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/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | have h2 (y) : g (y - x) + 2 * g y = (2 + 1) * g (y + x) := by
have h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y) := by
rw [mul_sub_one, add_sub_left_comm, sub_add_cancel_left, ← sub_eq_add_neg,
hg.toShiftGood23.alt_good, h0, sub_self, zero_mul, zero_add]
have h3 := hg.toShiftGood23.is_good (x + 1) (x * (y - 1))
rw [h0, sub_self, zero_mul, zero_add, ← add_rotate, ← mul_add_one x, sub_add_cancel,
← mul_assoc, add_one_mul x, ← mul_add_one x, mul_assoc, hg.toShiftGood23.is_good,
h2, hg.toShiftGood23.is_good, h, zero_sub, neg_one_mul, neg_one_mul, neg_sub,
neg_sub, sub_add, sub_add, sub_right_inj, sub_eq_iff_eq_add] at h3
replace h2 := hg.Eq4 (x + 1) (y - 1)
rw [h0, one_mul, mul_sub, mul_one_add (α := S), ← sub_sub, sub_left_inj,
eq_sub_iff_add_eq, h3, mul_add, add_assoc, ← mul_add_one (α := S),
add_add_sub_cancel, ← hg.Eq1, ← sub_add, ← add_assoc, ← add_rotate,
add_left_inj, sub_sub, add_sub_add_right_eq_sub, ← neg_sub, mul_sub,
hg.toShiftGood23.map_even, ← add_sub_assoc, sub_eq_iff_eq_add'] at h2
rw [h2, add_one_mul (α := S), add_comm x] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
⊢ x = 0 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | refine hg.period_imp_zero₀ λ y ↦ ?_ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
⊢ x = 0 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
y : R
⊢ g (y + x) = g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | have h3 := h2 (-y) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
y : R
⊢ g (y + x) = g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
y : R
h3 : g (-y - x) + 2 * g (-y) = (2 + 1) * g (-y + x)
⊢ g (y + x) = g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
y : R
⊢ g (y + x) = g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | rw [← neg_add', hg.toShiftGood23.map_even, hg.toShiftGood23.map_even,
neg_add_eq_sub, ← neg_sub, hg.toShiftGood23.map_even] at h3 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
y : R
h3 : g (-y - x) + 2 * g (-y) = (2 + 1) * g (-y + x)
⊢ g (y + x) = g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
y : R
h3 : g (y + x) + 2 * g y = (2 + 1) * g (y - x)
⊢ g (y + x) = g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
h2 : ∀ (y : R), g (y - x) + 2 * g y = (2 + 1) * g (y + x)
y : R
h3 : g (-y - x) + 2 * g (-y) = (2 + 1) * g (-y + x)
⊢ g (y + x) = g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | rw [add_sub_add_right_eq_sub, ← mul_sub, ← neg_sub, neg_eq_iff_add_eq_zero,
← one_add_mul (α := S), mul_eq_zero, add_left_comm, one_add_one_eq_two,
← two_mul, mul_self_eq_zero, or_iff_right hg.Schar_ne_two, sub_eq_zero] at h2 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g (y + x) + 2 * g y = (2 + 1) * g (y - x)
h2 : g (y + x) + 2 * g y - (g (y - x) + 2 * g y) = (2 + 1) * g (y - x) - (2 + 1) * g (y + x)
⊢ g (y + x) = g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g (y + x) + 2 * g y = (2 + 1) * g (y - x)
h2 : g (y - x) = g (y + x)
⊢ g (y + x) = g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g (y + x) + 2 * g y = (2 + 1) * g (y - x)
h2 : g (y + x) + 2 * g y - (g (y - x) + 2 * g y) = (2 + 1) * g (y - x) - (2 + 1) * g (y + x)
⊢ g (y + x) = g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | rwa [h2, add_one_mul (α := S), add_comm, add_left_inj, ← sub_eq_zero, ← mul_sub,
mul_eq_zero, or_iff_right hg.Schar_ne_two, sub_eq_zero, eq_comm] at h3 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g (y + x) + 2 * g y = (2 + 1) * g (y - x)
h2 : g (y - x) = g (y + x)
⊢ g (y + x) = g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g (y + x) + 2 * g y = (2 + 1) * g (y - x)
h2 : g (y - x) = g (y + x)
⊢ g (y + x) = g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | have h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y) := by
rw [mul_sub_one, add_sub_left_comm, sub_add_cancel_left, ← sub_eq_add_neg,
hg.toShiftGood23.alt_good, h0, sub_self, zero_mul, zero_add] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | have h3 := hg.toShiftGood23.is_good (x + 1) (x * (y - 1)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
h3 : g ((x + 1) * (x * (y - 1)) + 1) = (g (x + 1) - 1) * (g (x * (y - 1)) - 1) + g (x + 1 + x * (y - 1))
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | rw [h0, sub_self, zero_mul, zero_add, ← add_rotate, ← mul_add_one x, sub_add_cancel,
← mul_assoc, add_one_mul x, ← mul_add_one x, mul_assoc, hg.toShiftGood23.is_good,
h2, hg.toShiftGood23.is_good, h, zero_sub, neg_one_mul, neg_one_mul, neg_sub,
neg_sub, sub_add, sub_add, sub_right_inj, sub_eq_iff_eq_add] at h3 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
h3 : g ((x + 1) * (x * (y - 1)) + 1) = (g (x + 1) - 1) * (g (x * (y - 1)) - 1) + g (x + 1 + x * (y - 1))
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
h3 : g ((x + 1) * (x * (y - 1)) + 1) = (g (x + 1) - 1) * (g (x * (y - 1)) - 1) + g (x + 1 + x * (y - 1))
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | replace h2 := hg.Eq4 (x + 1) (y - 1) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
h2 :
2 * (g ((x + 1) * (y - 1)) - g (x + 1) * g (y - 1)) =
g (x + 1 + (y - 1)) + g (x + 1 - (y - 1)) - 2 * (g (x + 1) + g (y - 1))
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h2 : g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | rw [h0, one_mul, mul_sub, mul_one_add (α := S), ← sub_sub, sub_left_inj,
eq_sub_iff_add_eq, h3, mul_add, add_assoc, ← mul_add_one (α := S),
add_add_sub_cancel, ← hg.Eq1, ← sub_add, ← add_assoc, ← add_rotate,
add_left_inj, sub_sub, add_sub_add_right_eq_sub, ← neg_sub, mul_sub,
hg.toShiftGood23.map_even, ← add_sub_assoc, sub_eq_iff_eq_add'] at h2 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
h2 :
2 * (g ((x + 1) * (y - 1)) - g (x + 1) * g (y - 1)) =
g (x + 1 + (y - 1)) + g (x + 1 - (y - 1)) - 2 * (g (x + 1) + g (y - 1))
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
h2 : g (y - x) + 2 * g y = 2 * g (x + y) + g (x + y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
h2 :
2 * (g ((x + 1) * (y - 1)) - g (x + 1) * g (y - 1)) =
g (x + 1 + (y - 1)) + g (x + 1 - (y - 1)) - 2 * (g (x + 1) + g (y - 1))
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | rw [h2, add_one_mul (α := S), add_comm x] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
h2 : g (y - x) + 2 * g y = 2 * g (x + y) + g (x + y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
h3 : g ((x + 1) * (y - 1)) = g y - g (x + y) + g (x + 1 - y)
h2 : g (y - x) + 2 * g y = 2 * g (x + y) + g (x + y)
⊢ g (y - x) + 2 * g y = (2 + 1) * g (y + x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq6 | [570, 1] | [607, 75] | rw [mul_sub_one, add_sub_left_comm, sub_add_cancel_left, ← sub_eq_add_neg,
hg.toShiftGood23.alt_good, h0, sub_self, zero_mul, zero_add] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
⊢ g (x + (x + 1) * (y - 1)) = g (x + 1 - y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x : R
h : g x = 0
h0 : g (x + 1) = 1
h1 : g (x - 1) = 1
y : R
⊢ g (x + (x + 1) * (y - 1)) = g (x + 1 - y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | have X : (2 : S) ^ 4 ≠ 0 := pow_ne_zero 4 hg.Schar_ne_two | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
⊢ ∀ (x y : R), x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
⊢ ∀ (x y : R), x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
⊢ ∀ (x y : R), x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | have h (x y) : g (x * y) = g (y * x) := by
have h := hg.Eq3 x y
rw [add_comm, ← neg_sub y, hg.toShiftGood23.map_even, hg.Eq3,
← sub_eq_zero, ← mul_sub, mul_eq_zero, or_iff_right X] at h
exact (eq_of_sub_eq_zero h).symm | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
⊢ ∀ (x y : R), x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
⊢ ∀ (x y : R), x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
⊢ ∀ (x y : R), x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | replace h (x y) : (x * x - y * y) * (x * y - y * x) = 0 := hg.Eq6 <| by
have h0 := hg.Eq3 (x * x - y * y) (x * y - y * x)
rwa [sub_add_sub_comm, ← mul_add, ← mul_add, add_comm y, ← sub_mul, h,
sub_sub_sub_comm, ← mul_sub, ← mul_sub, ← neg_sub x, mul_neg, sub_neg_eq_add,
add_mul, sub_self, sq, zero_mul, zero_eq_mul, or_iff_right X] at h0 | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
⊢ ∀ (x y : R), x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
⊢ ∀ (x y : R), x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
⊢ ∀ (x y : R), x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | have h0 (x y : R) : (x + 1) * y - y * (x + 1) = x * y - y * x := by
rw [add_one_mul x, mul_add_one y, add_sub_add_right_eq_sub] | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
⊢ ∀ (x y : R), x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
⊢ ∀ (x y : R), x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
⊢ ∀ (x y : R), x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | replace h (x y : R) : (x * 2 + 1) * (x * y - y * x) = 0 := by
have h1 := h (x + 1) y
rwa [h0, add_one_mul x, mul_add_one x, add_assoc, add_sub_right_comm,
add_mul, h, zero_add, ← add_assoc, ← mul_two] at h1 | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
⊢ ∀ (x y : R), x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
⊢ ∀ (x y : R), x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
⊢ ∀ (x y : R), x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | intro x y | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
⊢ ∀ (x y : R), x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
⊢ x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
⊢ ∀ (x y : R), x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | have h1 := h (x + 1) y | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
⊢ x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
h1 : ((x + 1) * 2 + 1) * ((x + 1) * y - y * (x + 1)) = 0
⊢ x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
⊢ x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | rw [h0, add_one_mul x, add_right_comm, add_mul, h, zero_add] at h1 | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
h1 : ((x + 1) * 2 + 1) * ((x + 1) * y - y * (x + 1)) = 0
⊢ x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
h1 : 2 * (x * y - y * x) = 0
⊢ x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
h1 : ((x + 1) * 2 + 1) * ((x + 1) * y - y * (x + 1)) = 0
⊢ x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | specialize h x y | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
h1 : 2 * (x * y - y * x) = 0
⊢ x * y = y * x | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
h1 : 2 * (x * y - y * x) = 0
h : (x * 2 + 1) * (x * y - y * x) = 0
⊢ x * y = y * x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
h : ∀ (x y : R), (x * 2 + 1) * (x * y - y * x) = 0
x y : R
h1 : 2 * (x * y - y * x) = 0
⊢ x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | rwa [add_one_mul (α := R), mul_assoc, h1, mul_zero, zero_add, sub_eq_zero] at h | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
h1 : 2 * (x * y - y * x) = 0
h : (x * 2 + 1) * (x * y - y * x) = 0
⊢ x * y = y * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
h1 : 2 * (x * y - y * x) = 0
h : (x * 2 + 1) * (x * y - y * x) = 0
⊢ x * y = y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | have h := hg.Eq3 x y | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
⊢ g (x * y) = g (y * x) | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
h : (g (x + y) - g (x - y)) ^ 2 = 2 ^ 4 * g (x * y)
⊢ g (x * y) = g (y * x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
⊢ g (x * y) = g (y * x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | rw [add_comm, ← neg_sub y, hg.toShiftGood23.map_even, hg.Eq3,
← sub_eq_zero, ← mul_sub, mul_eq_zero, or_iff_right X] at h | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
h : (g (x + y) - g (x - y)) ^ 2 = 2 ^ 4 * g (x * y)
⊢ g (x * y) = g (y * x) | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
h : g (y * x) - g (x * y) = 0
⊢ g (x * y) = g (y * x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
h : (g (x + y) - g (x - y)) ^ 2 = 2 ^ 4 * g (x * y)
⊢ g (x * y) = g (y * x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | exact (eq_of_sub_eq_zero h).symm | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
h : g (y * x) - g (x * y) = 0
⊢ g (x * y) = g (y * x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
x y : R
h : g (y * x) - g (x * y) = 0
⊢ g (x * y) = g (y * x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | have h0 := hg.Eq3 (x * x - y * y) (x * y - y * x) | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
x y : R
⊢ g ((x * x - y * y) * (x * y - y * x)) = 0 | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
x y : R
h0 :
(g (x * x - y * y + (x * y - y * x)) - g (x * x - y * y - (x * y - y * x))) ^ 2 =
2 ^ 4 * g ((x * x - y * y) * (x * y - y * x))
⊢ g ((x * x - y * y) * (x * y - y * x)) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
x y : R
⊢ g ((x * x - y * y) * (x * y - y * x)) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | rwa [sub_add_sub_comm, ← mul_add, ← mul_add, add_comm y, ← sub_mul, h,
sub_sub_sub_comm, ← mul_sub, ← mul_sub, ← neg_sub x, mul_neg, sub_neg_eq_add,
add_mul, sub_self, sq, zero_mul, zero_eq_mul, or_iff_right X] at h0 | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
x y : R
h0 :
(g (x * x - y * y + (x * y - y * x)) - g (x * x - y * y - (x * y - y * x))) ^ 2 =
2 ^ 4 * g ((x * x - y * y) * (x * y - y * x))
⊢ g ((x * x - y * y) * (x * y - y * x)) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), g (x * y) = g (y * x)
x y : R
h0 :
(g (x * x - y * y + (x * y - y * x)) - g (x * x - y * y - (x * y - y * x))) ^ 2 =
2 ^ 4 * g ((x * x - y * y) * (x * y - y * x))
⊢ g ((x * x - y * y) * (x * y - y * x)) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | rw [add_one_mul x, mul_add_one y, add_sub_add_right_eq_sub] | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
x y : R
⊢ (x + 1) * y - y * (x + 1) = x * y - y * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
x y : R
⊢ (x + 1) * y - y * (x + 1) = x * y - y * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | have h1 := h (x + 1) y | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
⊢ (x * 2 + 1) * (x * y - y * x) = 0 | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
h1 : ((x + 1) * (x + 1) - y * y) * ((x + 1) * y - y * (x + 1)) = 0
⊢ (x * 2 + 1) * (x * y - y * x) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
⊢ (x * 2 + 1) * (x * y - y * x) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Rcomm | [609, 1] | [634, 82] | rwa [h0, add_one_mul x, mul_add_one x, add_assoc, add_sub_right_comm,
add_mul, h, zero_add, ← add_assoc, ← mul_two] at h1 | S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
h1 : ((x + 1) * (x + 1) - y * y) * ((x + 1) * y - y * (x + 1)) = 0
⊢ (x * 2 + 1) * (x * y - y * x) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type ?u.222622
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
X : 2 ^ 4 ≠ 0
h : ∀ (x y : R), (x * x - y * y) * (x * y - y * x) = 0
h0 : ∀ (x y : R), (x + 1) * y - y * (x + 1) = x * y - y * x
x y : R
h1 : ((x + 1) * (x + 1) - y * y) * ((x + 1) * y - y * (x + 1)) = 0
⊢ (x * 2 + 1) * (x * y - y * x) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | replace h (x y) : x * y = 0 ∨ g (x * y) = g x * g y := by
have h0 := h 1 (x * y)
rw [one_mul, ← mul_assoc, hg.Rcomm _ x, ← mul_assoc, h, mul_assoc, hg.Rcomm x,
hg.Rcomm x, h, mul_assoc, hg.Rcomm y, ← sub_eq_zero, ← mul_sub, mul_eq_zero] at h0
exact h0.imp hg.Eq6 (λ h0 ↦ (eq_of_sub_eq_zero h0).symm) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
⊢ ∀ (x y : R), g (x * y) = g x * g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
⊢ ∀ (x y : R), g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
⊢ ∀ (x y : R), g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | intro x y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
⊢ ∀ (x y : R), g (x * y) = g x * g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
⊢ ∀ (x y : R), g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | refine (h x y).elim (λ h0 ↦ ?_) id | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
⊢ g (x * y) = g x * g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | refine (h (x + 1) y).elim (λ h1 ↦ ?_) (λ h1 ↦ ?_) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
⊢ g (x * y) = g x * g y | case refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : (x + 1) * y = 0
⊢ g (x * y) = g x * g y
case refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | refine (h (x - 1) y).elim (λ h2 ↦ ?_) (λ h2 ↦ ?_) | case refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
⊢ g (x * y) = g x * g y | case refine_2.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : (x - 1) * y = 0
⊢ g (x * y) = g x * g y
case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : g ((x - 1) * y) = g (x - 1) * g y
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [add_one_mul x, h0, zero_add] at h1 | case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : g ((x - 1) * y) = g (x - 1) * g y
⊢ g (x * y) = g x * g y | case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g ((x - 1) * y) = g (x - 1) * g y
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : g ((x - 1) * y) = g (x - 1) * g y
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [sub_one_mul, h0, zero_sub, hg.toShiftGood23.map_even] at h2 | case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g ((x - 1) * y) = g (x - 1) * g y
⊢ g (x * y) = g x * g y | case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g y = g (x - 1) * g y
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g ((x - 1) * y) = g (x - 1) * g y
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [← two_mul, ← add_mul, hg.Eq1, mul_add_one (α := S), add_mul, self_eq_add_left,
mul_assoc, mul_eq_zero, or_iff_right hg.Schar_ne_two] at h3 | case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g y = g (x - 1) * g y
h3 : g y + g y = g (x + 1) * g y + g (x - 1) * g y
⊢ g (x * y) = g x * g y | case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g y = g (x - 1) * g y
h3 : g x * g y = 0
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g y = g (x - 1) * g y
h3 : g y + g y = g (x + 1) * g y + g (x - 1) * g y
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [h0, hg.toShiftGood23.map_zero, h3] | case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g y = g (x - 1) * g y
h3 : g x * g y = 0
⊢ g (x * y) = g x * g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g y = g (x + 1) * g y
h2 : g y = g (x - 1) * g y
h3 : g x * g y = 0
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | suffices g (x * y + y) + g (x * y - y) = 2 * (g (x * y) + g y) by
rw [← sub_eq_zero, ← hg.Eq4, mul_eq_zero] at this
exact eq_of_sub_eq_zero (this.resolve_left hg.Schar_ne_two) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
⊢ g (x * y * y) = g (x * y) * g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
⊢ g (x * y + y) + g (x * y - y) = 2 * (g (x * y) + g y) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
⊢ g (x * y * y) = g (x * y) * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rwa [← mul_add, sub_add_sub_comm, ← add_mul, sub_add_sub_comm, add_add_add_comm,
hg.Eq1, add_right_comm x, ← add_sub_right_comm x y, hg.Eq1, add_sub_right_comm x,
sub_right_comm, hg.Eq1, ← mul_add, ← mul_add, ← mul_sub, add_add_add_comm,
add_add_add_comm (g _) (g y), hg.Eq1, ← sub_eq_zero, ← mul_sub, mul_eq_zero,
or_iff_right hg.Schar_ne_two, sub_eq_zero, sub_eq_iff_eq_add, one_add_one_eq_two,
mul_add_one (α := S), ← two_mul, add_right_comm (2 * g x), add_sub_add_right_eq_sub,
← mul_add, ← hg.Eq4, ← mul_add_one (α := S), mul_assoc, ← mul_add, add_one_mul (g x),
add_comm _ (g y), sub_add_add_cancel, add_one_mul x, sub_one_mul] at h | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
h :
2 * (g ((x + 1) * y) - g (x + 1) * g y) + 2 * (g ((x - 1) * y) - g (x - 1) * g y) =
g (x + 1 + y) + g (x + 1 - y) - 2 * (g (x + 1) + g y) + (g (x - 1 + y) + g (x - 1 - y) - 2 * (g (x - 1) + g y))
⊢ g (x * y + y) + g (x * y - y) = 2 * (g (x * y) + g y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
h :
2 * (g ((x + 1) * y) - g (x + 1) * g y) + 2 * (g ((x - 1) * y) - g (x - 1) * g y) =
g (x + 1 + y) + g (x + 1 - y) - 2 * (g (x + 1) + g y) + (g (x - 1 + y) + g (x - 1 - y) - 2 * (g (x - 1) + g y))
⊢ g (x * y + y) + g (x * y - y) = 2 * (g (x * y) + g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [← sub_eq_zero, ← hg.Eq4, mul_eq_zero] at this | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
this : g (x * y + y) + g (x * y - y) = 2 * (g (x * y) + g y)
⊢ g (x * y * y) = g (x * y) * g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
this : 2 = 0 ∨ g (x * y * y) - g (x * y) * g y = 0
⊢ g (x * y * y) = g (x * y) * g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
this : g (x * y + y) + g (x * y - y) = 2 * (g (x * y) + g y)
⊢ g (x * y * y) = g (x * y) * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | exact eq_of_sub_eq_zero (this.resolve_left hg.Schar_ne_two) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
this : 2 = 0 ∨ g (x * y * y) - g (x * y) * g y = 0
⊢ g (x * y * y) = g (x * y) * g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
this : 2 = 0 ∨ g (x * y * y) - g (x * y) * g y = 0
⊢ g (x * y * y) = g (x * y) * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | have h0 := h 1 (x * y) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
⊢ x * y = 0 ∨ g (x * y) = g x * g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
h0 : g (1 * (x * y) * (x * y)) = g (1 * (x * y)) * g (x * y)
⊢ x * y = 0 ∨ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
⊢ x * y = 0 ∨ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [one_mul, ← mul_assoc, hg.Rcomm _ x, ← mul_assoc, h, mul_assoc, hg.Rcomm x,
hg.Rcomm x, h, mul_assoc, hg.Rcomm y, ← sub_eq_zero, ← mul_sub, mul_eq_zero] at h0 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
h0 : g (1 * (x * y) * (x * y)) = g (1 * (x * y)) * g (x * y)
⊢ x * y = 0 ∨ g (x * y) = g x * g y | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
h0 : g (x * y) = 0 ∨ g x * g y - g (x * y) = 0
⊢ x * y = 0 ∨ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
h0 : g (1 * (x * y) * (x * y)) = g (1 * (x * y)) * g (x * y)
⊢ x * y = 0 ∨ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | exact h0.imp hg.Eq6 (λ h0 ↦ (eq_of_sub_eq_zero h0).symm) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
h0 : g (x * y) = 0 ∨ g x * g y - g (x * y) = 0
⊢ x * y = 0 ∨ g (x * y) = g x * g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), g (x * y * y) = g (x * y) * g y
x y : R
h0 : g (x * y) = 0 ∨ g x * g y - g (x * y) = 0
⊢ x * y = 0 ∨ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [add_one_mul x, h0, zero_add] at h1 | case refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : (x + 1) * y = 0
⊢ g (x * y) = g x * g y | case refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : y = 0
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : (x + 1) * y = 0
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [h1, mul_zero, hg.toShiftGood23.map_zero, mul_zero] | case refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : y = 0
⊢ g (x * y) = g x * g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : y = 0
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [sub_one_mul x, h0, zero_sub, neg_eq_zero] at h2 | case refine_2.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : (x - 1) * y = 0
⊢ g (x * y) = g x * g y | case refine_2.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : y = 0
⊢ g (x * y) = g x * g y | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : (x - 1) * y = 0
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq7 | [636, 1] | [673, 41] | rw [h2, mul_zero, hg.toShiftGood23.map_zero, mul_zero] | case refine_2.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : y = 0
⊢ g (x * y) = g x * g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
h : ∀ (x y : R), x * y = 0 ∨ g (x * y) = g x * g y
x y : R
h0 : x * y = 0
h1 : g ((x + 1) * y) = g (x + 1) * g y
h2 : y = 0
⊢ g (x * y) = g x * g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.Eq8 | [675, 1] | [677, 81] | rw [two_nsmul, ← two_mul, ← sub_eq_zero, ← hg.Eq4, hg.Eq7, sub_self, mul_zero] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
⊢ g (x + y) + g (x - y) = 2 • (g x + g y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
x y : R
⊢ g (x + y) + g (x - y) = 2 • (g x + g y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | refine ⟨R, CommRing.mk hg.Rcomm, RingHom.id R, ?_⟩ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
⊢ ∃ R' x φ ι, ∀ (x_1 : R), g x_1 = ι (RestrictedSq (φ x_1)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
⊢ ∃ R' x φ ι, ∀ (x_1 : R), g x_1 = ι (RestrictedSq (φ x_1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | let hR := CommRing.mk hg.Rcomm | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | have hS (x y : S) (h : 2 • x = 2 • y) : x = y := by
rwa [two_nsmul, ← two_mul, two_nsmul, ← two_mul, ← sub_eq_zero, ← mul_sub,
mul_eq_zero, or_iff_right hg.Schar_ne_two, sub_eq_zero] at h | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | let φ := BilinMap hS hg.Eq8 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | let ρ := φ 1 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | have h : ∀ x, φ x x = 2 • g x := BilinMap_eq_two_nsmul _ _ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | have h0 (x y) : φ x y = ρ (x * y) :=
hS _ _ <| by rw [two_nsmul_BilinMap_eq, two_nsmul_BilinMap_eq,
← hg.Eq2, add_comm, ← neg_sub (x * y), hg.toShiftGood23.map_even] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | let R₂ := AddSubgroup.closure (Set.range λ x : R ↦ x ^ 2) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | obtain ⟨ι, h1⟩ : ∃ ι : SqSubring R →+ S, ∀ a : SqSubring R, ρ a = 2 • ι a :=
suffices ∃ ι : SqSubring R → S, ∀ a : SqSubring R, ρ a = 2 • ι a by
rcases this with ⟨ι, h1⟩
have h3 (x y) : ι (x + y) = ι x + ι y := hS _ _ <| by
rw [← h1, Subring.coe_add, ρ.map_add, h1, h1, nsmul_add]
exact ⟨AddMonoidHom.mk' ι h3, h1⟩
suffices ∀ r ∈ R₂, ∃ s, ρ r = 2 • s
from Classical.axiomOfChoice λ a ↦ this a.1 a.2
λ r h2 ↦ AddSubgroup.closure_induction h2
(λ y ⟨x, h3⟩ ↦ ⟨g x, by rw [← h, h0 x, ← sq, ← h3]⟩)
⟨0, by rw [ρ.map_zero, nsmul_zero]⟩
(λ x y ⟨s, hs⟩ ⟨t, ht⟩ ↦ ⟨s + t, by rw [ρ.map_add, hs, ht, nsmul_add]⟩)
(λ x ⟨s, hs⟩ ↦ ⟨-s, by rw [ρ.map_neg, hs, nsmul_eq_mul, ← mul_neg, nsmul_eq_mul]⟩) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | suffices ∀ x y, ι (x * y) = ι x * ι y by
have h2 : ι 1 = 1 := hS _ _ <| by
rw [← h1, Subring.coe_one, h, hg.toShiftGood23.map_one]
refine ⟨⟨⟨⟨ι, h2⟩, this⟩, ι.map_zero, ι.map_add⟩, λ x ↦ hS _ _ ?_⟩
change 2 • g x = 2 • ι (RestrictedSq x)
rw [← h, ← h1, RestrictedSq_coe, sq, h0] | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | have X (x y : S) : (2 • x) * (2 • y) = 2 • 2 • (x * y) := by
rw [two_nsmul, two_nsmul, add_mul, mul_add, ← two_nsmul, ← two_nsmul] | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
⊢ ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | suffices ∀ a b, a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b
from λ x y ↦ hS _ _ <| hS _ _ <| by
rw [← h1, Subring.coe_mul, this _ _ x.2 y.2, h1, h1, X] | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
⊢ ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
⊢ ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
⊢ ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | replace h (x) : ρ (x ^ 2) = 2 • g x := by rw [← h, sq, ← h0] | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
⊢ ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
⊢ ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
⊢ ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | intro a b ha hb | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
⊢ ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ 2 • ρ (a * b) = ρ a * ρ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
⊢ ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | refine AddSubgroup.closure_induction₂ ha hb ?_ ?_ ?_ ?_ ?_ ?_ ?_ | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ 2 • ρ (a * b) = ρ a * ρ b | case intro.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ x ∈ Set.range fun x => x ^ 2, ∀ y ∈ Set.range fun x => x ^ 2, 2 • ρ (x * y) = ρ x * ρ y
case intro.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x : R), 2 • ρ (0 * x) = ρ 0 * ρ x
case intro.refine_3
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x : R), 2 • ρ (x * 0) = ρ x * ρ 0
case intro.refine_4
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x₁ x₂ y : R), 2 • ρ (x₁ * y) = ρ x₁ * ρ y → 2 • ρ (x₂ * y) = ρ x₂ * ρ y → 2 • ρ ((x₁ + x₂) * y) = ρ (x₁ + x₂) * ρ y
case intro.refine_5
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y₁ y₂ : R), 2 • ρ (x * y₁) = ρ x * ρ y₁ → 2 • ρ (x * y₂) = ρ x * ρ y₂ → 2 • ρ (x * (y₁ + y₂)) = ρ x * ρ (y₁ + y₂)
case intro.refine_6
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y : R), 2 • ρ (x * y) = ρ x * ρ y → 2 • ρ (-x * y) = ρ (-x) * ρ y
case intro.refine_7
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y : R), 2 • ρ (x * y) = ρ x * ρ y → 2 • ρ (x * -y) = ρ x * ρ (-y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ 2 • ρ (a * b) = ρ a * ρ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rwa [two_nsmul, ← two_mul, two_nsmul, ← two_mul, ← sub_eq_zero, ← mul_sub,
mul_eq_zero, or_iff_right hg.Schar_ne_two, sub_eq_zero] at h | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
x y : S
h : 2 • x = 2 • y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
x y : S
h : 2 • x = 2 • y
⊢ x = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [two_nsmul_BilinMap_eq, two_nsmul_BilinMap_eq,
← hg.Eq2, add_comm, ← neg_sub (x * y), hg.toShiftGood23.map_even] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
x y : R
⊢ 2 • (φ x) y = 2 • ρ (x * y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
x y : R
⊢ 2 • (φ x) y = 2 • ρ (x * y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rcases this with ⟨ι, h1⟩ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
this : ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
this : ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | have h3 (x y) : ι (x + y) = ι x + ι y := hS _ _ <| by
rw [← h1, Subring.coe_add, ρ.map_add, h1, h1, nsmul_add] | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
h3 : ∀ (x y : ↥(SqSubring R)), ι (x + y) = ι x + ι y
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | exact ⟨AddMonoidHom.mk' ι h3, h1⟩ | case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
h3 : ∀ (x y : ↥(SqSubring R)), ι (x + y) = ι x + ι y
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
h3 : ∀ (x y : ↥(SqSubring R)), ι (x + y) = ι x + ι y
⊢ ∃ ι, ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [← h1, Subring.coe_add, ρ.map_add, h1, h1, nsmul_add] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
x y : ↥(SqSubring R)
⊢ 2 • ι (x + y) = 2 • (ι x + ι y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) → S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
x y : ↥(SqSubring R)
⊢ 2 • ι (x + y) = 2 • (ι x + ι y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [← h, h0 x, ← sq, ← h3] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
y : R
x✝ : y ∈ Set.range fun x => x ^ 2
x : R
h3 : (fun x => x ^ 2) x = y
⊢ ρ y = 2 • g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
y : R
x✝ : y ∈ Set.range fun x => x ^ 2
x : R
h3 : (fun x => x ^ 2) x = y
⊢ ρ y = 2 • g x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [ρ.map_zero, nsmul_zero] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
⊢ ρ 0 = 2 • 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
⊢ ρ 0 = 2 • 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [ρ.map_add, hs, ht, nsmul_add] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
x y : R
x✝¹ : ∃ s, ρ x = 2 • s
x✝ : ∃ s, ρ y = 2 • s
s : S
hs : ρ x = 2 • s
t : S
ht : ρ y = 2 • t
⊢ ρ (x + y) = 2 • (s + t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
x y : R
x✝¹ : ∃ s, ρ x = 2 • s
x✝ : ∃ s, ρ y = 2 • s
s : S
hs : ρ x = 2 • s
t : S
ht : ρ y = 2 • t
⊢ ρ (x + y) = 2 • (s + t)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [ρ.map_neg, hs, nsmul_eq_mul, ← mul_neg, nsmul_eq_mul] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
x : R
x✝ : ∃ s, ρ x = 2 • s
s : S
hs : ρ x = 2 • s
⊢ ρ (-x) = 2 • -s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
r : R
h2 : r ∈ R₂
x : R
x✝ : ∃ s, ρ x = 2 • s
s : S
hs : ρ x = 2 • s
⊢ ρ (-x) = 2 • -s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | have h2 : ι 1 = 1 := hS _ _ <| by
rw [← h1, Subring.coe_one, h, hg.toShiftGood23.map_one] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | refine ⟨⟨⟨⟨ι, h2⟩, this⟩, ι.map_zero, ι.map_add⟩, λ x ↦ hS _ _ ?_⟩ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
x : R
⊢ 2 • g x =
2 •
{ toFun := ⇑ι, map_one' := h2, map_mul' := this, map_zero' := ⋯, map_add' := ⋯ } (RestrictedSq ((RingHom.id R) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
⊢ ∃ ι, ∀ (x : R), g x = ι (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | change 2 • g x = 2 • ι (RestrictedSq x) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
x : R
⊢ 2 • g x =
2 •
{ toFun := ⇑ι, map_one' := h2, map_mul' := this, map_zero' := ⋯, map_add' := ⋯ } (RestrictedSq ((RingHom.id R) x)) | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
x : R
⊢ 2 • g x = 2 • ι (RestrictedSq x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
x : R
⊢ 2 • g x =
2 •
{ toFun := ⇑ι, map_one' := h2, map_mul' := this, map_zero' := ⋯, map_add' := ⋯ } (RestrictedSq ((RingHom.id R) x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [← h, ← h1, RestrictedSq_coe, sq, h0] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
x : R
⊢ 2 • g x = 2 • ι (RestrictedSq x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
h2 : ι 1 = 1
x : R
⊢ 2 • g x = 2 • ι (RestrictedSq x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [← h1, Subring.coe_one, h, hg.toShiftGood23.map_one] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
⊢ 2 • ι 1 = 2 • 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
this : ∀ (x y : ↥(SqSubring R)), ι (x * y) = ι x * ι y
⊢ 2 • ι 1 = 2 • 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [two_nsmul, two_nsmul, add_mul, mul_add, ← two_nsmul, ← two_nsmul] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
x y : S
⊢ 2 • x * 2 • y = 2 • 2 • (x * y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
x y : S
⊢ 2 • x * 2 • y = 2 • 2 • (x * y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [← h1, Subring.coe_mul, this _ _ x.2 y.2, h1, h1, X] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
this : ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b
x y : ↥(SqSubring R)
⊢ 2 • 2 • ι (x * y) = 2 • 2 • (ι x * ι y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
this : ∀ (a b : R), a ∈ R₂ → b ∈ R₂ → 2 • ρ (a * b) = ρ a * ρ b
x y : ↥(SqSubring R)
⊢ 2 • 2 • ι (x * y) = 2 • 2 • (ι x * ι y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [← h, sq, ← h0] | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
x : R
⊢ ρ (x ^ 2) = 2 • g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h : ∀ (x : R), (φ x) x = 2 • g x
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
x : R
⊢ ρ (x ^ 2) = 2 • g x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rintro _ ⟨c, rfl⟩ _ ⟨d, rfl⟩ | case intro.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ x ∈ Set.range fun x => x ^ 2, ∀ y ∈ Set.range fun x => x ^ 2, 2 • ρ (x * y) = ρ x * ρ y | case intro.refine_1.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
c d : R
⊢ 2 • ρ ((fun x => x ^ 2) c * (fun x => x ^ 2) d) = ρ ((fun x => x ^ 2) c) * ρ ((fun x => x ^ 2) d) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ x ∈ Set.range fun x => x ^ 2, ∀ y ∈ Set.range fun x => x ^ 2, 2 • ρ (x * y) = ρ x * ρ y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [← mul_pow, h, h, h, X, hg.Eq7] | case intro.refine_1.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
c d : R
⊢ 2 • ρ ((fun x => x ^ 2) c * (fun x => x ^ 2) d) = ρ ((fun x => x ^ 2) c) * ρ ((fun x => x ^ 2) d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
c d : R
⊢ 2 • ρ ((fun x => x ^ 2) c * (fun x => x ^ 2) d) = ρ ((fun x => x ^ 2) c) * ρ ((fun x => x ^ 2) d)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | intro x | case intro.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x : R), 2 • ρ (0 * x) = ρ 0 * ρ x | case intro.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x : R
⊢ 2 • ρ (0 * x) = ρ 0 * ρ x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x : R), 2 • ρ (0 * x) = ρ 0 * ρ x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [zero_mul, ρ.map_zero, zero_mul, nsmul_zero] | case intro.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x : R
⊢ 2 • ρ (0 * x) = ρ 0 * ρ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x : R
⊢ 2 • ρ (0 * x) = ρ 0 * ρ x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | intro x | case intro.refine_3
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x : R), 2 • ρ (x * 0) = ρ x * ρ 0 | case intro.refine_3
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x : R
⊢ 2 • ρ (x * 0) = ρ x * ρ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_3
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x : R), 2 • ρ (x * 0) = ρ x * ρ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [mul_zero, ρ.map_zero, mul_zero, nsmul_zero] | case intro.refine_3
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x : R
⊢ 2 • ρ (x * 0) = ρ x * ρ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_3
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x : R
⊢ 2 • ρ (x * 0) = ρ x * ρ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | intro x₁ x₂ y hx₁ hx₂ | case intro.refine_4
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x₁ x₂ y : R), 2 • ρ (x₁ * y) = ρ x₁ * ρ y → 2 • ρ (x₂ * y) = ρ x₂ * ρ y → 2 • ρ ((x₁ + x₂) * y) = ρ (x₁ + x₂) * ρ y | case intro.refine_4
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x₁ x₂ y : R
hx₁ : 2 • ρ (x₁ * y) = ρ x₁ * ρ y
hx₂ : 2 • ρ (x₂ * y) = ρ x₂ * ρ y
⊢ 2 • ρ ((x₁ + x₂) * y) = ρ (x₁ + x₂) * ρ y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_4
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x₁ x₂ y : R), 2 • ρ (x₁ * y) = ρ x₁ * ρ y → 2 • ρ (x₂ * y) = ρ x₂ * ρ y → 2 • ρ ((x₁ + x₂) * y) = ρ (x₁ + x₂) * ρ y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [add_mul, ρ.map_add, nsmul_add, hx₁, hx₂, ρ.map_add, add_mul] | case intro.refine_4
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x₁ x₂ y : R
hx₁ : 2 • ρ (x₁ * y) = ρ x₁ * ρ y
hx₂ : 2 • ρ (x₂ * y) = ρ x₂ * ρ y
⊢ 2 • ρ ((x₁ + x₂) * y) = ρ (x₁ + x₂) * ρ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_4
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x₁ x₂ y : R
hx₁ : 2 • ρ (x₁ * y) = ρ x₁ * ρ y
hx₂ : 2 • ρ (x₂ * y) = ρ x₂ * ρ y
⊢ 2 • ρ ((x₁ + x₂) * y) = ρ (x₁ + x₂) * ρ y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | intro x y₁ y₂ hy₁ hy₂ | case intro.refine_5
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y₁ y₂ : R), 2 • ρ (x * y₁) = ρ x * ρ y₁ → 2 • ρ (x * y₂) = ρ x * ρ y₂ → 2 • ρ (x * (y₁ + y₂)) = ρ x * ρ (y₁ + y₂) | case intro.refine_5
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y₁ y₂ : R
hy₁ : 2 • ρ (x * y₁) = ρ x * ρ y₁
hy₂ : 2 • ρ (x * y₂) = ρ x * ρ y₂
⊢ 2 • ρ (x * (y₁ + y₂)) = ρ x * ρ (y₁ + y₂) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_5
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y₁ y₂ : R), 2 • ρ (x * y₁) = ρ x * ρ y₁ → 2 • ρ (x * y₂) = ρ x * ρ y₂ → 2 • ρ (x * (y₁ + y₂)) = ρ x * ρ (y₁ + y₂)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [mul_add, ρ.map_add, nsmul_add, hy₁, hy₂, ρ.map_add, mul_add] | case intro.refine_5
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y₁ y₂ : R
hy₁ : 2 • ρ (x * y₁) = ρ x * ρ y₁
hy₂ : 2 • ρ (x * y₂) = ρ x * ρ y₂
⊢ 2 • ρ (x * (y₁ + y₂)) = ρ x * ρ (y₁ + y₂) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_5
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y₁ y₂ : R
hy₁ : 2 • ρ (x * y₁) = ρ x * ρ y₁
hy₂ : 2 • ρ (x * y₂) = ρ x * ρ y₂
⊢ 2 • ρ (x * (y₁ + y₂)) = ρ x * ρ (y₁ + y₂)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | intro x y h2 | case intro.refine_6
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y : R), 2 • ρ (x * y) = ρ x * ρ y → 2 • ρ (-x * y) = ρ (-x) * ρ y | case intro.refine_6
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y : R
h2 : 2 • ρ (x * y) = ρ x * ρ y
⊢ 2 • ρ (-x * y) = ρ (-x) * ρ y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_6
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y : R), 2 • ρ (x * y) = ρ x * ρ y → 2 • ρ (-x * y) = ρ (-x) * ρ y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [neg_mul, ρ.map_neg, ρ.map_neg, neg_mul, smul_neg, h2] | case intro.refine_6
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y : R
h2 : 2 • ρ (x * y) = ρ x * ρ y
⊢ 2 • ρ (-x * y) = ρ (-x) * ρ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_6
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y : R
h2 : 2 • ρ (x * y) = ρ x * ρ y
⊢ 2 • ρ (-x * y) = ρ (-x) * ρ y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | intro x y h2 | case intro.refine_7
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y : R), 2 • ρ (x * y) = ρ x * ρ y → 2 • ρ (x * -y) = ρ x * ρ (-y) | case intro.refine_7
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y : R
h2 : 2 • ρ (x * y) = ρ x * ρ y
⊢ 2 • ρ (x * -y) = ρ x * ρ (-y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_7
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
⊢ ∀ (x y : R), 2 • ρ (x * y) = ρ x * ρ y → 2 • ρ (x * -y) = ρ x * ρ (-y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RShiftGood23.solution | [682, 1] | [738, 76] | rw [mul_neg, ρ.map_neg, ρ.map_neg, mul_neg, smul_neg, h2] | case intro.refine_7
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y : R
h2 : 2 • ρ (x * y) = ρ x * ρ y
⊢ 2 • ρ (x * -y) = ρ x * ρ (-y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_7
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : RShiftGood23 g
hR : CommRing R := CommRing.mk ⋯
hS : ∀ (x y : S), 2 • x = 2 • y → x = y
φ : R →+ R →+ S := BilinMap hS ⋯
ρ : R →+ S := φ 1
h0 : ∀ (x y : R), (φ x) y = ρ (x * y)
R₂ : AddSubgroup R := AddSubgroup.closure (Set.range fun x => x ^ 2)
ι : ↥(SqSubring R) →+ S
h1 : ∀ (a : ↥(SqSubring R)), ρ ↑a = 2 • ι a
X : ∀ (x y : S), 2 • x * 2 • y = 2 • 2 • (x * y)
h : ∀ (x : R), ρ (x ^ 2) = 2 • g x
a b : R
ha : a ∈ R₂
hb : b ∈ R₂
x y : R
h2 : 2 • ρ (x * y) = ρ x * ρ y
⊢ 2 • ρ (x * -y) = ρ x * ρ (-y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase23.solution | [754, 1] | [760, 40] | rcases (RShiftGood23.shift_mk_iff.mpr ⟨hf, h⟩).solution with ⟨R', _, φ, ι, h0⟩ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
⊢ ∃ R' x φ ι, ∀ (x_1 : R), f x_1 = ι (RestrictedSq (φ x_1) - 1) | case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
⊢ ∃ R' x φ ι, ∀ (x_1 : R), f x_1 = ι (RestrictedSq (φ x_1) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
⊢ ∃ R' x φ ι, ∀ (x_1 : R), f x_1 = ι (RestrictedSq (φ x_1) - 1)
TACTIC:
|
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