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https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_left
[26, 1]
[31, 22]
rw [abs_mul, mul_nsmul', nsmul_eq_mul ⌈|r|⌉.natAbs]
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • |r * ε| ≤ (k * ⌈|r|⌉.natAbs) • |ε|
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • (|r| * |ε|) ≤ k • (↑⌈|r|⌉.natAbs * |ε|)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • |r * ε| ≤ (k * ⌈|r|⌉.natAbs) • |ε| TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_left
[26, 1]
[31, 22]
refine nsmul_le_nsmul_right (mul_le_mul_of_nonneg_right ?_ (abs_nonneg ε)) k
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • (|r| * |ε|) ≤ k • (↑⌈|r|⌉.natAbs * |ε|)
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ |r| ≤ ↑⌈|r|⌉.natAbs
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • (|r| * |ε|) ≤ k • (↑⌈|r|⌉.natAbs * |ε|) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_left
[26, 1]
[31, 22]
rw [← Int.cast_natCast ⌈|r|⌉.natAbs, ← Int.ceil_le]
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ |r| ≤ ↑⌈|r|⌉.natAbs
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ |r| ≤ ↑⌈|r|⌉.natAbs TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_left
[26, 1]
[31, 22]
exact Int.le_natAbs
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_right
[33, 1]
[38, 22]
apply (hε (k * ⌈|r|⌉.natAbs)).trans_le'
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • |ε * r| < 1
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • |ε * r| ≤ (k * ⌈|r|⌉.natAbs) • |ε|
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • |ε * r| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_right
[33, 1]
[38, 22]
rw [abs_mul, mul_nsmul', nsmul_eq_mul' |ε|]
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • |ε * r| ≤ (k * ⌈|r|⌉.natAbs) • |ε|
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • (|ε| * |r|) ≤ k • (|ε| * ↑⌈|r|⌉.natAbs)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • |ε * r| ≤ (k * ⌈|r|⌉.natAbs) • |ε| TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_right
[33, 1]
[38, 22]
refine nsmul_le_nsmul_right (mul_le_mul_of_nonneg_left ?_ (abs_nonneg ε)) k
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • (|ε| * |r|) ≤ k • (|ε| * ↑⌈|r|⌉.natAbs)
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ |r| ≤ ↑⌈|r|⌉.natAbs
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ k • (|ε| * |r|) ≤ k • (|ε| * ↑⌈|r|⌉.natAbs) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_right
[33, 1]
[38, 22]
rw [← Int.cast_natCast ⌈|r|⌉.natAbs, ← Int.ceil_le]
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ |r| ≤ ↑⌈|r|⌉.natAbs
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ |r| ≤ ↑⌈|r|⌉.natAbs TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_mul_right
[33, 1]
[38, 22]
exact Int.le_natAbs
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R hε : Infinitesimal ε r : R k : ℕ ⊢ ⌈|r|⌉ ≤ ↑⌈|r|⌉.natAbs TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁
[40, 1]
[44, 80]
rcases le_total 0 ε with h0 | h0
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ ⊢ k • |ε| < 1
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : 0 ≤ ε ⊢ k • |ε| < 1 case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : ε ≤ 0 ⊢ k • |ε| < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ ⊢ k • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁
[40, 1]
[44, 80]
rw [abs_eq_self.mpr h0, nsmul_eq_mul]
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : 0 ≤ ε ⊢ k • |ε| < 1
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : 0 ≤ ε ⊢ ↑k * ε < 1
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : 0 ≤ ε ⊢ k • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁
[40, 1]
[44, 80]
exact h _
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : 0 ≤ ε ⊢ ↑k * ε < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : 0 ≤ ε ⊢ ↑k * ε < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁
[40, 1]
[44, 80]
rw [abs_eq_neg_self.mpr h0, nsmul_eq_mul, mul_neg, ← neg_mul]
case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : ε ≤ 0 ⊢ k • |ε| < 1
case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : ε ≤ 0 ⊢ -↑k * ε < 1
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : ε ≤ 0 ⊢ k • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₁
[40, 1]
[44, 80]
exact h _
case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : ε ≤ 0 ⊢ -↑k * ε < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), r * ε < 1 k : ℕ h0 : ε ≤ 0 ⊢ -↑k * ε < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁
[46, 1]
[50, 81]
rcases le_total 0 ε with h0 | h0
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ ⊢ k • |ε| < 1
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : 0 ≤ ε ⊢ k • |ε| < 1 case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : ε ≤ 0 ⊢ k • |ε| < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ ⊢ k • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁
[46, 1]
[50, 81]
rw [abs_eq_self.mpr h0, nsmul_eq_mul']
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : 0 ≤ ε ⊢ k • |ε| < 1
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : 0 ≤ ε ⊢ ε * ↑k < 1
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : 0 ≤ ε ⊢ k • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁
[46, 1]
[50, 81]
exact h _
case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : 0 ≤ ε ⊢ ε * ↑k < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : 0 ≤ ε ⊢ ε * ↑k < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁
[46, 1]
[50, 81]
rw [abs_eq_neg_self.mpr h0, nsmul_eq_mul', neg_mul, ← mul_neg]
case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : ε ≤ 0 ⊢ k • |ε| < 1
case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : ε ≤ 0 ⊢ ε * -↑k < 1
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : ε ≤ 0 ⊢ k • |ε| < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₁
[46, 1]
[50, 81]
exact h _
case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : ε ≤ 0 ⊢ ε * -↑k < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R h : ∀ (r : R), ε * r < 1 k : ℕ h0 : ε ≤ 0 ⊢ ε * -↑k < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂
[52, 1]
[55, 73]
refine FloorRing_iff_mul_left₁.trans ⟨λ h ↦ ⟨1, h⟩, λ ⟨α, h⟩ r ↦ ?_⟩
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R ⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), r * ε < α
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h : ∀ (r : R), r * ε < α ⊢ r * ε < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R ⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), r * ε < α TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂
[52, 1]
[55, 73]
have h0 : 0 < α := (zero_mul ε).symm.trans_lt (h 0)
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h : ∀ (r : R), r * ε < α ⊢ r * ε < 1
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h : ∀ (r : R), r * ε < α h0 : 0 < α ⊢ r * ε < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h : ∀ (r : R), r * ε < α ⊢ r * ε < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂
[52, 1]
[55, 73]
specialize h (α * r)
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h : ∀ (r : R), r * ε < α h0 : 0 < α ⊢ r * ε < 1
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h0 : 0 < α h : α * r * ε < α ⊢ r * ε < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h : ∀ (r : R), r * ε < α h0 : 0 < α ⊢ r * ε < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_left₂
[52, 1]
[55, 73]
rwa [mul_assoc, mul_lt_iff_lt_one_right h0] at h
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h0 : 0 < α h : α * r * ε < α ⊢ r * ε < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), r * ε < α r α : R h0 : 0 < α h : α * r * ε < α ⊢ r * ε < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂
[57, 1]
[60, 74]
refine FloorRing_iff_mul_right₁.trans ⟨λ h ↦ ⟨1, h⟩, λ ⟨α, h⟩ r ↦ ?_⟩
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R ⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), ε * r < α
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h : ∀ (r : R), ε * r < α ⊢ ε * r < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R ⊢ Infinitesimal ε ↔ ∃ α, ∀ (r : R), ε * r < α TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂
[57, 1]
[60, 74]
have h0 : 0 < α := (mul_zero ε).symm.trans_lt (h 0)
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h : ∀ (r : R), ε * r < α ⊢ ε * r < 1
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h : ∀ (r : R), ε * r < α h0 : 0 < α ⊢ ε * r < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h : ∀ (r : R), ε * r < α ⊢ ε * r < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂
[57, 1]
[60, 74]
specialize h (r * α)
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h : ∀ (r : R), ε * r < α h0 : 0 < α ⊢ ε * r < 1
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h0 : 0 < α h : ε * (r * α) < α ⊢ ε * r < 1
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h : ∀ (r : R), ε * r < α h0 : 0 < α ⊢ ε * r < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/Infinitesimal/FloorRing.lean
IMOSL.Extra.Infinitesimal.FloorRing_iff_mul_right₂
[57, 1]
[60, 74]
rwa [← mul_assoc, mul_lt_iff_lt_one_left h0] at h
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h0 : 0 < α h : ε * (r * α) < α ⊢ ε * r < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R ε : R x✝ : ∃ α, ∀ (r : R), ε * r < α r α : R h0 : 0 < α h : ε * (r * α) < α ⊢ ε * r < 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4
[48, 1]
[50, 80]
change x (2 * 2 * k) = xor k.bodd (x k)
k : ℕ ⊢ x (4 * k) = xor k.bodd (x k)
k : ℕ ⊢ x (2 * 2 * k) = xor k.bodd (x k)
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (4 * k) = xor k.bodd (x k) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4
[48, 1]
[50, 80]
rw [mul_assoc, x_mul2, x_mul2, ← Nat.bit0_val, Nat.bodd_bit0, Bool.false_xor]
k : ℕ ⊢ x (2 * 2 * k) = xor k.bodd (x k)
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (2 * 2 * k) = xor k.bodd (x k) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4_add1
[52, 1]
[55, 51]
change x (2 * 2 * k + 1) = !x (2 * 2 * k)
k : ℕ ⊢ x (4 * k + 1) = !x (4 * k)
k : ℕ ⊢ x (2 * 2 * k + 1) = !x (2 * 2 * k)
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (4 * k + 1) = !x (4 * k) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4_add1
[52, 1]
[55, 51]
rw [mul_assoc, x_mul2, x_mul2_add1, ← Nat.bit0_val, Nat.bodd_bit0, Bool.false_xor]
k : ℕ ⊢ x (2 * 2 * k + 1) = !x (2 * 2 * k)
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (2 * 2 * k + 1) = !x (2 * 2 * k) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4_add2
[57, 1]
[60, 64]
change x (2 * 2 * k + 2) = x k
k : ℕ ⊢ x (4 * k + 2) = x k
k : ℕ ⊢ x (2 * 2 * k + 2) = x k
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (4 * k + 2) = x k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4_add2
[57, 1]
[60, 64]
rw [mul_assoc, ← mul_add_one, x_mul2, x_mul2_add1, ← Nat.bit1_val, Nat.bodd_bit1, Bool.true_xor, Bool.not_not]
k : ℕ ⊢ x (2 * 2 * k + 2) = x k
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (2 * 2 * k + 2) = x k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4_add3
[62, 1]
[64, 72]
change x (2 * 2 * k + 2 + 1) = x k
k : ℕ ⊢ x (4 * k + 3) = x k
k : ℕ ⊢ x (2 * 2 * k + 2 + 1) = x k
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (4 * k + 3) = x k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.x_mul4_add3
[62, 1]
[64, 72]
rw [mul_assoc, ← mul_add_one, x_mul2_add1, x_mul2_add1, Bool.not_not]
k : ℕ ⊢ x (2 * 2 * k + 2 + 1) = x k
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ x (2 * 2 * k + 2 + 1) = x k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_mul4_add2
[79, 1]
[81, 34]
rw [S_succ, S_succ, add_assoc, add_right_eq_self, x_mul4_add1]
k : ℕ ⊢ S (4 * k + 2) = S (4 * k)
k : ℕ ⊢ ((bif x (4 * k) then -1 else 1) + bif !x (4 * k) then -1 else 1) = 0
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ S (4 * k + 2) = S (4 * k) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_mul4_add2
[79, 1]
[81, 34]
exact (x (4 * k)).recOn rfl rfl
k : ℕ ⊢ ((bif x (4 * k) then -1 else 1) + bif !x (4 * k) then -1 else 1) = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ ((bif x (4 * k) then -1 else 1) + bif !x (4 * k) then -1 else 1) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_mul4
[83, 1]
[87, 52]
change S (4 * k + 4) = 2 * S k.succ
k : ℕ h : S (4 * k) = 2 * S k ⊢ S (4 * k.succ) = 2 * S k.succ
k : ℕ h : S (4 * k) = 2 * S k ⊢ S (4 * k + 4) = 2 * S k.succ
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ h : S (4 * k) = 2 * S k ⊢ S (4 * k.succ) = 2 * S k.succ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_mul4
[83, 1]
[87, 52]
rw [S_succ, x_mul4_add3, S_succ, x_mul4_add2, S_mul4_add2, h, add_assoc, ← two_mul, ← mul_add, ← S_succ]
k : ℕ h : S (4 * k) = 2 * S k ⊢ S (4 * k + 4) = 2 * S k.succ
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ h : S (4 * k) = 2 * S k ⊢ S (4 * k + 4) = 2 * S k.succ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_parity
[89, 1]
[92, 28]
rw [Nat.bodd_succ, S_succ, Int.bodd_add, h, ← Bool.xor_true]
k : ℕ h : (S k).bodd = k.bodd ⊢ (S k.succ).bodd = k.succ.bodd
k : ℕ h : (S k).bodd = k.bodd ⊢ xor k.bodd (bif x k then -1 else 1).bodd = xor k.bodd true
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ h : (S k).bodd = k.bodd ⊢ (S k.succ).bodd = k.succ.bodd TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_parity
[89, 1]
[92, 28]
exact (x k).rec rfl rfl
k : ℕ h : (S k).bodd = k.bodd ⊢ xor k.bodd (bif x k then -1 else 1).bodd = xor k.bodd true
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ h : (S k).bodd = k.bodd ⊢ xor k.bodd (bif x k then -1 else 1).bodd = xor k.bodd true TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
refine ⟨fun h0 ↦ (and_iff_right_of_imp ?_).mpr ?_, fun h0 ↦ ?_⟩
q r : ℕ h : r < 4 ⊢ S (4 * q + r) = 0 ↔ S q = 0 ∧ (r = 0 ∨ r = 2)
case refine_1 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 → S q = 0 case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: q r : ℕ h : r < 4 ⊢ S (4 * q + r) = 0 ↔ S q = 0 ∧ (r = 0 ∨ r = 2) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
replace h : (2 : ℤ) ≠ 0 := two_ne_zero
case refine_1 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 → S q = 0 case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_1 q r : ℕ h0 : S (4 * q + r) = 0 h : 2 ≠ 0 ⊢ r = 0 ∨ r = 2 → S q = 0 case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 → S q = 0 case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rintro (rfl | rfl)
case refine_1 q r : ℕ h0 : S (4 * q + r) = 0 h : 2 ≠ 0 ⊢ r = 0 ∨ r = 2 → S q = 0 case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_1.inl q : ℕ h : 2 ≠ 0 h0 : S (4 * q + 0) = 0 ⊢ S q = 0 case refine_1.inr q : ℕ h : 2 ≠ 0 h0 : S (4 * q + 2) = 0 ⊢ S q = 0 case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 q r : ℕ h0 : S (4 * q + r) = 0 h : 2 ≠ 0 ⊢ r = 0 ∨ r = 2 → S q = 0 case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
apply_fun Int.bodd at h0
case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h : r < 4 h0 : (S (4 * q + r)).bodd = Int.bodd 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h : r < 4 h0 : S (4 * q + r) = 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rw [Int.bodd_zero, S_parity, Nat.bodd_add, Nat.bodd_mul] at h0
case refine_2 q r : ℕ h : r < 4 h0 : (S (4 * q + r)).bodd = Int.bodd 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h : r < 4 h0 : xor (Nat.bodd 4 && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h : r < 4 h0 : (S (4 * q + r)).bodd = Int.bodd 0 ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
change xor (false && q.bodd) r.bodd = false at h0
case refine_2 q r : ℕ h : r < 4 h0 : xor (Nat.bodd 4 && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h : r < 4 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h : r < 4 h0 : xor (Nat.bodd 4 && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
iterate 3 rw [Nat.lt_succ_iff_lt_or_eq] at h
case refine_2 q r : ℕ h : r < 4 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h : ((r < 1 ∨ r = 1) ∨ r = 2) ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h : r < 4 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rw [Nat.lt_one_iff, or_assoc, or_or_or_comm] at h
case refine_2 q r : ℕ h : ((r < 1 ∨ r = 1) ∨ r = 2) ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h : (r = 0 ∨ r = 2) ∨ r = 1 ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h : ((r < 1 ∨ r = 1) ∨ r = 2) ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
revert h
case refine_2 q r : ℕ h : (r = 0 ∨ r = 2) ∨ r = 1 ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h0 : xor (false && q.bodd) r.bodd = false ⊢ (r = 0 ∨ r = 2) ∨ r = 1 ∨ r = 3 → r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h : (r = 0 ∨ r = 2) ∨ r = 1 ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
refine' (or_iff_left _).mp
case refine_2 q r : ℕ h0 : xor (false && q.bodd) r.bodd = false ⊢ (r = 0 ∨ r = 2) ∨ r = 1 ∨ r = 3 → r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h0 : xor (false && q.bodd) r.bodd = false ⊢ ¬(r = 1 ∨ r = 3) case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h0 : xor (false && q.bodd) r.bodd = false ⊢ (r = 0 ∨ r = 2) ∨ r = 1 ∨ r = 3 → r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rintro (rfl | rfl) <;> exact Bool.true_eq_false_eq_False h0
case refine_2 q r : ℕ h0 : xor (false && q.bodd) r.bodd = false ⊢ ¬(r = 1 ∨ r = 3) case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h0 : xor (false && q.bodd) r.bodd = false ⊢ ¬(r = 1 ∨ r = 3) case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rcases h0 with ⟨h0, rfl | rfl⟩
case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_3.intro.inl q : ℕ h0 : S q = 0 h : 0 < 4 ⊢ S (4 * q + 0) = 0 case refine_3.intro.inr q : ℕ h0 : S q = 0 h : 2 < 4 ⊢ S (4 * q + 2) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rwa [add_zero, S_mul4, mul_eq_zero, or_iff_right h] at h0
case refine_1.inl q : ℕ h : 2 ≠ 0 h0 : S (4 * q + 0) = 0 ⊢ S q = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1.inl q : ℕ h : 2 ≠ 0 h0 : S (4 * q + 0) = 0 ⊢ S q = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rwa [S_mul4_add2, S_mul4, mul_eq_zero, or_iff_right h] at h0
case refine_1.inr q : ℕ h : 2 ≠ 0 h0 : S (4 * q + 2) = 0 ⊢ S q = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1.inr q : ℕ h : 2 ≠ 0 h0 : S (4 * q + 2) = 0 ⊢ S q = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rw [Nat.lt_succ_iff_lt_or_eq] at h
case refine_2 q r : ℕ h : (r < 2 ∨ r = 2) ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
case refine_2 q r : ℕ h : ((r < 1 ∨ r = 1) ∨ r = 2) ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 q r : ℕ h : (r < 2 ∨ r = 2) ∨ r = 3 h0 : xor (false && q.bodd) r.bodd = false ⊢ r = 0 ∨ r = 2 case refine_3 q r : ℕ h : r < 4 h0 : S q = 0 ∧ (r = 0 ∨ r = 2) ⊢ S (4 * q + r) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rw [add_zero, S_mul4, h0]
case refine_3.intro.inl q : ℕ h0 : S q = 0 h : 0 < 4 ⊢ S (4 * q + 0) = 0
case refine_3.intro.inl q : ℕ h0 : S q = 0 h : 0 < 4 ⊢ 2 * 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_3.intro.inl q : ℕ h0 : S q = 0 h : 0 < 4 ⊢ S (4 * q + 0) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rfl
case refine_3.intro.inl q : ℕ h0 : S q = 0 h : 0 < 4 ⊢ 2 * 0 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_3.intro.inl q : ℕ h0 : S q = 0 h : 0 < 4 ⊢ 2 * 0 = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rw [S_mul4_add2, S_mul4, h0]
case refine_3.intro.inr q : ℕ h0 : S q = 0 h : 2 < 4 ⊢ S (4 * q + 2) = 0
case refine_3.intro.inr q : ℕ h0 : S q = 0 h : 2 < 4 ⊢ 2 * 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: case refine_3.intro.inr q : ℕ h0 : S q = 0 h : 2 < 4 ⊢ S (4 * q + 2) = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.S_four_mul_add_eq_zero_iff
[94, 1]
[113, 38]
rfl
case refine_3.intro.inr q : ℕ h0 : S q = 0 h : 2 < 4 ⊢ 2 * 0 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_3.intro.inr q : ℕ h0 : S q = 0 h : 2 < 4 ⊢ 2 * 0 = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
intro k h
⊢ ∀ (k : ℕ), S k = 0 → x k = false
k : ℕ h : S k = 0 ⊢ x k = false
Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (k : ℕ), S k = 0 → x k = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
induction' k using Nat.strong_induction_on with k k_ih
k : ℕ h : S k = 0 ⊢ x k = false
case h k : ℕ k_ih : ∀ m < k, S m = 0 → x m = false h : S k = 0 ⊢ x k = false
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ h : S k = 0 ⊢ x k = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
obtain ⟨q, r, h0, rfl⟩ : ∃ q r : ℕ, r < 4 ∧ 4 * q + r = k := ⟨k / 4, k % 4, Nat.mod_lt k four_pos, Nat.div_add_mod k 4⟩
case h k : ℕ k_ih : ∀ m < k, S m = 0 → x m = false h : S k = 0 ⊢ x k = false
case h.intro.intro.intro q r : ℕ h0 : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S (4 * q + r) = 0 ⊢ x (4 * q + r) = false
Please generate a tactic in lean4 to solve the state. STATE: case h k : ℕ k_ih : ∀ m < k, S m = 0 → x m = false h : S k = 0 ⊢ x k = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [S_four_mul_add_eq_zero_iff q h0, or_comm] at h
case h.intro.intro.intro q r : ℕ h0 : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S (4 * q + r) = 0 ⊢ x (4 * q + r) = false
case h.intro.intro.intro q r : ℕ h0 : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S q = 0 ∧ (r = 2 ∨ r = 0) ⊢ x (4 * q + r) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro q r : ℕ h0 : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S (4 * q + r) = 0 ⊢ x (4 * q + r) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
clear h0
case h.intro.intro.intro q r : ℕ h0 : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S q = 0 ∧ (r = 2 ∨ r = 0) ⊢ x (4 * q + r) = false
case h.intro.intro.intro q r : ℕ k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S q = 0 ∧ (r = 2 ∨ r = 0) ⊢ x (4 * q + r) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro q r : ℕ h0 : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S q = 0 ∧ (r = 2 ∨ r = 0) ⊢ x (4 * q + r) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rcases h with ⟨h, rfl | rfl⟩
case h.intro.intro.intro q r : ℕ k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S q = 0 ∧ (r = 2 ∨ r = 0) ⊢ x (4 * q + r) = false
case h.intro.intro.intro.intro.inl q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 2, S m = 0 → x m = false ⊢ x (4 * q + 2) = false case h.intro.intro.intro.intro.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false ⊢ x (4 * q + 0) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro q r : ℕ k_ih : ∀ m < 4 * q + r, S m = 0 → x m = false h : S q = 0 ∧ (r = 2 ∨ r = 0) ⊢ x (4 * q + r) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rcases q.eq_zero_or_pos with (rfl | h0)
case h.intro.intro.intro.intro.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false ⊢ x (4 * q + 0) = false
case h.intro.intro.intro.intro.inr.inl h : S 0 = 0 k_ih : ∀ m < 4 * 0 + 0, S m = 0 → x m = false ⊢ x (4 * 0 + 0) = false case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false h0 : q > 0 ⊢ x (4 * q + 0) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false ⊢ x (4 * q + 0) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [add_zero, MulZeroClass.mul_zero, x_zero]
case h.intro.intro.intro.intro.inr.inl h : S 0 = 0 k_ih : ∀ m < 4 * 0 + 0, S m = 0 → x m = false ⊢ x (4 * 0 + 0) = false case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false h0 : q > 0 ⊢ x (4 * q + 0) = false
case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false h0 : q > 0 ⊢ x (4 * q + 0) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inr.inl h : S 0 = 0 k_ih : ∀ m < 4 * 0 + 0, S m = 0 → x m = false ⊢ x (4 * 0 + 0) = false case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false h0 : q > 0 ⊢ x (4 * q + 0) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
replace k_ih := k_ih q (lt_mul_left h0 <| Nat.succ_lt_succ <| Nat.succ_pos 2) h
case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false h0 : q > 0 ⊢ x (4 * q + 0) = false
case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 h0 : q > 0 k_ih : x q = false ⊢ x (4 * q + 0) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 0, S m = 0 → x m = false h0 : q > 0 ⊢ x (4 * q + 0) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
apply_fun Int.bodd at h
case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 h0 : q > 0 k_ih : x q = false ⊢ x (4 * q + 0) = false
case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : (S q).bodd = Int.bodd 0 ⊢ x (4 * q + 0) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inr.inr q : ℕ h : S q = 0 h0 : q > 0 k_ih : x q = false ⊢ x (4 * q + 0) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [Int.bodd_zero, S_parity] at h
case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : (S q).bodd = Int.bodd 0 ⊢ x (4 * q + 0) = false
case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : q.bodd = false ⊢ x (4 * q + 0) = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : (S q).bodd = Int.bodd 0 ⊢ x (4 * q + 0) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [add_zero, x_mul4, h, k_ih]
case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : q.bodd = false ⊢ x (4 * q + 0) = false
case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : q.bodd = false ⊢ xor false false = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : q.bodd = false ⊢ x (4 * q + 0) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rfl
case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : q.bodd = false ⊢ xor false false = false
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inr.inr q : ℕ h0 : q > 0 k_ih : x q = false h : q.bodd = false ⊢ xor false false = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
refine Nat.rec (Int.le_refl 0) (λ k h ↦ ?_)
this : ∀ (k : ℕ), S k = 0 → x k = false ⊢ ∀ (k : ℕ), 0 ≤ S k
this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 ≤ S k ⊢ 0 ≤ S k.succ
Please generate a tactic in lean4 to solve the state. STATE: this : ∀ (k : ℕ), S k = 0 → x k = false ⊢ ∀ (k : ℕ), 0 ≤ S k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [le_iff_lt_or_eq, Int.lt_iff_add_one_le, zero_add, or_comm] at h
this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 ≤ S k ⊢ 0 ≤ S k.succ
this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ∨ 1 ≤ S k ⊢ 0 ≤ S k.succ
Please generate a tactic in lean4 to solve the state. STATE: this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 ≤ S k ⊢ 0 ≤ S k.succ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [S_succ]
this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ∨ 1 ≤ S k ⊢ 0 ≤ S k.succ
this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ∨ 1 ≤ S k ⊢ 0 ≤ S k + bif x k then -1 else 1
Please generate a tactic in lean4 to solve the state. STATE: this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ∨ 1 ≤ S k ⊢ 0 ≤ S k.succ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rcases h with h | h
this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ∨ 1 ≤ S k ⊢ 0 ≤ S k + bif x k then -1 else 1
case inl this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ⊢ 0 ≤ S k + bif x k then -1 else 1 case inr this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 1 ≤ S k ⊢ 0 ≤ S k + bif x k then -1 else 1
Please generate a tactic in lean4 to solve the state. STATE: this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ∨ 1 ≤ S k ⊢ 0 ≤ S k + bif x k then -1 else 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [← h, zero_add, this k h.symm]
case inl this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ⊢ 0 ≤ S k + bif x k then -1 else 1
case inl this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ⊢ 0 ≤ bif false then -1 else 1
Please generate a tactic in lean4 to solve the state. STATE: case inl this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ⊢ 0 ≤ S k + bif x k then -1 else 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
exact Int.zero_lt_one.le
case inl this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ⊢ 0 ≤ bif false then -1 else 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 0 = S k ⊢ 0 ≤ bif false then -1 else 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [← add_neg_self (1 : ℤ)]
case inr this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 1 ≤ S k ⊢ 0 ≤ S k + bif x k then -1 else 1
case inr this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 1 ≤ S k ⊢ 1 + -1 ≤ S k + bif x k then -1 else 1
Please generate a tactic in lean4 to solve the state. STATE: case inr this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 1 ≤ S k ⊢ 0 ≤ S k + bif x k then -1 else 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
exact add_le_add h ((x k).rec (neg_le_self zero_le_one) (-1).le_refl)
case inr this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 1 ≤ S k ⊢ 1 + -1 ≤ S k + bif x k then -1 else 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr this : ∀ (k : ℕ), S k = 0 → x k = false k : ℕ h : 1 ≤ S k ⊢ 1 + -1 ≤ S k + bif x k then -1 else 1 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
rw [x_mul4_add2]
case h.intro.intro.intro.intro.inl q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 2, S m = 0 → x m = false ⊢ x (4 * q + 2) = false
case h.intro.intro.intro.intro.inl q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 2, S m = 0 → x m = false ⊢ x q = false
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inl q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 2, S m = 0 → x m = false ⊢ x (4 * q + 2) = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution
[122, 1]
[145, 38]
exact k_ih q (lt_add_of_le_of_pos (Nat.le_mul_of_pos_left _ four_pos) two_pos) h
case h.intro.intro.intro.intro.inl q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 2, S m = 0 → x m = false ⊢ x q = false
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.intro.inl q : ℕ h : S q = 0 k_ih : ∀ m < 4 * q + 2, S m = 0 → x m = false ⊢ x q = false TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
induction' k using Nat.strong_induction_on with k k_ih
k : ℕ ⊢ S k = 0 ↔ ∀ c ∈ Nat.digits 4 k, c = 0 ∨ c = 2
case h k : ℕ k_ih : ∀ m < k, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S k = 0 ↔ ∀ c ∈ Nat.digits 4 k, c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: k : ℕ ⊢ S k = 0 ↔ ∀ c ∈ Nat.digits 4 k, c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
obtain ⟨q, r, h, rfl⟩ : ∃ q r : ℕ, r < 4 ∧ 4 * q + r = k := ⟨k / 4, k % 4, Nat.mod_lt k four_pos, Nat.div_add_mod k 4⟩
case h k : ℕ k_ih : ∀ m < k, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S k = 0 ↔ ∀ c ∈ Nat.digits 4 k, c = 0 ∨ c = 2
case h.intro.intro.intro q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S (4 * q + r) = 0 ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h k : ℕ k_ih : ∀ m < k, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S k = 0 ↔ ∀ c ∈ Nat.digits 4 k, c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rw [S_four_mul_add_eq_zero_iff q h]
case h.intro.intro.intro q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S (4 * q + r) = 0 ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S (4 * q + r) = 0 ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rcases q.eq_zero_or_pos with (rfl | h0)
case h.intro.intro.intro q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S 0 = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * 0 + r), c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rw [S_zero, eq_self_iff_true, true_and_iff, MulZeroClass.mul_zero, zero_add]
case h.intro.intro.intro.inl r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S 0 = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * 0 + r), c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ S 0 = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * 0 + r), c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rcases r.eq_zero_or_pos with (rfl | h0)
case h.intro.intro.intro.inl r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ 0 = 0 ∨ 0 = 2 ↔ ∀ c ∈ Nat.digits 4 0, c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rw [eq_self_iff_true, true_or_iff, true_iff_iff, Nat.digits_zero]
case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ 0 = 0 ∨ 0 = 2 ↔ ∀ c ∈ Nat.digits 4 0, c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ ∀ c ∈ [], c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ 0 = 0 ∨ 0 = 2 ↔ ∀ c ∈ Nat.digits 4 0, c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
intro c h0
case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ ∀ c ∈ [], c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 c : ℕ h0 : c ∈ [] ⊢ c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 ⊢ ∀ c ∈ [], c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
exact absurd h0 (by exact List.not_mem_nil c)
case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 c : ℕ h0 : c ∈ [] ⊢ c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl.inl h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 c : ℕ h0 : c ∈ [] ⊢ c = 0 ∨ c = 2 case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rw [Nat.digits_def' (Nat.succ_lt_succ <| Nat.succ_pos 2) h0, Nat.mod_eq_of_lt h, Nat.div_eq_of_lt h, Nat.digits_zero]
case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ [r], c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ Nat.digits 4 r, c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
simp_rw [List.mem_singleton]
case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ [r], c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ (c : ℕ), c = r → c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ c ∈ [r], c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rw [forall_eq]
case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ (c : ℕ), c = r → c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inl.inr r : ℕ h : r < 4 k_ih : ∀ m < 4 * 0 + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : r > 0 ⊢ r = 0 ∨ r = 2 ↔ ∀ (c : ℕ), c = r → c = 0 ∨ c = 2 case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
replace k_ih := k_ih q (Nat.lt_add_right _ <| lt_mul_left h0 <| Nat.succ_lt_succ <| Nat.succ_pos 2)
case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inr q r : ℕ h : r < 4 k_ih : ∀ m < 4 * q + r, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 h0 : q > 0 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rw [k_ih, add_comm, Nat.digits_add 4 (Nat.succ_lt_succ <| Nat.succ_pos 2) r q h <| Or.inr h0.ne.symm]
case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2
case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ (∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2) ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ r :: Nat.digits 4 q, c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ S q = 0 ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ Nat.digits 4 (4 * q + r), c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
simp_rw [List.mem_cons]
case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ (∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2) ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ r :: Nat.digits 4 q, c = 0 ∨ c = 2
case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ (∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2) ∧ (r = 0 ∨ r = 2) ↔ ∀ (c : ℕ), c = r ∨ c ∈ Nat.digits 4 q → c = 0 ∨ c = 2
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ (∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2) ∧ (r = 0 ∨ r = 2) ↔ ∀ c ∈ r :: Nat.digits 4 q, c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
rw [forall_eq_or_imp, and_comm]
case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ (∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2) ∧ (r = 0 ∨ r = 2) ↔ ∀ (c : ℕ), c = r ∨ c ∈ Nat.digits 4 q → c = 0 ∨ c = 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro.intro.inr q r : ℕ h : r < 4 h0 : q > 0 k_ih : S q = 0 ↔ ∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2 ⊢ (∀ c ∈ Nat.digits 4 q, c = 0 ∨ c = 2) ∧ (r = 0 ∨ r = 2) ↔ ∀ (c : ℕ), c = r ∨ c ∈ Nat.digits 4 q → c = 0 ∨ c = 2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A4/A4.lean
IMOSL.IMO2010A4.final_solution_extra
[148, 1]
[168, 59]
exact List.not_mem_nil c
h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 c : ℕ h0 : c ∈ [] ⊢ c ∉ []
no goals
Please generate a tactic in lean4 to solve the state. STATE: h : 0 < 4 k_ih : ∀ m < 4 * 0 + 0, S m = 0 ↔ ∀ c ∈ Nat.digits 4 m, c = 0 ∨ c = 2 c : ℕ h0 : c ∈ [] ⊢ c ∉ [] TACTIC: