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/Case3.lean | IMOSL.IMO2012A5.Case3.SCharTwo.solution | [153, 1] | [161, 60] | exact ρ.congr_arg (CommCase.SCharTwo_map_add_one hf' x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : CharTwo S'
⊢ ρ (f' (x + 1)) = ρ (f' 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
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : CharTwo S'
⊢ ρ (f' (x + 1)) = ρ (f' x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | rcases CommSubring.oneVarCommLiftDomain_exists hf x with
⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, -, f', h, hf'⟩ | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | have R'char := pullback_of_inj φ.toAddMonoidHom hφ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | have S'char : (2 : S') ≠ 0 := λ h1 ↦ hS <| by rw [← map_ofNat ρ 2, h1, ρ.map_zero] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | rw [h, ← φ.map_one, ← φ.map_add, h, ← ρ.map_zero] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
⊢ (ρ (f' x) = ρ 0 ∨ ρ (f' (x + 1)) = ρ 0) ∨ ρ (f' x) + ρ (f' (x + 1)) = 1 ∧ ρ (f' x) * ρ (f' (x + 1)) = -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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
⊢ (f (φ x) = 0 ∨ f (φ x + 1) = 0) ∨ f (φ x) + f (φ x + 1) = 1 ∧ f (φ x) * f (φ x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | refine (CommCase.SCharNeTwo_cases hf' S'char x).imp
(Or.imp ρ.congr_arg ρ.congr_arg) (And.imp (λ h0 ↦ ?_) (λ h0 ↦ ?_)) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
⊢ (ρ (f' x) = ρ 0 ∨ ρ (f' (x + 1)) = ρ 0) ∨ ρ (f' x) + ρ (f' (x + 1)) = 1 ∧ ρ (f' x) * ρ (f' (x + 1)) = -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
h0 : f' x + f' (x + 1) = 1
⊢ ρ (f' x) + ρ (f' (x + 1)) = 1
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
h0 : f' x * f' (x + 1) = -1
⊢ ρ (f' x) * ρ (f' (x + 1)) = -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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
⊢ (ρ (f' x) = ρ 0 ∨ ρ (f' (x + 1)) = ρ 0) ∨ ρ (f' x) + ρ (f' (x + 1)) = 1 ∧ ρ (f' x) * ρ (f' (x + 1)) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | rw [← map_ofNat ρ 2, h1, ρ.map_zero] | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
h1 : 2 = 0
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
h1 : 2 = 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | rw [← 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_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
h0 : f' x + f' (x + 1) = 1
⊢ ρ (f' x) + ρ (f' (x + 1)) = 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_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
h0 : f' x + f' (x + 1) = 1
⊢ ρ (f' x) + ρ (f' (x + 1)) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.main_cases | [170, 1] | [181, 47] | rw [← ρ.map_mul, h0, 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_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
h0 : f' x * f' (x + 1) = -1
⊢ ρ (f' x) * ρ (f' (x + 1)) = -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_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : 2 ≠ 0
h0 : f' x * f' (x + 1) = -1
⊢ ρ (f' x) * ρ (f' (x + 1)) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | refine ⟨λ h0 ↦ ?_, λ h0 ↦ ?_⟩ | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) = 0 ↔ f x ^ 2 = 1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
⊢ f x ^ 2 = 1
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) = 0 ↔ f x ^ 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | have h1 := Eq3_v2 hf x | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
⊢ f x ^ 2 = 1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
h1 : f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x
⊢ f x ^ 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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
⊢ f x ^ 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | rw [h0, zero_mul, zero_mul, add_zero, zero_eq_mul] at h1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
h1 : f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x
⊢ f x ^ 2 = 1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
h1 : f x ^ 2 - 1 = 0 ∨ f x - 1 = 0
⊢ f x ^ 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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
h1 : f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x
⊢ f x ^ 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | exact h1.elim eq_of_sub_eq_zero λ h1 ↦ eq_of_sub_eq_zero h1 ▸ one_pow 2 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
h1 : f x ^ 2 - 1 = 0 ∨ f x - 1 = 0
⊢ f x ^ 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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f (x + 1) = 0
h1 : f x ^ 2 - 1 = 0 ∨ f x - 1 = 0
⊢ f x ^ 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | rcases main_cases hS hf x with (h1 | h1) | ⟨h1, h2⟩ | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
⊢ f (x + 1) = 0 | case refine_2.inl.inl
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x = 0
⊢ f (x + 1) = 0
case refine_2.inl.inr
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f (x + 1) = 0
⊢ f (x + 1) = 0
case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x + f (x + 1) = 1
h2 : f x * f (x + 1) = -1
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | rw [h1, sq, zero_mul] at h0 | case refine_2.inl.inl
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x = 0
⊢ f (x + 1) = 0 | case refine_2.inl.inl
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : 0 = 1
h1 : f x = 0
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl.inl
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x = 0
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | rw [← mul_one (f _), ← h0, mul_zero] | case refine_2.inl.inl
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : 0 = 1
h1 : f x = 0
⊢ f (x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl.inl
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : 0 = 1
h1 : f x = 0
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | exact h1 | case refine_2.inl.inr
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f (x + 1) = 0
⊢ f (x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl.inr
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f (x + 1) = 0
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | rw [← mul_assoc, ← sq, h0, one_mul, mul_neg_one] at h2 | case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x + f (x + 1) = 1
h2 : f x * (f x * f (x + 1)) = f x * -1
⊢ f (x + 1) = 0 | case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x + f (x + 1) = 1
h2 : f (x + 1) = -f x
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x + f (x + 1) = 1
h2 : f x * (f x * f (x + 1)) = f x * -1
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | rw [h2, add_neg_self] at h1 | case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x + f (x + 1) = 1
h2 : f (x + 1) = -f x
⊢ f (x + 1) = 0 | case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : 0 = 1
h2 : f (x + 1) = -f x
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : f x + f (x + 1) = 1
h2 : f (x + 1) = -f x
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_iff_map_eq | [183, 1] | [195, 43] | rw [← mul_one (f _), ← h1, mul_zero] | case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : 0 = 1
h2 : f (x + 1) = -f x
⊢ f (x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 = 1
h1 : 0 = 1
h2 : f (x + 1) = -f x
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem1 | [210, 1] | [211, 81] | rw [Eq2_v2 hf, map_eq_neg_one_imp_map_add_one hS hf h, sq, zero_mul, zero_sub] | R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
⊢ f (r * r) = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
⊢ f (r * r) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem2 | [213, 1] | [216, 72] | have h0 := Eq3 hf r | R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
⊢ f (r * r + r) = -1 | R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
h0 : f r * f (r * r + r) = (f (r + 1) ^ 2 - 1) * (f (r + 1) - 1) + f r * f (r + 1)
⊢ f (r * r + r) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
⊢ f (r * r + r) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem2 | [213, 1] | [216, 72] | rwa [map_eq_neg_one_imp_map_add_one hS hf h, mul_zero, add_zero,
sq, mul_zero, zero_sub, mul_neg_one, h, neg_one_mul, neg_inj] at h0 | R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
h0 : f r * f (r * r + r) = (f (r + 1) ^ 2 - 1) * (f (r + 1) - 1) + f r * f (r + 1)
⊢ f (r * r + r) = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
h0 : f r * f (r * r + r) = (f (r + 1) ^ 2 - 1) * (f (r + 1) - 1) + f r * f (r + 1)
⊢ f (r * r + r) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem3 | [218, 1] | [219, 50] | rw [hf.is_good, h, neg_one_mul, neg_add_eq_sub] | R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f (r * x + 1) = f (r + x) - f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f (r * x + 1) = f (r + x) - f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem4 | [221, 1] | [223, 82] | rcases h0 with ⟨x, rfl⟩ | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : ∃ y, x = r * y + 1
⊢ f (r * r + x) = -f x | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f (r * r + (r * x + 1)) = -f (r * x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : ∃ y, x = r * y + 1
⊢ f (r * r + x) = -f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem4 | [221, 1] | [223, 82] | rw [Lem3 hf h, ← add_assoc, ← mul_add, Lem3 hf h, add_add_cancel_left, neg_sub] | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f (r * r + (r * x + 1)) = -f (r * x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f (r * r + (r * x + 1)) = -f (r * x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem5 | [225, 1] | [230, 89] | rcases h0 with ⟨x, rfl⟩ | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : ∃ y, x = r * r * y + 1
⊢ f ((r * r + r) * (r * r + r) + x) = f x | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f ((r * r + r) * (r * r + r) + (r * r * x + 1)) = f (r * r * x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : ∃ y, x = r * r * y + 1
⊢ f ((r * r + r) * (r * r + r) + x) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem5 | [225, 1] | [230, 89] | have h1 : Commute (r * r) r := mul_assoc r r r | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f ((r * r + r) * (r * r + r) + (r * r * x + 1)) = f (r * r * x + 1) | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ f ((r * r + r) * (r * r + r) + (r * r * x + 1)) = f (r * r * x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f ((r * r + r) * (r * r + r) + (r * r * x + 1)) = f (r * r * x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem5 | [225, 1] | [230, 89] | rw [add_mul_self_of_Commute h1, add_assoc, Lem4 hf (Lem1 hS hf h), Lem4 hf h, neg_neg] | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ f ((r * r + r) * (r * r + r) + (r * r * x + 1)) = f (r * r * x + 1) | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ ∃ y, r * r * x + 1 = r * y + 1
case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ ∃ y, r * r + (r * r * x + 1) = r * r * y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ f ((r * r + r) * (r * r + r) + (r * r * x + 1)) = f (r * r * x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem5 | [225, 1] | [230, 89] | exacts [⟨r * x, by rw [mul_assoc]⟩, ⟨1 + x, by rw [← add_assoc, mul_one_add (r * r)]⟩] | case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ ∃ y, r * r * x + 1 = r * y + 1
case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ ∃ y, r * r + (r * r * x + 1) = r * r * y + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ ∃ y, r * r * x + 1 = r * y + 1
case intro
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ ∃ y, r * r + (r * r * x + 1) = r * r * y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem5 | [225, 1] | [230, 89] | rw [mul_assoc] | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ r * r * x + 1 = r * (r * x) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ r * r * x + 1 = r * (r * x) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem5 | [225, 1] | [230, 89] | rw [← add_assoc, mul_one_add (r * r)] | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ r * r + (r * r * x + 1) = r * r * (1 + x) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h1 : Commute (r * r) r
⊢ r * r + (r * r * x + 1) = r * r * (1 + x) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | have h0 : ∃ y, r * r * (r + 1) * x + 1 = r * r * y + 1 :=
⟨(r + 1) * x, by rw [← mul_assoc]⟩ | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f (r * r * (r + 1) * x + 1) = 0 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : ∃ y, r * r * (r + 1) * x + 1 = r * r * y + 1
⊢ f (r * r * (r + 1) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ f (r * r * (r + 1) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | apply Lem5 hS hf h at h0 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : ∃ y, r * r * (r + 1) * x + 1 = r * r * y + 1
⊢ f (r * r * (r + 1) * x + 1) = 0 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : ∃ y, r * r * (r + 1) * x + 1 = r * r * y + 1
⊢ f (r * r * (r + 1) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | have h1 : ∃ y, r * r * (r + 1) * x + 1 = (r * r + r) * y + 1 :=
⟨r * x, by rw [← mul_assoc, add_mul, mul_add_one (r * r)]⟩ | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
h1 : ∃ y, r * r * (r + 1) * x + 1 = (r * r + r) * y + 1
⊢ f (r * r * (r + 1) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | apply Lem4 hf (Lem2 hS hf h) at h1 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
h1 : ∃ y, r * r * (r + 1) * x + 1 = (r * r + r) * y + 1
⊢ f (r * r * (r + 1) * x + 1) = 0 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
h1 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = -f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
h1 : ∃ y, r * r * (r + 1) * x + 1 = (r * r + r) * y + 1
⊢ f (r * r * (r + 1) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | rw [h1, neg_eq_iff_add_eq_zero, ← two_mul, mul_eq_zero] at h0 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
h1 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = -f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0 | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : 2 = 0 ∨ f (r * r * (r + 1) * x + 1) = 0
h1 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = -f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
h1 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = -f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | exact h0.resolve_left hS | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : 2 = 0 ∨ f (r * r * (r + 1) * x + 1) = 0
h1 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = -f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : 2 = 0 ∨ f (r * r * (r + 1) * x + 1) = 0
h1 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = -f (r * r * (r + 1) * x + 1)
⊢ f (r * r * (r + 1) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | rw [← mul_assoc] | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ r * r * (r + 1) * x + 1 = r * r * ((r + 1) * x) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
⊢ r * r * (r + 1) * x + 1 = r * r * ((r + 1) * x) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.ReductionLemmas.Lem6 | [232, 1] | [244, 29] | rw [← mul_assoc, add_mul, mul_add_one (r * r)] | R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
⊢ r * r * (r + 1) * x + 1 = (r * r + r) * (r * x) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
r : R
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : NontrivialGood f
h : f r = -1
x : R
h0 : f ((r * r + r) * (r * r + r) + (r * r * (r + 1) * x + 1)) = f (r * r * (r + 1) * x + 1)
⊢ r * r * (r + 1) * x + 1 = (r * r + r) * (r * x) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | let hf' := hf.toNontrivialGood | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
⊢ r = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | have h0 : (r * r + r) * (r * r + r) = 0 := by
refine (QuasiPeriodic.reduced_eq_zero_iff hf ?_).mpr
(ReductionLemmas.Lem1 hS hf' (ReductionLemmas.Lem2 hS hf' h))
have h0 : Commute (r * r) r := mul_assoc r r r
rw [add_mul_self_of_Commute h0, ← mul_add_one (r * r), ← add_one_mul_self, ← mul_assoc]
exact (ReductionLemmas.Lem6 hS hf' h).mul_right hf' (r + 1) | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ r = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | replace h0 : r * r + r = 0 := by
refine (QuasiPeriodic.reduced_eq_zero_iff hf <|
(QuasiPeriodic.iff_left2 hf').mpr λ x ↦ ?_).mpr (ReductionLemmas.Lem2 hS hf' h)
have h1 := ReductionLemmas.Lem4 hf' (ReductionLemmas.Lem2 hS hf' h) ⟨x, rfl⟩
rw [h0, zero_add, eq_neg_iff_add_eq_zero, ← two_mul, mul_eq_zero] at h1
exact h1.resolve_left hS | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
⊢ r = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | refine (QuasiPeriodic.reduced_eq_zero_iff hf <|
(QuasiPeriodic.iff_left2 hf').mpr λ x ↦ ?_).mpr h | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
⊢ r = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
x : R
⊢ f (r * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | have h1 := hf.is_good (r + 1) (r * (x - 1)) | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
x : R
⊢ f (r * x + 1) = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
x : R
h1 : f ((r + 1) * (r * (x - 1)) + 1) = f (r + 1) * f (r * (x - 1)) + f (r + 1 + r * (x - 1))
⊢ f (r * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
x : R
⊢ f (r * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | rwa [map_eq_neg_one_imp_map_add_one hS hf' h, zero_mul, zero_add,
← mul_assoc, add_one_mul r, h0, zero_mul, zero_add, hf.map_one,
← add_rotate, ← mul_add_one r, sub_add_cancel, eq_comm] at h1 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
x : R
h1 : f ((r + 1) * (r * (x - 1)) + 1) = f (r + 1) * f (r * (x - 1)) + f (r + 1 + r * (x - 1))
⊢ f (r * 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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : r * r + r = 0
x : R
h1 : f ((r + 1) * (r * (x - 1)) + 1) = f (r + 1) * f (r * (x - 1)) + f (r + 1 + r * (x - 1))
⊢ f (r * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | refine (QuasiPeriodic.reduced_eq_zero_iff hf ?_).mpr
(ReductionLemmas.Lem1 hS hf' (ReductionLemmas.Lem2 hS hf' h)) | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ (r * r + r) * (r * r + r) = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ QuasiPeriodic f ((r * r + r) * (r * r + r)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ (r * r + r) * (r * r + r) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | have h0 : Commute (r * r) r := mul_assoc r r r | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ QuasiPeriodic f ((r * r + r) * (r * r + r)) | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : Commute (r * r) r
⊢ QuasiPeriodic f ((r * r + r) * (r * r + r)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
⊢ QuasiPeriodic f ((r * r + r) * (r * r + r))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | rw [add_mul_self_of_Commute h0, ← mul_add_one (r * r), ← add_one_mul_self, ← mul_assoc] | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : Commute (r * r) r
⊢ QuasiPeriodic f ((r * r + r) * (r * r + r)) | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : Commute (r * r) r
⊢ QuasiPeriodic f (r * r * (r + 1) * (r + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : Commute (r * r) r
⊢ QuasiPeriodic f ((r * r + r) * (r * r + r))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | exact (ReductionLemmas.Lem6 hS hf' h).mul_right hf' (r + 1) | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : Commute (r * r) r
⊢ QuasiPeriodic f (r * r * (r + 1) * (r + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : Commute (r * r) r
⊢ QuasiPeriodic f (r * r * (r + 1) * (r + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | refine (QuasiPeriodic.reduced_eq_zero_iff hf <|
(QuasiPeriodic.iff_left2 hf').mpr λ x ↦ ?_).mpr (ReductionLemmas.Lem2 hS hf' h) | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
⊢ r * r + r = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
⊢ f ((r * r + r) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
⊢ r * r + r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | have h1 := ReductionLemmas.Lem4 hf' (ReductionLemmas.Lem2 hS hf' h) ⟨x, rfl⟩ | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
⊢ f ((r * r + r) * x + 1) = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
h1 : f ((r * r + r) * (r * r + r) + ((r * r + r) * x + 1)) = -f ((r * r + r) * x + 1)
⊢ f ((r * r + r) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
⊢ f ((r * r + r) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | rw [h0, zero_add, eq_neg_iff_add_eq_zero, ← two_mul, mul_eq_zero] at h1 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
h1 : f ((r * r + r) * (r * r + r) + ((r * r + r) * x + 1)) = -f ((r * r + r) * x + 1)
⊢ f ((r * r + r) * x + 1) = 0 | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
h1 : 2 = 0 ∨ f ((r * r + r) * x + 1) = 0
⊢ f ((r * r + r) * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
h1 : f ((r * r + r) * (r * r + r) + ((r * r + r) * x + 1)) = -f ((r * r + r) * x + 1)
⊢ f ((r * r + r) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_eq_neg_one_reduced_imp | [259, 1] | [281, 66] | exact h1.resolve_left hS | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
h1 : 2 = 0 ∨ f ((r * r + r) * x + 1) = 0
⊢ f ((r * r + r) * 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✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r = -1
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : (r * r + r) * (r * r + r) = 0
x : R
h1 : 2 = 0 ∨ f ((r * r + r) * x + 1) = 0
⊢ f ((r * r + r) * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.map_add_one_eq_zero_imp | [283, 1] | [285, 63] | rw [Eq2_v2 hf.toNontrivialGood, h, sq, zero_mul, zero_sub] | R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
x : R
h : f (x + 1) = 0
⊢ f (x * x) = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
x : R
h : f (x + 1) = 0
⊢ f (x * x) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.reduced_𝔽₂ε_iff | [290, 1] | [292, 37] | rw [reduced_eq_zero_iff hS hf, Eq2_v2 hf.toNontrivialGood,
sub_eq_neg_self, sq_eq_zero_iff] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ r * r = 0 ↔ f (r + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ r * r = 0 ↔ f (r + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.reduced_𝔽₄_iff | [294, 1] | [299, 39] | rw [reduced_eq_zero_iff hS hf, Eq1 hf.toNontrivialGood, iff_and_self] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ r * (r + 1) + 1 = 0 ↔ f r + f (r + 1) = 1 ∧ f r * f (r + 1) = -1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ f r * f (r + 1) = -1 → f r + f (r + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ r * (r + 1) + 1 = 0 ↔ f r + f (r + 1) = 1 ∧ f r * f (r + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.reduced_𝔽₄_iff | [294, 1] | [299, 39] | intro h | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ f r * f (r + 1) = -1 → f r + f (r + 1) = 1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
⊢ f r + f (r + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ f r * f (r + 1) = -1 → f r + f (r + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.reduced_𝔽₄_iff | [294, 1] | [299, 39] | refine (main_cases hS hf.toNontrivialGood r).elim (λ h0 ↦ ?_) And.left | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
⊢ f r + f (r + 1) = 1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
h0 : f r = 0 ∨ f (r + 1) = 0
⊢ f r + f (r + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
⊢ f r + f (r + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.reduced_𝔽₄_iff | [294, 1] | [299, 39] | rw [← mul_eq_zero, h, neg_eq_zero] at h0 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
h0 : f r = 0 ∨ f (r + 1) = 0
⊢ f r + f (r + 1) = 1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
h0 : 1 = 0
⊢ f r + f (r + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
h0 : f r = 0 ∨ f (r + 1) = 0
⊢ f r + f (r + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.reduced_𝔽₄_iff | [294, 1] | [299, 39] | rw [← mul_one (_ + _), h0, mul_zero] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
h0 : 1 = 0
⊢ f r + f (r + 1) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
h : f r * f (r + 1) = -1
h0 : 1 = 0
⊢ f r + f (r + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_cases | [301, 1] | [305, 44] | rw [reduced_𝔽₂ε_iff hS hf, reduced_𝔽₂ε_iff hS hf,
reduced_𝔽₄_iff hS hf, add_add_cancel_right] | R : Type u_1
S : Type ?u.155523
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ ((r + 1) * (r + 1) = 0 ∨ r * r = 0) ∨ r * (r + 1) + 1 = 0 | R : Type u_1
S : Type ?u.155523
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ (f r = 0 ∨ f (r + 1) = 0) ∨ f r + f (r + 1) = 1 ∧ f r * f (r + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.155523
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ ((r + 1) * (r + 1) = 0 ∨ r * r = 0) ∨ r * (r + 1) + 1 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_cases | [301, 1] | [305, 44] | exact main_cases hS hf.toNontrivialGood r | R : Type u_1
S : Type ?u.155523
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ (f r = 0 ∨ f (r + 1) = 0) ∨ f r + f (r + 1) = 1 ∧ f r * f (r + 1) = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.155523
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
⊢ (f r = 0 ∨ f (r + 1) = 0) ∨ f r + f (r + 1) = 1 ∧ f r * f (r + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [reduced_𝔽₂ε_iff hS hf, hf.is_good] at h | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : r * s * (r * s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : f r * f s + f (r + s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : r * s * (r * s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [reduced_𝔽₂ε_iff hS hf, map_add_one_eq_zero_iff_map_eq hS hf.toNontrivialGood,
_root_.sq_eq_one_iff, or_comm] at hr hs | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : f r * f s + f (r + s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : f r = -1 ∨ f r = 1
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : f r * f s + f (r + s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | have h0 {x} : f x = -1 → x = 0 := map_eq_neg_one_reduced_imp hS hf | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : f r = -1 ∨ f r = 1
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : f r = -1 ∨ f r = 1
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
⊢ r = 0 ∨ s = 0 ∨ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : f r = -1 ∨ f r = 1
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
⊢ r = 0 ∨ s = 0 ∨ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | revert hr | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : f r = -1 ∨ f r = 1
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
⊢ r = 0 ∨ s = 0 ∨ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
⊢ f r = -1 ∨ f r = 1 → r = 0 ∨ s = 0 ∨ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : f r = -1 ∨ f r = 1
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
⊢ r = 0 ∨ s = 0 ∨ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | refine Or.imp h0 λ hr ↦ ?_ | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
⊢ f r = -1 ∨ f r = 1 → r = 0 ∨ s = 0 ∨ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
⊢ s = 0 ∨ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
⊢ f r = -1 ∨ f r = 1 → r = 0 ∨ s = 0 ∨ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | revert hs | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
⊢ s = 0 ∨ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
⊢ f s = -1 ∨ f s = 1 → s = 0 ∨ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hs : f s = -1 ∨ f s = 1
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
⊢ s = 0 ∨ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | refine Or.imp h0 λ hs ↦ ?_ | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
⊢ f s = -1 ∨ f s = 1 → s = 0 ∨ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
hs : f s = 1
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
⊢ f s = -1 ∨ f s = 1 → s = 0 ∨ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [hr, hs, mul_one, ← neg_eq_iff_add_eq_zero, eq_comm] at h | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
hs : f s = 1
⊢ r = s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f (r + s) = -1
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
hs : f s = 1
⊢ r = s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f r * f s + f (r + s) = 0
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
hs : f s = 1
⊢ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | exact add_eq_zero_iff_eq.mp (h0 h) | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f (r + s) = -1
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
hs : f s = 1
⊢ r = s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
h : f (r + s) = -1
h0 : ∀ {x : R}, f x = -1 → x = 0
hr : f r = 1
hs : f s = 1
⊢ r = s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | have h : (r + s) * (r + s) = r * s + s * r := by
rw [add_mul, mul_add, hr, zero_add, mul_add, hs, add_zero] | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
⊢ r * s * (r * s) = 0 | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rcases R_elts_cases hS hf (r + s) with (h0 | h0) | h0 | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
⊢ r * s * (r * s) = 0 | case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s + 1) * (r + s + 1) = 0
⊢ r * s * (r * s) = 0
case inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s) * (r + s) = 0
⊢ r * s * (r * s) = 0
case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s) * (r + s + 1) + 1 = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [add_mul, mul_add, hr, zero_add, mul_add, hs, add_zero] | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
⊢ (r + s) * (r + s) = r * s + s * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
⊢ (r + s) * (r + s) = r * s + s * r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [add_one_mul_self, h, add_eq_zero_iff_eq] at h0 | case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s + 1) * (r + s + 1) = 0
⊢ r * s * (r * s) = 0 | case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s + 1) * (r + s + 1) = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | have h1 : (r * s) * (r * s) = r * s := by
apply congrArg (λ x ↦ r * x * s) at h0
rwa [mul_add, ← mul_assoc, hr, zero_mul, zero_add,
mul_one, mul_assoc, mul_assoc, ← mul_assoc] at h0 | case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
⊢ r * s * (r * s) = 0 | case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rcases R_elts_cases hS hf (r * s) with (h2 | h2) | h2 | case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
⊢ r * s * (r * s) = 0 | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : (r * s + 1) * (r * s + 1) = 0
⊢ r * s * (r * s) = 0
case inl.inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s * (r * s) = 0
⊢ r * s * (r * s) = 0
case inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s * (r * s + 1) + 1 = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | apply congrArg (λ x ↦ r * x * s) at h0 | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
⊢ r * s * (r * s) = r * s | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * (r * s + s * r) * s = r * 1 * s
⊢ r * s * (r * s) = r * s | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
⊢ r * s * (r * s) = r * s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rwa [mul_add, ← mul_assoc, hr, zero_mul, zero_add,
mul_one, mul_assoc, mul_assoc, ← mul_assoc] at h0 | R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * (r * s + s * r) * s = r * 1 * s
⊢ r * s * (r * s) = r * s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * (r * s + s * r) * s = r * 1 * s
⊢ r * s * (r * s) = r * s
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [add_one_mul_self, h1, add_eq_zero_iff_eq] at h2 | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : (r * s + 1) * (r * s + 1) = 0
⊢ r * s * (r * s) = 0 | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : (r * s + 1) * (r * s + 1) = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [h2, add_right_eq_self] at h0 | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
⊢ r * s * (r * s) = 0 | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : s * r = 0
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [mul_zero, ← mul_assoc, h2, one_mul] at h0 | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
h0 : r * (s * r) = r * 0
⊢ r * s * (r * s) = 0 | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
h0 : r = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
h0 : r * (s * r) = r * 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [h0, zero_mul, zero_mul] | case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
h0 : r = 0
⊢ r * s * (r * s) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inl.inl
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h1 : r * s * (r * s) = r * s
h2 : r * s = 1
h0 : r = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | exact h2 | case inl.inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s * (r * s) = 0
⊢ r * s * (r * s) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s * (r * s) = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [mul_add_one (r * s), h1, add_self_eq_zero, zero_add] at h2 | case inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s * (r * s + 1) + 1 = 0
⊢ r * s * (r * s) = 0 | case inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : 1 = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : r * s * (r * s + 1) + 1 = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [← mul_one (r * s * _), h2, mul_zero] | case inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : 1 = 0
⊢ r * s * (r * s) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r = 1
h1 : r * s * (r * s) = r * s
h2 : 1 = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [h, add_eq_zero_iff_eq] at h0 | case inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s) * (r + s) = 0
⊢ r * s * (r * s) = 0 | case inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s = s * r
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s) * (r + s) = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [← mul_assoc, mul_assoc r, ← h0, mul_assoc, mul_assoc, hs, mul_zero, mul_zero] | case inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s = s * r
⊢ r * s * (r * s) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s = s * r
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [mul_add_one (r + s), h] at h0 | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s) * (r + s + 1) + 1 = 0
⊢ r * s * (r * s) = 0 | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : r * s + s * r + (r + s) + 1 = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h : (r + s) * (r + s) = r * s + s * r
h0 : (r + s) * (r + s + 1) + 1 = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [mul_zero, mul_add_one r, mul_add, mul_add, ← mul_assoc,
mul_add, hr, zero_mul, zero_add, zero_add, ← mul_assoc] at h | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * (r * s + s * r + (r + s) + 1) = r * 0
⊢ r * s * (r * s) = 0 | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s * r + r * s + r = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * (r * s + s * r + (r + s) + 1) = r * 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [zero_mul, add_mul, add_mul, mul_assoc, hr, add_zero, mul_zero, zero_add] at h1 | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s * r + r * s + r = 0
h1 : (r * s * r + r * s + r) * r = 0 * r
⊢ r * s * (r * s) = 0 | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s * r + r * s + r = 0
h1 : r * s * r = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s * r + r * s + r = 0
h1 : (r * s * r + r * s + r) * r = 0 * r
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [h1, zero_add, add_eq_zero_iff_eq] at h | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s * r + r * s + r = 0
h1 : r * s * r = 0
⊢ r * s * (r * s) = 0 | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s = r
h1 : r * s * r = 0
⊢ r * s * (r * s) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s * r + r * s + r = 0
h1 : r * s * r = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim1 | [307, 1] | [347, 37] | rw [h, hr] | case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s = r
h1 : r * s * r = 0
⊢ r * s * (r * s) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.159546
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * s = 0
h0 : r * s + s * r + (r + s) + 1 = 0
h : r * s = r
h1 : r * s * r = 0
⊢ r * s * (r * s) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rcases R_elts_cases hS hf (r + s) with (h | h) | h | R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
⊢ r = 0 | case inl.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
⊢ r = 0
case inl.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
⊢ r = 0
case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s + 1) + 1 = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rcases R_elts_claim1 hS hf hr h with hr | h | h0 | case inl.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
⊢ r = 0 | case inl.inl.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
hr : r = 0
⊢ r = 0
case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r + s + 1 = 0
⊢ r = 0
case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : r = r + s + 1
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | exact hr | case inl.inl.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
hr : r = 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
hr : r = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [add_assoc, add_eq_zero_iff_eq] at h | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r + s + 1 = 0
⊢ r = 0 | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r + s + 1 = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [h, add_one_mul s, add_eq_zero_iff_eq] at hr | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [hr, add_add_cancel_right] at hs | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0
TACTIC:
|
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