Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
xet
Community
1
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/Extra/Infinitesimal/Basic.lean
IMOSL.Extra.Infinitesimal.mul_of_abs_le_one_left
[53, 1]
[56, 87]
exact mul_lt_one_of_nonneg_of_lt_one_right hr (nsmul_nonneg (abs_nonneg ε) k) (hε k)
R : Type u_1 inst✝ : LinearOrderedRing R ε r : R hε : Infinitesimal ε hr : |r| ≤ 1 k : ℕ ⊢ |r| * k • |ε| < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedRing R ε r : R hε : Infinitesimal ε hr : |r| ≤ 1 k : ℕ ⊢ |r| * k • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/Basic.lean
IMOSL.Extra.Infinitesimal.mul_of_abs_le_one_right
[58, 1]
[61, 86]
rw [abs_mul, nsmul_eq_mul, ← mul_assoc, ← nsmul_eq_mul]
R : Type u_1 inst✝ : LinearOrderedRing R ε r : R hε : Infinitesimal ε hr : |r| ≤ 1 k : ℕ ⊢ k • |ε * r| < 1
R : Type u_1 inst✝ : LinearOrderedRing R ε r : R hε : Infinitesimal ε hr : |r| ≤ 1 k : ℕ ⊢ k • |ε| * |r| < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedRing R ε r : R hε : Infinitesimal ε hr : |r| ≤ 1 k : ℕ ⊢ k • |ε * r| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/Basic.lean
IMOSL.Extra.Infinitesimal.mul_of_abs_le_one_right
[58, 1]
[61, 86]
exact mul_lt_one_of_nonneg_of_lt_one_left (nsmul_nonneg (abs_nonneg ε) k) (hε k) hr
R : Type u_1 inst✝ : LinearOrderedRing R ε r : R hε : Infinitesimal ε hr : |r| ≤ 1 k : ℕ ⊢ k • |ε| * |r| < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedRing R ε r : R hε : Infinitesimal ε hr : |r| ≤ 1 k : ℕ ⊢ k • |ε| * |r| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/Basic.lean
IMOSL.Extra.Infinitesimal.nsmul
[67, 1]
[68, 43]
rw [abs_nsmul, ← mul_nsmul']
R : Type u_1 inst✝ : LinearOrderedRing R ε : R hε : Infinitesimal ε n k : ℕ ⊢ k • |n • ε| < 1
R : Type u_1 inst✝ : LinearOrderedRing R ε : R hε : Infinitesimal ε n k : ℕ ⊢ (k * n) • |ε| < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedRing R ε : R hε : Infinitesimal ε n k : ℕ ⊢ k • |n • ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/Basic.lean
IMOSL.Extra.Infinitesimal.nsmul
[67, 1]
[68, 43]
exact hε _
R : Type u_1 inst✝ : LinearOrderedRing R ε : R hε : Infinitesimal ε n k : ℕ ⊢ (k * n) • |ε| < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedRing R ε : R hε : Infinitesimal ε n k : ℕ ⊢ (k * n) • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/Basic.lean
IMOSL.Extra.Infinitesimal.iff_nsmul_Nat_bdd
[70, 1]
[72, 90]
rw [← mul_nsmul, nsmul_one]
R : Type u_1 inst✝ : LinearOrderedRing R ε : R x✝ : ∃ N, ∀ (k : ℕ), k • |ε| < ↑N k N : ℕ h : ∀ (k : ℕ), k • |ε| < ↑N ⊢ N • k • |ε| < N • 1
R : Type u_1 inst✝ : LinearOrderedRing R ε : R x✝ : ∃ N, ∀ (k : ℕ), k • |ε| < ↑N k N : ℕ h : ∀ (k : ℕ), k • |ε| < ↑N ⊢ (k * N) • |ε| < ↑N
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedRing R ε : R x✝ : ∃ N, ∀ (k : ℕ), k • |ε| < ↑N k N : ℕ h : ∀ (k : ℕ), k • |ε| < ↑N ⊢ N • k • |ε| < N • 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/Basic.lean
IMOSL.Extra.Infinitesimal.iff_nsmul_Nat_bdd
[70, 1]
[72, 90]
exact h _
R : Type u_1 inst✝ : LinearOrderedRing R ε : R x✝ : ∃ N, ∀ (k : ℕ), k • |ε| < ↑N k N : ℕ h : ∀ (k : ℕ), k • |ε| < ↑N ⊢ (k * N) • |ε| < ↑N
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedRing R ε : R x✝ : ∃ N, ∀ (k : ℕ), k • |ε| < ↑N k N : ℕ h : ∀ (k : ℕ), k • |ε| < ↑N ⊢ (k * N) • |ε| < ↑N TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.apply_eq
[40, 1]
[42, 33]
rw [← zero_add c, h, zero_add]
R : Type u_1 S : Sort u_2 inst✝ : AddCommMonoid R f : R → S c d : R h : (PeriodEquiv f) c d ⊢ f c = f d
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Sort u_2 inst✝ : AddCommMonoid R f : R → S c d : R h : (PeriodEquiv f) c d ⊢ f c = f d TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.zero_right
[44, 1]
[46, 39]
rw [add_zero]
R : Type u_1 S : Sort u_2 inst✝ : AddCommMonoid R f : R → S c x : R ⊢ f (x + c) = f (x + 0) ↔ f (x + c) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Sort u_2 inst✝ : AddCommMonoid R f : R → S c x : R ⊢ f (x + c) = f (x + 0) ↔ f (x + c) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.zero_right'
[48, 1]
[50, 59]
rw [add_comm]
R : Type u_1 S : Sort u_2 inst✝ : AddCommMonoid R f : R → S c x : R ⊢ f (x + c) = f x ↔ f (c + x) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Sort u_2 inst✝ : AddCommMonoid R f : R → S c x : R ⊢ f (x + c) = f x ↔ f (c + x) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.iff_sub
[52, 1]
[55, 59]
rw [← add_comm_sub, h, sub_add_cancel, add_zero]
R : Type u_1 S : Sort u_2 inst✝ : AddCommGroup R f : R → S c d : R h : (PeriodEquiv f) c d x : R ⊢ f (x + (c - d)) = f (x + 0)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Sort u_2 inst✝ : AddCommGroup R f : R → S c d : R h : (PeriodEquiv f) c d x : R ⊢ f (x + (c - d)) = f (x + 0) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.iff_sub
[52, 1]
[55, 59]
rw [← add_add_sub_cancel x c d, h, add_zero]
R : Type u_1 S : Sort u_2 inst✝ : AddCommGroup R f : R → S c d : R h : (PeriodEquiv f) (c - d) 0 x : R ⊢ f (x + c) = f (x + d)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Sort u_2 inst✝ : AddCommGroup R f : R → S c d : R h : (PeriodEquiv f) (c - d) 0 x : R ⊢ f (x + c) = f (x + d) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.equiv_zero_iff
[61, 1]
[67, 31]
rw [zero_right, QuasiPeriodic.iff_right hf]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R ⊢ (PeriodEquiv f) c 0 ↔ QuasiPeriodic f c ∧ f c = -1
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R ⊢ (∀ (x : R), f (x + c) = f x) ↔ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R ⊢ (PeriodEquiv f) c 0 ↔ QuasiPeriodic f c ∧ f c = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.equiv_zero_iff
[61, 1]
[67, 31]
refine ⟨λ h ↦ ?_, λ h x ↦ by rw [h.1, h.2, neg_neg, mul_one]⟩
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R ⊢ (∀ (x : R), f (x + c) = f x) ↔ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x ⊢ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R ⊢ (∀ (x : R), f (x + c) = f x) ↔ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.equiv_zero_iff
[61, 1]
[67, 31]
have h0 : f c = -1 := by rw [← zero_add c, h, hf.map_zero]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x ⊢ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x h0 : f c = -1 ⊢ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x ⊢ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.equiv_zero_iff
[61, 1]
[67, 31]
refine ⟨λ x ↦ ?_, h0⟩
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x h0 : f c = -1 ⊢ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x h0 : f c = -1 x : R ⊢ f (x + c) = f x * -f c
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x h0 : f c = -1 ⊢ (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.equiv_zero_iff
[61, 1]
[67, 31]
rw [h0, neg_neg, mul_one, h]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x h0 : f c = -1 x : R ⊢ f (x + c) = f x * -f c
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x h0 : f c = -1 x : R ⊢ f (x + c) = f x * -f c TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.equiv_zero_iff
[61, 1]
[67, 31]
rw [h.1, h.2, neg_neg, mul_one]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1 x : R ⊢ f (x + c) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (∀ (x : R), f (x + c) = f x * -f c) ∧ f c = -1 x : R ⊢ f (x + c) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.equiv_zero_iff
[61, 1]
[67, 31]
rw [← zero_add c, h, hf.map_zero]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x ⊢ f c = -1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : ∀ (x : R), f (x + c) = f x ⊢ f c = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
have h0 (d) : QuasiPeriodic f (d * c) := ((equiv_zero_iff hf).mp h).1.mul_left hf d
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
have h1 := zero_right'.mp h
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
have h2 (d x) : f (d * c) = -1 ∨ f (d * x + 1) = 0 := by rw [eq_neg_iff_add_eq_zero, ← mul_eq_zero, add_one_mul (f _), ← neg_eq_iff_add_eq_zero, ← neg_mul, ← (h0 d).imp_left hf, ← add_assoc, ← mul_add, hf.is_good, hf.is_good, h1, add_left_comm, h1]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
refine λ d ↦ (equiv_zero_iff hf).mpr ⟨h0 d, (h2 d (-d)).elim id λ h3 ↦ ?_⟩
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 ⊢ f (d * c) = -1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 ⊢ ∀ (d : R), (PeriodEquiv f) (d * c) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
refine (h2 (d - 1) 1).elim (λ h4 ↦ ?_) (λ h4 ↦ ?_)
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 ⊢ f (d * c) = -1
case refine_1 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f ((d - 1) * c) = -1 ⊢ f (d * c) = -1 case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f ((d - 1) * 1 + 1) = 0 ⊢ f (d * c) = -1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 ⊢ f (d * c) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
rw [eq_neg_iff_add_eq_zero, ← mul_eq_zero, add_one_mul (f _), ← neg_eq_iff_add_eq_zero, ← neg_mul, ← (h0 d).imp_left hf, ← add_assoc, ← mul_add, hf.is_good, hf.is_good, h1, add_left_comm, h1]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x d x : R ⊢ f (d * c) = -1 ∨ f (d * x + 1) = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x d x : R ⊢ f (d * c) = -1 ∨ f (d * x + 1) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
rwa [← h1, ← one_add_mul _ c, add_sub_cancel] at h4
case refine_1 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f ((d - 1) * c) = -1 ⊢ f (d * c) = -1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f ((d - 1) * c) = -1 ⊢ f (d * c) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
rw [mul_one, sub_add_cancel] at h4
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f ((d - 1) * 1 + 1) = 0 ⊢ f (d * c) = -1
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f d = 0 ⊢ f (d * c) = -1
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f ((d - 1) * 1 + 1) = 0 ⊢ f (d * c) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
rw [hf.is_good, h4, zero_mul, zero_add, add_neg_self, hf.map_zero] at h3
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f d = 0 ⊢ f (d * c) = -1
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : -1 = 0 h4 : f d = 0 ⊢ f (d * c) = -1
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : f (d * -d + 1) = 0 h4 : f d = 0 ⊢ f (d * c) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv_zero
[69, 1]
[85, 58]
rw [h3, ← neg_neg (f _), ← neg_one_mul, h3, zero_mul]
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : -1 = 0 h4 : f d = 0 ⊢ f (d * c) = -1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (d * c) h1 : ∀ (x : R), f (c + x) = f x h2 : ∀ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 d : R h3 : -1 = 0 h4 : f d = 0 ⊢ f (d * c) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
have h0 (d) : QuasiPeriodic f (c * d) := ((equiv_zero_iff hf).mp h).1.mul_right hf d
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
have h1 := zero_right.mp h
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
have h2 (d x) : f (c * d) = -1 ∨ f (x * d + 1) = 0 := by rw [eq_neg_iff_add_eq_zero, or_comm, ← mul_eq_zero, mul_add_one (f _), ← neg_eq_iff_add_eq_zero, ← mul_neg, ← (h0 d).imp_right hf, add_right_comm, ← add_mul, hf.is_good, hf.is_good, h1, add_right_comm, h1]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
refine λ d ↦ (equiv_zero_iff hf).mpr ⟨h0 d, (h2 d (-d)).elim id λ h3 ↦ ?_⟩
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 ⊢ f (c * d) = -1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 ⊢ ∀ (d : R), (PeriodEquiv f) (c * d) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
refine (h2 (d - 1) 1).elim (λ h4 ↦ ?_) (λ h4 ↦ ?_)
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 ⊢ f (c * d) = -1
case refine_1 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f (c * (d - 1)) = -1 ⊢ f (c * d) = -1 case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f (1 * (d - 1) + 1) = 0 ⊢ f (c * d) = -1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 ⊢ f (c * d) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
rw [eq_neg_iff_add_eq_zero, or_comm, ← mul_eq_zero, mul_add_one (f _), ← neg_eq_iff_add_eq_zero, ← mul_neg, ← (h0 d).imp_right hf, add_right_comm, ← add_mul, hf.is_good, hf.is_good, h1, add_right_comm, h1]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x d x : R ⊢ f (c * d) = -1 ∨ f (x * d + 1) = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x d x : R ⊢ f (c * d) = -1 ∨ f (x * d + 1) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
rwa [← h1, ← mul_add_one c, sub_add_cancel] at h4
case refine_1 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f (c * (d - 1)) = -1 ⊢ f (c * d) = -1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f (c * (d - 1)) = -1 ⊢ f (c * d) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
rw [one_mul, sub_add_cancel] at h4
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f (1 * (d - 1) + 1) = 0 ⊢ f (c * d) = -1
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f d = 0 ⊢ f (c * d) = -1
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f (1 * (d - 1) + 1) = 0 ⊢ f (c * d) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
rw [hf.is_good, h4, mul_zero, zero_add, neg_add_self, hf.map_zero] at h3
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f d = 0 ⊢ f (c * d) = -1
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : -1 = 0 h4 : f d = 0 ⊢ f (c * d) = -1
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : f (-d * d + 1) = 0 h4 : f d = 0 ⊢ f (c * d) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv_zero
[87, 1]
[103, 58]
rw [h3, ← neg_neg (f _), ← neg_one_mul, h3, zero_mul]
case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : -1 = 0 h4 : f d = 0 ⊢ f (c * d) = -1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : ∀ (d : R), QuasiPeriodic f (c * d) h1 : ∀ (x : R), f (x + c) = f x h2 : ∀ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 d : R h3 : -1 = 0 h4 : f d = 0 ⊢ f (c * d) = -1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv
[105, 1]
[108, 48]
rw [iff_sub, ← mul_sub]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) (a * c) (a * d)
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) (a * (c - d)) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) (a * c) (a * d) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_left_equiv
[105, 1]
[108, 48]
exact mul_left_equiv_zero hf (iff_sub.mp h) a
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) (a * (c - d)) 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) (a * (c - d)) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv
[110, 1]
[113, 49]
rw [iff_sub, ← sub_mul]
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) (c * a) (d * a)
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) ((c - d) * a) 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) (c * a) (d * a) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5General/A5Periodic.lean
IMOSL.IMO2012A5.PeriodEquiv.mul_right_equiv
[110, 1]
[113, 49]
exact mul_right_equiv_zero hf (iff_sub.mp h) a
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) ((c - d) * a) 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f c d : R h : (PeriodEquiv f) c d a : R ⊢ (PeriodEquiv f) ((c - d) * a) 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.map_even_of_map_one
[31, 1]
[34, 54]
specialize hf (x + 1) (-1)
R : Type u_1 S : Type u_2 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : good f h : f (-1) = 0 x : R ⊢ f (-x) = f x
R : Type u_1 S : Type u_2 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S h : f (-1) = 0 x : R hf : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1) ⊢ f (-x) = f x
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : good f h : f (-1) = 0 x : R ⊢ f (-x) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.map_even_of_map_one
[31, 1]
[34, 54]
rwa [h, mul_zero, zero_add, add_neg_cancel_right, mul_neg_one, neg_add, neg_add_cancel_right] at hf
R : Type u_1 S : Type u_2 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S h : f (-1) = 0 x : R hf : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1) ⊢ f (-x) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S h : f (-1) = 0 x : R hf : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1) ⊢ f (-x) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq1
[39, 1]
[40, 84]
rw [← h y, sub_eq_add_neg x, ← hf.is_good, mul_neg, neg_add_eq_sub, ← neg_sub, h]
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x y : R ⊢ f (x * y - 1) = f x * f y + f (x - y)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x y : R ⊢ f (x * y - 1) = f x * f y + f (x - y) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq2
[43, 1]
[46, 59]
have h0 := hf.is_good (x - 1) (1 + 1)
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1)
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f ((x - 1) * (1 + 1) + 1) = f (x - 1) * f (1 + 1) + f (x - 1 + (1 + 1)) ⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq2
[43, 1]
[46, 59]
rwa [sub_add_add_cancel, one_add_one_eq_two, mul_two, add_assoc, sub_add_cancel, ← add_sub_right_comm, ← mul_two] at h0
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f ((x - 1) * (1 + 1) + 1) = f (x - 1) * f (1 + 1) + f (x - 1 + (1 + 1)) ⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f ((x - 1) * (1 + 1) + 1) = f (x - 1) * f (1 + 1) + f (x - 1 + (1 + 1)) ⊢ f (x * 2 - 1) = f (x - 1) * f 2 + f (x + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq3
[49, 1]
[51, 82]
have h0 := Eq2 hf (-x)
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (-x * 2 - 1) = f (-x - 1) * f 2 + f (-x + 1) ⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq3
[49, 1]
[51, 82]
rwa [neg_mul, ← neg_add', h, ← neg_add', h, neg_add_eq_sub, ← neg_sub, h] at h0
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (-x * 2 - 1) = f (-x - 1) * f 2 + f (-x + 1) ⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (-x * 2 - 1) = f (-x - 1) * f 2 + f (-x + 1) ⊢ f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq5
[54, 1]
[64, 80]
rw [two_mul, ← add_assoc, add_left_comm, this, sub_add_cancel_right, h]
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 this : ∀ (y : R), f (x + y + 1) = f (x - y) y : R ⊢ f (y + (2 * x + 1)) = f y
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 this : ∀ (y : R), f (x + y + 1) = f (x - y) y : R ⊢ f (y + (2 * x + 1)) = f y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq5
[54, 1]
[64, 80]
have h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) := by rw [add_one_mul x, mul_add, add_one_mul x]
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R ⊢ f (x + y + 1) = f (x - y)
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) ⊢ f (x + y + 1) = f (x - y)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R ⊢ f (x + y + 1) = f (x - y) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq5
[54, 1]
[64, 80]
have h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1 := by rw [mul_add_one _ y, add_sub_assoc, add_sub_cancel_right, add_comm]
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) ⊢ f (x + y + 1) = f (x - y)
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1 ⊢ f (x + y + 1) = f (x - y)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) ⊢ f (x + y + 1) = f (x - y) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq5
[54, 1]
[64, 80]
rwa [hf.is_good, h3, Eq1 hf h, hf.is_good, ← add_rotate, ← mul_add_one x, hf.is_good, h0, h1, zero_mul, zero_add, zero_mul, zero_add, zero_add, zero_mul, zero_add, add_sub_add_right_eq_sub, ← add_assoc, eq_comm] at h2
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1 ⊢ f (x + y + 1) = f (x - y)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) h3 : x + (x + 1) * y = (x + 1) * (y + 1) - 1 ⊢ f (x + y + 1) = f (x - y) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq5
[54, 1]
[64, 80]
rw [add_one_mul x, mul_add, add_one_mul x]
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R ⊢ f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R ⊢ f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.Eq5
[54, 1]
[64, 80]
rw [mul_add_one _ y, add_sub_assoc, add_sub_cancel_right, add_comm]
R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) ⊢ x + (x + 1) * y = (x + 1) * (y + 1) - 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝¹ : NonAssocRing R inst✝ : NonAssocSemiring S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x = 0 h1 : f (x + 1) = 0 y : R h2 : f (x * ((x + 1) * y) + 1) = f ((x + 1) * (x * y) + 1) ⊢ x + (x + 1) * y = (x + 1) * (y + 1) - 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq4
[75, 1]
[83, 58]
have h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1 := by rw [mul_add_one (x - 1), add_assoc, sub_add_cancel, sub_one_mul, ← add_sub_right_comm, add_comm, add_sub_add_right_eq_sub]
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1 ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq4
[75, 1]
[83, 58]
apply congrArg f at h0
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1 ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (x * (x + 1) - 1) = f ((x - 1) * (x + 1 + 1) + 1) ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1 ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq4
[75, 1]
[83, 58]
rw [Eq1 hf h, hf.is_good, sub_add_cancel_left, h, hf.map_one, sub_add_add_cancel, add_zero, add_assoc, one_add_one_eq_two, ← add_assoc, ← mul_two] at h0
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (x * (x + 1) - 1) = f ((x - 1) * (x + 1 + 1) + 1) ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x * f (x + 1) = f (x - 1) * f (x + 2) + f (x * 2 + 1) ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (x * (x + 1) - 1) = f ((x - 1) * (x + 1 + 1) + 1) ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq4
[75, 1]
[83, 58]
rw [Eq2 hf, mul_add, h0, ← add_assoc, add_one_mul (f _), add_left_inj, mul_left_comm, ← mul_add, ← hf.is_good]
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x * f (x + 1) = f (x - 1) * f (x + 2) + f (x * 2 + 1) ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f x * f (x + 1) = f (x - 1) * f (x + 2) + f (x * 2 + 1) ⊢ f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq4
[75, 1]
[83, 58]
rw [mul_add_one (x - 1), add_assoc, sub_add_cancel, sub_one_mul, ← add_sub_right_comm, add_comm, add_sub_add_right_eq_sub]
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ x * (x + 1) - 1 = (x - 1) * (x + 1 + 1) + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq4_alt
[86, 1]
[88, 85]
have h0 := Eq4 hf h (-x)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (-x) * f (-x * 2 - 1) = (f (-x - 1) + 1) * f (-x * 2 + 1) ⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R ⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq4_alt
[86, 1]
[88, 85]
rwa [h, neg_mul, ← neg_add', h, ← neg_add', h, neg_add_eq_sub, ← neg_sub, h] at h0
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (-x) * f (-x * 2 - 1) = (f (-x - 1) + 1) * f (-x * 2 + 1) ⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x x : R h0 : f (-x) * f (-x * 2 - 1) = (f (-x - 1) + 1) * f (-x * 2 + 1) ⊢ f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
rw [hf.is_good, h0, mul_neg_one, neg_add_eq_sub]
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 x : R ⊢ f (x * 2 + 1) = f (x + 2) - f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 x : R ⊢ f (x * 2 + 1) = f (x + 2) - f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
rw [Eq2 hf, Eq3 hf h, h0, mul_neg_one, mul_neg_one, neg_add_rev, neg_neg]
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x x : R ⊢ f (x * 2 + 1) = -f (x * 2 - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x x : R ⊢ f (x * 2 + 1) = -f (x * 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
have h3 := Eq4_alt hf h x
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R h3 : f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
rw [h2, mul_neg, neg_eq_iff_add_eq_zero, ← add_mul, mul_eq_zero, ← add_assoc] at h3
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R h3 : f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R h3 : f x + f (x + 1) + 1 = 0 ∨ f (x * 2 - 1) = 0 ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R h3 : f x * f (x * 2 + 1) = (f (x + 1) + 1) * f (x * 2 - 1) ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
exact h3.imp_left eq_neg_of_add_eq_zero_left
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R h3 : f x + f (x + 1) + 1 = 0 ∨ f (x * 2 - 1) = 0 ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) x : R h3 : f x + f (x + 1) + 1 = 0 ∨ f (x * 2 - 1) = 0 ⊢ f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
rwa [mul_two, add_sub_assoc, add_sub_cancel_right, add_right_comm, ← mul_two, h1, sub_eq_zero] at h4
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 x : R h4 : f ((x + 1) * 2 - 1) = 0 ⊢ f (x + 2) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 x : R h4 : f ((x + 1) * 2 - 1) = 0 ⊢ f (x + 2) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
rwa [← neg_eq_zero, ← h2, h1, sub_eq_zero] at h5
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 x : R h4 : f (x + 1) + f (x + 1 + 1) = -1 h5 : f (x * 2 - 1) = 0 ⊢ f (x + 2) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 x : R h4 : f (x + 1) + f (x + 1 + 1) = -1 h5 : f (x * 2 - 1) = 0 ⊢ f (x + 2) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.two_periodic_of_map_two
[91, 1]
[107, 90]
rwa [← h5, add_comm, add_left_inj, add_assoc, one_add_one_eq_two] at h4
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 x : R h4 : f (x + 1) + f (x + 1 + 1) = -1 h5 : f x + f (x + 1) = -1 ⊢ f (x + 2) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 = -1 h1 : ∀ (x : R), f (x * 2 + 1) = f (x + 2) - f x h2 : ∀ (x : R), f (x * 2 + 1) = -f (x * 2 - 1) h3 : ∀ (x : R), f x + f (x + 1) = -1 ∨ f (x * 2 - 1) = 0 x : R h4 : f (x + 1) + f (x + 1 + 1) = -1 h5 : f x + f (x + 1) = -1 ⊢ f (x + 2) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6_ring_id
[109, 1]
[111, 57]
ring
R : Type ?u.45782 S✝ : Type ?u.45785 inst✝³ : Ring R inst✝² : CommRing S✝ inst✝¹ : NoZeroDivisors S✝ f : R → S✝ hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x S : Type u_1 inst✝ : CommRing S a b c d : S ⊢ a * (c * d + b) - a * (b * d + c) - ((c + 1) * (b * d + c) - (b + 1) * (c * d + b)) = (b + c - (a + 1) * (d - 1)) * (b - c)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type ?u.45782 S✝ : Type ?u.45785 inst✝³ : Ring R inst✝² : CommRing S✝ inst✝¹ : NoZeroDivisors S✝ f : R → S✝ hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x S : Type u_1 inst✝ : CommRing S a b c d : S ⊢ a * (c * d + b) - a * (b * d + c) - ((c + 1) * (b * d + c) - (b + 1) * (c * d + b)) = (b + c - (a + 1) * (d - 1)) * (b - c) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
intro x
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) ⊢ ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) x : R ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) ⊢ ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
refine (h1 x).elim id λ h2 ↦ ?_
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) x : R ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) x : R h2 : f (x + 1) = f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) x : R ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
specialize h1 (x + 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) x : R h2 : f (x + 1) = f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 1 + 1) = f (x + 1 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : ∀ (x : R), f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) x : R h2 : f (x + 1) = f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [add_sub_cancel_right, add_assoc, one_add_one_eq_two] at h1
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 1 + 1) = f (x + 1 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 2) = f x ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 1 + 1) + f (x + 1 - 1) = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 1 + 1) = f (x + 1 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rcases h1 with h1 | h1
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 2) = f x ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ∨ f (x + 2) = f x ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [Eq2 hf, Eq3 hf h, ← sub_eq_zero, Eq6_ring_id, mul_eq_zero] at h1
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h1 : f x * f (x * 2 - 1) - f x * f (x * 2 + 1) = (f (x - 1) + 1) * f (x * 2 + 1) - (f (x + 1) + 1) * f (x * 2 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h1 : f (x + 1) + f (x - 1) - (f x + 1) * (f 2 - 1) = 0 ∨ f (x + 1) - f (x - 1) = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h1 : f x * f (x * 2 - 1) - f x * f (x * 2 + 1) = (f (x - 1) + 1) * f (x * 2 + 1) - (f (x + 1) + 1) * f (x * 2 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
exact h1.imp eq_of_sub_eq_zero eq_of_sub_eq_zero
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h1 : f (x + 1) + f (x - 1) - (f x + 1) * (f 2 - 1) = 0 ∨ f (x + 1) - f (x - 1) = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h1 : f (x + 1) + f (x - 1) - (f x + 1) * (f 2 - 1) = 0 ∨ f (x + 1) - f (x - 1) = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) ∨ f (x + 1) = f (x - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
have h2 := Eq3 hf h x
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2✝ : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) h2 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [hf.is_good, eq_sub_of_add_eq h1, add_sub_left_comm, ← mul_sub_one, add_one_mul (f _), add_assoc, ← one_add_mul (f x), mul_sub_one, ← add_sub_right_comm, add_sub_assoc, add_right_inj, add_comm] at h2
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2✝ : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) h2 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2✝ : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) h2 : (f x + 1) * (f 2 - 1) - f (x + 1) = f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2✝ : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) h2 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
exact (eq_add_of_sub_eq' h2).symm
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2✝ : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) h2 : (f x + 1) * (f 2 - 1) - f (x + 1) = f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2✝ : f (x + 1) = f (x - 1) h1 : f (x + 2) + f x = (f (x + 1) + 1) * (f 2 - 1) h2 : (f x + 1) * (f 2 - 1) - f (x + 1) = f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
have h0 : f 2 + 1 ≠ 0 := λ X ↦ h0 (eq_neg_of_add_eq_zero_left X)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
have h3 := Eq3 hf h x
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [← h2, hf.is_good, h1, ← mul_add_one (f x), ← mul_add_one (f _), ← sub_eq_zero, ← sub_mul, mul_eq_zero, or_iff_left h0, sub_eq_zero] at h3
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x * 2 + 1) = f (x + 1) * f 2 + f (x - 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
have h4 := Eq4 hf h x
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) h4 : f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [Eq3 hf h, Eq2 hf, ← h2, ← h3, ← sub_eq_zero, ← sub_mul, sub_add_cancel_left, neg_one_mul, neg_eq_zero, ← mul_add_one (f x), mul_eq_zero, or_iff_left h0] at h4
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) h4 : f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) h4 : f x * f (x * 2 - 1) = (f (x - 1) + 1) * f (x * 2 + 1) ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [eq_comm, h4] at h3
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f x = f (x + 1) h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [eq_comm, h3] at h2
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x + 1) = f (x - 1) h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
have h5 := Eq5 hf h h4 h3 0
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 h5 : f (0 + (2 * x + 1)) = f 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [zero_add, hf.is_good, add_comm 2 x, h1, h4, mul_zero, add_zero, hf.map_zero, eq_comm, neg_eq_zero] at h5
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 h5 : f (0 + (2 * x + 1)) = f 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 h5 : 1 = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 h5 : f (0 + (2 * x + 1)) = f 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq6
[114, 1]
[143, 56]
rw [← sub_eq_zero, ← one_mul (_ - _), h5, zero_mul]
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 h5 : 1 = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0✝ : f 2 ≠ -1 x : R h2 : f (x - 1) = 0 h1 : f (x + 2) = f x h0 : f 2 + 1 ≠ 0 h3 : f (x + 1) = 0 h4 : f x = 0 h5 : 1 = 0 ⊢ f (x + 1) + f (x - 1) = (f x + 1) * (f 2 - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
rw [← mul_assoc, ← sq, mul_left_comm, Eq4 hf h, ← mul_assoc, ← sub_eq_zero, ← sub_mul, mul_eq_zero] at h2
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x * (f x * f (x * 2 + 1)) = f x * ((f (x + 1) + 1) * f (x * 2 - 1)) ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ∨ f (x * 2 + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x * (f x * f (x * 2 + 1)) = f x * ((f (x + 1) + 1) * f (x * 2 - 1)) ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
rcases h2 with h2 | h2
R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ∨ f (x * 2 + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ∨ f (x * 2 + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
exact (eq_of_sub_eq_zero h2).symm
case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f x ^ 2 - (f (x + 1) + 1) * (f (x - 1) + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
rw [← mul_assoc, ← sq, mul_left_comm, Eq4_alt hf h, ← mul_assoc, ← sub_eq_zero, ← sub_mul, mul_eq_zero] at h3
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x * (f x * f (x * 2 - 1)) = f x * ((f (x - 1) + 1) * f (x * 2 + 1)) ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ∨ f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x * (f x * f (x * 2 - 1)) = f x * ((f (x - 1) + 1) * f (x * 2 + 1)) ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
rcases h3 with h3 | h3
case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ∨ f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr.inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ∨ f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
rwa [sub_eq_zero, eq_comm, mul_comm] at h3
case inr.inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case inr.inl R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f x ^ 2 - (f (x - 1) + 1) * (f (x + 1) + 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
rw [Eq3 hf h] at h2
case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x + 1) * f 2 + f (x - 1) = 0 h3 : f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x * 2 + 1) = 0 h3 : f (x * 2 - 1) = 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean
IMOSL.IMO2012A5.Case2.CommCase.Eq7
[146, 1]
[169, 76]
rw [Eq2 hf, add_add_add_comm, add_zero, ← add_mul, add_comm (f _), ← mul_add_one (α := S), mul_eq_zero, or_iff_left (h0 ∘ eq_neg_of_add_eq_zero_left), ← eq_neg_iff_add_eq_zero] at h3
case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x + 1) * f 2 + f (x - 1) = 0 h3 : f (x + 1) * f 2 + f (x - 1) + f (x * 2 - 1) = 0 + 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x + 1) * f 2 + f (x - 1) = 0 h3 : f (x - 1) = -f (x + 1) ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr R : Type u_2 S : Type u_1 inst✝² : Ring R inst✝¹ : CommRing S inst✝ : NoZeroDivisors S f : R → S hf : NontrivialGood f h : ∀ (x : R), f (-x) = f x h0 : f 2 ≠ -1 h1 : f 2 ≠ 1 x : R h2 : f (x + 1) * f 2 + f (x - 1) = 0 h3 : f (x + 1) * f 2 + f (x - 1) + f (x * 2 - 1) = 0 + 0 ⊢ (f (x + 1) + 1) * (f (x - 1) + 1) = f x ^ 2 TACTIC: