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/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | rw [← nsmul_add, add_sub_cancel] | G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n : ℕ
⊢ 2 • c D ≤ 2 • b D + 2 • (c D - b D) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n : ℕ
⊢ 2 • c D ≤ 2 • b D + 2 • (c D - b D)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | rcases c_bdd h h0 h1 h4 h6 with ⟨h7, h8⟩ | case inl
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : c n ≤ 2 • b n
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D)) | case inl.intro
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : c n ≤ 2 • b n
h7 : b (n + 1) = b n
h8 : c (n + 1) ≤ 2 • b (n + 1)
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D)) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : c n ≤ 2 • b n
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | rwa [max_eq_right h8, h7, ← max_eq_right h6] | case inl.intro
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : c n ≤ 2 • b n
h7 : b (n + 1) = b n
h8 : c (n + 1) ≤ 2 • b (n + 1)
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : c n ≤ 2 • b n
h7 : b (n + 1) = b n
h8 : c (n + 1) ≤ 2 • b (n + 1)
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | have h7 := c_succ_eq_D_of_b_bdd h h0 h1 h2 h4 h6 | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D)) | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
h7 : c (n + 1) = c D
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D)) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | refine max_le (h7 ▸ h3.trans' (le_max_left _ _))
(le_max_of_le_right <| nsmul_le_nsmul_right ((h _ h4).trans ?_) 2) | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
h7 : c (n + 1) = c D
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D)) | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
h7 : c (n + 1) = c D
⊢ max (b n) (c n - b n) ≤ c D - b D | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
h7 : c (n + 1) = c D
⊢ max (c (n + 1)) (2 • b (n + 1)) ≤ max (2 • b D) (2 • (c D - b D))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | rw [two_nsmul, ← lt_sub_iff_add_lt] at h6 | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
h7 : c (n + 1) = c D
⊢ max (b n) (c n - b n) ≤ c D - b D | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : b n < c n - b n
h7 : c (n + 1) = c D
⊢ max (b n) (c n - b n) ≤ c D - b D | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : 2 • b n < c n
h7 : c (n + 1) = c D
⊢ max (b n) (c n - b n) ≤ c D - b D
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | rw [max_eq_right h6.le, ← h7] | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : b n < c n - b n
h7 : c (n + 1) = c D
⊢ max (b n) (c n - b n) ≤ c D - b D | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : b n < c n - b n
h7 : c (n + 1) = c D
⊢ c n - b n ≤ c (n + 1) - b D | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : b n < c n - b n
h7 : c (n + 1) = c D
⊢ max (b n) (c n - b n) ≤ c D - b D
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | exact sub_le_sub (h2 n.le_succ) (h1 h4) | case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : b n < c n - b n
h7 : c (n + 1) = c D
⊢ c n - b n ≤ c (n + 1) - b D | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
G : Type u_1
D : ℕ
inst✝ : LinearOrderedAddCommGroup G
b c : ℕ → G
h : ∀ (n : ℕ), D ≤ n → b (n + 1) ≤ max (b n) (c n - b n)
h0 : ∀ (n : ℕ), D ≤ n → c (n + 1) ≤ max (c n) (2 • b n)
h1 : Monotone b
h2 : Monotone c
n✝ : ℕ
h3 : max (c D) (2 • b D) ≤ max (2 • b D) (2 • (c D - b D))
n : ℕ
h4 : D ≤ n
h5 : max (c n) (2 • b n) ≤ max (2 • b D) (2 • (c D - b D))
h6 : b n < c n - b n
h7 : c (n + 1) = c D
⊢ c n - b n ≤ c (n + 1) - b D
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.main_answer_is_good | [32, 1] | [35, 58] | rw [id, id, add_right_comm, add_left_inj, two_nsmul, mul_add,
add_mul, mul_comm y, add_assoc, ← add_mul, add_assoc] | R : Type u_1
inst✝ : NonUnitalCommRing R
C x y : R
⊢ id ((fun x => x * x + C) (x + y)) = (fun x => x * x + C) x + (2 • x + y) * id y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonUnitalCommRing R
C x y : R
⊢ id ((fun x => x * x + C) (x + y)) = (fun x => x * x + C) x + (2 • x + y) * id y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | have X (z : R) : φ.toEquiv z = (φ : R ≃+* S) z := rfl | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | intro x y | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
⊢ ∀ {x y : R},
g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | simp only [φ.conj_apply, X] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ x + φ y)))))) = φ (f (φ.symm (φ x))) + (2 • φ x + φ y) * φ (g (φ.symm (φ y))) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ.conj g (φ.conj f (φ.toEquiv x + φ.toEquiv y)) =
φ.conj f (φ.toEquiv x) + (2 • φ.toEquiv x + φ.toEquiv y) * φ.conj g (φ.toEquiv y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | rw [← φ.map_add, ← map_nsmul, ← φ.map_add] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ x + φ y)))))) = φ (f (φ.symm (φ x))) + (2 • φ x + φ y) * φ (g (φ.symm (φ y))) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ (x + y))))))) = φ (f (φ.symm (φ x))) + φ (2 • x + y) * φ (g (φ.symm (φ y))) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ x + φ y)))))) = φ (f (φ.symm (φ x))) + (2 • φ x + φ y) * φ (g (φ.symm (φ y)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | simp only [φ.toEquiv_eq_coe, EquivLike.coe_symm_apply_apply] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ (x + y))))))) = φ (f (φ.symm (φ x))) + φ (2 • x + y) * φ (g (φ.symm (φ y))) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x) + φ (2 • x + y) * φ (g y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔
φ (g (φ.symm (φ (f (φ.symm (φ (x + y))))))) = φ (f (φ.symm (φ x))) + φ (2 • x + y) * φ (g (φ.symm (φ y)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | rw [← φ.map_mul, ← φ.map_add] | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x) + φ (2 • x + y) * φ (g y) | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x + (2 • x + y) * g y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x) + φ (2 • x + y) * φ (g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.good_Equiv_conj_iff | [45, 1] | [53, 33] | exact φ.apply_eq_iff_eq.symm | R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x + (2 • x + y) * g y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f g : R → R
φ : R ≃+* S
X : ∀ (z : R), φ.toEquiv z = φ z
x y : R
⊢ g (f (x + y)) = f x + (2 • x + y) * g y ↔ φ (g (f (x + y))) = φ (f x + (2 • x + y) * g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1_1 | [57, 1] | [62, 43] | rw [nsmul_sub, sub_eq_sub_iff_add_eq_add, ← add_sub_right_comm, ← add_sub_assoc,
sub_eq_sub_iff_add_eq_add, two_nsmul, ← add_mul, add_assoc, ← add_mul,
← two_nsmul, ← h, add_comm _ (f y), add_assoc, two_nsmul, ← add_mul,
← add_mul, ← two_nsmul, ← h, add_comm] | R : Type u_1
inst✝ : NonUnitalNonAssocRing R
f g : R → R
h : good f g
x y : R
⊢ f x - x * g x - (f y - y * g y) = 2 • (y * g x - x * g y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
f g : R → R
h : good f g
x y : R
⊢ f x - x * g x - (f y - y * g y) = 2 • (y * g x - x * g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | obtain ⟨A, h0⟩ : ∃ A, ∀ x, g x = x * A + g 0 := ⟨g 1 - g 0, λ x ↦ by
have h0 : _ + _ = _ + _ := congrArg₂ _ (step1_1 h x 1) (step1_1 h 1 0)
rw [sub_add_sub_cancel, step1_1 h, ← nsmul_add] at h0
replace h0 := hR _ _ h0
rwa [zero_mul, zero_sub, one_mul, zero_mul, one_mul, zero_sub, eq_add_neg_iff_add_eq,
eq_sub_iff_add_eq', ← add_assoc, ← sub_eq_add_neg, ← mul_sub, eq_comm] at h0⟩ | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | refine ⟨A, g 0, f 0, funext λ x ↦ ?_, funext h0⟩ | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
⊢ f x = x * (x * A) - x * g 0 + f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
⊢ ∃ A B C, (f = fun x => x * (x * A) - x * B + C) ∧ g = fun x => x * A + B
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | have h1 := step1_1 h x 0 | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
⊢ f x = x * (x * A) - x * g 0 + f 0 | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
h1 : f x - x * g x - (f 0 - 0 * g 0) = 2 • (0 * g x - x * g 0)
⊢ f x = x * (x * A) - x * g 0 + f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
⊢ f x = x * (x * A) - x * g 0 + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | rwa [zero_mul, sub_zero, zero_mul, zero_sub, sub_eq_iff_eq_add,
sub_eq_iff_eq_add', h0, mul_add, two_nsmul, add_assoc _ _ (f 0),
add_add_add_comm, add_neg_cancel_left, ← sub_eq_add_neg] at h1 | case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
h1 : f x - x * g x - (f 0 - 0 * g 0) = 2 • (0 * g x - x * g 0)
⊢ f x = x * (x * A) - x * g 0 + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
A : R
h0 : ∀ (x : R), g x = x * A + g 0
x : R
h1 : f x - x * g x - (f 0 - 0 * g 0) = 2 • (0 * g x - x * g 0)
⊢ f x = x * (x * A) - x * g 0 + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | have h0 : _ + _ = _ + _ := congrArg₂ _ (step1_1 h x 1) (step1_1 h 1 0) | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
⊢ g x = x * (g 1 - g 0) + g 0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 :
f x - x * g x - (f 1 - 1 * g 1) + (f 1 - 1 * g 1 - (f 0 - 0 * g 0)) =
2 • (1 * g x - x * g 1) + 2 • (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | rw [sub_add_sub_cancel, step1_1 h, ← nsmul_add] at h0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 :
f x - x * g x - (f 1 - 1 * g 1) + (f 1 - 1 * g 1 - (f 0 - 0 * g 0)) =
2 • (1 * g x - x * g 1) + 2 • (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 2 • (0 * g x - x * g 0) = 2 • (1 * g x - x * g 1 + (0 * g 1 - 1 * g 0))
⊢ g x = x * (g 1 - g 0) + g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 :
f x - x * g x - (f 1 - 1 * g 1) + (f 1 - 1 * g 1 - (f 0 - 0 * g 0)) =
2 • (1 * g x - x * g 1) + 2 • (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | replace h0 := hR _ _ h0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 2 • (0 * g x - x * g 0) = 2 • (1 * g x - x * g 1 + (0 * g 1 - 1 * g 0))
⊢ g x = x * (g 1 - g 0) + g 0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 0 * g x - x * g 0 = 1 * g x - x * g 1 + (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 2 • (0 * g x - x * g 0) = 2 • (1 * g x - x * g 1 + (0 * g 1 - 1 * g 0))
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step1 | [64, 1] | [79, 67] | rwa [zero_mul, zero_sub, one_mul, zero_mul, one_mul, zero_sub, eq_add_neg_iff_add_eq,
eq_sub_iff_add_eq', ← add_assoc, ← sub_eq_add_neg, ← mul_sub, eq_comm] at h0 | R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 0 * g x - x * g 0 = 1 * g x - x * g 1 + (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
x : R
h0 : 0 * g x - x * g 0 = 1 * g x - x * g 1 + (0 * g 1 - 1 * g 0)
⊢ g x = x * (g 1 - g 0) + g 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h0 (x) : (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C := by
specialize h x (-(2 • x)); simp only at h
rwa [add_right_neg, zero_mul, add_zero, two_nsmul, neg_add, add_neg_cancel_comm_assoc,
neg_mul x A, neg_mul_neg, neg_mul, sub_neg_eq_add] at h | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h1 := h0 0 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : (0 * (0 * A) + 0 * B + C) * A + B = 0 * (0 * A) - 0 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [zero_mul, zero_mul, zero_add, sub_zero, zero_add] at h1 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : (0 * (0 * A) + 0 * B + C) * A + B = 0 * (0 * A) - 0 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : (0 * (0 * A) + 0 * B + C) * A + B = 0 * (0 * A) - 0 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h2 := h0 1 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (1 * (1 * A) + 1 * B + C) * A + B = 1 * (1 * A) - 1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [one_mul, one_mul, one_mul, add_mul, add_assoc, h1, add_left_inj] at h2 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (1 * (1 * A) + 1 * B + C) * A + B = 1 * (1 * A) - 1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (1 * (1 * A) + 1 * B + C) * A + B = 1 * (1 * A) - 1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | have h3 := h0 (-1) | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (-1 * (-1 * A) + -1 * B + C) * A + B = -1 * (-1 * A) - -1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [neg_one_mul, neg_one_mul, neg_one_mul, neg_neg,
← sub_eq_add_neg, sub_neg_eq_add, add_mul, add_assoc, h1, add_left_inj] at h3 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (-1 * (-1 * A) + -1 * B + C) * A + B = -1 * (-1 * A) - -1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (A - B) * A = A + B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (-1 * (-1 * A) + -1 * B + C) * A + B = -1 * (-1 * A) - -1 * B + C
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [← sub_mul, add_sub_sub_cancel, add_mul, ← two_nsmul, ← sub_sub,
sub_sub_cancel_left, sub_eq_add_neg, ← two_nsmul] at h3 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (A + B) * A - (A - B) * A = A - B - (A + B)
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : 2 • (B * A) = 2 • -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : (A + B) * A - (A - B) * A = A - B - (A + B)
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | replace h3 := hR _ _ h3 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : 2 • (B * A) = 2 • -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : 2 • (B * A) = 2 • -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [add_mul, h3, sub_eq_add_neg, add_left_inj] at h2 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h0 : ∀ (x : R), (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
h1 : C * A + B = C
h2 : (A + B) * A = A - B
h3 : B * A = -B
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [add_mul, mul_assoc, h2, add_right_eq_self, h3, neg_eq_zero] at h0 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : (C * A + B) * A = C * A
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : (C * A + B) * A = C * A
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rw [h0, add_zero] at h1 | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A + B = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | exact ⟨h0, h2, h1⟩ | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
h1 : C * A = C
h2 : A * A = A
h3 : B * A = -B
h0 : B = 0
⊢ B = 0 ∧ A * A = A ∧ C * A = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | specialize h x (-(2 • x)) | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
x : R
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
(fun x => x * A + B) ((fun x => x * (x * A) - x * B + C) (x + -(2 • x))) =
(fun x => x * (x * A) - x * B + C) x + (2 • x + -(2 • x)) * (fun x => x * A + B) (-(2 • x))
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
x : R
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | simp only at h | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
(fun x => x * A + B) ((fun x => x * (x * A) - x * B + C) (x + -(2 • x))) =
(fun x => x * (x * A) - x * B + C) x + (2 • x + -(2 • x)) * (fun x => x * A + B) (-(2 • x))
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
((x + -(2 • x)) * ((x + -(2 • x)) * A) - (x + -(2 • x)) * B + C) * A + B =
x * (x * A) - x * B + C + (2 • x + -(2 • x)) * (-(2 • x) * A + B)
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
(fun x => x * A + B) ((fun x => x * (x * A) - x * B + C) (x + -(2 • x))) =
(fun x => x * (x * A) - x * B + C) x + (2 • x + -(2 • x)) * (fun x => x * A + B) (-(2 • x))
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.step2 | [81, 1] | [103, 46] | rwa [add_right_neg, zero_mul, add_zero, two_nsmul, neg_add, add_neg_cancel_comm_assoc,
neg_mul x A, neg_mul_neg, neg_mul, sub_neg_eq_add] at h | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
((x + -(2 • x)) * ((x + -(2 • x)) * A) - (x + -(2 • x)) * B + C) * A + B =
x * (x * A) - x * B + C + (2 • x + -(2 • x)) * (-(2 • x) * A + B)
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C x : R
h :
((x + -(2 • x)) * ((x + -(2 • x)) * A) - (x + -(2 • x)) * B + C) * A + B =
x * (x * A) - x * B + C + (2 • x + -(2 • x)) * (-(2 • x) * A + B)
⊢ (x * (x * A) + x * B + C) * A + B = x * (x * A) - x * B + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | rcases step1 hR h with ⟨A, B, C, rfl, rfl⟩ | R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A C, A * A = A ∧ C * A = C ∧ (f = fun x => x * x * A + C) ∧ g = fun x => x * A | case intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * B + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + B) = fun x => x * A_1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ A C, A * A = A ∧ C * A = C ∧ (f = fun x => x * x * A + C) ∧ g = fun x => x * A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | rcases step2 hR h with ⟨rfl, h0, h1⟩ | case intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * B + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + B) = fun x => x * A_1 | case intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + 0) = fun x => x * A_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A B C : R
h : good (fun x => x * (x * A) - x * B + C) fun x => x * A + B
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * B + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + B) = fun x => x * A_1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | refine ⟨A, C, h0, h1, ?_, ?_⟩ | case intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + 0) = fun x => x * A_1 | case intro.intro.intro.intro.intro.intro.refine_1
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A + C
case intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * A + 0) = fun x => x * A | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ ∃ A_1 C_1,
A_1 * A_1 = A_1 ∧
C_1 * A_1 = C_1 ∧
((fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A_1 + C_1) ∧ (fun x => x * A + 0) = fun x => x * A_1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | funext x | case intro.intro.intro.intro.intro.intro.refine_1
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A + C | case intro.intro.intro.intro.intro.intro.refine_1.h
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
x : R
⊢ x * (x * A) - x * 0 + C = x * x * A + C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_1
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * (x * A) - x * 0 + C) = fun x => x * x * A + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | rw [mul_zero, sub_zero, ← mul_assoc] | case intro.intro.intro.intro.intro.intro.refine_1.h
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
x : R
⊢ x * (x * A) - x * 0 + C = x * x * A + C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_1.h
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
x : R
⊢ x * (x * A) - x * 0 + C = x * x * A + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.relation_summary | [105, 1] | [112, 25] | simp only [add_zero] | case intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * A + 0) = fun x => x * A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
inst✝ : Ring R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h : good (fun x => x * (x * A) - x * 0 + C) fun x => x * A + 0
h0 : A * A = A
h1 : C * A = C
⊢ (fun x => x * A + 0) = fun x => x * A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_self_comp_idempotent_self_mul_part | [133, 1] | [136, 74] | rw [mul_neg, ← neg_mul, neg_sub, sub_one_mul, mul_comm] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = (1 - a) * -x | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = x * a - x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = (1 - a) * -x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_self_comp_idempotent_self_mul_part | [133, 1] | [136, 74] | rfl | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = x * a - x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_self_mul_part h) ⟦x⟧ - x = x * a - x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_compl_comp_idempotent_compl_mul_part | [152, 1] | [156, 62] | change x * (1 - a) - x = _ | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_compl_mul_part h) ⟦x⟧ - x = a * -x | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ x * (1 - a) - x = a * -x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_compl_mul_part h) ⟦x⟧ - x = a * -x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_compl_comp_idempotent_compl_mul_part | [152, 1] | [156, 62] | rw [mul_one_sub, sub_sub_cancel_left, mul_neg, mul_comm] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ x * (1 - a) - x = a * -x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ x * (1 - a) - x = a * -x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.idempotent_decomp_AddHom_map_quot_self | [173, 1] | [175, 77] | rw [idempotent_decomp_AddHom_map_quot, ← mul_add, add_sub_cancel, mul_one] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_decomp_AddHom h) ((Ideal.Quotient.mk (Ideal.span {1 - a})) x, (Ideal.Quotient.mk (Ideal.span {a})) x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
x : R
⊢ (idempotent_decomp_AddHom h) ((Ideal.Quotient.mk (Ideal.span {1 - a})) x, (Ideal.Quotient.mk (Ideal.span {a})) x) = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_self_comp_idempotent_decomp_AddHom | [177, 1] | [182, 56] | rw [idempotent_decomp_AddHom, AddMonoidHom.coprod_apply,
map_add, quot_self_comp_idempotent_self_mul_part,
quot_self_comp_idempotent_compl_mul_part, add_zero] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {1 - a})) ((idempotent_decomp_AddHom h) p) = p.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {1 - a})) ((idempotent_decomp_AddHom h) p) = p.1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.quot_compl_comp_idempotent_decomp_AddHom | [184, 1] | [189, 57] | rw [idempotent_decomp_AddHom, AddMonoidHom.coprod_apply,
map_add, quot_compl_comp_idempotent_self_mul_part,
quot_compl_comp_idempotent_compl_mul_part, zero_add] | R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {a})) ((idempotent_decomp_AddHom h) p) = p.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
a : R
h : a * a = a
p : (R ⧸ Ideal.span {1 - a}) × R ⧸ Ideal.span {a}
⊢ (Ideal.Quotient.mk (Ideal.span {a})) ((idempotent_decomp_AddHom h) p) = p.2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rcases relation_summary hR h with ⟨A, C, h0, h1, rfl, rfl⟩ | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0) | case intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
⊢ ∃ R₁ R₂ x x_1 φ c,
(fun x => x * x * A + C) = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧
(fun x => x * A) = φ.conj (Prod.map id 0) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
h : good f g
⊢ ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | refine ⟨R ⧸ Ideal.span {1 - A}, R ⧸ Ideal.span {A},
Ideal.Quotient.commRing _, Ideal.Quotient.commRing _,
(idempotent_decomp h0).symm, idempotent_self_mul_part h0 C,
funext λ x ↦ ?_, funext λ x ↦ ?_⟩ | case intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
⊢ ∃ R₁ R₂ x x_1 φ c,
(fun x => x * x * A + C) = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧
(fun x => x * A) = φ.conj (Prod.map id 0) | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm.conj
(Prod.map
(fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
0)
x
case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = (idempotent_decomp h0).symm.conj (Prod.map id 0) x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
⊢ ∃ R₁ R₂ x x_1 φ c,
(fun x => x * x * A + C) = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧
(fun x => x * A) = φ.conj (Prod.map id 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | change _ = (idempotent_decomp h0).symm (_, 0) | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm.conj
(Prod.map
(fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
0)
x | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm
((fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
((Ideal.Quotient.mk (Ideal.span {1 - A})) x),
0) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm.conj
(Prod.map
(fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
0)
x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rw [idempotent_decomp_symm_apply, idempotent_self_mul_part_map_quot, map_zero,
add_zero, h1, map_add, idempotent_self_mul_part_map_quot, ← map_mul,
idempotent_self_mul_part_map_quot, h1] | case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm
((fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
((Ideal.Quotient.mk (Ideal.span {1 - A})) x),
0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * x * A + C =
(idempotent_decomp h0).symm
((fun x =>
x * x +
(Ideal.Quotient.mk (Ideal.span {1 - A}))
((idempotent_self_mul_part h0) ((Ideal.Quotient.mk (Ideal.span {1 - A})) C)))
((Ideal.Quotient.mk (Ideal.span {1 - A})) x),
0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | change x * A = x * A + 0 * (1 - A) | case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = (idempotent_decomp h0).symm.conj (Prod.map id 0) x | case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = x * A + 0 * (1 - A) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = (idempotent_decomp h0).symm.conj (Prod.map id 0) x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rw [zero_mul, add_zero] | case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = x * A + 0 * (1 - A) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.refine_2
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
A C : R
h0 : A * A = A
h1 : C * A = C
h : good (fun x => x * x * A + C) fun x => x * A
x : R
⊢ x * A = x * A + 0 * (1 - A)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | rw [h, h0, ← good_Equiv_conj_iff] | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good f g | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good (Prod.map (fun x => x * x + c) 0) (Prod.map id 0) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good f g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A3/A3.lean | IMOSL.IMO2011A3.final_solution | [213, 1] | [231, 61] | exact prod_is_good (main_answer_is_good c) zero_is_good | R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good (Prod.map (fun x => x * x + c) 0) (Prod.map id 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
inst✝ : CommRing R
hR : ∀ (x y : R), 2 • x = 2 • y → x = y
f g : R → R
x✝ : ∃ R₁ R₂ x x_1 φ c, f = φ.conj (Prod.map (fun x_2 => x_2 * x_2 + c) 0) ∧ g = φ.conj (Prod.map id 0)
R₁ R₂ : Type u
hR₁ : CommRing R₁
hR₂ : CommRing R₂
φ : R₁ × R₂ ≃+* R
c : R₁
h : f = φ.conj (Prod.map (fun x => x * x + c) 0)
h0 : g = φ.conj (Prod.map id 0)
⊢ good (Prod.map (fun x => x * x + c) 0) (Prod.map id 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | have h0 : f (c + 1) = 0 := by cases h with
| Left h => rw [h, hf.map_one, mul_zero]
| Right h => rw [add_comm, h, hf.map_one, zero_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (-c) = f c | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
⊢ f (-c) = f c | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (-c) = f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | have h1 := hf.is_good (c + 1) (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
⊢ f (-c) = f c | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
h1 : f ((c + 1) * -1 + 1) = f (c + 1) * f (-1) + f (c + 1 + -1)
⊢ f (-c) = f c | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
⊢ f (-c) = f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | rwa [h0, zero_mul, zero_add, add_neg_cancel_right,
mul_neg_one, neg_add_rev, neg_add_cancel_comm] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
h1 : f ((c + 1) * -1 + 1) = f (c + 1) * f (-1) + f (c + 1 + -1)
⊢ f (-c) = f c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
h1 : f ((c + 1) * -1 + 1) = f (c + 1) * f (-1) + f (c + 1 + -1)
⊢ f (-c) = f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | cases h with
| Left h => rw [h, hf.map_one, mul_zero]
| Right h => rw [add_comm, h, hf.map_one, zero_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | rw [h, hf.map_one, mul_zero] | case Left
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f (c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Left
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f (c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | rw [add_comm, h, hf.map_one, zero_mul] | case Right
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f (c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Right
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f (c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | cases h with
| Left h => specialize h (-c); rwa [add_neg_self, h0, neg_mul] at h
| Right h => specialize h (-c); rwa [neg_add_self, h0, mul_neg] at h | R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
⊢ f 0 = -(f c * f c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | specialize h (-c) | case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f 0 = -(f c * f c) | case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (c + -c) = -f c * f (-c)
⊢ f 0 = -(f c * f c) | Please generate a tactic in lean4 to solve the state.
STATE:
case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | rwa [add_neg_self, h0, neg_mul] at h | case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (c + -c) = -f c * f (-c)
⊢ f 0 = -(f c * f c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (c + -c) = -f c * f (-c)
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | specialize h (-c) | case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f 0 = -(f c * f c) | case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (-c + c) = f (-c) * -f c
⊢ f 0 = -(f c * f c) | Please generate a tactic in lean4 to solve the state.
STATE:
case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | rwa [neg_add_self, h0, mul_neg] at h | case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (-c + c) = f (-c) * -f c
⊢ f 0 = -(f c * f c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (-c + c) = f (-c) * -f c
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | rwa [hf.map_zero, neg_inj, eq_comm] at h1 | R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
h1 : f 0 = -(f c * f c)
⊢ f c * f c = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
h1 : f 0 = -(f c * f c)
⊢ f c * f c = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | cases map_eq_one_or_neg_one hf h with
| inl h => rw [h]; exact Commute.neg_one_left (f x)
| inr h => rw [h, neg_neg]; exact Commute.one_left (f x) | R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ Commute (-f c) (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ Commute (-f c) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | rw [h] | case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-f c) (f x) | case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-1) (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-f c) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | exact Commute.neg_one_left (f x) | case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-1) (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-1) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | rw [h, neg_neg] | case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute (-f c) (f x) | case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute 1 (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute (-f c) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | exact Commute.one_left (f x) | case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute 1 (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute 1 (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.imp_left | [62, 1] | [63, 63] | rw [add_comm, h0, map_commute hf h] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (x + c) = f x * -f c
x : R
⊢ f (c + x) = -f c * f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (x + c) = f x * -f c
x : R
⊢ f (c + x) = -f c * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.imp_right | [65, 1] | [66, 65] | rw [add_comm, h0, map_commute hf h] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (c + x) = -f c * f x
x : R
⊢ f (x + c) = f x * -f c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (c + x) = -f c * f x
x : R
⊢ f (x + c) = f x * -f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.iff_left2 | [74, 1] | [76, 63] | rw [neg_mul, hf.is_good, eq_neg_iff_add_eq_zero, add_comm] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (c + x) = -f c * f x ↔ f (c * x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (c + x) = -f c * f x ↔ f (c * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.iff_right2 | [78, 1] | [80, 63] | rw [mul_neg, hf.is_good, eq_neg_iff_add_eq_zero, add_comm] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (x + c) = f x * -f c ↔ f (x * c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (x + c) = f x * -f c ↔ f (x * c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | rw [iff_right2 hf] at h ⊢ | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (d * c) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d : R
⊢ ∀ (x : R), f (x * (d * c) + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (d * c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | intro x | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d : R
⊢ ∀ (x : R), f (x * (d * c) + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * (d * c) + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d : R
⊢ ∀ (x : R), f (x * (d * c) + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | rw [← mul_assoc] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * (d * c) + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * d * c + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * (d * c) + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | exact h (x * d) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * d * c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * d * c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | rw [iff_left2 hf] at h ⊢ | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (c * d) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d : R
⊢ ∀ (x : R), f (c * d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (c * d)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | intro x | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d : R
⊢ ∀ (x : R), f (c * d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d : R
⊢ ∀ (x : R), f (c * d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | rw [mul_assoc] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * (d * x) + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | exact h (d * x) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * (d * x) + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * (d * x) + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_eq_zero_iff | [112, 1] | [114, 79] | rw [add_zero, (iff_right hf.toNontrivialGood).mp h, h0, neg_neg, mul_one] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0✝ : c ≠ 0
h0 : f c = -1
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✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0✝ : c ≠ 0
h0 : f c = -1
x : R
⊢ f (x + c) = f (x + 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_QuasiPeriodic_eq | [120, 1] | [127, 60] | have h3 (c) := (iff_right hf.toNontrivialGood (c := c)).mp | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
⊢ f (x + d) = f (x + c) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
h3 : ∀ (c : R), QuasiPeriodic f c → ∀ (x : R), f (x + c) = f x * -f c
⊢ f (x + d) = f (x + c) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
⊢ f (x + d) = f (x + c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_QuasiPeriodic_eq | [120, 1] | [127, 60] | rw [h3 c h, h3 d h1, h2, reduced_map_eq_one hf h h0] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
h3 : ∀ (c : R), QuasiPeriodic f c → ∀ (x : R), f (x + c) = f x * -f c
⊢ f (x + d) = f (x + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
h3 : ∀ (c : R), QuasiPeriodic f c → ∀ (x : R), f (x + c) = f x * -f c
⊢ f (x + d) = f (x + c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [iff_left2 hf.toNontrivialGood] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ QuasiPeriodic f d | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ ∀ (x : R), f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ QuasiPeriodic f d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | intro x | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ ∀ (x : R), f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ ∀ (x : R), f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | have h2 := reduced_map_eq_one hf h h0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
⊢ f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | have h3 : f (d * (c + x) + 1) = -f (d * x + 1) := by
rw [iff_left hf.toNontrivialGood] at h
rw [hf.is_good, h, add_left_comm, h, h2, neg_one_mul,
neg_one_mul, mul_neg, ← neg_add, hf.is_good] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : f (d * (c + x) + 1) = -f (d * x + 1)
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [mul_add, h1, zero_add, eq_neg_iff_add_eq_zero, ← two_mul, mul_eq_zero] at h3 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : f (d * (c + x) + 1) = -f (d * x + 1)
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : 2 = 0 ∨ f (d * x + 1) = 0
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : f (d * (c + x) + 1) = -f (d * x + 1)
⊢ f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | refine h3.resolve_left λ h3 ↦ h0 ?_ | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : 2 = 0 ∨ f (d * x + 1) = 0
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3✝ : 2 = 0 ∨ f (d * x + 1) = 0
h3 : 2 = 0
⊢ c = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : 2 = 0 ∨ f (d * x + 1) = 0
⊢ f (d * x + 1) = 0
TACTIC:
|
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