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.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:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase23.solution | [754, 1] | [760, 40] | refine ⟨R', _, φ, ι, λ x ↦ ?_⟩ | 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) | 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))
x : R
⊢ f x = ι (RestrictedSq (φ x) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase23.solution | [754, 1] | [760, 40] | rw [ι.map_sub, ← h0, ι.map_one] | 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))
x : R
⊢ f x = ι (RestrictedSq (φ x) - 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))
x : R
⊢ f x = (f + 1) x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
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))
x : R
⊢ f x = ι (RestrictedSq (φ x) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase23.solution | [754, 1] | [760, 40] | exact (add_sub_cancel_right _ _).symm | 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))
x : R
⊢ f x = (f + 1) x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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))
x : R
⊢ f x = (f + 1) x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | refine hf.period_imp_zero λ x ↦ ?_ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
⊢ 2 = 0 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
x : R
⊢ f (x + 2) = f 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
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rcases CommSubring.oneVarCommLiftDomain_exists hf.toNontrivialGood x with
⟨R', R'comm, φ, -, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h1, hf'⟩ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
x : R
⊢ f (x + 2) = f x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.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 : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 2) = f (φ 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
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
x : R
⊢ f (x + 2) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rw [← map_ofNat φ 2, ← φ.map_add, h1, h1, CommCase.two_periodic_of_map_two hf'] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.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 : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 2) = f (φ x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 2) = f (φ x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | refine map_even_of_map_one hf'.is_good (hρ ?_) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' (-1)) = ρ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rw [← h1, φ.map_neg, φ.map_one, h, ρ.map_zero] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' (-1)) = ρ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' (-1)) = ρ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | apply hρ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 = -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0.a
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' 2) = ρ (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rw [← h1, map_ofNat, h0, ρ.map_neg, ρ.map_one] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0.a
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' 2) = ρ (-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0.a
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' 2) = ρ (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.admissible_mem_sq_mul | [32, 1] | [36, 62] | have h1 := h z z h0 h0 (k - 2) | R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
⊢ k * z ^ 2 ∈ A | R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
h1 : z ^ 2 + (k - 2) * z * z + z ^ 2 ∈ A
⊢ k * z ^ 2 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
⊢ k * z ^ 2 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.admissible_mem_sq_mul | [32, 1] | [36, 62] | rwa [mul_assoc, ← sq, ← one_add_mul (α := R), ← add_one_mul (α := R),
add_right_comm, one_add_one_eq_two, add_sub_cancel] at h1 | R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
h1 : z ^ 2 + (k - 2) * z * z + z ^ 2 ∈ A
⊢ k * z ^ 2 ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
h1 : z ^ 2 + (k - 2) * z * z + z ^ 2 ∈ A
⊢ k * z ^ 2 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | specialize h (Ideal.span {x, y}) (admissible_ideal _)
(Ideal.subset_span (Set.mem_insert x _))
(Ideal.subset_span (Set.mem_insert_of_mem x rfl)) 1 | R : Type u_1
inst✝ : CommRing R
x y : R
h : ∀ (A : Set R), admissible A → x ∈ A → y ∈ A → ∀ (z : R), z ∈ A
⊢ IsCoprime x y | R : Type u_1
inst✝ : CommRing R
x y : R
h : 1 ∈ ↑(Ideal.span {x, y})
⊢ IsCoprime x y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : ∀ (A : Set R), admissible A → x ∈ A → y ∈ A → ∀ (z : R), z ∈ A
⊢ IsCoprime x y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | rwa [SetLike.mem_coe, Ideal.mem_span_pair] at h | R : Type u_1
inst✝ : CommRing R
x y : R
h : 1 ∈ ↑(Ideal.span {x, y})
⊢ IsCoprime x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : 1 ∈ ↑(Ideal.span {x, y})
⊢ IsCoprime x y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | suffices 1 ∈ A from λ z ↦ mul_one z ▸
one_pow (M := R) 2 ▸ admissible_mem_sq_mul h0 this z | R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ ∀ (z : R), z ∈ A | R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ 1 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ ∀ (z : R), z ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | obtain ⟨a, b, h⟩ : IsCoprime (x ^ 2) (y ^ 2) := IsCoprime.pow h | R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ 1 ∈ A | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ 1 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ 1 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | rw [← one_pow 2, ← h] | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ 1 ∈ A | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ (a * x ^ 2 + b * y ^ 2) ^ 2 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ 1 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | refine admissible_add_sq h0 (admissible_mem_sq_mul h0 h1 a)
(admissible_mem_sq_mul h0 h2 b) | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ (a * x ^ 2 + b * y ^ 2) ^ 2 ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ (a * x ^ 2 + b * y ^ 2) ^ 2 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SqSubOneMap.lean | IMOSL.IMO2012A5.sq_sub_one_is_good | [50, 1] | [51, 18] | ring | R : Type u_1
inst✝ : CommRing R
x✝¹ x✝ : R
⊢ (fun x => x ^ 2 - 1) (x✝¹ * x✝ + 1) =
(fun x => x ^ 2 - 1) x✝¹ * (fun x => x ^ 2 - 1) x✝ + (fun x => x ^ 2 - 1) (x✝¹ + x✝) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x✝¹ x✝ : R
⊢ (fun x => x ^ 2 - 1) (x✝¹ * x✝ + 1) =
(fun x => x ^ 2 - 1) x✝¹ * (fun x => x ^ 2 - 1) x✝ + (fun x => x ^ 2 - 1) (x✝¹ + x✝)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/toList.lean | IMOSL.IMO2012A5.𝔽₂X.toList_inj | [33, 1] | [35, 23] | rw [← 𝔽₂X.toFinset_toList, h, 𝔽₂X.toFinset_toList] | P Q : 𝔽₂X
h : P.toList = Q.toList
⊢ P.toFinset = Q.toFinset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P Q : 𝔽₂X
h : P.toList = Q.toList
⊢ P.toFinset = Q.toFinset
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_left | [26, 1] | [31, 22] | apply (hε (k * ⌈|r|⌉.natAbs)).trans_le' | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| < 1 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| ≤ (k * ⌈|r|⌉.natAbs) • |ε| | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_left | [26, 1] | [31, 22] | rw [abs_mul, mul_nsmul', nsmul_eq_mul ⌈|r|⌉.natAbs] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| ≤ (k * ⌈|r|⌉.natAbs) • |ε| | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • (|r| * |ε|) ≤ k • (↑⌈|r|⌉.natAbs * |ε|) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| ≤ (k * ⌈|r|⌉.natAbs) • |ε|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_left | [26, 1] | [31, 22] | refine nsmul_le_nsmul_right (mul_le_mul_of_nonneg_right ?_ (abs_nonneg ε)) k | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • (|r| * |ε|) ≤ k • (↑⌈|r|⌉.natAbs * |ε|) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ |r| ≤ ↑⌈|r|⌉.natAbs | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • (|r| * |ε|) ≤ k • (↑⌈|r|⌉.natAbs * |ε|)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_left | [26, 1] | [31, 22] | rw [← Int.cast_natCast ⌈|r|⌉.natAbs, ← Int.ceil_le] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ |r| ≤ ↑⌈|r|⌉.natAbs | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ |r| ≤ ↑⌈|r|⌉.natAbs
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_left | [26, 1] | [31, 22] | exact Int.le_natAbs | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_right | [33, 1] | [38, 22] | apply (hε (k * ⌈|r|⌉.natAbs)).trans_le' | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |ε * r| < 1 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |ε * r| ≤ (k * ⌈|r|⌉.natAbs) • |ε| | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |ε * r| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_right | [33, 1] | [38, 22] | rw [abs_mul, mul_nsmul', nsmul_eq_mul' |ε|] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |ε * r| ≤ (k * ⌈|r|⌉.natAbs) • |ε| | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • (|ε| * |r|) ≤ k • (|ε| * ↑⌈|r|⌉.natAbs) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |ε * r| ≤ (k * ⌈|r|⌉.natAbs) • |ε|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_right | [33, 1] | [38, 22] | refine nsmul_le_nsmul_right (mul_le_mul_of_nonneg_left ?_ (abs_nonneg ε)) k | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • (|ε| * |r|) ≤ k • (|ε| * ↑⌈|r|⌉.natAbs) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ |r| ≤ ↑⌈|r|⌉.natAbs | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • (|ε| * |r|) ≤ k • (|ε| * ↑⌈|r|⌉.natAbs)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_right | [33, 1] | [38, 22] | rw [← Int.cast_natCast ⌈|r|⌉.natAbs, ← Int.ceil_le] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ |r| ≤ ↑⌈|r|⌉.natAbs | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ |r| ≤ ↑⌈|r|⌉.natAbs
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_right | [33, 1] | [38, 22] | exact Int.le_natAbs | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁ | [40, 1] | [44, 80] | rcases le_total 0 ε with h0 | h0 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
⊢ k • |ε| < 1 | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : 0 ≤ ε
⊢ k • |ε| < 1
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : ε ≤ 0
⊢ k • |ε| < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
⊢ k • |ε| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁ | [40, 1] | [44, 80] | rw [abs_eq_self.mpr h0, nsmul_eq_mul] | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : 0 ≤ ε
⊢ k • |ε| < 1 | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : 0 ≤ ε
⊢ ↑k * ε < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : 0 ≤ ε
⊢ k • |ε| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁ | [40, 1] | [44, 80] | exact h _ | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : 0 ≤ ε
⊢ ↑k * ε < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : 0 ≤ ε
⊢ ↑k * ε < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁ | [40, 1] | [44, 80] | rw [abs_eq_neg_self.mpr h0, nsmul_eq_mul, mul_neg, ← neg_mul] | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : ε ≤ 0
⊢ k • |ε| < 1 | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : ε ≤ 0
⊢ -↑k * ε < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : ε ≤ 0
⊢ k • |ε| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁ | [40, 1] | [44, 80] | exact h _ | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : ε ≤ 0
⊢ -↑k * ε < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), r * ε < 1
k : ℕ
h0 : ε ≤ 0
⊢ -↑k * ε < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁ | [46, 1] | [50, 81] | rcases le_total 0 ε with h0 | h0 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
⊢ k • |ε| < 1 | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : 0 ≤ ε
⊢ k • |ε| < 1
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : ε ≤ 0
⊢ k • |ε| < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
⊢ k • |ε| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁ | [46, 1] | [50, 81] | rw [abs_eq_self.mpr h0, nsmul_eq_mul'] | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : 0 ≤ ε
⊢ k • |ε| < 1 | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : 0 ≤ ε
⊢ ε * ↑k < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : 0 ≤ ε
⊢ k • |ε| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁ | [46, 1] | [50, 81] | exact h _ | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : 0 ≤ ε
⊢ ε * ↑k < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : 0 ≤ ε
⊢ ε * ↑k < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁ | [46, 1] | [50, 81] | rw [abs_eq_neg_self.mpr h0, nsmul_eq_mul', neg_mul, ← mul_neg] | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : ε ≤ 0
⊢ k • |ε| < 1 | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : ε ≤ 0
⊢ ε * -↑k < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : ε ≤ 0
⊢ k • |ε| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁ | [46, 1] | [50, 81] | exact h _ | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : ε ≤ 0
⊢ ε * -↑k < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
h : ∀ (r : R), ε * r < 1
k : ℕ
h0 : ε ≤ 0
⊢ ε * -↑k < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂ | [52, 1] | [55, 73] | refine FloorRing_iff_mul_left₁.trans ⟨λ h ↦ ⟨1, h⟩, λ ⟨α, h⟩ r ↦ ?_⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), r * ε < α | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h : ∀ (r : R), r * ε < α
⊢ r * ε < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), r * ε < α
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂ | [52, 1] | [55, 73] | have h0 : 0 < α := (zero_mul ε).symm.trans_lt (h 0) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h : ∀ (r : R), r * ε < α
⊢ r * ε < 1 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h : ∀ (r : R), r * ε < α
h0 : 0 < α
⊢ r * ε < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h : ∀ (r : R), r * ε < α
⊢ r * ε < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂ | [52, 1] | [55, 73] | specialize h (α * r) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h : ∀ (r : R), r * ε < α
h0 : 0 < α
⊢ r * ε < 1 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h0 : 0 < α
h : α * r * ε < α
⊢ r * ε < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h : ∀ (r : R), r * ε < α
h0 : 0 < α
⊢ r * ε < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂ | [52, 1] | [55, 73] | rwa [mul_assoc, mul_lt_iff_lt_one_right h0] at h | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h0 : 0 < α
h : α * r * ε < α
⊢ r * ε < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), r * ε < α
r α : R
h0 : 0 < α
h : α * r * ε < α
⊢ r * ε < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂ | [57, 1] | [60, 74] | refine FloorRing_iff_mul_right₁.trans ⟨λ h ↦ ⟨1, h⟩, λ ⟨α, h⟩ r ↦ ?_⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), ε * r < α | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h : ∀ (r : R), ε * r < α
⊢ ε * r < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), ε * r < α
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂ | [57, 1] | [60, 74] | have h0 : 0 < α := (mul_zero ε).symm.trans_lt (h 0) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h : ∀ (r : R), ε * r < α
⊢ ε * r < 1 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h : ∀ (r : R), ε * r < α
h0 : 0 < α
⊢ ε * r < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h : ∀ (r : R), ε * r < α
⊢ ε * r < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂ | [57, 1] | [60, 74] | specialize h (r * α) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h : ∀ (r : R), ε * r < α
h0 : 0 < α
⊢ ε * r < 1 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h0 : 0 < α
h : ε * (r * α) < α
⊢ ε * r < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h : ∀ (r : R), ε * r < α
h0 : 0 < α
⊢ ε * r < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂ | [57, 1] | [60, 74] | rwa [← mul_assoc, mul_lt_iff_lt_one_left h0] at h | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h0 : 0 < α
h : ε * (r * α) < α
⊢ ε * r < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
x✝ : ∃ α, ∀ (r : R), ε * r < α
r α : R
h0 : 0 < α
h : ε * (r * α) < α
⊢ ε * r < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.le_max_one_sq | [48, 1] | [51, 76] | apply (le_abs_self a).trans | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ a ≤ max 1 (a ^ 2) | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ max 1 (a ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ a ≤ max 1 (a ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.le_max_one_sq | [48, 1] | [51, 76] | rw [← sq_abs, le_max_iff] | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ max 1 (a ^ 2) | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ 1 ∨ |a| ≤ |a| ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ max 1 (a ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.le_max_one_sq | [48, 1] | [51, 76] | exact (le_total |a| 1).imp_right λ h ↦ le_self_pow h (Nat.succ_ne_zero 1) | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ 1 ∨ |a| ≤ |a| ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ 1 ∨ |a| ≤ |a| ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rcases le_total a 0 with h | h | R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
case inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rcases le_total b 0 with h0 | h0 | case inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rw [posPart_eq_self.mpr h, posPart_eq_self.mpr h0, eq_comm] | case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ = a * b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | refine (congr_arg _ (min_eq_left <| le_min ?_ ?_)).trans
(posPart_eq_self.mpr (mul_nonneg h h0)) | case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ = a * b | case inr.inr.refine_1
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max a (a * b ^ 2)
case inr.inr.refine_2
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max b (a ^ 2 * b) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ = a * b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rw [posPart_eq_zero.mpr h, zero_mul, eq_comm, posPart_eq_zero, min_le_iff, min_le_iff] | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | right | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | left | case inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | exact max_le h (mul_nonpos_of_nonpos_of_nonneg h (sq_nonneg b)) | case inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rw [posPart_eq_zero.mpr h0, mul_zero, eq_comm, posPart_eq_zero, min_le_iff, min_le_iff] | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | right | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inr.inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | right | case inr.inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inr.inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
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