢ x * f x ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (t x : R), 0 ≤ (f t - x) * f x + (f x - t) * f t
x : R
h1 : 0 ≤ (f x - (f x + f x)) * f (f x + f x) + (f (f x + f x) - x) * f x
⊢ x * f x ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | have h1 := h x 0 | R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (x : R), x * f x ≤ 0
x : R
⊢ f x ≤ f (f x) | R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (x : R), x * f x ≤ 0
x : R
h1 : f (x + 0) ≤ 0 * f x + f (f x)
⊢ f x ≤ f (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (x : R), x * f x ≤ 0
x : R
⊢ f x ≤ f (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | rwa [add_zero, zero_mul, zero_add] at h1 | R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (x : R), x * f x ≤ 0
x : R
h1 : f (x + 0) ≤ 0 * f x + f (f x)
⊢ f x ≤ f (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h0 : ∀ (x : R), x * f x ≤ 0
x : R
h1 : f (x + 0) ≤ 0 * f x + f (f x)
⊢ f x ≤ f (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | exact h0 x h2 | case inl
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
x : R
h2✝ : x ≤ 0
h2 : x < 0
⊢ f x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
x : R
h2✝ : x ≤ 0
h2 : x < 0
⊢ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | apply (h1 0).antisymm | case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
⊢ f 0 = 0 | case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
⊢ 0 ≤ f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
⊢ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | specialize h (-1) 0 | case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
⊢ 0 ≤ f 0 | case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
h : f (-1 + 0) ≤ 0 * f (-1) + f (f (-1))
⊢ 0 ≤ f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h : ∀ (x y : R), f (x + y) ≤ y * f x + f (f x)
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
⊢ 0 ≤ f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A6/A6.lean | IMOSL.IMO2011A6.final_solution | [22, 1] | [48, 66] | rwa [add_zero, zero_mul, zero_add, h0 _ neg_one_lt_zero] at h | case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
h : f (-1 + 0) ≤ 0 * f (-1) + f (f (-1))
⊢ 0 ≤ f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝ : LinearOrderedCommRing R
f : R → R
h1 : ∀ (x : R), f x ≤ 0
h0 : ∀ (x : R), x < 0 → f x = 0
h2 : 0 ≤ 0
h : f (-1 + 0) ≤ 0 * f (-1) + f (f (-1))
⊢ 0 ≤ f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.map_zero | [27, 1] | [30, 45] | specialize h 0 0 | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
⊢ f 0 = 0 | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : f (0 + 0) + f (0 - 0) = 2 • (f 0 + f 0)
⊢ f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
⊢ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.map_zero | [27, 1] | [30, 45] | rw [add_zero, sub_zero, two_nsmul, self_eq_add_right, ← two_nsmul] at h | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : f (0 + 0) + f (0 - 0) = 2 • (f 0 + f 0)
⊢ f 0 = 0 | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : 2 • f 0 = 0
⊢ f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : f (0 + 0) + f (0 - 0) = 2 • (f 0 + f 0)
⊢ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.map_zero | [27, 1] | [30, 45] | exact hH _ _ (h.trans (nsmul_zero 2).symm) | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : 2 • f 0 = 0
⊢ f 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : 2 • f 0 = 0
⊢ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.map_even | [32, 1] | [34, 84] | have h0 := h 0 x | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x : G
⊢ f (-x) = f x | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x : G
h0 : f (0 + x) + f (0 - x) = 2 • (f 0 + f x)
⊢ f (-x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x : G
⊢ f (-x) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.map_even | [32, 1] | [34, 84] | rwa [zero_add, zero_sub, map_zero hH h, zero_add, two_nsmul, add_right_inj] at h0 | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x : G
h0 : f (0 + x) + f (0 - x) = 2 • (f 0 + f x)
⊢ f (-x) = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x : G
h0 : f (0 + x) + f (0 - x) = 2 • (f 0 + f x)
⊢ f (-x) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.map_triple | [36, 1] | [40, 77] | rw [nsmul_add, ← h, ← h, add_add_add_comm, add_assoc x, add_sub_cancel_left,
add_comm (f (y + z)), add_add_add_comm, ← two_nsmul, nsmul_add, add_left_inj, ← h,
add_sub_add_left_eq_sub, add_left_inj, add_right_comm, add_add_add_comm] | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x y z : G
⊢ 2 • (f (x + y + z) + f x + (f y + f z)) = 2 • (f (x + y) + f (x + z) + f (y + z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCancelCommMonoid H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x y z : G
⊢ 2 • (f (x + y + z) + f x + (f y + f z)) = 2 • (f (x + y) + f (x + z) + f (y + z))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.BilinMap_eq_two_nsmul | [75, 1] | [77, 58] | rw [two_nsmul, ← neg_inj, ← map_neg, BilinMap_def,
add_neg_self, map_zero hH h, map_even hH h, zero_sub] | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCommGroup H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x : G
⊢ ((BilinMap hH h) x) x = 2 • f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCommGroup H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x : G
⊢ ((BilinMap hH h) x) x = 2 • f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/SquareLike.lean | IMOSL.Extra.SquareLike.two_nsmul_BilinMap_eq | [79, 1] | [81, 72] | rw [BilinMap_def, nsmul_sub, ← h, two_nsmul, add_sub_add_left_eq_sub] | G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCommGroup H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x y : G
⊢ 2 • ((BilinMap hH h) x) y = f (x + y) - f (x - y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
H : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : AddCommGroup H
hH : ∀ (x y : H), 2 • x = 2 • y → x = y
f : G → H
h : ∀ (x y : G), f (x + y) + f (x - y) = 2 • (f x + f y)
x y : G
⊢ 2 • ((BilinMap hH h) x) y = f (x + y) - f (x - y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.main_answer_is_good | [32, 1] | [35, 58] | rw [id, id, add_right_comm, add_left_inj, two_nsmul, mul_add,
add_mul, mul_comm y, add_assoc, ← add_mul, add_assoc] | R : Type u_1
inst✝ : NonUnitalCommRing R
C x y : R
⊢ id ((fun x => x * x + C) (x + y)) = (fun x => x * x + C) x + (2 • x + y) * id y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonUnitalCommRing R
C x y : R
⊢ id ((fun x => x * x + C) (x + y)) = (fun x => x * x + C) x + (2 • x + y) * id y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | have X (z : R) : φ.toEquiv z = (φ : R ≃+* S) z := rfl | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | intro x y | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | simp only [φ.conj_apply, X] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ x + φ y)))))) = φ (f (φ.symm (φ x))) + (2 • φ x + φ y) * φ (g (φ.symm (φ y))) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | rw [← φ.map_add, ← map_nsmul, ← φ.map_add] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ x + φ y)))))) = φ (f (φ.symm (φ x))) + (2 • φ x + φ y) * φ (g (φ.symm (φ y))) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ (x + y))))))) = φ (f (φ.symm (φ x))) + φ (2 • x + y) * φ (g (φ.symm (φ y))) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ x + φ y)))))) = φ (f (φ.symm (φ x))) + (2 • φ x + φ y) * φ (g (φ.symm (φ y)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | simp only [φ.toEquiv_eq_coe, EquivLike.coe_symm_apply_apply] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ (x + y))))))) = φ (f (φ.symm (φ x))) + φ (2 • x + y) * φ (g (φ.symm (φ y))) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x) + φ (2 • x + y) * φ (g y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ (x + y))))))) = φ (f (φ.symm (φ x))) + φ (2 • x + y) * φ (g (φ.symm (φ y)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | rw [← φ.map_mul, ← φ.map_add] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x) + φ (2 • x + y) * φ (g y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x + (2 • x + y) * g y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x) + φ (2 • x + y) * φ (g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | exact φ.apply_eq_iff_eq.symm | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x + (2 • x + y) * g y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x + (2 • x + y) * g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1_1 | [57, 1] | [62, 43] | rw [nsmul_sub, sub_eq_sub_iff_add_eq_add, ← add_sub_right_comm, ← add_sub_assoc,
sub_eq_sub_iff_add_eq_add, two_nsmul, ← add_mul, add_assoc, ← add_mul,
← two_nsmul, ← h, add_comm _ (f y), add_assoc, two_nsmul, ← add_mul,
← add_mul, ← two_nsmul, ← h, add_comm] | R : Type u_1
inst✝ : NonUnitalNonAssocRing R
f g : R → R
h : good f g
x y : R
⊢ f x - x * g x - (f y - y * g y) = 2 • (y * g x - x * g y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
f g : R → R
h : good f g
x y : R
⊢ f x - x * g x - (f y - y * g y) = 2 • (y * g x - x * g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | obtain ⟨A, h0⟩ : ∃ A, ∀ x, g x = x * A + g 0 := ⟨g 1 - g 0, λ x ↦ by
have h0 : _ + _ = _ + _ := congrArg₂ _ (step1_1 h x 1) (step1_1 h 1 0)
rw [sub_add_sub_cancel, step1_1 h, ← nsmul_add] at h0
replace h0 := hR _ _ h0
rwa [zero_mul, zero_sub, one_mul, zero_mul, one_mul, zero_sub, eq_add_neg_iff_add_eq,
eq_sub_iff_add_eq', ← add_assoc, ← sub_eq_add_neg, ← mul_sub, eq_comm] at h0⟩ | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | refine ⟨A, g 0, f 0, funext λ x ↦ ?_, funext h0⟩ | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
⊢ f x = x * (x * A) - x * g 0 + f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | have h1 := step1_1 h x 0 | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
⊢ f x = x * (x * A) - x * g 0 + f 0 | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
h1 : f x - x * g x - (f 0 - 0 * g 0) = 2 • (0 * g x - x * g 0)
⊢ f x = x * (x * A) - x * g 0 + f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
⊢ f x = x * (x * A) - x * g 0 + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | rwa [zero_mul, sub_zero, zero_mul, zero_sub, sub_eq_iff_eq_add,
sub_eq_iff_eq_add', h0, mul_add, two_nsmul, add_assoc _ _ (f 0),
add_add_add_comm, add_neg_cancel_left, ← sub_eq_add_neg] at h1 | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
h1 : f x - x * g x - (f 0 - 0 * g 0) = 2 • (0 * g x - x * g 0)
⊢ f x = x * (x * A) - x * g 0 + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
h1 : f x - x * g x - (f 0 - 0 * g 0) = 2 • (0 * g x - x * g 0)
⊢ f x = x * (x * A) - x * g 0 + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | have h0 : _ + _ = _ + _ := congrArg₂ _ (step1_1 h x 1) (step1_1 h 1 0) | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
⊢ g x = x * (g 1 - g 0) + g 0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 :
f x - x * g x - (f 1 - 1 * g 1) + (f 1 - 1 * g 1 - (f 0 - 0 * g 0)) =
2 • (1 * g x - x * g 1) + 2 • (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | rw [sub_add_sub_cancel, step1_1 h, ← nsmul_add] at h0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 :
f x - x * g x - (f 1 - 1 * g 1) + (f 1 - 1 * g 1 - (f 0 - 0 * g 0)) =
2 • (1 * g x - x * g 1) + 2 • (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 2 • (0 * g x - x * g 0) = 2 • (1 * g x - x * g 1 + (0 * g 1 - 1 * g 0))
⊢ g x = x * (g 1 - g 0) + g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 :
f x - x * g x - (f 1 - 1 * g 1) + (f 1 - 1 * g 1 - (f 0 - 0 * g 0)) =
2 • (1 * g x - x * g 1) + 2 • (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | replace h0 := hR _ _ h0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 2 • (0 * g x - x * g 0) = 2 • (1 * g x - x * g 1 + (0 * g 1 - 1 * g 0))
⊢ g x = x * (g 1 - g 0) + g 0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 0 * g x - x * g 0 = 1 * g x - x * g 1 + (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 2 • (0 * g x - x * g 0) = 2 • (1 * g x - x * g 1 + (0 * g 1 - 1 * g 0))
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | rwa [zero_mul, zero_sub, one_mul, zero_mul, one_mul, zero_sub, eq_add_neg_iff_add_eq,
eq_sub_iff_add_eq', ← add_assoc, ← sub_eq_add_neg, ← mul_sub, eq_comm] at h0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 0 * g x - x * g 0 = 1 * g x - x * g 1 + (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 0 * g x - x * g 0 = 1 * g x - x * g 1 + (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h0 (x) : (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C := by
specialize h x (-(2 • x)); simp only at h
rwa [add_right_neg, zero_mul, add_zero, two_nsmul, neg_add, add_neg_cancel_comm_assoc,
neg_mul x A, neg_mul_neg, neg_mul, sub_neg_eq_add] at h | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h1 := h0 0 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : (0 * (0 * A) + 0 * B + C) * A + B = 0 * (0 * A) - 0 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [zero_mul, zero_mul, zero_add, sub_zero, zero_add] at h1 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : (0 * (0 * A) + 0 * B + C) * A + B = 0 * (0 * A) - 0 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : (0 * (0 * A) + 0 * B + C) * A + B = 0 * (0 * A) - 0 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h2 := h0 1 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (1 * (1 * A) + 1 * B + C) * A + B = 1 * (1 * A) - 1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [one_mul, one_mul, one_mul, add_mul, add_assoc, h1, add_left_inj] at h2 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (1 * (1 * A) + 1 * B + C) * A + B = 1 * (1 * A) - 1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (1 * (1 * A) + 1 * B + C) * A + B = 1 * (1 * A) - 1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h3 := h0 (-1) | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (-1 * (-1 * A) + -1 * B + C) * A + B = -1 * (-1 * A) - -1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [neg_one_mul, neg_one_mul, neg_one_mul, neg_neg,
← sub_eq_add_neg, sub_neg_eq_add, add_mul, add_assoc, h1, add_left_inj] at h3 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (-1 * (-1 * A) + -1 * B + C) * A + B = -1 * (-1 * A) - -1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (A - B) * A = A + B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (-1 * (-1 * A) + -1 * B + C) * A + B = -1 * (-1 * A) - -1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [← sub_mul, add_sub_sub_cancel, add_mul, ← two_nsmul, ← sub_sub,
sub_sub_cancel_left, sub_eq_add_neg, ← two_nsmul] at h3 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (A + B) * A - (A - B) * A = A - B - (A + B)
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : 2 • (B * A) = 2 • -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (A + B) * A - (A - B) * A = A - B - (A + B)
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | replace h3 := hR _ _ h3 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : 2 • (B * A) = 2 • -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : 2 • (B * A) = 2 • -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [add_mul, h3, sub_eq_add_neg, add_left_inj] at h2 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [add_mul, mul_assoc, h2, add_right_eq_self, h3, neg_eq_zero] at h0 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : (C * A + B) * A = C * A
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : (C * A + B) * A = C * A
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [h0, add_zero] at h1 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | exact ⟨h0, h2, h1⟩ | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | specialize h x (-(2 • x)) | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
x : R
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
(fun x => x * A + B) ((fun x => x * (x * A) - x * B + C) (x + -(2 • x))) =
(fun x => x * (x * A) - x * B + C) x + (2 • x + -(2 • x)) * (fun x => x * A + B) (-(2 • x))
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
x : R
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | simp only at h | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
(fun x => x * A + B) ((fun x => x * (x * A) - x * B + C) (x + -(2 • x))) =
(fun x => x * (x * A) - x * B + C) x + (2 • x + -(2 • x)) * (fun x => x * A + B) (-(2 • x))
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
((x + -(2 • x)) * ((x + -(2 • x)) * A) - (x + -(2 • x)) * B + C) * A + B =
x * (x * A) - x * B + C + (2 • x + -(2 • x)) * (-(2 • x) * A + B)
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
(fun x => x * A + B) ((fun x => x * (x * A) - x * B + C) (x + -(2 • x))) =
(fun x => x * (x * A) - x * B + C) x + (2 • x + -(2 • x)) * (fun x => x * A + B) (-(2 • x))
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rwa [add_right_neg, zero_mul, add_zero, two_nsmul, neg_add, add_neg_cancel_comm_assoc,
neg_mul x A, neg_mul_neg, neg_mul, sub_neg_eq_add] at h | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
((x + -(2 • x)) * ((x + -(2 • x)) * A) - (x + -(2 • x)) * B + C) * A + B =
x * (x * A) - x * B + C + (2 • x + -(2 • x)) * (-(2 • x) * A + B)
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
((x + -(2 • x)) * ((x + -(2 • x)) * A) - (x + -(2 • x)) * B + C) * A + B =
x * (x * A) - x * B + C + (2 • x + -(2 • x)) * (-(2 • x) * A + B)
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | rcases step1 hR h with ⟨A, B, C, rfl, rfl⟩ | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A C, A * A = A ∧ C * A = C ∧ (f = fun x => x * x * A + C) ∧ g = fun x => x * A | case intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * B + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + B) = fun x => x * A_1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A C, A * A = A ∧ C * A = C ∧ (f = fun x => x * x * A + C) ∧ g = fun x => x * A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | rcases step2 hR h with ⟨rfl, h0, h1⟩ | case intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * B + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + B) = fun x => x * A_1 | case intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + 0) = fun x => x * A_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * B + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + B) = fun x => x * A_1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | refine ⟨A, C, h0, h1, ?_, ?_⟩ | case intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + 0) = fun x => x * A_1 | case intro.intro.intro.intro.intro.intro.refine_1
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A + C
case intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * A + 0) = fun x => x * A | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + 0) = fun x => x * A_1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | funext x | case intro.intro.intro.intro.intro.intro.refine_1
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A + C | case intro.intro.intro.intro.intro.intro.refine_1.h
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
x : R
⊢ x * (x * A) - x * 0 + C = x * x * A + C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_1
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | rw [mul_zero, sub_zero, ← mul_assoc] | case intro.intro.intro.intro.intro.intro.refine_1.h
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
x : R
⊢ x * (x * A) - x * 0 + C = x * x * A + C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_1.h
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
x : R
⊢ x * (x * A) - x * 0 + C = x * x * A + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | simp only [add_zero] | case intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * A + 0) = fun x => x * A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * A + 0) = fun x => x * A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_self_comp_idempotent_self_mul_part | [133, 1] | [136, 74] | rw [mul_neg, ← neg_mul, neg_sub, sub_one_mul, mul_comm] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = (1 - a) * -x | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = x * a - x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = (1 - a) * -x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_self_comp_idempotent_self_mul_part | [133, 1] | [136, 74] | rfl | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = x * a - x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = x * a - x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_compl_comp_idempotent_compl_mul_part | [152, 1] | [156, 62] | change x * (1 - a) - x = _ | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_compl_mul_part h) ⟦x⟧ - x = a * -x | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ x * (1 - a) - x = a * -x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_compl_mul_part h) ⟦x⟧ - x = a * -x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_compl_comp_idempotent_compl_mul_part | [152, 1] | [156, 62] | rw [mul_one_sub, sub_sub_cancel_left, mul_neg, mul_comm] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ x * (1 - a) - x = a * -x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ x * (1 - a) - x = a * -x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.idempotent_decomp_AddHom_map_quot_self | [173, 1] | [175, 77] | rw [idempotent_decomp_AddHom_map_quot, ← mul_add, add_sub_cancel, mul_one] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_decomp_AddHom h) ((Ideal.Quotient.mk (Ideal.span {1 - a})) x, (Ideal.Quotient.mk (Ideal.span {a})) x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_decomp_AddHom h) ((Ideal.Quotient.mk (Ideal.span {1 - a})) x, (Ideal.Quotient.mk (Ideal.span {a})) x) = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_self_comp_idempotent_decomp_AddHom | [177, 1] | [182, 56] | rw [idempotent_decomp_AddHom, AddMonoidHom.coprod_apply,
map_add, quot_self_comp_idempotent_self_mul_part,
quot_self_comp_idempotent_compl_mul_part, add_zero] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {1 - a})) ((idempotent_decomp_AddHom h) p) = p.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {1 - a})) ((idempotent_decomp_AddHom h) p) = p.1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_compl_comp_idempotent_decomp_AddHom | [184, 1] | [189, 57] | rw [idempotent_decomp_AddHom, AddMonoidHom.coprod_apply,
map_add, quot_compl_comp_idempotent_self_mul_part,
quot_compl_comp_idempotent_compl_mul_part, zero_add] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {a})) ((idempotent_decomp_AddHom h) p) = p.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {a})) ((idempotent_decomp_AddHom h) p) = p.2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rcases relation_summary hR h with ⟨A, C, h0, h1, rfl, rfl⟩ | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0) | case intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
⊢ ∃ R₁ R₂ x x_1 φ c,
(fun x => x * x * A + C) = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧
(fun x => x * A) = φ.conj (Prod.map id 0) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | refine ⟨R ⧸ Ideal.span {1 - A}, R ⧸ Ideal.span {A},
Ideal.Quotient.commRing _, Ideal.Quotient.commRing _,
(idempotent_decomp h0).symm, idempotent_self_mul_part h0 C,
funext λ x ↦ ?_, funext λ x ↦ ?_⟩ | case intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
⊢ ∃ R₁ R₂ x x_1 φ c,
(fun x => x * x * A + C) = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧
(fun x => x * A) = φ.conj (Prod.map id 0) | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm.conj
(Prod.map
(fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
0)
x
case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = (idempotent_decomp h0).symm.conj (Prod.map id 0) x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
⊢ ∃ R₁ R₂ x x_1 φ c,
(fun x => x * x * A + C) = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧
(fun x => x * A) = φ.conj (Prod.map id 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | change _ = (idempotent_decomp h0).symm (_, 0) | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm.conj
(Prod.map
(fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
0)
x | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm
((fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
((Ideal.Quotient.mk (Ideal.span {1 - A})) x),
0) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm.conj
(Prod.map
(fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
0)
x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rw [idempotent_decomp_symm_apply, idempotent_self_mul_part_map_quot, map_zero,
add_zero, h1, map_add, idempotent_self_mul_part_map_quot, ← map_mul,
idempotent_self_mul_part_map_quot, h1] | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm
((fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
((Ideal.Quotient.mk (Ideal.span {1 - A})) x),
0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm
((fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
((Ideal.Quotient.mk (Ideal.span {1 - A})) x),
0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | change x * A = x * A + 0 * (1 - A) | case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = (idempotent_decomp h0).symm.conj (Prod.map id 0) x | case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = x * A + 0 * (1 - A) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = (idempotent_decomp h0).symm.conj (Prod.map id 0) x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rw [zero_mul, add_zero] | case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = x * A + 0 * (1 - A) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = x * A + 0 * (1 - A)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rw [h, h0, ← good_Equiv_conj_iff] | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good f g | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good (Prod.map (fun x => x * x + c) 0) (Prod.map id 0) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good f g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | exact prod_is_good (main_answer_is_good c) zero_is_good | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good (Prod.map (fun x => x * x + c) 0) (Prod.map id 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good (Prod.map (fun x => x * x + c) 0) (Prod.map id 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part1 | [35, 1] | [43, 40] | apply (le_add_of_nonneg_left (sub_nonneg_of_le (h h0))).trans | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ a k - a m ≤ 2 • seqMax (fun i => |x i - a i|) n | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ x m - x k + (a k - a m) ≤ 2 • seqMax (fun i => |x i - a i|) n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ a k - a m ≤ 2 • seqMax (fun i => |x i - a i|) n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part1 | [35, 1] | [43, 40] | rw [← add_comm_sub, sub_add, sub_sub_sub_comm, two_nsmul] | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ x m - x k + (a k - a m) ≤ 2 • seqMax (fun i => |x i - a i|) n | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ x m - a m - (x k - a k) ≤ seqMax (fun i => |x i - a i|) n + seqMax (fun i => |x i - a i|) n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ x m - x k + (a k - a m) ≤ 2 • seqMax (fun i => |x i - a i|) n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part1 | [35, 1] | [43, 40] | have X {i} : i ≤ n → |x i - a i| ≤ seqMax (λ i ↦ |x i - a i|) n :=
le_seqMax_of_le (λ i ↦ |x i - a i|) | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ x m - a m - (x k - a k) ≤ seqMax (fun i => |x i - a i|) n + seqMax (fun i => |x i - a i|) n | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
X : ∀ {i : ℕ}, i ≤ n → |x i - a i| ≤ seqMax (fun i => |x i - a i|) n
⊢ x m - a m - (x k - a k) ≤ seqMax (fun i => |x i - a i|) n + seqMax (fun i => |x i - a i|) n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
⊢ x m - a m - (x k - a k) ≤ seqMax (fun i => |x i - a i|) n + seqMax (fun i => |x i - a i|) n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part1 | [35, 1] | [43, 40] | exact (le_abs_self _).trans <| (abs_sub _ _).trans <|
add_le_add (X h1) (X (h0.trans h1)) | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
X : ∀ {i : ℕ}, i ≤ n → |x i - a i| ≤ seqMax (fun i => |x i - a i|) n
⊢ x m - a m - (x k - a k) ≤ seqMax (fun i => |x i - a i|) n + seqMax (fun i => |x i - a i|) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n k m : ℕ
x : ℕ → G
h : Monotone x
h0 : k ≤ m
h1 : m ≤ n
X : ∀ {i : ℕ}, i ≤ n → |x i - a i| ≤ seqMax (fun i => |x i - a i|) n
⊢ x m - a m - (x k - a k) ≤ seqMax (fun i => |x i - a i|) n + seqMax (fun i => |x i - a i|) n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | apply le_antisymm | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n = g | case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n ≤ g
case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n = g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | rcases exists_map_eq_seqMax (λ i ↦ |seqMax a i - g - a i|) n with ⟨i, h0, h1⟩ | case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n ≤ g | case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
h1 : |seqMax a i - g - a i| = seqMax (fun i => |seqMax a i - g - a i|) n
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n ≤ g | Please generate a tactic in lean4 to solve the state.
STATE:
case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n ≤ g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | rw [← h1, sub_right_comm, abs_le] | case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
h1 : |seqMax a i - g - a i| = seqMax (fun i => |seqMax a i - g - a i|) n
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n ≤ g | case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
h1 : |seqMax a i - g - a i| = seqMax (fun i => |seqMax a i - g - a i|) n
⊢ -g ≤ seqMax a i - a i - g ∧ seqMax a i - a i - g ≤ g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
h1 : |seqMax a i - g - a i| = seqMax (fun i => |seqMax a i - g - a i|) n
⊢ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n ≤ g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | clear h1 | case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
h1 : |seqMax a i - g - a i| = seqMax (fun i => |seqMax a i - g - a i|) n
⊢ -g ≤ seqMax a i - a i - g ∧ seqMax a i - a i - g ≤ g | case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ -g ≤ seqMax a i - a i - g ∧ seqMax a i - a i - g ≤ g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
h1 : |seqMax a i - g - a i| = seqMax (fun i => |seqMax a i - g - a i|) n
⊢ -g ≤ seqMax a i - a i - g ∧ seqMax a i - a i - g ≤ g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | refine ⟨?_, ?_⟩ | case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ -g ≤ seqMax a i - a i - g ∧ seqMax a i - a i - g ≤ g | case a.intro.intro.refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ -g ≤ seqMax a i - a i - g
case a.intro.intro.refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ seqMax a i - a i - g ≤ g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ -g ≤ seqMax a i - a i - g ∧ seqMax a i - a i - g ≤ g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | rw [le_sub_iff_add_le, neg_add_self, sub_nonneg] | case a.intro.intro.refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ -g ≤ seqMax a i - a i - g | case a.intro.intro.refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ a i ≤ seqMax a i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ -g ≤ seqMax a i - a i - g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | exact le_seqMax_self a i | case a.intro.intro.refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ a i ≤ seqMax a i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ a i ≤ seqMax a i
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | rcases exists_map_eq_seqMax a i with ⟨j, h1, h2⟩ | case a.intro.intro.refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ seqMax a i - a i - g ≤ g | case a.intro.intro.refine_2.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
j : ℕ
h1 : j ≤ i
h2 : a j = seqMax a i
⊢ seqMax a i - a i - g ≤ g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
⊢ seqMax a i - a i - g ≤ g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | rw [← h2, sub_le_iff_le_add, ← two_nsmul] | case a.intro.intro.refine_2.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
j : ℕ
h1 : j ≤ i
h2 : a j = seqMax a i
⊢ seqMax a i - a i - g ≤ g | case a.intro.intro.refine_2.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
j : ℕ
h1 : j ≤ i
h2 : a j = seqMax a i
⊢ a j - a i ≤ 2 • g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.refine_2.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
j : ℕ
h1 : j ≤ i
h2 : a j = seqMax a i
⊢ seqMax a i - a i - g ≤ g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | exact h j i h1 h0 | case a.intro.intro.refine_2.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
j : ℕ
h1 : j ≤ i
h2 : a j = seqMax a i
⊢ a j - a i ≤ 2 • g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro.refine_2.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
i : ℕ
h0 : i ≤ n
j : ℕ
h1 : j ≤ i
h2 : a j = seqMax a i
⊢ a j - a i ≤ 2 • g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | apply (le_seqMax_of_le _ n.zero_le).trans' | case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n | case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ |(fun i => seqMax a i - g) 0 - a 0| | Please generate a tactic in lean4 to solve the state.
STATE:
case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ seqMax (fun i => |(fun i => seqMax a i - g) i - a i|) n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | rw [sub_sub, seqMax, sub_add_cancel_right, abs_neg] | case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ |(fun i => seqMax a i - g) 0 - a 0| | case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ |g| | Please generate a tactic in lean4 to solve the state.
STATE:
case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ |(fun i => seqMax a i - g) 0 - a 0|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2007/A1/A1.lean | IMOSL.IMO2007A1.final_solution_part2 | [46, 1] | [61, 27] | exact le_abs_self g | case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ |g| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
g : G
h : ∀ (k m : ℕ), k ≤ m → m ≤ n → a k - a m ≤ 2 • g
⊢ g ≤ |g|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.surjective_iff | [45, 1] | [47, 55] | rw [← Set.range_iff_surjective, ← compl_inj_iff,
← h.rangeCompl_spec, Set.compl_univ, coe_eq_empty] | α : Type u_1
f : α → α
h : FinChainFn f
⊢ Surjective f ↔ h.rangeCompl = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : α → α
h : FinChainFn f
⊢ Surjective f ↔ h.rangeCompl = ∅
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | wlog h3 : m < n | α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
⊢ f^[m] a ≠ f^[n] b | case inr
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
this :
∀ {α : Type u_1} {f : α → α} (h : FinChainFn f) {m n : ℕ} {a b : α},
m ≠ n → a ∈ h.rangeCompl → b ∈ h.rangeCompl → m < n → f^[m] a ≠ f^[n] b
h3 : ¬m < n
⊢ f^[m] a ≠ f^[n] b
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
⊢ f^[m] a ≠ f^[n] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | exact (this h h0.symm h2 h1 <| (le_of_not_lt h3).lt_of_ne h0.symm).symm | case inr
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
this :
∀ {α : Type u_1} {f : α → α} (h : FinChainFn f) {m n : ℕ} {a b : α},
m ≠ n → a ∈ h.rangeCompl → b ∈ h.rangeCompl → m < n → f^[m] a ≠ f^[n] b
h3 : ¬m < n
⊢ f^[m] a ≠ f^[n] b
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
this :
∀ {α : Type u_1} {f : α → α} (h : FinChainFn f) {m n : ℕ} {a b : α},
m ≠ n → a ∈ h.rangeCompl → b ∈ h.rangeCompl → m < n → f^[m] a ≠ f^[n] b
h3 : ¬m < n
⊢ f^[m] a ≠ f^[n] b
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rcases Nat.exists_eq_add_of_le h3.le with ⟨k, rfl⟩ | α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : m < m + k
⊢ f^[m] a ≠ f^[m + k] b | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [Nat.lt_add_right_iff_pos] at h3 | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : m < m + k
⊢ f^[m] a ≠ f^[m + k] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ f^[m] a ≠ f^[m + k] b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : m < m + k
⊢ f^[m] a ≠ f^[m + k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [f.iterate_add_apply m k b, (h.injective.iterate m).ne_iff] | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ f^[m] a ≠ f^[m + k] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ a ≠ f^[k] b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ f^[m] a ≠ f^[m + k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rintro rfl | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ a ≠ f^[k] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : f^[k] b ∈ h.rangeCompl
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ a ≠ f^[k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [mem_rangeCompl_iff, Set.mem_range] at h1 | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : f^[k] b ∈ h.rangeCompl
⊢ False | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : f^[k] b ∈ h.rangeCompl
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | refine h1 ⟨f^[k.pred] b, ?_⟩ | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ False | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ f (f^[k.pred] b) = f^[k] b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [← f.iterate_succ_apply', Nat.succ_pred_eq_of_pos h3] | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ f (f^[k.pred] b) = f^[k] b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ f (f^[k.pred] b) = f^[k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_spec | [88, 1] | [92, 72] | rw [exactIterRange, coe_image, h.rangeCompl_spec, iterate_succ,
Set.range_comp _ f, Set.range_diff_image (h.injective.iterate n)] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
a : α
⊢ a ∈ ↑(h.exactIterRange n) ↔ a ∈ Set.range f^[n] \ Set.range f^[n + 1] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
a : α
⊢ a ∈ ↑(h.exactIterRange n) ↔ a ∈ Set.range f^[n] \ Set.range f^[n + 1]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | rw [disjoint_iff_ne] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ Disjoint (h.exactIterRange m) (h.exactIterRange n) | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ ∀ a ∈ h.exactIterRange m, ∀ b ∈ h.exactIterRange n, a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ Disjoint (h.exactIterRange m) (h.exactIterRange n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | intro a h1 b h2 | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ ∀ a ∈ h.exactIterRange m, ∀ b ∈ h.exactIterRange n, a ≠ b | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : a ∈ h.exactIterRange m
b : α
h2 : b ∈ h.exactIterRange n
⊢ a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ ∀ a ∈ h.exactIterRange m, ∀ b ∈ h.exactIterRange n, a ≠ b
TACTIC:
|
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