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.CommCase.Eq7 | [146, 1] | [169, 76] | have X : f 2 - 1 ≠ 0 := h1 ∘ eq_of_sub_eq_zero | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h3, ← sub_eq_add_neg, ← mul_sub_one, mul_eq_zero, or_iff_left X] at h2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) * f 2 + f (x - 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h2, neg_zero] at h3 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = -f (x + 1)
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | have h4 := Eq6 hf h h0 x | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h2, h3, add_zero, zero_eq_mul, or_iff_left X] at h4 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f x + 1 = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CommCase.Eq7 | [146, 1] | [169, 76] | rw [h2, h3, zero_add, one_mul, eq_neg_of_add_eq_zero_left h4, neg_one_sq] | case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f x + 1 = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (-x) = f x
h0 : f 2 ≠ -1
h1 : f 2 ≠ 1
x : R
h2 : f (x + 1) = 0
h3 : f (x - 1) = 0
X : f 2 - 1 ≠ 0
h4 : f x + 1 = 0
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Rtwo_ne_zero | [195, 1] | [196, 56] | rw [h, hf.map_zero] | R : Type u_1
S : Type ?u.70700
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : 2 = 0
⊢ f 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.70700
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : 2 = 0
⊢ f 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | rcases CommSubring.oneVarCommLiftDomain_exists hf.toNontrivialGood c with
⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h0, hf'⟩ | R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
c : R
⊢ ∃ R' x φ,
∃ (_ : Function.Injective ⇑φ) (_ : c ∈ Set.range ⇑φ),
∃ S' x_3,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'), f (φ a) = ρ (f' a)), GoodCase2 f' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'_1), f (φ_1 a) = ρ (f' a)), GoodCase2 f' | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
c : R
⊢ ∃ R' x φ,
∃ (_ : Function.Injective ⇑φ) (_ : c ∈ Set.range ⇑φ),
∃ S' x_3,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'), f (φ a) = ρ (f' a)), GoodCase2 f'
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | refine ⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h0, hf', ?_, ?_⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'_1), f (φ_1 a) = ρ (f' a)), GoodCase2 f' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 ≠ -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ f', ∃ (_ : ∀ (a : R'_1), f (φ_1 a) = ρ (f' a)), GoodCase2 f'
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | intro x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ f' (-x) = f' x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | apply hρ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ f' (-x) = f' x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.a
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ ρ (f' (-x)) = ρ (f' x) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ f' (-x) = f' x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | rw [← h0, ← h0, φ.map_neg, hf.map_even] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.a
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ ρ (f' (-x)) = ρ (f' x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.a
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
x : R'
⊢ ρ (f' (-x)) = ρ (f' x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | intro h1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 ≠ -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 ≠ -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | apply hf.map_two_ne_neg_one | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ False | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ f 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.oneVarLift_exists | [199, 1] | [214, 55] | rw [← map_ofNat φ 2, h0, h1, ρ.map_neg, ρ.map_one] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ f 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R✝ : Type ?u.71956
S✝ : Type ?u.71959
inst✝⁵ : Ring R✝
inst✝⁴ : Ring S✝
inst✝³ : NoZeroDivisors S✝
f✝ : R✝ → S✝
hf✝ : GoodCase2 f✝
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
h1 : f' 2 = -1
⊢ f 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq6 | [228, 1] | [234, 78] | rcases oneVarLift_exists hf x with
⟨R', R'comm, φ, -, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, -, f', h0, hf'⟩ | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
x : R
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ f (φ x + 1) + f (φ x - 1) = (f (φ x) + 1) * (f 2 - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
x : R
⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq6 | [228, 1] | [234, 78] | rw [h0, ← φ.map_one, ← φ.map_sub, ← φ.map_add, h0, h0, ← map_ofNat φ 2, h0,
← ρ.map_one, ← ρ.map_add, ← ρ.map_add, ← ρ.map_sub, ← ρ.map_mul] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ f (φ x + 1) + f (φ x - 1) = (f (φ x) + 1) * (f 2 - 1) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ρ (f' (x + 1) + f' (x - 1)) = ρ ((f' x + 1) * (f' (OfNat.ofNat 2) - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ f (φ x + 1) + f (φ x - 1) = (f (φ x) + 1) * (f 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq6 | [228, 1] | [234, 78] | exact congrArg ρ
(CommCase.Eq6 hf'.toNontrivialGood hf'.map_even hf'.map_two_ne_neg_one x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ρ (f' (x + 1) + f' (x - 1)) = ρ ((f' x + 1) * (f' (OfNat.ofNat 2) - 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ρ (f' (x + 1) + f' (x - 1)) = ρ ((f' x + 1) * (f' (OfNat.ofNat 2) - 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | rcases oneVarLift_exists hf x with
⟨R', R'comm, φ, -, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, -, f', h0, hf'⟩ | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
x : R
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
x : R
⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | have h2 : f' 2 ≠ 1 := λ h2 ↦ h1 <| by rw [← map_ofNat φ 2, h0, h2, ρ.map_one] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | rw [h0, ← φ.map_one, ← φ.map_sub, ← φ.map_add, h0, h0, ← ρ.map_one,
← ρ.map_add, ← ρ.map_add, ← ρ.map_mul, ← ρ.map_pow] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ ρ ((f' (x + 1) + 1) * (f' (x - 1) + 1)) = ρ (f' x ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ (f (φ x + 1) + 1) * (f (φ x - 1) + 1) = f (φ x) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | exact congrArg ρ
(CommCase.Eq7 hf'.toNontrivialGood hf'.map_even hf'.map_two_ne_neg_one h2 x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ ρ ((f' (x + 1) + 1) * (f' (x - 1) + 1)) = ρ (f' x ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 ≠ 1
⊢ ρ ((f' (x + 1) + 1) * (f' (x - 1) + 1)) = ρ (f' x ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.Eq7 | [237, 1] | [244, 81] | rw [← map_ofNat φ 2, h0, h2, ρ.map_one] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 = 1
⊢ f 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h1 : f 2 ≠ 1
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h0 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : GoodCase2 f'
h2 : f' 2 = 1
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | have h := Eq1 hf (2 + 1) 2 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f ((2 + 1) * 2 - 1) = f (2 + 1) * f 2 + f (2 + 1 - 2)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [add_sub_cancel_left, hf.map_one, add_zero,
mul_two, add_sub_assoc, add_sub_cancel_right] at h | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f ((2 + 1) * 2 - 1) = f (2 + 1) * f 2 + f (2 + 1 - 2)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f ((2 + 1) * 2 - 1) = f (2 + 1) * f 2 + f (2 + 1 - 2)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | have h0 := Eq6 hf (2 + 1 + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1 + 1 + 1) + f (2 + 1 + 1 - 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [add_sub_cancel_right, add_assoc, one_add_one_eq_two, h, ← mul_add_one (f _)] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1 + 1 + 1) + f (2 + 1 + 1 - 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1 + 1 + 1) + f (2 + 1 + 1 - 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | replace h := Eq6 hf (2 + 1) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + f (2 + 1 - 1) = (f (2 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1 + 2) = f (2 + 1) * f 2
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [add_sub_cancel_right, add_one_mul (α := S), add_sub_left_comm,
add_comm, add_right_inj, eq_sub_iff_add_eq] at h | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + f (2 + 1 - 1) = (f (2 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + f (2 + 1 - 1) = (f (2 + 1) + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [h, mul_assoc, eq_comm, ← sub_eq_zero, ← mul_sub, mul_eq_zero] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) * (f 2 + 1) = (f (2 + 1 + 1) + 1) * (f 2 - 1)
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | clear h | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
h : f (2 + 1 + 1) + 1 = f (2 + 1) * (f 2 - 1)
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | revert h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0 → f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h0 : f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | refine Or.imp (λ h ↦ ?_) (λ h ↦ ?_) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0 → f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
⊢ f 2 = 1
case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
⊢ f (2 + 1) = 0 ∨ (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0 → f 2 = 1 ∨ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | have h0 := Eq1 hf 2 (1 + 1) | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
⊢ f 2 = 1 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f (2 * (1 + 1) - 1) = f 2 * f (1 + 1) + f (2 - (1 + 1))
⊢ f 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [two_mul, ← add_assoc, add_sub_cancel_right, one_add_one_eq_two, sub_self,
hf.map_zero, h, eq_comm, add_neg_eq_zero, mul_self_eq_one_iff] at h0 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f (2 * (1 + 1) - 1) = f 2 * f (1 + 1) + f (2 - (1 + 1))
⊢ f 2 = 1 | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f 2 = 1 ∨ f 2 = -1
⊢ f 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f (2 * (1 + 1) - 1) = f 2 * f (1 + 1) + f (2 - (1 + 1))
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | exact h0.resolve_right hf.map_two_ne_neg_one | case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f 2 = 1 ∨ f 2 = -1
⊢ f 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f (2 + 1) = 0
h0 : f 2 = 1 ∨ f 2 = -1
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | rw [mul_sub_one, sub_one_mul, sub_sub, sub_add_add_cancel, ← two_mul,
sub_sub, ← one_add_mul 2 (f 2), ← sub_mul, mul_eq_zero, add_comm] at h | case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 0 ∨ f 2 = 3 | case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f 2 - (2 + 1) = 0 ∨ f 2 = 0
⊢ f 2 = 0 ∨ f 2 = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : (f 2 - 1) * (f 2 - 1) - (f 2 + 1) = 0
⊢ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.GoodCase2.map_two_cases | [246, 1] | [263, 82] | exact h.symm.imp_right λ h ↦ (eq_of_sub_eq_zero h).trans two_add_one_eq_three | case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f 2 - (2 + 1) = 0 ∨ f 2 = 0
⊢ f 2 = 0 ∨ f 2 = 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodCase2 f
h : f 2 - (2 + 1) = 0 ∨ f 2 = 0
⊢ f 2 = 0 ∨ f 2 = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | have h0 := hf.toGoodCase2.Eq6 (x + 1) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
x : R
⊢ f (x + 2) = -f x | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
x : R
h0 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1)
⊢ f (x + 2) = -f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : 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.RGoodSubcase21.solution | [273, 1] | [302, 78] | rw [h, sub_self, mul_zero, add_sub_cancel_right, add_assoc, one_add_one_eq_two] at h0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
x : R
h0 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1)
⊢ f (x + 2) = -f x | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
x : R
h0 : f (x + 2) + f x = 0
⊢ f (x + 2) = -f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
x : R
h0 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1)
⊢ f (x + 2) = -f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | exact eq_neg_iff_add_eq_zero.mpr h0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
x : R
h0 : f (x + 2) + f x = 0
⊢ f (x + 2) = -f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
x : R
h0 : f (x + 2) + f x = 0
⊢ f (x + 2) = -f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | have h1 : (2 : R) + 2 = 4 := by norm_num | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
x : R
⊢ f (x + 4) = f x | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
x : R
h1 : 2 + 2 = 4
⊢ f (x + 4) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
x : R
⊢ f (x + 4) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | rw [← h1, ← add_assoc, h0, h0, neg_neg] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
x : R
h1 : 2 + 2 = 4
⊢ f (x + 4) = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
x : R
h1 : 2 + 2 = 4
⊢ f (x + 4) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | norm_num | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
x : R
⊢ 2 + 2 = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
x : R
⊢ 2 + 2 = 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | rw [h0, h, mul_neg_one] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
x : R
⊢ f (x + 2) = f x * -f 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
x : R
⊢ f (x + 2) = f x * -f 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | rwa [two_add_one_eq_three, eq_neg_iff_add_eq_zero, three_add_one_eq_four] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
h2 : QuasiPeriodic f 2
⊢ 2 + 1 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
h2 : QuasiPeriodic f 2
⊢ 2 + 1 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | change f 0 = ((-1 : ℤ) : S) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
hSurj : Function.Surjective ℤ₄.cast
h2 : Function.Bijective ℤ₄.cast
⊢ f ℤ₄.ℤ₄0.cast = ↑(ℤ₄Map ℤ₄.ℤ₄0) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
hSurj : Function.Surjective ℤ₄.cast
h2 : Function.Bijective ℤ₄.cast
⊢ f 0 = ↑(-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
hSurj : Function.Surjective ℤ₄.cast
h2 : Function.Bijective ℤ₄.cast
⊢ f ℤ₄.ℤ₄0.cast = ↑(ℤ₄Map ℤ₄.ℤ₄0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase21.solution | [273, 1] | [302, 78] | rw [hf.map_zero, Int.cast_neg, Int.cast_one] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
hSurj : Function.Surjective ℤ₄.cast
h2 : Function.Bijective ℤ₄.cast
⊢ f 0 = ↑(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 1
h0 : ∀ (x : R), f (x + 2) = -f x
h1 : 4 = 0
hInj : Function.Injective ℤ₄.cast
hSurj : Function.Surjective ℤ₄.cast
h2 : Function.Bijective ℤ₄.cast
⊢ f 0 = ↑(-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Eq1 | [316, 1] | [317, 87] | rw [hf.toGoodCase2.Eq6, h, zero_sub, mul_neg_one, neg_add_rev, neg_add_cancel_right] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
⊢ f (x + 1) + f (x - 1) + f x = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
⊢ f (x + 1) + f (x - 1) + f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Rchar | [319, 1] | [324, 69] | rw [← two_add_one_eq_three, ← eq_neg_iff_add_eq_zero] | R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
⊢ 3 = 0 | R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
⊢ 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Rchar | [319, 1] | [324, 69] | refine hf.period_imp_eq _ _ λ x ↦ ?_ | R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
⊢ 2 = -1 | R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
⊢ f (x + 2) = f (x + -1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
⊢ 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Rchar | [319, 1] | [324, 69] | have h0 := Eq1 hf h (x + 1) | R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
⊢ f (x + 2) = f (x + -1) | R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f (x + 1 + 1) + f (x + 1 - 1) + f (x + 1) = -1
⊢ f (x + 2) = f (x + -1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
⊢ f (x + 2) = f (x + -1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Rchar | [319, 1] | [324, 69] | rwa [add_sub_cancel_right, add_assoc x, one_add_one_eq_two, ← Eq1 hf h x,
← add_rotate, add_left_inj, add_right_inj, sub_eq_add_neg] at h0 | R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f (x + 1 + 1) + f (x + 1 - 1) + f (x + 1) = -1
⊢ f (x + 2) = f (x + -1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.109330
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f (x + 1 + 1) + f (x + 1 - 1) + f (x + 1) = -1
⊢ f (x + 2) = f (x + -1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Rchar' | [326, 1] | [327, 64] | rw [eq_neg_iff_add_eq_zero, two_add_one_eq_three, Rchar hf h] | R : Type u_1
S : Type ?u.112996
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
⊢ 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.112996
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
⊢ 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Eq2 | [330, 1] | [334, 70] | have h0 : f 2 ≠ 1 := λ h0 ↦ hf.map_two_ne_neg_one <| by rw [h, zero_eq_neg, ← h, ← h0] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
⊢ f (x + 1) * f (x - 1) = (f x + 1) * f x | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f 2 ≠ 1
⊢ f (x + 1) * f (x - 1) = (f x + 1) * f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
⊢ f (x + 1) * f (x - 1) = (f x + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Eq2 | [330, 1] | [334, 70] | rw [add_one_mul (f x), ← sq, ← hf.toGoodCase2.Eq7 h0, add_one_mul (f _),
mul_add_one (f _), add_assoc, add_assoc, self_eq_add_right,
add_right_comm, ← add_assoc, ← add_assoc, Eq1 hf h, neg_add_self] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f 2 ≠ 1
⊢ f (x + 1) * f (x - 1) = (f x + 1) * f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f 2 ≠ 1
⊢ f (x + 1) * f (x - 1) = (f x + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.Eq2 | [330, 1] | [334, 70] | rw [h, zero_eq_neg, ← h, ← h0] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f 2 = 1
⊢ f 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f 2 = 1
⊢ f 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | rw [sub_eq_add_neg, ← Rchar' hf h, ← one_add_one_eq_two, ← add_assoc] at h1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0 ∧ f (x - 1) = 0
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0 ∧ f (x + 1 + 1) = 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0 ∧ f (x - 1) = 0
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | replace h1 := hf.toReducedGood.period_imp_zero
(Eq5 hf.toNontrivialGood hf.map_even h1.1 h1.2) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0 ∧ f (x + 1 + 1) = 0
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : 2 * (x + 1) + 1 = 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0 ∧ f (x + 1 + 1) = 0
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | rwa [Rchar' hf h, neg_one_mul, neg_add_rev, neg_add_cancel_comm, neg_eq_zero] at h1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : 2 * (x + 1) + 1 = 0
⊢ x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : 2 * (x + 1) + 1 = 0
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | have h1 : f (x + 1) + f (x - 1) = 0 := by
rw [eq_sub_of_add_eq (Eq1 hf h x), h0, sub_self] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | obtain h2 | h2 : f (x + 1) = 0 ∨ f (x - 1) = 0 := by
rw [← mul_eq_zero, Eq2 hf h, h0, neg_add_self, zero_mul] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
h2 : f (x + 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
h2 : f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | rw [eq_sub_of_add_eq (Eq1 hf h x), h0, sub_self] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
⊢ f (x + 1) + f (x - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
⊢ f (x + 1) + f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | rw [← mul_eq_zero, Eq2 hf h, h0, neg_add_self, zero_mul] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
⊢ f (x + 1) = 0 ∨ f (x - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
⊢ f (x + 1) = 0 ∨ f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | rw [h2, zero_add] at h1 | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
h2 : f (x + 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x - 1) = 0
h2 : f (x + 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
h2 : f (x + 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | exact ⟨h2, h1⟩ | case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x - 1) = 0
h2 : f (x + 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x - 1) = 0
h2 : f (x + 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | rw [h2, add_zero] at h1 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
h2 : f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0
h2 : f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) + f (x - 1) = 0
h2 : f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_imp | [336, 1] | [349, 86] | exact ⟨h1, h2⟩ | case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0
h2 : f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
x : R
h0 : f x = -1
h1 : f (x + 1) = 0
h2 : f (x - 1) = 0
⊢ f (x + 1) = 0 ∧ f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | have Rchar' := Rchar' hf h | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | have h1 : f (x * x + 1) = (f x + 1) * f x := by
rw [hf.is_good, ← two_mul, Rchar', neg_one_mul, hf.map_even, add_one_mul (f x)] | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
⊢ f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | have h2 : f (x * x - 1) = f x * f x - 1 := by
rw [Case2.Eq1 hf.toNontrivialGood hf.map_even, sub_self, hf.map_zero, sub_eq_add_neg] | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
⊢ f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | have h3 : f (x * x) = (f x + 1 + 1) * f x := by
have h2 := hf.is_good (x + 1) (x - 1)
rw [Eq2 hf h, add_one_mul x, mul_sub_one, sub_add_sub_cancel, sub_add_cancel,
add_add_sub_cancel, ← two_mul, Rchar', neg_one_mul, hf.map_even] at h2
exact h2.trans (add_one_mul _ (f x)).symm | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
⊢ f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | have h4 := Eq1 hf h (x * x) | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
h4 : f (x * x + 1) + f (x * x - 1) + f (x * x) = -1
⊢ f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | rw [h1, h2, h3, ← add_sub_assoc, ← add_mul, ← add_sub_right_comm,
sub_eq_neg_self, ← add_mul, mul_eq_zero, add_add_add_comm] at h4 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
h4 : f (x * x + 1) + f (x * x - 1) + f (x * x) = -1
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
h4 : f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0
⊢ f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
h4 : f (x * x + 1) + f (x * x - 1) + f (x * x) = -1
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | clear h1 h2 h3 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
h4 : f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h4 : f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0
⊢ f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
h3 : f (x * x) = (f x + 1 + 1) * f x
h4 : f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | revert h4 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h4 : f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0
⊢ f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0 → f x = -1 ∨ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h4 : f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0
⊢ f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | refine Or.imp_left λ h1 ↦ ?_ | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0 → f x = -1 ∨ f x = 0 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f x + 1 + (f x + 1) + (f x + 1) = 0
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f x + 1 + (f x + 1) + (f x + 1) = 0 ∨ f x = 0 → f x = -1 ∨ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | rw [← two_mul, ← add_one_mul 2 (f x + 1), mul_eq_zero, two_add_one_eq_three] at h1 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f x + 1 + (f x + 1) + (f x + 1) = 0
⊢ f x = -1 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : 3 = 0 ∨ f x + 1 = 0
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f x + 1 + (f x + 1) + (f x + 1) = 0
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | exact eq_neg_of_add_eq_zero_left (h1.resolve_left h0) | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : 3 = 0 ∨ f x + 1 = 0
⊢ f x = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : 3 = 0 ∨ f x + 1 = 0
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | rw [hf.is_good, ← two_mul, Rchar', neg_one_mul, hf.map_even, add_one_mul (f x)] | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f (x * x + 1) = (f x + 1) * f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
⊢ f (x * x + 1) = (f x + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | rw [Case2.Eq1 hf.toNontrivialGood hf.map_even, sub_self, hf.map_zero, sub_eq_add_neg] | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
⊢ f (x * x - 1) = f x * f x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
⊢ f (x * x - 1) = f x * f x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | have h2 := hf.is_good (x + 1) (x - 1) | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
⊢ f (x * x) = (f x + 1 + 1) * f x | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2✝ : f (x * x - 1) = f x * f x - 1
h2 : f ((x + 1) * (x - 1) + 1) = f (x + 1) * f (x - 1) + f (x + 1 + (x - 1))
⊢ f (x * x) = (f x + 1 + 1) * f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2 : f (x * x - 1) = f x * f x - 1
⊢ f (x * x) = (f x + 1 + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | rw [Eq2 hf h, add_one_mul x, mul_sub_one, sub_add_sub_cancel, sub_add_cancel,
add_add_sub_cancel, ← two_mul, Rchar', neg_one_mul, hf.map_even] at h2 | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2✝ : f (x * x - 1) = f x * f x - 1
h2 : f ((x + 1) * (x - 1) + 1) = f (x + 1) * f (x - 1) + f (x + 1 + (x - 1))
⊢ f (x * x) = (f x + 1 + 1) * f x | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2✝ : f (x * x - 1) = f x * f x - 1
h2 : f (x * x) = (f x + 1) * f x + f x
⊢ f (x * x) = (f x + 1 + 1) * f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2✝ : f (x * x - 1) = f x * f x - 1
h2 : f ((x + 1) * (x - 1) + 1) = f (x + 1) * f (x - 1) + f (x + 1 + (x - 1))
⊢ f (x * x) = (f x + 1 + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.map_eq_neg_one_or_zero | [351, 1] | [370, 56] | exact h2.trans (add_one_mul _ (f x)).symm | R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2✝ : f (x * x - 1) = f x * f x - 1
h2 : f (x * x) = (f x + 1) * f x + f x
⊢ f (x * x) = (f x + 1 + 1) * f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.124008
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
Rchar' : 2 = -1
h1 : f (x * x + 1) = (f x + 1) * f x
h2✝ : f (x * x - 1) = f x * f x - 1
h2 : f (x * x) = (f x + 1) * f x + f x
⊢ f (x * x) = (f x + 1 + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.value_bash | [372, 1] | [380, 22] | have h1 (x) : x = 0 ∨ f x = 0 :=
(map_eq_neg_one_or_zero hf h h0 x).imp_left (map_eq_neg_one_imp hf h) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
⊢ x = 0 ∨ x = 1 ∨ x = -1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
⊢ x = 0 ∨ x = 1 ∨ x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
⊢ x = 0 ∨ x = 1 ∨ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.value_bash | [372, 1] | [380, 22] | refine (h1 x).imp_right λ h2 ↦ (h1 (x - 1)).imp eq_of_sub_eq_zero λ h3 ↦
(h1 (x + 1)).elim eq_neg_of_add_eq_zero_left λ h4 ↦ ?_ | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
⊢ x = 0 ∨ x = 1 ∨ x = -1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
⊢ x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
⊢ x = 0 ∨ x = 1 ∨ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.value_bash | [372, 1] | [380, 22] | have h5 := Eq1 hf h x | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
⊢ x = -1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : f (x + 1) + f (x - 1) + f x = -1
⊢ x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
⊢ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.value_bash | [372, 1] | [380, 22] | rw [h2, h3, h4, add_zero, add_zero, eq_comm, neg_eq_zero] at h5 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : f (x + 1) + f (x - 1) + f x = -1
⊢ x = -1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : 1 = 0
⊢ x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : f (x + 1) + f (x - 1) + f x = -1
⊢ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.value_bash | [372, 1] | [380, 22] | rw [← mul_one 3, h5, mul_zero] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : 1 = 0
⊢ x = -1 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 0 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : 1 = 0
⊢ x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : 1 = 0
⊢ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.value_bash | [372, 1] | [380, 22] | exact absurd rfl h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 0 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : 1 = 0
⊢ x = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 0 ≠ 0
x : R
h1 : ∀ (x : R), x = 0 ∨ f x = 0
h2 : f x = 0
h3 : f (x - 1) = 0
h4 : f (x + 1) = 0
h5 : 1 = 0
⊢ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.solution | [382, 1] | [396, 78] | have h0 := hf.map_one | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0 : 3 ≠ 0
h : 1 = 0
⊢ 3 = 0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0✝ : 3 ≠ 0
h : 1 = 0
h0 : f 1 = 0
⊢ 3 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0 : 3 ≠ 0
h : 1 = 0
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.solution | [382, 1] | [396, 78] | rw [h, hf.map_zero, neg_eq_zero] at h0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0✝ : 3 ≠ 0
h : 1 = 0
h0 : f 1 = 0
⊢ 3 = 0 | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0✝ : 3 ≠ 0
h : 1 = 0
h0 : 1 = 0
⊢ 3 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0✝ : 3 ≠ 0
h : 1 = 0
h0 : f 1 = 0
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.solution | [382, 1] | [396, 78] | rw [← mul_one 3, h0, mul_zero] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0✝ : 3 ≠ 0
h : 1 = 0
h0 : 1 = 0
⊢ 3 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h✝ : f 2 = 0
h0✝ : 3 ≠ 0
h : 1 = 0
h0 : 1 = 0
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.solution | [382, 1] | [396, 78] | change f 0 = ((-1 : ℤ) : S) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
h1 : Function.Bijective 𝔽₃.cast
⊢ f 𝔽₃.𝔽₃0.cast = ↑(𝔽₃Map2 𝔽₃.𝔽₃0) | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
h1 : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
h1 : Function.Bijective 𝔽₃.cast
⊢ f 𝔽₃.𝔽₃0.cast = ↑(𝔽₃Map2 𝔽₃.𝔽₃0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase22.solution | [382, 1] | [396, 78] | rw [hf.map_zero, Int.cast_neg, Int.cast_one] | R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
h1 : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 0
h0 : 3 ≠ 0
h1 : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.ShiftGood23.oneVarLift_exists | [414, 1] | [430, 73] | rcases hg.shift_good.oneVarLift_exists c with
⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h0, hf'⟩ | R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
c : R
⊢ ∃ R' x φ,
∃ (_ : Function.Injective ⇑φ) (_ : c ∈ Set.range ⇑φ),
∃ S' x_3,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ g', ∃ (_ : ∀ (a : R'), g (φ a) = ρ (g' a)), ShiftGood23 g' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ g', ∃ (_ : ∀ (a : R'_1), g (φ_1 a) = ρ (g' a)), ShiftGood23 g' | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
c : R
⊢ ∃ R' x φ,
∃ (_ : Function.Injective ⇑φ) (_ : c ∈ Set.range ⇑φ),
∃ S' x_3,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ g', ∃ (_ : ∀ (a : R'), g (φ a) = ρ (g' a)), ShiftGood23 g'
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.ShiftGood23.oneVarLift_exists | [414, 1] | [430, 73] | refine ⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f' + 1, ?_, ?_⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ g', ∃ (_ : ∀ (a : R'_1), g (φ_1 a) = ρ (g' a)), ShiftGood23 g' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ∀ (a : R'), g (φ a) = ρ ((f' + 1) a)
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ShiftGood23 (f' + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ∃ R'_1 x_1 φ_1,
∃ (_ : Function.Injective ⇑φ_1) (_ : φ x ∈ Set.range ⇑φ_1),
∃ S' x,
∃ (_ : NoZeroDivisors S'),
∃ ρ, ∃ (_ : Function.Injective ⇑ρ), ∃ g', ∃ (_ : ∀ (a : R'_1), g (φ_1 a) = ρ (g' a)), ShiftGood23 g'
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.ShiftGood23.oneVarLift_exists | [414, 1] | [430, 73] | intro x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ∀ (a : R'), g (φ a) = ρ ((f' + 1) a) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
x : R'
⊢ g (φ x) = ρ ((f' + 1) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
⊢ ∀ (a : R'), g (φ a) = ρ ((f' + 1) a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.ShiftGood23.oneVarLift_exists | [414, 1] | [430, 73] | rw [Pi.add_apply, Pi.one_apply, ρ.map_add, ← h0, ρ.map_one] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
x : R'
⊢ g (φ x) = ρ ((f' + 1) x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
x : R'
⊢ g (φ x) = (g - 1) (φ x) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
x : R'
⊢ g (φ x) = ρ ((f' + 1) x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.ShiftGood23.oneVarLift_exists | [414, 1] | [430, 73] | exact (sub_add_cancel _ _).symm | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
x : R'
⊢ g (φ x) = (g - 1) (φ x) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
S : Type v
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
g : R → S
hg : ShiftGood23 g
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x✝ : R'
S' : Type v
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h0 : ∀ (a : R'), (g - 1) (φ a) = ρ (f' a)
hf' : GoodCase2 f'
x : R'
⊢ g (φ x) = (g - 1) (φ x) + 1
TACTIC:
|
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