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/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [h1, ← h1 0, zero_add] | N : ℤ
h : N ≠ 0
f : ℤ → ℤ
h0 : ∀ (a b : ℤ), f (N * a) + N * f b = f (f (a + b))
h1 : ∀ (n : ℤ), N * (f (n + 1) - f n) = f N - f 0
n : ℤ
⊢ N * (f (n + 1) - f n) = N * (f 1 - f 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℤ
h : N ≠ 0
f : ℤ → ℤ
h0 : ∀ (a b : ℤ), f (N * a) + N * f b = f (f (a + b))
h1 : ∀ (n : ℤ), N * (f (n + 1) - f n) = f N - f 0
n : ℤ
⊢ N * (f (n + 1) - f n) = N * (f 1 - f 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rwa [h2] | N : ℤ
h : N ≠ 0
f : ℤ → ℤ
h0 : ∀ (a b : ℤ), f (N * a) + N * f b = f (f (a + b))
q : ℤ
h1 : ∀ (n : ℤ), f n = q * n + f 0
h2 : N = q
⊢ ∀ (x : ℤ), f x = N * x + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℤ
h : N ≠ 0
f : ℤ → ℤ
h0 : ∀ (a b : ℤ), f (N * a) + N * f b = f (f (a + b))
q : ℤ
h1 : ∀ (n : ℤ), f n = q * n + f 0
h2 : N = q
⊢ ∀ (x : ℤ), f x = N * x + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rcases h0 with rfl | ⟨c, rfl⟩ | N : ℤ
h : N ≠ 0
f : ℤ → ℤ
h0 : f = 0 ∨ ∃ c, f = fun x => N * x + c
a b : ℤ
⊢ f (N * a) + N * f b = f (f (a + b)) | case inl
N : ℤ
h : N ≠ 0
a b : ℤ
⊢ 0 (N * a) + N * 0 b = 0 (0 (a + b))
case inr.intro
N : ℤ
h : N ≠ 0
a b c : ℤ
⊢ (fun x => N * x + c) (N * a) + N * (fun x => N * x + c) b = (fun x => N * x + c) ((fun x => N * x + c) (a + b)) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℤ
h : N ≠ 0
f : ℤ → ℤ
h0 : f = 0 ∨ ∃ c, f = fun x => N * x + c
a b : ℤ
⊢ f (N * a) + N * f b = f (f (a + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | exact (N * 0).zero_add.trans N.mul_zero | case inl
N : ℤ
h : N ≠ 0
a b : ℤ
⊢ 0 (N * a) + N * 0 b = 0 (0 (a + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
N : ℤ
h : N ≠ 0
a b : ℤ
⊢ 0 (N * a) + N * 0 b = 0 (0 (a + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [add_right_comm, ← mul_add, ← add_assoc, ← mul_add] | case inr.intro
N : ℤ
h : N ≠ 0
a b c : ℤ
⊢ (fun x => N * x + c) (N * a) + N * (fun x => N * x + c) b = (fun x => N * x + c) ((fun x => N * x + c) (a + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
N : ℤ
h : N ≠ 0
a b c : ℤ
⊢ (fun x => N * x + c) (N * a) + N * (fun x => N * x + c) b = (fun x => N * x + c) ((fun x => N * x + c) (a + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.map_even_of_map_one | [31, 1] | [34, 54] | specialize hf (x + 1) (-1) | R : Type u_1
S : Type u_2
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : good f
h : f (-1) = 0
x : R
⊢ f (-x) = f x | R : Type u_1
S : Type u_2
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
h : f (-1) = 0
x : R
hf : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : good f
h : f (-1) = 0
x : R
⊢ f (-x) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.map_even_of_map_one | [31, 1] | [34, 54] | rwa [h, mul_zero, zero_add, add_neg_cancel_right,
mul_neg_one, neg_add, neg_add_cancel_right] at hf | R : Type u_1
S : Type u_2
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
h : f (-1) = 0
x : R
hf : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
h : f (-1) = 0
x : R
hf : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq1 | [39, 1] | [40, 84] | rw [← h y, sub_eq_add_neg x, ← hf.is_good, mul_neg, neg_add_eq_sub, ← neg_sub, h] | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x y : R
⊢ f (x * y - 1) = f x * f y + f (x - y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x y : R
⊢ f (x * y - 1) = f x * f y + f (x - y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq2 | [43, 1] | [46, 59] | have h0 := hf.is_good (x - 1) (1 + 1) | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1) | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f ((x - 1) * (1 + 1) + 1) = f (x - 1) * f (1 + 1) + f (x - 1 + (1 + 1))
⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq2 | [43, 1] | [46, 59] | rwa [sub_add_add_cancel, one_add_one_eq_two, mul_two, add_assoc,
sub_add_cancel, ← add_sub_right_comm, ← mul_two] at h0 | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f ((x - 1) * (1 + 1) + 1) = f (x - 1) * f (1 + 1) + f (x - 1 + (1 + 1))
⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f ((x - 1) * (1 + 1) + 1) = f (x - 1) * f (1 + 1) + f (x - 1 + (1 + 1))
⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq3 | [49, 1] | [51, 82] | have h0 := Eq2 hf (-x) | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (-x * 2 - 1) = f (-x - 1) * f 2 + f (-x + 1)
⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq3 | [49, 1] | [51, 82] | rwa [neg_mul, ← neg_add', h, ← neg_add', h, neg_add_eq_sub, ← neg_sub, h] at h0 | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (-x * 2 - 1) = f (-x - 1) * f 2 + f (-x + 1)
⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (-x * 2 - 1) = f (-x - 1) * f 2 + f (-x + 1)
⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq5 | [54, 1] | [64, 80] | rw [two_mul, ← add_assoc, add_left_comm, this, sub_add_cancel_right, h] | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
this : ∀ (y : R), f (x + y + 1) = f (x - y)
y : R
⊢ f (y + (2 * x + 1)) = f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
this : ∀ (y : R), f (x + y + 1) = f (x - y)
y : R
⊢ f (y + (2 * x + 1)) = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq5 | [54, 1] | [64, 80] | have h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) := by
rw [add_one_mul x, mul_add, add_one_mul x] | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
⊢ f (x + y + 1) = f (x - y) | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
⊢ f (x + y + 1) = f (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
⊢ f (x + y + 1) = f (x - y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq5 | [54, 1] | [64, 80] | have h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1 := by
rw [mul_add_one _ y, add_sub_assoc, add_sub_cancel_right, add_comm] | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
⊢ f (x + y + 1) = f (x - y) | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1
⊢ f (x + y + 1) = f (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
⊢ f (x + y + 1) = f (x - y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq5 | [54, 1] | [64, 80] | rwa [hf.is_good, h3, Eq1 hf h, hf.is_good, ← add_rotate, ← mul_add_one x,
hf.is_good, h0, h1, zero_mul, zero_add, zero_mul, zero_add, zero_add,
zero_mul, zero_add, add_sub_add_right_eq_sub, ← add_assoc, eq_comm] at h2 | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1
⊢ f (x + y + 1) = f (x - y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1
⊢ f (x + y + 1) = f (x - y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq5 | [54, 1] | [64, 80] | rw [add_one_mul x, mul_add, add_one_mul x] | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
⊢ f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
⊢ f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.Eq5 | [54, 1] | [64, 80] | rw [mul_add_one _ y, add_sub_assoc, add_sub_cancel_right, add_comm] | R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
⊢ x + (x + 1) * y = (x + 1) * (y + 1) - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝¹ : NonAssocRing R
inst✝ : NonAssocSemiring S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x = 0
h1 : f (x + 1) = 0
y : R
h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
⊢ x + (x + 1) * y = (x + 1) * (y + 1) - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq4 | [75, 1] | [83, 58] | have h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1 := by
rw [mul_add_one (x - 1), add_assoc, sub_add_cancel, sub_one_mul,
← add_sub_right_comm, add_comm, add_sub_add_right_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq4 | [75, 1] | [83, 58] | apply congrArg f at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (x * (x + 1) - 1) = f ((x - 1) * (x + 1 + 1) + 1)
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq4 | [75, 1] | [83, 58] | rw [Eq1 hf h, hf.is_good, sub_add_cancel_left, h, hf.map_one, sub_add_add_cancel,
add_zero, add_assoc, one_add_one_eq_two, ← add_assoc, ← mul_two] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (x * (x + 1) - 1) = f ((x - 1) * (x + 1 + 1) + 1)
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x * f (x + 1) = f (x - 1) * f (x + 2) + f (x * 2 + 1)
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (x * (x + 1) - 1) = f ((x - 1) * (x + 1 + 1) + 1)
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq4 | [75, 1] | [83, 58] | rw [Eq2 hf, mul_add, h0, ← add_assoc, add_one_mul (f _),
add_left_inj, mul_left_comm, ← mul_add, ← hf.is_good] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x * f (x + 1) = f (x - 1) * f (x + 2) + f (x * 2 + 1)
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f x * f (x + 1) = f (x - 1) * f (x + 2) + f (x * 2 + 1)
⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq4 | [75, 1] | [83, 58] | rw [mul_add_one (x - 1), add_assoc, sub_add_cancel, sub_one_mul,
← add_sub_right_comm, add_comm, add_sub_add_right_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq4_alt | [86, 1] | [88, 85] | have h0 := Eq4 hf h (-x) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (-x) * f (-x * 2 - 1) = (f (-x - 1) + 1) * f (-x * 2 + 1)
⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq4_alt | [86, 1] | [88, 85] | rwa [h, neg_mul, ← neg_add', h, ← neg_add', h, neg_add_eq_sub, ← neg_sub, h] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (-x) * f (-x * 2 - 1) = (f (-x - 1) + 1) * f (-x * 2 + 1)
⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
x : R
h0 : f (-x) * f (-x * 2 - 1) = (f (-x - 1) + 1) * f (-x * 2 + 1)
⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | rw [hf.is_good, h0, mul_neg_one, neg_add_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
x : R
⊢ f (x * 2 + 1) = f (x + 2) - 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 : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
x : R
⊢ f (x * 2 + 1) = f (x + 2) - f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | rw [Eq2 hf, Eq3 hf h, h0, mul_neg_one, mul_neg_one, neg_add_rev, neg_neg] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
x : R
⊢ f (x * 2 + 1) = -f (x * 2 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
x : R
⊢ f (x * 2 + 1) = -f (x * 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | have h3 := Eq4_alt hf h x | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
h3 : f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | rw [h2, mul_neg, neg_eq_iff_add_eq_zero, ← add_mul, mul_eq_zero, ← add_assoc] at h3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
h3 : f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
h3 : f x + f (x + 1) + 1 = 0 ∨ f (x * 2 - 1) = 0
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
h3 : f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | exact h3.imp_left eq_neg_of_add_eq_zero_left | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
h3 : f x + f (x + 1) + 1 = 0 ∨ f (x * 2 - 1) = 0
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 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 : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
x : R
h3 : f x + f (x + 1) + 1 = 0 ∨ f (x * 2 - 1) = 0
⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | rwa [mul_two, add_sub_assoc, add_sub_cancel_right,
add_right_comm, ← mul_two, h1, sub_eq_zero] at h4 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
x : R
h4 : f ((x + 1) * 2 - 1) = 0
⊢ f (x + 2) = 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 : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
x : R
h4 : f ((x + 1) * 2 - 1) = 0
⊢ f (x + 2) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | rwa [← neg_eq_zero, ← h2, h1, sub_eq_zero] at h5 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
x : R
h4 : f (x + 1) + f (x + 1 + 1) = -1
h5 : f (x * 2 - 1) = 0
⊢ f (x + 2) = 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 : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
x : R
h4 : f (x + 1) + f (x + 1 + 1) = -1
h5 : f (x * 2 - 1) = 0
⊢ f (x + 2) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two | [91, 1] | [107, 90] | rwa [← h5, add_comm, add_left_inj, add_assoc, one_add_one_eq_two] at h4 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
x : R
h4 : f (x + 1) + f (x + 1 + 1) = -1
h5 : f x + f (x + 1) = -1
⊢ f (x + 2) = 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 : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 = -1
h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x
h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1)
h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
x : R
h4 : f (x + 1) + f (x + 1 + 1) = -1
h5 : f x + f (x + 1) = -1
⊢ f (x + 2) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6_ring_id | [109, 1] | [111, 57] | ring | R : Type ?u.45782
S✝ : Type ?u.45785
inst✝³ : Ring R
inst✝² : CommRing S✝
inst✝¹ : NoZeroDivisors S✝
f : R → S✝
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
S : Type u_1
inst✝ : CommRing S
a b c d : S
⊢ a * (c * d + b) - a * (b * d + c) - ((c + 1) * (b * d + c) - (b + 1) * (c * d + b)) =
(b + c - (a + 1) * (d - 1)) * (b - c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.45782
S✝ : Type ?u.45785
inst✝³ : Ring R
inst✝² : CommRing S✝
inst✝¹ : NoZeroDivisors S✝
f : R → S✝
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
S : Type u_1
inst✝ : CommRing S
a b c d : S
⊢ a * (c * d + b) - a * (b * d + c) - ((c + 1) * (b * d + c) - (b + 1) * (c * d + b)) =
(b + c - (a + 1) * (d - 1)) * (b - c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | intro x | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
⊢ ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
x : R
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
⊢ ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | refine (h1 x).elim id λ h2 ↦ ?_ | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
x : R
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
x : R
h2 : f (x + 1) = f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
x : R
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | specialize h1 (x + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
x : R
h2 : f (x + 1) = f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 1 + 1) = f (x + 1 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
x : R
h2 : f (x + 1) = f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [add_sub_cancel_right, add_assoc, one_add_one_eq_two] at h1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 1 + 1) = f (x + 1 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 2) = f x
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 1 + 1) = f (x + 1 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rcases h1 with h1 | h1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 2) = f x
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 2) = f x
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [Eq2 hf, Eq3 hf h, ← sub_eq_zero, Eq6_ring_id, mul_eq_zero] at h1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h1 : f x * f (x * 2 - 1) - f x * f (x * 2 + 1) = (f (x - 1) + 1) * f (x * 2 + 1) - (f (x + 1) + 1) * f (x * 2 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h1 : f (x + 1) + f (x - 1) - (f x + 1) * (f 2 - 1) = 0 ∨ f (x + 1) - f (x - 1) = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h1 : f x * f (x * 2 - 1) - f x * f (x * 2 + 1) = (f (x - 1) + 1) * f (x * 2 + 1) - (f (x + 1) + 1) * f (x * 2 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | exact h1.imp eq_of_sub_eq_zero eq_of_sub_eq_zero | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h1 : f (x + 1) + f (x - 1) - (f x + 1) * (f 2 - 1) = 0 ∨ f (x + 1) - f (x - 1) = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h1 : f (x + 1) + f (x - 1) - (f x + 1) * (f 2 - 1) = 0 ∨ f (x + 1) - f (x - 1) = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | have h2 := Eq3 hf h x | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2✝ : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
h2 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [hf.is_good, eq_sub_of_add_eq h1, add_sub_left_comm, ← mul_sub_one,
add_one_mul (f _), add_assoc, ← one_add_mul (f x), mul_sub_one,
← add_sub_right_comm, add_sub_assoc, add_right_inj, add_comm] at h2 | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2✝ : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
h2 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2✝ : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
h2 : (f x + 1) * (f 2 - 1) - f (x + 1) = f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2✝ : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
h2 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | exact (eq_add_of_sub_eq' h2).symm | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2✝ : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
h2 : (f x + 1) * (f 2 - 1) - f (x + 1) = f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2✝ : f (x + 1) = f (x - 1)
h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1)
h2 : (f x + 1) * (f 2 - 1) - f (x + 1) = f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | have h0 : f 2 + 1 ≠ 0 := λ X ↦ h0 (eq_neg_of_add_eq_zero_left X) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | have h3 := Eq3 hf h x | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [← h2, hf.is_good, h1, ← mul_add_one (f x), ← mul_add_one (f _),
← sub_eq_zero, ← sub_mul, mul_eq_zero, or_iff_left h0, sub_eq_zero] at h3 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | have h4 := Eq4 hf h x | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
h4 : f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [Eq3 hf h, Eq2 hf, ← h2, ← h3, ← sub_eq_zero, ← sub_mul, sub_add_cancel_left,
neg_one_mul, neg_eq_zero, ← mul_add_one (f x), mul_eq_zero, or_iff_left h0] at h4 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
h4 : f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
h4 : f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [eq_comm, h4] at h3 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f x = f (x + 1)
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [eq_comm, h3] at h2 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x + 1) = f (x - 1)
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | have h5 := Eq5 hf h h4 h3 0 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
h5 : f (0 + (2 * x + 1)) = f 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [zero_add, hf.is_good, add_comm 2 x, h1, h4, mul_zero,
add_zero, hf.map_zero, eq_comm, neg_eq_zero] at h5 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
h5 : f (0 + (2 * x + 1)) = f 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
h5 : 1 = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
h5 : f (0 + (2 * x + 1)) = f 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq6 | [114, 1] | [143, 56] | rw [← sub_eq_zero, ← one_mul (_ - _), h5, zero_mul] | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
h5 : 1 = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0✝ : f 2 ≠ -1
x : R
h2 : f (x - 1) = 0
h1 : f (x + 2) = f x
h0 : f 2 + 1 ≠ 0
h3 : f (x + 1) = 0
h4 : f x = 0
h5 : 1 = 0
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [← mul_assoc, ← sq, mul_left_comm, Eq4 hf h,
← mul_assoc, ← sub_eq_zero, ← sub_mul, mul_eq_zero] at h2 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x * (f x * f (x * 2 + 1)) = f x * ((f (x + 1) + 1) * f (x * 2 - 1))
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ∨ f (x * 2 + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x * (f x * f (x * 2 + 1)) = f x * ((f (x + 1) + 1) * f (x * 2 - 1))
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rcases h2 with h2 | h2 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ∨ f (x * 2 + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ∨ f (x * 2 + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | exact (eq_of_sub_eq_zero h2).symm | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [← mul_assoc, ← sq, mul_left_comm, Eq4_alt hf h,
← mul_assoc, ← sub_eq_zero, ← sub_mul, mul_eq_zero] at h3 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x * (f x * f (x * 2 - 1)) = f x * ((f (x - 1) + 1) * f (x * 2 + 1))
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ∨ f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x * (f x * f (x * 2 - 1)) = f x * ((f (x - 1) + 1) * f (x * 2 + 1))
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rcases h3 with h3 | h3 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ∨ f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ∨ f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rwa [sub_eq_zero, eq_comm, mul_comm] at h3 | case inr.inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [Eq3 hf h] at h2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x * 2 + 1) = 0
h3 : f (x * 2 - 1) = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [Eq2 hf, add_add_add_comm, add_zero, ← add_mul,
add_comm (f _), ← mul_add_one (α := S), mul_eq_zero,
or_iff_left (h0 ∘ eq_neg_of_add_eq_zero_left), ← eq_neg_iff_add_eq_zero] at h3 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x + 1) * f 2 + f (x - 1) + f (x * 2 - 1) = 0 + 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x + 1) * f 2 + f (x - 1) + f (x * 2 - 1) = 0 + 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | have X : f 2 - 1 ≠ 0 := h1 ∘ eq_of_sub_eq_zero | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h3, ← sub_eq_add_neg, ← mul_sub_one, mul_eq_zero, or_iff_left X] at h2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h2, neg_zero] at h3 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | have h4 := Eq6 hf h h0 x | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h2, h3, add_zero, zero_eq_mul, or_iff_left X] at h4 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f x + 1 = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h2, h3, zero_add, one_mul, eq_neg_of_add_eq_zero_left h4, neg_one_sq] | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f x + 1 = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f x + 1 = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Rtwo_ne_zero | [195, 1] | [196, 56] | rw [h, hf.map_zero] | R : Type u_1
S : Type ?u.70700
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : 2 = 0
⊢ f 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.70700
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : 2 = 0
⊢ f 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | rcases CommSubring.oneVarCommLiftDomain_exists hf.toNontrivialGood c with
⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h0, hf'⟩ | R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
c : R
⊢ ∃ R' x φ,
∃ (_ : Function.Injective ⇑φ) (_ : c ∈ Set.range ⇑φ),
∃ S' x_3,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'), f (φ a) = ρ (f' a)), GoodCase2 f' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'_1), f (φ_1 a) = ρ (f' a)), GoodCase2 f' | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
c : R
⊢ ∃ R' x φ,
∃ (_ : Function.Injective ⇑φ) (_ : c ∈ Set.range ⇑φ),
∃ S' x_3,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'), f (φ a) = ρ (f' a)), GoodCase2 f'
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | refine ⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h0, hf', ?_, ?_⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'_1), f (φ_1 a) = ρ (f' a)), GoodCase2 f' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 ≠ -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'_1), f (φ_1 a) = ρ (f' a)), GoodCase2 f'
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | intro x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ f' (-x) = f' x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | apply hρ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ f' (-x) = f' x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.a
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ ρ (f' (-x)) = ρ (f' x) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ f' (-x) = f' x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | rw [← h0, ← h0, φ.map_neg, hf.map_even] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.a
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ ρ (f' (-x)) = ρ (f' x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.a
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ ρ (f' (-x)) = ρ (f' x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | intro h1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 ≠ -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 ≠ -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | apply hf.map_two_ne_neg_one | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ f 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | rw [← map_ofNat φ 2, h0, h1, ρ.map_neg, ρ.map_one] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ f 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ f 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq6 | [228, 1] | [234, 78] | rcases oneVarLift_exists hf x with
⟨R', R'comm, φ, -, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, -, f', h0, hf'⟩ | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
x : R
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ f (φ x + 1) + f (φ x - 1) = (f (φ x) + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
x : R
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq6 | [228, 1] | [234, 78] | rw [h0, ← φ.map_one, ← φ.map_sub, ← φ.map_add, h0, h0, ← map_ofNat φ 2, h0,
← ρ.map_one, ← ρ.map_add, ← ρ.map_add, ← ρ.map_sub, ← ρ.map_mul] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ f (φ x + 1) + f (φ x - 1) = (f (φ x) + 1) * (f 2 - 1) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ρ (f' (x + 1) + f' (x - 1)) = ρ ((f' x + 1) * (f' (OfNat.ofNat 2) - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ f (φ x + 1) + f (φ x - 1) = (f (φ x) + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq6 | [228, 1] | [234, 78] | exact congrArg ρ
(CommCase.Eq6 hf'.toNontrivialGood hf'.map_even hf'.map_two_ne_neg_one x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ρ (f' (x + 1) + f' (x - 1)) = ρ ((f' x + 1) * (f' (OfNat.ofNat 2) - 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ρ (f' (x + 1) + f' (x - 1)) = ρ ((f' x + 1) * (f' (OfNat.ofNat 2) - 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | rcases oneVarLift_exists hf x with
⟨R', R'comm, φ, -, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, -, f', h0, hf'⟩ | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
x : R
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
x : R
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | have h2 : f' 2 ≠ 1 := λ h2 ↦ h1 <| by rw [← map_ofNat φ 2, h0, h2, ρ.map_one] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | rw [h0, ← φ.map_one, ← φ.map_sub, ← φ.map_add, h0, h0, ← ρ.map_one,
← ρ.map_add, ← ρ.map_add, ← ρ.map_mul, ← ρ.map_pow] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ ρ ((f' (x + 1) + 1) * (f' (x - 1) + 1)) = ρ (f' x ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | exact congrArg ρ
(CommCase.Eq7 hf'.toNontrivialGood hf'.map_even hf'.map_two_ne_neg_one h2 x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ ρ ((f' (x + 1) + 1) * (f' (x - 1) + 1)) = ρ (f' x ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ ρ ((f' (x + 1) + 1) * (f' (x - 1) + 1)) = ρ (f' x ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | rw [← map_ofNat φ 2, h0, h2, ρ.map_one] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 = 1
⊢ f 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 = 1
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | have h := Eq1 hf (2 + 1) 2 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f ((2 + 1) * 2 - 1) = f (2 + 1) * f 2 + f (2 + 1 - 2)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [add_sub_cancel_left, hf.map_one, add_zero,
mul_two, add_sub_assoc, add_sub_cancel_right] at h | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f ((2 + 1) * 2 - 1) = f (2 + 1) * f 2 + f (2 + 1 - 2)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f ((2 + 1) * 2 - 1) = f (2 + 1) * f 2 + f (2 + 1 - 2)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | have h0 := Eq6 hf (2 + 1 + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1 + 1 + 1) + f (2 + 1 + 1 - 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [add_sub_cancel_right, add_assoc, one_add_one_eq_two, h, ← mul_add_one (f _)] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1 + 1 + 1) + f (2 + 1 + 1 - 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1 + 1 + 1) + f (2 + 1 + 1 - 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | replace h := Eq6 hf (2 + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + f (2 + 1 - 1) = (f (2 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [add_sub_cancel_right, add_one_mul (α := S), add_sub_left_comm,
add_comm, add_right_inj, eq_sub_iff_add_eq] at h | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + f (2 + 1 - 1) = (f (2 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + f (2 + 1 - 1) = (f (2 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [h, mul_assoc, eq_comm, ← sub_eq_zero, ← mul_sub, mul_eq_zero] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | clear h | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | revert h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0 → f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | refine Or.imp (λ h ↦ ?_) (λ h ↦ ?_) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0 → f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
⊢ f 2 = 1
case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0 → f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | have h0 := Eq1 hf 2 (1 + 1) | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
⊢ f 2 = 1 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f (2 * (1 + 1) - 1) = f 2 * f (1 + 1) + f (2 - (1 + 1))
⊢ f 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [two_mul, ← add_assoc, add_sub_cancel_right, one_add_one_eq_two, sub_self,
hf.map_zero, h, eq_comm, add_neg_eq_zero, mul_self_eq_one_iff] at h0 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f (2 * (1 + 1) - 1) = f 2 * f (1 + 1) + f (2 - (1 + 1))
⊢ f 2 = 1 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f 2 = 1 ∨ f 2 = -1
⊢ f 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f (2 * (1 + 1) - 1) = f 2 * f (1 + 1) + f (2 - (1 + 1))
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | exact h0.resolve_right hf.map_two_ne_neg_one | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f 2 = 1 ∨ f 2 = -1
⊢ f 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f 2 = 1 ∨ f 2 = -1
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [mul_sub_one, sub_one_mul, sub_sub, sub_add_add_cancel, ← two_mul,
sub_sub, ← one_add_mul 2 (f 2), ← sub_mul, mul_eq_zero, add_comm] at h | case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 0 ∨ f 2 = 3 | case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f 2 - (2 + 1) = 0 ∨ f 2 = 0
⊢ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | exact h.symm.imp_right λ h ↦ (eq_of_sub_eq_zero h).trans two_add_one_eq_three | case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f 2 - (2 + 1) = 0 ∨ f 2 = 0
⊢ f 2 = 0 ∨ f 2 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f 2 - (2 + 1) = 0 ∨ f 2 = 0
⊢ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
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