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pkuAI4M
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https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.c_eq_b_sub_one_mul_map_zero
[105, 1]
[108, 64]
have h3 := h 0 0
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 ⊒ c = (b - 1) * f 0
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h3 : f (0 + f 0) - f 0 = f (b * 0) - f 0 + c ⊒ c = (b - 1) * f 0
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 ⊒ c = (b - 1) * f 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.c_eq_b_sub_one_mul_map_zero
[105, 1]
[108, 64]
rwa [zero_add, mul_zero, sub_self, zero_add, map_is_linear h h0 h1, add_sub_cancel_right, eq_comm] at h3
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h3 : f (0 + f 0) - f 0 = f (b * 0) - f 0 + c ⊒ c = (b - 1) * f 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h3 : f (0 + f 0) - f 0 = f (b * 0) - f 0 + c ⊒ c = (b - 1) * f 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.final_solution_case2
[125, 1]
[134, 35]
rcases h1 with ⟨k, rfl⟩
b c : β„€ h : 1 < b.natAbs h0 : c β‰  0 f : β„€ β†’ β„€ h1 : b - 1 ∣ c h2 : good b c f ⊒ f = fun x => (b - 1) * x + c / (b - 1)
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1)
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ h : 1 < b.natAbs h0 : c β‰  0 f : β„€ β†’ β„€ h1 : b - 1 ∣ c h2 : good b c f ⊒ f = fun x => (b - 1) * x + c / (b - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.final_solution_case2
[125, 1]
[134, 35]
have h1 := c_eq_b_sub_one_mul_map_zero h2 h h0
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1)
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1)
Please generate a tactic in lean4 to solve the state. STATE: case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.final_solution_case2
[125, 1]
[134, 35]
have h3 := (mul_ne_zero_iff.mp h0).1
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1)
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1)
Please generate a tactic in lean4 to solve the state. STATE: case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.final_solution_case2
[125, 1]
[134, 35]
rw [Int.mul_ediv_cancel_left _ h3]
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1)
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + k
Please generate a tactic in lean4 to solve the state. STATE: case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + (b - 1) * k / (b - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.final_solution_case2
[125, 1]
[134, 35]
rw [mul_eq_mul_left_iff, or_iff_left h3] at h1
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + k
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : k = f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + k
Please generate a tactic in lean4 to solve the state. STATE: case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : (b - 1) * k = (b - 1) * f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.final_solution_case2
[125, 1]
[134, 35]
rw [h1]
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : k = f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + k
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : k = f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + f 0
Please generate a tactic in lean4 to solve the state. STATE: case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : k = f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.final_solution_case2
[125, 1]
[134, 35]
exact funext (map_is_linear h2 h h0)
case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : k = f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + f 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro b : β„€ h : 1 < b.natAbs f : β„€ β†’ β„€ k : β„€ h0 : (b - 1) * k β‰  0 h2 : good b ((b - 1) * k) f h1 : k = f 0 h3 : b - 1 β‰  0 ⊒ f = fun x => (b - 1) * x + f 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond1
[25, 1]
[32, 62]
rcases (x a).eq_or_eq_not (x b) with h6 | h6
n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2
case inl n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2 case inr n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2
Please generate a tactic in lean4 to solve the state. STATE: n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond1
[25, 1]
[32, 62]
rcases (x c).eq_or_eq_not (x b) with h7 | h7
case inr n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2
case inr.inl n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b h7 : x c = x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2 case inr.inr n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b h7 : x c = !x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case inr n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond1
[25, 1]
[32, 62]
exact ⟨a, b, h, h0, h1.le.trans h2, h6, k, h3⟩
case inl n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond1
[25, 1]
[32, 62]
exact ⟨b, c, h.trans h0.le, h1, h2, h7.symm, m, h5⟩
case inr.inl n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b h7 : x c = x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr.inl n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b h7 : x c = x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond1
[25, 1]
[32, 62]
exact ⟨a, c, h, h0.trans h1, h2, h6.trans h7.symm, l, h4⟩
case inr.inr n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b h7 : x c = !x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr n a b c k l m : β„• h : n ≀ a h0 : a < b h1 : b < c h2 : c ≀ 2 * n h3 : a + b = k ^ 2 h4 : a + c = l ^ 2 h5 : b + c = m ^ 2 x : β„• β†’ Bool h6 : x a = !x b h7 : x c = !x b ⊒ βˆƒ a b, n ≀ a ∧ a < b ∧ b ≀ 2 * n ∧ x a = x b ∧ βˆƒ k, a + b = k ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
refine good_cond1 (b := 2 * k ^ 2 + 8 * k + 9) (k := 2 * k + 3) (l := 2 * k + 4) (m := 2 * k + 5) h ?_ ?_ (Nat.mul_le_mul_left 2 h0) ?_ ?_ ?_
n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ good n
case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k < 2 * k ^ 2 + 8 * k + 9 case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 8 * k + 9 < 2 * (k ^ 2 + 6 * k + 8) case refine_3 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k + (2 * k ^ 2 + 8 * k + 9) = (2 * k + 3) ^ 2 case refine_4 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 4) ^ 2 case refine_5 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 8 * k + 9 + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 5) ^ 2
Please generate a tactic in lean4 to solve the state. STATE: n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ good n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rw [Nat.add_assoc, Nat.add_lt_add_iff_left, Nat.lt_succ_iff]
case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k < 2 * k ^ 2 + 8 * k + 9
case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 4 * k ≀ 8 * k + 8
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k < 2 * k ^ 2 + 8 * k + 9 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
refine (Nat.mul_le_mul_right k ?_).trans (Nat.le_add_right _ _)
case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 4 * k ≀ 8 * k + 8
case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 4 ≀ 8
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 4 * k ≀ 8 * k + 8 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
exact Nat.le_mul_of_pos_left 4 (Nat.succ_pos 1)
case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 4 ≀ 8
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 4 ≀ 8 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rw [Nat.mul_add, Nat.mul_add, ← Nat.mul_assoc, Nat.lt_succ_iff, Nat.add_assoc, Nat.add_assoc, Nat.add_le_add_iff_left, Nat.add_le_add_iff_right]
case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 8 * k + 9 < 2 * (k ^ 2 + 6 * k + 8)
case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 8 * k ≀ 2 * 6 * k + 6
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 8 * k + 9 < 2 * (k ^ 2 + 6 * k + 8) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
refine (Nat.mul_le_mul_right k ?_).trans (Nat.le_add_right _ _)
case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 8 * k ≀ 2 * 6 * k + 6
case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 8 ≀ 2 * 6
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 8 * k ≀ 2 * 6 * k + 6 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
exact Nat.mul_le_mul_left 4 (Nat.le_succ 2)
case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 8 ≀ 2 * 6
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 8 ≀ 2 * 6 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rw [← Nat.add_assoc, Nat.add_add_add_comm, ← Nat.add_mul, ← Nat.add_mul, add_sq, Nat.mul_pow, Nat.mul_right_comm, ← Nat.mul_assoc]
case refine_3 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k + (2 * k ^ 2 + 8 * k + 9) = (2 * k + 3) ^ 2
case refine_3 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (4 + 8) * k + 9 = 2 ^ 2 * k ^ 2 + 2 * 3 * 2 * k + 3 ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case refine_3 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k + (2 * k ^ 2 + 8 * k + 9) = (2 * k + 3) ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rfl
case refine_3 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (4 + 8) * k + 9 = 2 ^ 2 * k ^ 2 + 2 * 3 * 2 * k + 3 ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_3 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (4 + 8) * k + 9 = 2 ^ 2 * k ^ 2 + 2 * 3 * 2 * k + 3 ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rw [Nat.mul_add, ← Nat.add_assoc, Nat.mul_add, Nat.add_add_add_comm, ← Nat.add_mul, ← Nat.mul_assoc, ← Nat.add_mul, add_sq, Nat.mul_pow, Nat.mul_right_comm, ← Nat.mul_assoc]
case refine_4 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 4) ^ 2
case refine_4 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (4 + 2 * 6) * k + 2 * 8 = 2 ^ 2 * k ^ 2 + 2 * 4 * 2 * k + 4 ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case refine_4 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 4 * k + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 4) ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rfl
case refine_4 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (4 + 2 * 6) * k + 2 * 8 = 2 ^ 2 * k ^ 2 + 2 * 4 * 2 * k + 4 ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_4 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (4 + 2 * 6) * k + 2 * 8 = 2 ^ 2 * k ^ 2 + 2 * 4 * 2 * k + 4 ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rw [Nat.mul_add, Nat.add_add_add_comm, Nat.mul_add, ← Nat.mul_assoc, Nat.add_add_add_comm (2 * k ^ 2), ← Nat.add_mul, ← Nat.add_mul, add_sq, Nat.mul_pow, Nat.mul_right_comm, ← Nat.mul_assoc]
case refine_5 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 8 * k + 9 + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 5) ^ 2
case refine_5 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (8 + 2 * 6) * k + (9 + 2 * 8) = 2 ^ 2 * k ^ 2 + 2 * 5 * 2 * k + 5 ^ 2
Please generate a tactic in lean4 to solve the state. STATE: case refine_5 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ 2 * k ^ 2 + 8 * k + 9 + 2 * (k ^ 2 + 6 * k + 8) = (2 * k + 5) ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2
[34, 1]
[62, 8]
rfl
case refine_5 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (8 + 2 * 6) * k + (9 + 2 * 8) = 2 ^ 2 * k ^ 2 + 2 * 5 * 2 * k + 5 ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_5 n k : β„• h : n ≀ 2 * k ^ 2 + 4 * k h0 : k ^ 2 + 6 * k + 8 ≀ n ⊒ (2 + 2) * k ^ 2 + (8 + 2 * 6) * k + (9 + 2 * 8) = 2 ^ 2 * k ^ 2 + 2 * 5 * 2 * k + 5 ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rcases h0 with ⟨k, h0, h1⟩
n : β„• h : 99 ≀ n h0 : βˆƒ k, n ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1
case intro.intro n : β„• h : 99 ≀ n k : β„• h0 : n ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1
Please generate a tactic in lean4 to solve the state. STATE: n : β„• h : 99 ≀ n h0 : βˆƒ k, n ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rw [le_iff_lt_or_eq] at h0
case intro.intro n : β„• h : 99 ≀ n k : β„• h0 : n ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1
case intro.intro n : β„• h : 99 ≀ n k : β„• h0 : n < 2 * k ^ 2 + 4 * k ∨ n = 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro n : β„• h : 99 ≀ n k : β„• h0 : n ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rcases h0 with h0 | rfl
case intro.intro n : β„• h : 99 ≀ n k : β„• h0 : n < 2 * k ^ 2 + 4 * k ∨ n = 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1
case intro.intro.inl n : β„• h : 99 ≀ n k : β„• h1 : k ^ 2 + 6 * k + 8 ≀ n h0 : n < 2 * k ^ 2 + 4 * k ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1 case intro.intro.inr k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ βˆƒ k_1, 2 * k ^ 2 + 4 * k + 1 ≀ 2 * k_1 ^ 2 + 4 * k_1 ∧ k_1 ^ 2 + 6 * k_1 + 8 ≀ 2 * k ^ 2 + 4 * k + 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro n : β„• h : 99 ≀ n k : β„• h0 : n < 2 * k ^ 2 + 4 * k ∨ n = 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ n ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
refine ⟨k + 1, ?_, ?_⟩
case intro.intro.inr k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ βˆƒ k_1, 2 * k ^ 2 + 4 * k + 1 ≀ 2 * k_1 ^ 2 + 4 * k_1 ∧ k_1 ^ 2 + 6 * k_1 + 8 ≀ 2 * k ^ 2 + 4 * k + 1
case intro.intro.inr.refine_1 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 1 ≀ 2 * (k + 1) ^ 2 + 4 * (k + 1) case intro.intro.inr.refine_2 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ βˆƒ k_1, 2 * k ^ 2 + 4 * k + 1 ≀ 2 * k_1 ^ 2 + 4 * k_1 ∧ k_1 ^ 2 + 6 * k_1 + 8 ≀ 2 * k ^ 2 + 4 * k + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
exact ⟨k, h0, Nat.le_succ_of_le h1⟩
case intro.intro.inl n : β„• h : 99 ≀ n k : β„• h1 : k ^ 2 + 6 * k + 8 ≀ n h0 : n < 2 * k ^ 2 + 4 * k ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inl n : β„• h : 99 ≀ n k : β„• h1 : k ^ 2 + 6 * k + 8 ≀ n h0 : n < 2 * k ^ 2 + 4 * k ⊒ βˆƒ k, n + 1 ≀ 2 * k ^ 2 + 4 * k ∧ k ^ 2 + 6 * k + 8 ≀ n + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rw [Nat.mul_succ, ← Nat.add_assoc]
case intro.intro.inr.refine_1 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 1 ≀ 2 * (k + 1) ^ 2 + 4 * (k + 1)
case intro.intro.inr.refine_1 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 1 ≀ 2 * (k + 1) ^ 2 + 4 * k + 4
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_1 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 1 ≀ 2 * (k + 1) ^ 2 + 4 * (k + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
exact Nat.add_le_add (Nat.add_le_add_right (Nat.mul_le_mul_left 2 <| Nat.pow_le_pow_of_le_left k.le_succ 2) _) (Nat.le_add_left 1 3)
case intro.intro.inr.refine_1 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 1 ≀ 2 * (k + 1) ^ 2 + 4 * k + 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_1 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 1 ≀ 2 * (k + 1) ^ 2 + 4 * k + 4 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
replace h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 := by rw [add_sq, Nat.mul_add, Nat.mul_add, Nat.mul_one, ← Nat.mul_assoc]; rfl
case intro.intro.inr.refine_2 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
case intro.intro.inr.refine_2 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_2 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rw [← Nat.add_le_add_iff_right (n := 2), h1, Nat.succ_le_iff, Nat.mul_comm, ← Nat.div_lt_iff_lt_mul (Nat.succ_pos 1), ← Nat.sqrt_lt'] at h
case intro.intro.inr.refine_2 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
case intro.intro.inr.refine_2 k : β„• h : ((99 + 1) / Nat.succ 1).sqrt < k + 1 h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_2 k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
change 8 ≀ k + 1 at h
case intro.intro.inr.refine_2 k : β„• h : ((99 + 1) / Nat.succ 1).sqrt < k + 1 h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_2 k : β„• h : ((99 + 1) / Nat.succ 1).sqrt < k + 1 h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rw [← Nat.add_le_add_iff_right (n := 1), h1, Nat.two_mul, Nat.add_assoc, Nat.add_assoc, Nat.add_le_add_iff_left, sq]
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 6 * (k + 1) + (8 + 1) ≀ (k + 1) * (k + 1)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ (k + 1) ^ 2 + 6 * (k + 1) + 8 ≀ 2 * k ^ 2 + 4 * k + 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
apply (Nat.mul_le_mul_right _ h).trans'
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 6 * (k + 1) + (8 + 1) ≀ (k + 1) * (k + 1)
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 6 * (k + 1) + (8 + 1) ≀ 8 * (k + 1)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 6 * (k + 1) + (8 + 1) ≀ (k + 1) * (k + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rw [Nat.add_mul 6 2, Nat.add_le_add_iff_left, Nat.two_mul]
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 6 * (k + 1) + (8 + 1) ≀ 8 * (k + 1)
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 8 + 1 ≀ k + 1 + (k + 1)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 6 * (k + 1) + (8 + 1) ≀ 8 * (k + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
exact Nat.add_le_add h (Nat.le_add_left 1 k)
case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 8 + 1 ≀ k + 1 + (k + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.inr.refine_2 k : β„• h1 : 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 h : 8 ≀ k + 1 ⊒ 8 + 1 ≀ k + 1 + (k + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rw [add_sq, Nat.mul_add, Nat.mul_add, Nat.mul_one, ← Nat.mul_assoc]
k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2
k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 2 = 2 * k ^ 2 + 2 * 2 * k + 2 * 1 ^ 2
Please generate a tactic in lean4 to solve the state. STATE: k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 2 = 2 * (k + 1) ^ 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2021/N2/N2.lean
IMOSL.IMO2021N2.good_cond2_k_witness
[64, 1]
[89, 51]
rfl
k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 2 = 2 * k ^ 2 + 2 * 2 * k + 2 * 1 ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : β„• h : 99 ≀ 2 * k ^ 2 + 4 * k h1 : k ^ 2 + 6 * k + 8 ≀ 2 * k ^ 2 + 4 * k ⊒ 2 * k ^ 2 + 4 * k + 2 = 2 * k ^ 2 + 2 * 2 * k + 2 * 1 ^ 2 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ S hf : NontrivialGood f c : R ⊒ (βˆ€ (x : R), f (x + c) = f x) ↔ (βˆ€ (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 β†’ 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 ↦ ?_, Ξ» 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 β†’ S hf : NontrivialGood f c : R ⊒ (βˆ€ (x : R), f (x + c) = f x) ↔ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (x : R), f (x + c) = f x ⊒ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R ⊒ (βˆ€ (x : R), f (x + c) = f x) ↔ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (x : R), f (x + c) = f x ⊒ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (x : R), f (x + c) = f x h0 : f c = -1 ⊒ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (x : R), f (x + c) = f x ⊒ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (x : R), f (x + c) = f x h0 : f c = -1 ⊒ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (x : R), f (x + c) = f x h0 : f c = -1 ⊒ (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x ⊒ βˆ€ (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 Ξ» d ↦ (equiv_zero_iff hf).mpr ⟨h0 d, (h2 d (-d)).elim id Ξ» h3 ↦ ?_⟩
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (d x : R), f (d * c) = -1 ∨ f (d * x + 1) = 0 ⊒ βˆ€ (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 (Ξ» h4 ↦ ?_) (Ξ» h4 ↦ ?_)
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (d * c) h1 : βˆ€ (x : R), f (c + x) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x ⊒ βˆ€ (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 Ξ» d ↦ (equiv_zero_iff hf).mpr ⟨h0 d, (h2 d (-d)).elim id Ξ» h3 ↦ ?_⟩
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 ⊒ βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (d x : R), f (c * d) = -1 ∨ f (x * d + 1) = 0 ⊒ βˆ€ (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 (Ξ» h4 ↦ ?_) (Ξ» h4 ↦ ?_)
R : Type u_1 S : Type u_2 inst✝² : Ring R inst✝¹ : NonAssocRing S inst✝ : NoZeroDivisors S f : R β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ S hf : NontrivialGood f c : R h : (PeriodEquiv f) c 0 h0 : βˆ€ (d : R), QuasiPeriodic f (c * d) h1 : βˆ€ (x : R), f (x + c) = f x h2 : βˆ€ (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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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 β†’ 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/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.map_zero
[27, 1]
[30, 45]
specialize h 0 0
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) ⊒ f 0 = 0
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : f (0 + 0) + f (0 - 0) = 2 β€’ (f 0 + f 0) ⊒ f 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) ⊒ f 0 = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.map_zero
[27, 1]
[30, 45]
rw [add_zero, sub_zero, two_nsmul, self_eq_add_right, ← two_nsmul] at h
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : f (0 + 0) + f (0 - 0) = 2 β€’ (f 0 + f 0) ⊒ f 0 = 0
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : 2 β€’ f 0 = 0 ⊒ f 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : f (0 + 0) + f (0 - 0) = 2 β€’ (f 0 + f 0) ⊒ f 0 = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.map_zero
[27, 1]
[30, 45]
exact hH _ _ (h.trans (nsmul_zero 2).symm)
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : 2 β€’ f 0 = 0 ⊒ f 0 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : 2 β€’ f 0 = 0 ⊒ f 0 = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.map_even
[32, 1]
[34, 84]
have h0 := h 0 x
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x : G ⊒ f (-x) = f x
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x : G h0 : f (0 + x) + f (0 - x) = 2 β€’ (f 0 + f x) ⊒ f (-x) = f x
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x : G ⊒ f (-x) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.map_even
[32, 1]
[34, 84]
rwa [zero_add, zero_sub, map_zero hH h, zero_add, two_nsmul, add_right_inj] at h0
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x : G h0 : f (0 + x) + f (0 - x) = 2 β€’ (f 0 + f x) ⊒ f (-x) = f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x : G h0 : f (0 + x) + f (0 - x) = 2 β€’ (f 0 + f x) ⊒ f (-x) = f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.map_triple
[36, 1]
[40, 77]
rw [nsmul_add, ← h, ← h, add_add_add_comm, add_assoc x, add_sub_cancel_left, add_comm (f (y + z)), add_add_add_comm, ← two_nsmul, nsmul_add, add_left_inj, ← h, add_sub_add_left_eq_sub, add_left_inj, add_right_comm, add_add_add_comm]
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x y z : G ⊒ 2 β€’ (f (x + y + z) + f x + (f y + f z)) = 2 β€’ (f (x + y) + f (x + z) + f (y + z))
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCancelCommMonoid H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x y z : G ⊒ 2 β€’ (f (x + y + z) + f x + (f y + f z)) = 2 β€’ (f (x + y) + f (x + z) + f (y + z)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.BilinMap_eq_two_nsmul
[75, 1]
[77, 58]
rw [two_nsmul, ← neg_inj, ← map_neg, BilinMap_def, add_neg_self, map_zero hH h, map_even hH h, zero_sub]
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCommGroup H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x : G ⊒ ((BilinMap hH h) x) x = 2 β€’ f x
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCommGroup H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x : G ⊒ ((BilinMap hH h) x) x = 2 β€’ f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/SquareLike.lean
IMOSL.Extra.SquareLike.two_nsmul_BilinMap_eq
[79, 1]
[81, 72]
rw [BilinMap_def, nsmul_sub, ← h, two_nsmul, add_sub_add_left_eq_sub]
G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCommGroup H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x y : G ⊒ 2 β€’ ((BilinMap hH h) x) y = f (x + y) - f (x - y)
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_2 H : Type u_1 inst✝¹ : AddCommGroup G inst✝ : AddCommGroup H hH : βˆ€ (x y : H), 2 β€’ x = 2 β€’ y β†’ x = y f : G β†’ H h : βˆ€ (x y : G), f (x + y) + f (x - y) = 2 β€’ (f x + f y) x y : G ⊒ 2 β€’ ((BilinMap hH h) x) y = f (x + y) - f (x - y) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
have h0 : βˆ€ t x : R, f (f t) - f (f x) ≀ (f t - x) * f x := Ξ» t x ↦ by rw [sub_le_iff_le_add] apply (h _ _).trans_eq' rw [add_sub_cancel]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h0 : βˆ€ t x : R, 0 ≀ (f t - x) * f x + (f x - t) * f t := Ξ» t x ↦ by rw [← sub_self (f (f t)), ← sub_add_sub_cancel _ (f (f x))] exact add_le_add (h0 t x) (h0 x t)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h0 : βˆ€ x : R, x * f x ≀ 0 := Ξ» x ↦ by have h1 := h0 x (f x + f x) rwa [sub_add_cancel_left, sub_mul, neg_mul, mul_comm, ← add_sub_assoc, neg_add_self, zero_sub, neg_nonneg] at h1
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), 0 ≀ (f t - x) * f x + (f x - t) * f t ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
have h1 : βˆ€ x : R, f x ≀ f (f x) := Ξ» x ↦ by have h1 := h x 0 rwa [add_zero, zero_mul, zero_add] at h1
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h1 : βˆ€ x : R, f x ≀ 0 := Ξ» x ↦ le_of_not_lt Ξ» h2 ↦ (h0 (f x)).not_lt <| mul_pos h2 (h2.trans_le (h1 x))
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ f (f x) ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
replace h0 : βˆ€ x : R, x < 0 β†’ f x = 0 := Ξ» x h2 ↦ (h1 x).antisymm (nonneg_of_mul_nonpos_right (h0 x) h2)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (x : R), x * f x ≀ 0 h1 : βˆ€ (x : R), f x ≀ 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
intros x h2
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2 : x ≀ 0 ⊒ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 ⊒ βˆ€ (x : R), x ≀ 0 β†’ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rcases h2.lt_or_eq with h2 | rfl
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2 : x ≀ 0 ⊒ f x = 0
case inl R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2✝ : x ≀ 0 h2 : x < 0 ⊒ f x = 0 case inr R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 h2 : 0 ≀ 0 ⊒ f 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h1 : βˆ€ (x : R), f x ≀ 0 h0 : βˆ€ (x : R), x < 0 β†’ f x = 0 x : R h2 : x ≀ 0 ⊒ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rw [sub_le_iff_le_add]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) - f (f x) ≀ (f t - x) * f x
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) ≀ (f t - x) * f x + f (f x)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) - f (f x) ≀ (f t - x) * f x TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
apply (h _ _).trans_eq'
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) ≀ (f t - x) * f x + f (f x)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) = f (x + (f t - x))
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) ≀ (f t - x) * f x + f (f x) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rw [add_sub_cancel]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) = f (x + (f t - x))
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) t x : R ⊒ f (f t) = f (x + (f t - x)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
rw [← sub_self (f (f t)), ← sub_add_sub_cancel _ (f (f x))]
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ 0 ≀ (f t - x) * f x + (f x - t) * f t
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ f (f t) - f (f x) + (f (f x) - f (f t)) ≀ (f t - x) * f x + (f x - t) * f t
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ 0 ≀ (f t - x) * f x + (f x - t) * f t TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2011/A6/A6.lean
IMOSL.IMO2011A6.final_solution
[22, 1]
[48, 66]
exact add_le_add (h0 t x) (h0 x t)
R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ f (f t) - f (f x) + (f (f x) - f (f t)) ≀ (f t - x) * f x + (f x - t) * f t
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝ : LinearOrderedCommRing R f : R β†’ R h : βˆ€ (x y : R), f (x + y) ≀ y * f x + f (f x) h0 : βˆ€ (t x : R), f (f t) - f (f x) ≀ (f t - x) * f x t x : R ⊒ f (f t) - f (f x) + (f (f x) - f (f t)) ≀ (f t - x) * f x + (f x - t) * f t TACTIC: