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/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | have h8 := h0 a (-b) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
⊢ f (a - b) = f (-(a - b)) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
h8 : f (f a * f (-b)) + f (a + -b) = f (a * -b)
⊢ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
⊢ f (a - b) = f (-(a - b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rwa [mul_neg, h5 _ _ h6, ← mul_neg, ← h0 b, h4, h5 a b h4,
add_right_inj, ← sub_eq_add_neg, ← sub_eq_add_neg, ← neg_sub a] at h8 | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
h8 : f (f a * f (-b)) + f (a + -b) = f (a * -b)
⊢ f (a - b) = f (-(a - b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
h8 : f (f a * f (-b)) + f (a + -b) = f (a * -b)
⊢ f (a - b) = f (-(a - b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [add_assoc, ← neg_one_mul, ← h0 (-1)] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + 1 + f y = f (-y) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + (1 + f y) = f (f (-1) * f y) + f (-1 + y) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + 1 + f y = f (-y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | refine congr_arg₂ _ ?_ ?_ | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + (1 + f y) = f (f (-1) * f y) + f (-1 + y) | case refine_1
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) = f (f (-1) * f y)
case refine_2
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ 1 + f y = f (-1 + y) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + (1 + f y) = f (f (-1) * f y) + f (-1 + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [← zero_sub, h2, h1, one_add_one_eq_two] | case refine_1
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) = f (f (-1) * f y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) = f (f (-1) * f y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [add_comm, ← h2, neg_add_eq_sub] | case refine_2
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ 1 + f y = f (-1 + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ 1 + f y = f (-1 + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rwa [← h3, self_eq_add_left, ← h2, good_map_eq_zero_iff h0 h1,
sub_eq_iff_eq_add, one_add_one_eq_two, mul_right_eq_self₀,
or_iff_left h, ← add_sub_cancel_right y 1, h2, add_left_eq_self,
good_map_add_one_eq_zero_iff h0 h1] at h4 | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
y : D
h4 : f y = f (-y)
⊢ y = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
y : D
h4 : f y = f (-y)
⊢ y = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [← h3, h5, h3] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
y z : D
h5 : f y = f z
⊢ f (-y) = f (-z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
y z : D
h5 : f y = f z
⊢ f (-y) = f (-z)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [← h0, ← h0 b, h4, add_comm a] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
⊢ f (a * b) = f (b * a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
⊢ f (a * b) = f (b * a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h2 := good_shift h0 h1 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
⊢ Function.Injective f | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
⊢ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
⊢ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h3 : ∀ c d : F, d ≠ 0 → f (c + 1) = f (d + 1) →
f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1) := λ c d h3 h4 ↦ by
rw [good_shift2 h0 h1, ← h0, h4, add_assoc, ← add_assoc (c + 1), h2,
good_special_equality h0 (mul_inv_cancel h3), zero_add, add_right_comm] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
⊢ Function.Injective f | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
⊢ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
⊢ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | intros a b h4 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
⊢ Function.Injective f | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f a = f b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
⊢ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [← h2 a, ← h2 b, add_left_inj] at h4 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f a = f b
⊢ a = b | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f a = f b
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h5 := good_map_add_one_eq_zero_iff h0 h1 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
⊢ a = b | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rcases eq_or_ne a 0 with rfl | ha | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
⊢ a = b | case inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
b : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
h4 : f (0 + 1) = f (b + 1)
⊢ 0 = b
case inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rcases eq_or_ne b 0 with rfl | hb | case inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
⊢ a = b | case inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
h4 : f (a + 1) = f (0 + 1)
⊢ a = 0
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹))
+ a * a⁻¹ * (b * b⁻¹) := by ring | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h7 : ∀ c d : F, (c + 1) * (d + 1) = c * d + c + d + 1 := λ c d ↦ by
rw [add_one_mul (α := F), mul_add_one (α := F), ← add_assoc] | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [mul_inv_cancel ha, mul_inv_cancel hb, mul_one, mul_one, mul_one,
add_comm b b⁻¹, add_add_add_comm, add_comm a⁻¹ b, ← add_assoc, ← h7] at h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) = (a + b⁻¹ + 1) * (b + a⁻¹ + 1)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | replace h6 := congr_arg f h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) = (a + b⁻¹ + 1) * (b + a⁻¹ + 1)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1)) = f ((a + b⁻¹ + 1) * (b + a⁻¹ + 1))
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) = (a + b⁻¹ + 1) * (b + a⁻¹ + 1)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [← h0, h3 a b hb h4, h3 b a ha h4.symm, h7, add_sub_cancel_right,
h7, add_sub_cancel_right, ← h0 (a + b⁻¹ + 1), add_right_inj] at h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1)) = f ((a + b⁻¹ + 1) * (b + a⁻¹ + 1))
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1)) = f ((a + b⁻¹ + 1) * (b + a⁻¹ + 1))
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | replace h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) =
a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1) := by ring | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [mul_inv_cancel ha, mul_inv_cancel hb, one_add_one_eq_two, h, zero_add] at h7 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [← h2, ← h7, add_add_add_comm, add_add_add_comm a, ← add_add_add_comm,
← h0, add_right_comm, add_left_eq_self, ← h2, add_assoc,
one_add_one_eq_two, h, add_zero, h5, mul_eq_zero, h5, h5] at h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | replace h3 : ∀ c : F, -c = c := λ c ↦ by
rw [neg_eq_iff_add_eq_zero, ← two_mul, h, zero_mul] | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rcases h6 with h6 | h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
⊢ a = b | case inr.inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : a + b = 0
⊢ a = b
case inr.inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : b⁻¹ + a⁻¹ = 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [good_shift2 h0 h1, ← h0, h4, add_assoc, ← add_assoc (c + 1), h2,
good_special_equality h0 (mul_inv_cancel h3), zero_add, add_right_comm] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
c d : F
h3 : d ≠ 0
h4 : f (c + 1) = f (d + 1)
⊢ f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
c d : F
h3 : d ≠ 0
h4 : f (c + 1) = f (d + 1)
⊢ f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [zero_add, good_map_one h0, eq_comm, h5, eq_comm] at h4 | case inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
b : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
h4 : f (0 + 1) = f (b + 1)
⊢ 0 = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
b : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
h4 : f (0 + 1) = f (b + 1)
⊢ 0 = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [zero_add, good_map_one h0, h5] at h4 | case inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
h4 : f (a + 1) = f (0 + 1)
⊢ a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
h4 : f (a + 1) = f (0 + 1)
⊢ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | ring | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [add_one_mul (α := F), mul_add_one (α := F), ← add_assoc] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
c d : F
⊢ (c + 1) * (d + 1) = c * d + c + d + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
c d : F
⊢ (c + 1) * (d + 1) = c * d + c + d + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | ring | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [neg_eq_iff_add_eq_zero, ← two_mul, h, zero_mul] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
c : F
⊢ -c = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
c : F
⊢ -c = c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [add_eq_zero_iff_eq_neg, h3] at h6 | case inr.inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : a + b = 0
⊢ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : a + b = 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [add_eq_zero_iff_eq_neg, h3, inv_inj, eq_comm] at h6 | case inr.inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : b⁻¹ + a⁻¹ = 0
⊢ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : b⁻¹ + a⁻¹ = 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Hom.lean | IMOSL.Extra.CharTwo.ofRingHom | [21, 1] | [24, 47] | rw [← one_add_one_eq_two, ← φ.map_one,
← φ.map_add, add_self_eq_zero, φ.map_zero] | R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CharTwo R
inst✝ : NonAssocSemiring S
φ : R →+* S
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CharTwo R
inst✝ : NonAssocSemiring S
φ : R →+* S
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | have h0 : ∃ n, ∃ x, f x = n := ⟨f 0, 0, rfl⟩ | f g : ℕ → ℕ
h : good f g
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | refine ⟨Nat.find h0, λ k ↦
⟨Nat.find_min' h0, Nat.le_induction (Nat.find_spec h0) ?_ k⟩⟩ | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k : ℕ
⊢ ∀ (n : ℕ), Nat.find h0 ≤ n → (∃ x, f x = n) → ∃ x, f x = n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | rintro n h1 ⟨x, rfl⟩ | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k : ℕ
⊢ ∀ (n : ℕ), Nat.find h0 ≤ n → (∃ x, f x = n) → ∃ x, f x = n + 1 | case intro
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k x : ℕ
h1 : Nat.find h0 ≤ f x
⊢ ∃ x_1, f x_1 = f x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k : ℕ
⊢ ∀ (n : ℕ), Nat.find h0 ≤ n → (∃ x, f x = n) → ∃ x, f x = n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | exact ⟨g x, h x⟩ | case intro
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k x : ℕ
h1 : Nat.find h0 ≤ f x
⊢ ∃ x_1, f x_1 = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k x : ℕ
h1 : Nat.find h0 ≤ f x
⊢ ∃ x_1, f x_1 = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | rcases h1 with ⟨x, rfl⟩ | f g : ℕ → ℕ
h : good f g
h0 : good g f
k m : ℕ
h1 : ∃ x, f x = k
h2 : ∃ y, f y = m
h3 : f k = f m
⊢ k = m | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
m : ℕ
h2 : ∃ y, f y = m
x : ℕ
h3 : f (f x) = f m
⊢ f x = m | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
k m : ℕ
h1 : ∃ x, f x = k
h2 : ∃ y, f y = m
h3 : f k = f m
⊢ k = m
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | rcases h2 with ⟨y, rfl⟩ | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
m : ℕ
h2 : ∃ y, f y = m
x : ℕ
h3 : f (f x) = f m
⊢ f x = m | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : f (f x) = f (f y)
⊢ f x = f y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
m : ℕ
h2 : ∃ y, f y = m
x : ℕ
h3 : f (f x) = f m
⊢ f x = m
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | replace h3 := lem2 h0 h3 | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : f (f x) = f (f y)
⊢ f x = f y | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g (f x) = g (f y)
⊢ f x = f y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : f (f x) = f (f y)
⊢ f x = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | rw [h0, h0, Nat.succ_inj'] at h3 | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g (f x) = g (f y)
⊢ f x = f y | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g x = g y
⊢ f x = f y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g (f x) = g (f y)
⊢ f x = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | exact lem2 h h3 | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g x = g y
⊢ f x = f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g x = g y
⊢ f x = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | wlog h3 : a ≤ b | f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
⊢ a = b | case inr
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
this :
∀ {f g : ℕ → ℕ},
good f g →
good g f → ∀ {a b : ℕ}, (∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k) → (∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k) → a ≤ b → a = b
h3 : ¬a ≤ b
⊢ a = b
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | refine h3.antisymm ((h2 _).mp ?_) | f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ a = b | f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | obtain ⟨k, h4⟩ := Nat.exists_eq_add_of_le (lem3 h0 h1) | f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ ∃ x, g x = a | case intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | obtain ⟨d, h5⟩ := (h1 (a + k)).mpr (a.le_add_right k) | case intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
⊢ ∃ x, g x = a | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
d : ℕ
h5 : f d = a + k
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | rw [Nat.succ_add, ← h5, ← h, eq_comm] at h4 | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
d : ℕ
h5 : f d = a + k
⊢ ∃ x, g x = a | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k d : ℕ
h4 : f (g d) = f a
h5 : f d = a + k
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
d : ℕ
h5 : f d = a + k
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | exact ⟨d, lem4 h h0 ((h1 _).mpr <| h3.trans <| (h2 _).mp ⟨d, rfl⟩)
((h1 a).mpr a.le_refl) h4⟩ | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k d : ℕ
h4 : f (g d) = f a
h5 : f d = a + k
⊢ ∃ x, g x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k d : ℕ
h4 : f (g d) = f a
h5 : f d = a + k
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | exact (this h0 h h2 h1 (le_of_not_le h3)).symm | case inr
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
this :
∀ {f g : ℕ → ℕ},
good f g →
good g f → ∀ {a b : ℕ}, (∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k) → (∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k) → a ≤ b → a = b
h3 : ¬a ≤ b
⊢ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
this :
∀ {f g : ℕ → ℕ},
good f g →
good g f → ∀ {a b : ℕ}, (∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k) → (∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k) → a ≤ b → a = b
h3 : ¬a ≤ b
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | refine (Nat.succ_le_of_lt (lem3 h0 h1)).eq_or_gt.resolve_right λ h3 ↦ ?_ | f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f a = a.succ | f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : a.succ < f a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f a = a.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | obtain ⟨t, h3⟩ := Nat.exists_eq_add_of_lt h3 | f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : a.succ < f a
⊢ False | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : a.succ < f a
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | obtain ⟨x, h4⟩ := (h1 (a + t)).mpr (a.le_add_right t) | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
⊢ False | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | obtain ⟨y, h5⟩ := (h1 (g x)).mpr <| (h2 _).mp ⟨x, rfl⟩ | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
⊢ False | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
y : ℕ
h5 : f y = g x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | rw [Nat.succ_add, ← h4, ← h, ← Nat.succ_eq_add_one, ← h, ← h5] at h3 | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
y : ℕ
h5 : f y = g x
⊢ False | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
y : ℕ
h5 : f y = g x
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | refine (lem4 h h0 ((h1 a).mpr a.le_refl)
((h1 _).mpr <| (h2 _).mp ⟨f y, rfl⟩) h3).not_lt ?_ | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ False | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ a < g (f y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | rw [h0, Nat.lt_succ_iff, ← h2] | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ a < g (f y) | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ ∃ x, g x = g y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ a < g (f y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | exact ⟨y, rfl⟩ | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ ∃ x, g x = g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ ∃ x, g x = g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | obtain ⟨a, h1, h2⟩ := lem6 h h0 | f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
⊢ f = g | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f = g | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
⊢ f = g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | suffices h3 : ∀ n, a ≤ n → f n = n.succ ∧ g n = n.succ by
ext x; rw [← Nat.succ_inj', ← h, (h3 _ <| (h2 _).mp ⟨x, rfl⟩).1] | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f = g | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f = g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | refine Nat.le_induction ⟨lem7 h h0 h1 h2, lem7 h0 h h2 h1⟩
(λ n _ h3 ↦ ⟨?_, ?_⟩) | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ | case intro.intro.refine_1
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ f (n + 1) = (n + 1).succ
case intro.intro.refine_2
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ g (n + 1) = (n + 1).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | ext x | f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
⊢ f = g | case h
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
x : ℕ
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
⊢ f = g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | rw [← Nat.succ_inj', ← h, (h3 _ <| (h2 _).mp ⟨x, rfl⟩).1] | case h
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
x : ℕ
⊢ f x = g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
x : ℕ
⊢ f x = g x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | rw [← Nat.succ_eq_add_one, ← h3.2, h, h3.1, h3.2] | case intro.intro.refine_1
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ f (n + 1) = (n + 1).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_1
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ f (n + 1) = (n + 1).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | rw [← Nat.succ_eq_add_one, ← h3.1, h0, h3.1, h3.2] | case intro.intro.refine_2
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ g (n + 1) = (n + 1).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ g (n + 1) = (n + 1).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Defs.lean | IMOSL.IMO2012A5.map_commute_of_commute | [31, 1] | [33, 69] | rw [← h, h0, h, add_comm x] | R : Type u_2
S : Type u_1
inst✝³ : NonAssocSemiring R
inst✝² : Add S
inst✝¹ : Mul S
x y : R
inst✝ : IsCancelAdd S
f : R → S
h : good f
h0 : x * y = y * x
⊢ f x * f y + f (x + y) = f y * f x + f (x + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : NonAssocSemiring R
inst✝² : Add S
inst✝¹ : Mul S
x y : R
inst✝ : IsCancelAdd S
f : R → S
h : good f
h0 : x * y = y * x
⊢ f x * f y + f (x + y) = f y * f x + f (x + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Defs.lean | IMOSL.IMO2012A5.ReducedGood.period_imp_zero | [50, 1] | [52, 49] | rw [h, add_zero] | R : Type u_1
S : Type u_2
inst✝⁴ : NonAssocSemiring R
inst✝³ : Add S
inst✝² : Mul S
inst✝¹ : One S
inst✝ : Zero S
c : R
f : R → S
hf : ReducedGood f
h : ∀ (x : R), f (x + c) = f x
x : R
⊢ f (x + c) = f (x + 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝⁴ : NonAssocSemiring R
inst✝³ : Add S
inst✝² : Mul S
inst✝¹ : One S
inst✝ : Zero S
c : R
f : R → S
hf : ReducedGood f
h : ∀ (x : R), f (x + c) = f x
x : R
⊢ f (x + c) = f (x + 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond1 | [25, 1] | [32, 62] | rcases (x a).eq_or_eq_not (x b) with h6 | h6 | n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2 | case inl
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2
case inr
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond1 | [25, 1] | [32, 62] | rcases (x c).eq_or_eq_not (x b) with h7 | h7 | case inr
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2 | case inr.inl
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
h7 : x c = x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2
case inr.inr
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
h7 : x c = !x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond1 | [25, 1] | [32, 62] | exact ⟨a, b, h, h0, h1.le.trans h2, h6, k, h3⟩ | case inl
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond1 | [25, 1] | [32, 62] | exact ⟨b, c, h.trans h0.le, h1, h2, h7.symm, m, h5⟩ | case inr.inl
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
h7 : x c = x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
h7 : x c = x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond1 | [25, 1] | [32, 62] | exact ⟨a, c, h, h0.trans h1, h2, h6.trans h7.symm, l, h4⟩ | case inr.inr
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
h7 : x c = !x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
n a b c k l m : ℕ
h : n ≤ a
h0 : a < b
h1 : b < c
h2 : c ≤ 2 * n
h3 : a + b = k ^ 2
h4 : a + c = l ^ 2
h5 : b + c = m ^ 2
x : ℕ → Bool
h6 : x a = !x b
h7 : x c = !x b
⊢ ∃ a b, n ≤ a ∧ a < b ∧ b ≤ 2 * n ∧ x a = x b ∧ ∃ k, a + b = k ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | refine good_cond1 (b := 2 * k ^ 2 + 8 * k + 9) (k := 2 * k + 3)
(l := 2 * k + 4) (m := 2 * k + 5) h ?_ ?_
(Nat.mul_le_mul_left 2 h0) ?_ ?_ ?_ | n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ good n | case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k < 2 * k ^ 2 + 8 * k + 9
case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 8 * k + 9 < 2 * (k ^ 2 + 6 * k + 8)
case refine_3
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k + (2 * k ^ 2 + 8 * k + 9) = (2 * k + 3) ^ 2
case refine_4
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 4) ^ 2
case refine_5
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 8 * k + 9 + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 5) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ good n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rw [Nat.add_assoc, Nat.add_lt_add_iff_left, Nat.lt_succ_iff] | case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k < 2 * k ^ 2 + 8 * k + 9 | case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 4 * k ≤ 8 * k + 8 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k < 2 * k ^ 2 + 8 * k + 9
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | refine (Nat.mul_le_mul_right k ?_).trans (Nat.le_add_right _ _) | case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 4 * k ≤ 8 * k + 8 | case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 4 ≤ 8 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 4 * k ≤ 8 * k + 8
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | exact Nat.le_mul_of_pos_left 4 (Nat.succ_pos 1) | case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 4 ≤ 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 4 ≤ 8
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rw [Nat.mul_add, Nat.mul_add, ← Nat.mul_assoc,
Nat.lt_succ_iff, Nat.add_assoc, Nat.add_assoc,
Nat.add_le_add_iff_left, Nat.add_le_add_iff_right] | case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 8 * k + 9 < 2 * (k ^ 2 + 6 * k + 8) | case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 8 * k ≤ 2 * 6 * k + 6 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 8 * k + 9 < 2 * (k ^ 2 + 6 * k + 8)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | refine (Nat.mul_le_mul_right k ?_).trans (Nat.le_add_right _ _) | case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 8 * k ≤ 2 * 6 * k + 6 | case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 8 ≤ 2 * 6 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 8 * k ≤ 2 * 6 * k + 6
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | exact Nat.mul_le_mul_left 4 (Nat.le_succ 2) | case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 8 ≤ 2 * 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 8 ≤ 2 * 6
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rw [← Nat.add_assoc, Nat.add_add_add_comm, ← Nat.add_mul, ← Nat.add_mul,
add_sq, Nat.mul_pow, Nat.mul_right_comm, ← Nat.mul_assoc] | case refine_3
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k + (2 * k ^ 2 + 8 * k + 9) = (2 * k + 3) ^ 2 | case refine_3
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (4 + 8) * k + 9 = 2 ^ 2 * k ^ 2 + 2 * 3 * 2 * k + 3 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k + (2 * k ^ 2 + 8 * k + 9) = (2 * k + 3) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rfl | case refine_3
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (4 + 8) * k + 9 = 2 ^ 2 * k ^ 2 + 2 * 3 * 2 * k + 3 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (4 + 8) * k + 9 = 2 ^ 2 * k ^ 2 + 2 * 3 * 2 * k + 3 ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rw [Nat.mul_add, ← Nat.add_assoc, Nat.mul_add, Nat.add_add_add_comm,
← Nat.add_mul, ← Nat.mul_assoc, ← Nat.add_mul, add_sq, Nat.mul_pow,
Nat.mul_right_comm, ← Nat.mul_assoc] | case refine_4
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 4) ^ 2 | case refine_4
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (4 + 2 * 6) * k + 2 * 8 = 2 ^ 2 * k ^ 2 + 2 * 4 * 2 * k + 4 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 4 * k + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 4) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rfl | case refine_4
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (4 + 2 * 6) * k + 2 * 8 = 2 ^ 2 * k ^ 2 + 2 * 4 * 2 * k + 4 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (4 + 2 * 6) * k + 2 * 8 = 2 ^ 2 * k ^ 2 + 2 * 4 * 2 * k + 4 ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rw [Nat.mul_add, Nat.add_add_add_comm, Nat.mul_add, ← Nat.mul_assoc,
Nat.add_add_add_comm (2 * k ^ 2), ← Nat.add_mul, ← Nat.add_mul,
add_sq, Nat.mul_pow, Nat.mul_right_comm, ← Nat.mul_assoc] | case refine_5
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 8 * k + 9 + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 5) ^ 2 | case refine_5
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (8 + 2 * 6) * k + (9 + 2 * 8) = 2 ^ 2 * k ^ 2 + 2 * 5 * 2 * k + 5 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_5
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ 2 * k ^ 2 + 8 * k + 9 + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 5) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2 | [34, 1] | [62, 8] | rfl | case refine_5
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (8 + 2 * 6) * k + (9 + 2 * 8) = 2 ^ 2 * k ^ 2 + 2 * 5 * 2 * k + 5 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_5
n k : ℕ
h : n ≤ 2 * k ^ 2 + 4 * k
h0 : k ^ 2 + 6 * k + 8 ≤ n
⊢ (2 + 2) * k ^ 2 + (8 + 2 * 6) * k + (9 + 2 * 8) = 2 ^ 2 * k ^ 2 + 2 * 5 * 2 * k + 5 ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | rcases h0 with ⟨k, h0, h1⟩ | n : ℕ
h : 99 ≤ n
h0 : ∃ k, n ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1 | case intro.intro
n : ℕ
h : 99 ≤ n
k : ℕ
h0 : n ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : 99 ≤ n
h0 : ∃ k, n ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | rw [le_iff_lt_or_eq] at h0 | case intro.intro
n : ℕ
h : 99 ≤ n
k : ℕ
h0 : n ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1 | case intro.intro
n : ℕ
h : 99 ≤ n
k : ℕ
h0 : n < 2 * k ^ 2 + 4 * k ∨ n = 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : ℕ
h : 99 ≤ n
k : ℕ
h0 : n ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | rcases h0 with h0 | rfl | case intro.intro
n : ℕ
h : 99 ≤ n
k : ℕ
h0 : n < 2 * k ^ 2 + 4 * k ∨ n = 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1 | case intro.intro.inl
n : ℕ
h : 99 ≤ n
k : ℕ
h1 : k ^ 2 + 6 * k + 8 ≤ n
h0 : n < 2 * k ^ 2 + 4 * k
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1
case intro.intro.inr
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ ∃ k_1, 2 * k ^ 2 + 4 * k + 1 ≤ 2 * k_1 ^ 2 + 4 * k_1 ∧ k_1 ^ 2 + 6 * k_1 + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : ℕ
h : 99 ≤ n
k : ℕ
h0 : n < 2 * k ^ 2 + 4 * k ∨ n = 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ n
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | refine ⟨k + 1, ?_, ?_⟩ | case intro.intro.inr
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ ∃ k_1, 2 * k ^ 2 + 4 * k + 1 ≤ 2 * k_1 ^ 2 + 4 * k_1 ∧ k_1 ^ 2 + 6 * k_1 + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | case intro.intro.inr.refine_1
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ 2 * k ^ 2 + 4 * k + 1 ≤ 2 * (k + 1) ^ 2 + 4 * (k + 1)
case intro.intro.inr.refine_2
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ ∃ k_1, 2 * k ^ 2 + 4 * k + 1 ≤ 2 * k_1 ^ 2 + 4 * k_1 ∧ k_1 ^ 2 + 6 * k_1 + 8 ≤ 2 * k ^ 2 + 4 * k + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | exact ⟨k, h0, Nat.le_succ_of_le h1⟩ | case intro.intro.inl
n : ℕ
h : 99 ≤ n
k : ℕ
h1 : k ^ 2 + 6 * k + 8 ≤ n
h0 : n < 2 * k ^ 2 + 4 * k
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
n : ℕ
h : 99 ≤ n
k : ℕ
h1 : k ^ 2 + 6 * k + 8 ≤ n
h0 : n < 2 * k ^ 2 + 4 * k
⊢ ∃ k, n + 1 ≤ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≤ n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | rw [Nat.mul_succ, ← Nat.add_assoc] | case intro.intro.inr.refine_1
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ 2 * k ^ 2 + 4 * k + 1 ≤ 2 * (k + 1) ^ 2 + 4 * (k + 1) | case intro.intro.inr.refine_1
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ 2 * k ^ 2 + 4 * k + 1 ≤ 2 * (k + 1) ^ 2 + 4 * k + 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_1
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ 2 * k ^ 2 + 4 * k + 1 ≤ 2 * (k + 1) ^ 2 + 4 * (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | exact Nat.add_le_add (Nat.add_le_add_right (Nat.mul_le_mul_left 2 <|
Nat.pow_le_pow_of_le_left k.le_succ 2) _) (Nat.le_add_left 1 3) | case intro.intro.inr.refine_1
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ 2 * k ^ 2 + 4 * k + 1 ≤ 2 * (k + 1) ^ 2 + 4 * k + 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_1
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ 2 * k ^ 2 + 4 * k + 1 ≤ 2 * (k + 1) ^ 2 + 4 * k + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | replace h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 := by
rw [add_sq, Nat.mul_add, Nat.mul_add, Nat.mul_one, ← Nat.mul_assoc]; rfl | case intro.intro.inr.refine_2
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | case intro.intro.inr.refine_2
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_2
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : k ^ 2 + 6 * k + 8 ≤ 2 * k ^ 2 + 4 * k
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | rw [← Nat.add_le_add_iff_right (n := 2), h1, Nat.succ_le_iff, Nat.mul_comm,
← Nat.div_lt_iff_lt_mul (Nat.succ_pos 1), ← Nat.sqrt_lt'] at h | case intro.intro.inr.refine_2
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | case intro.intro.inr.refine_2
k : ℕ
h : ((99 + 1) / Nat.succ 1).sqrt < k + 1
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_2
k : ℕ
h : 99 ≤ 2 * k ^ 2 + 4 * k
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | change 8 ≤ k + 1 at h | case intro.intro.inr.refine_2
k : ℕ
h : ((99 + 1) / Nat.succ 1).sqrt < k + 1
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_2
k : ℕ
h : ((99 + 1) / Nat.succ 1).sqrt < k + 1
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | rw [← Nat.add_le_add_iff_right (n := 1), h1, Nat.two_mul,
Nat.add_assoc, Nat.add_assoc, Nat.add_le_add_iff_left, sq] | case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1 | case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ 6 * (k + 1) + (8 + 1) ≤ (k + 1) * (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≤ 2 * k ^ 2 + 4 * k + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | apply (Nat.mul_le_mul_right _ h).trans' | case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ 6 * (k + 1) + (8 + 1) ≤ (k + 1) * (k + 1) | case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ 6 * (k + 1) + (8 + 1) ≤ 8 * (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ 6 * (k + 1) + (8 + 1) ≤ (k + 1) * (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/N2/N2.lean | IMOSL.IMO2021N2.good_cond2_k_witness | [64, 1] | [89, 51] | rw [Nat.add_mul 6 2, Nat.add_le_add_iff_left, Nat.two_mul] | case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ 6 * (k + 1) + (8 + 1) ≤ 8 * (k + 1) | case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ 8 + 1 ≤ k + 1 + (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.refine_2
k : ℕ
h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
h : 8 ≤ k + 1
⊢ 6 * (k + 1) + (8 + 1) ≤ 8 * (k + 1)
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.