Datasets:
pkuAI4M
/
lean_github

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2.09M
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.c_bdd
[87, 1]
[92, 69]
suffices b K = b (K + 1) from ⟨this.symm, (h0 K h2).trans_eq (this β–Έ max_eq_right h3)⟩
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K ≀ 2 β€’ b K ⊒ b (K + 1) = b K ∧ c (K + 1) ≀ 2 β€’ b (K + 1)
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K ≀ 2 β€’ b K ⊒ b K = b (K + 1)
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K ≀ 2 β€’ b K ⊒ b (K + 1) = b K ∧ c (K + 1) ≀ 2 β€’ b (K + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.c_bdd
[87, 1]
[92, 69]
rw [two_nsmul, ← sub_le_iff_le_add] at h3
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K ≀ 2 β€’ b K ⊒ b K = b (K + 1)
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K - b K ≀ b K ⊒ b K = b (K + 1)
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K ≀ 2 β€’ b K ⊒ b K = b (K + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.c_bdd
[87, 1]
[92, 69]
exact (h1 K.le_succ).antisymm ((h K h2).trans_eq (max_eq_left h3))
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K - b K ≀ b K ⊒ b K = b (K + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : D ≀ K h3 : c K - b K ≀ b K ⊒ b K = b (K + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd
[94, 1]
[100, 67]
have X {M : β„•} : D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M := Ξ» h3 h4 ↦ ((h0 M h3).trans_eq (max_eq_left_of_lt h4)).antisymm (h2 M.le_succ)
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K ⊒ 2 β€’ b K < c K β†’ c (K + 1) = c D
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M ⊒ 2 β€’ b K < c K β†’ c (K + 1) = c D
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K ⊒ 2 β€’ b K < c K β†’ c (K + 1) = c D TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd
[94, 1]
[100, 67]
refine Nat.le_induction (X D.le_refl) (Ξ» n h4 h5 h6 ↦ ?_) K h3
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M ⊒ 2 β€’ b K < c K β†’ c (K + 1) = c D
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M n : β„• h4 : D ≀ n h5 : 2 β€’ b n < c n β†’ c (n + 1) = c D h6 : 2 β€’ b (n + 1) < c (n + 1) ⊒ c (n + 1 + 1) = c D
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M ⊒ 2 β€’ b K < c K β†’ c (K + 1) = c D TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd
[94, 1]
[100, 67]
rw [X (Nat.le_step h4) h6]
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M n : β„• h4 : D ≀ n h5 : 2 β€’ b n < c n β†’ c (n + 1) = c D h6 : 2 β€’ b (n + 1) < c (n + 1) ⊒ c (n + 1 + 1) = c D
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M n : β„• h4 : D ≀ n h5 : 2 β€’ b n < c n β†’ c (n + 1) = c D h6 : 2 β€’ b (n + 1) < c (n + 1) ⊒ c n.succ = c D
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M n : β„• h4 : D ≀ n h5 : 2 β€’ b n < c n β†’ c (n + 1) = c D h6 : 2 β€’ b (n + 1) < c (n + 1) ⊒ c (n + 1 + 1) = c D TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd
[94, 1]
[100, 67]
exact h5 (lt_of_not_le Ξ» h7 ↦ h6.not_le (c_bdd h h0 h1 h4 h7).2)
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M n : β„• h4 : D ≀ n h5 : 2 β€’ b n < c n β†’ c (n + 1) = c D h6 : 2 β€’ b (n + 1) < c (n + 1) ⊒ c n.succ = c D
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b K : β„• h2 : Monotone c h3 : D ≀ K X : βˆ€ {M : β„•}, D ≀ M β†’ 2 β€’ b M < c M β†’ c (M + 1) = c M n : β„• h4 : D ≀ n h5 : 2 β€’ b n < c n β†’ c (n + 1) = c D h6 : 2 β€’ b (n + 1) < c (n + 1) ⊒ c n.succ = c D TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
have h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) := by refine max_le (le_max_of_two_nsmul_le_add ?_) (le_max_left _ _) rw [← nsmul_add, add_sub_cancel]
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• ⊒ max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) ⊒ max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• ⊒ max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
refine (le_total n D).elim (Ξ» h4 ↦ h3.trans' (max_le_max (h2 h4) (nsmul_le_nsmul_right (h1 h4) 2))) (Nat.le_induction h3 (Ξ» n h4 h5 ↦ ?_) n)
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) ⊒ max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) ⊒ max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
rcases le_or_lt (c n) (2 β€’ b n) with h6 | h6
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
case inl G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : c n ≀ 2 β€’ b n ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
refine max_le (le_max_of_two_nsmul_le_add ?_) (le_max_left _ _)
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• ⊒ max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• ⊒ 2 β€’ c D ≀ 2 β€’ b D + 2 β€’ (c D - b D)
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• ⊒ max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
rw [← nsmul_add, add_sub_cancel]
G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• ⊒ 2 β€’ c D ≀ 2 β€’ b D + 2 β€’ (c D - b D)
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n : β„• ⊒ 2 β€’ c D ≀ 2 β€’ b D + 2 β€’ (c D - b D) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
rcases c_bdd h h0 h1 h4 h6 with ⟨h7, h8⟩
case inl G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : c n ≀ 2 β€’ b n ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
case inl.intro G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : c n ≀ 2 β€’ b n h7 : b (n + 1) = b n h8 : c (n + 1) ≀ 2 β€’ b (n + 1) ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
Please generate a tactic in lean4 to solve the state. STATE: case inl G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : c n ≀ 2 β€’ b n ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
rwa [max_eq_right h8, h7, ← max_eq_right h6]
case inl.intro G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : c n ≀ 2 β€’ b n h7 : b (n + 1) = b n h8 : c (n + 1) ≀ 2 β€’ b (n + 1) ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl.intro G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : c n ≀ 2 β€’ b n h7 : b (n + 1) = b n h8 : c (n + 1) ≀ 2 β€’ b (n + 1) ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
have h7 := c_succ_eq_D_of_b_bdd h h0 h1 h2 h4 h6
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n h7 : c (n + 1) = c D ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
Please generate a tactic in lean4 to solve the state. STATE: case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
refine max_le (h7 β–Έ h3.trans' (le_max_left _ _)) (le_max_of_le_right <| nsmul_le_nsmul_right ((h _ h4).trans ?_) 2)
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n h7 : c (n + 1) = c D ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D))
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n h7 : c (n + 1) = c D ⊒ max (b n) (c n - b n) ≀ c D - b D
Please generate a tactic in lean4 to solve the state. STATE: case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n h7 : c (n + 1) = c D ⊒ max (c (n + 1)) (2 β€’ b (n + 1)) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
rw [two_nsmul, ← lt_sub_iff_add_lt] at h6
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n h7 : c (n + 1) = c D ⊒ max (b n) (c n - b n) ≀ c D - b D
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : b n < c n - b n h7 : c (n + 1) = c D ⊒ max (b n) (c n - b n) ≀ c D - b D
Please generate a tactic in lean4 to solve the state. STATE: case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : 2 β€’ b n < c n h7 : c (n + 1) = c D ⊒ max (b n) (c n - b n) ≀ c D - b D TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
rw [max_eq_right h6.le, ← h7]
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : b n < c n - b n h7 : c (n + 1) = c D ⊒ max (b n) (c n - b n) ≀ c D - b D
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : b n < c n - b n h7 : c (n + 1) = c D ⊒ c n - b n ≀ c (n + 1) - b D
Please generate a tactic in lean4 to solve the state. STATE: case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : b n < c n - b n h7 : c (n + 1) = c D ⊒ max (b n) (c n - b n) ≀ c D - b D TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2017/A4/A4.lean
IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd
[102, 1]
[120, 44]
exact sub_le_sub (h2 n.le_succ) (h1 h4)
case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : b n < c n - b n h7 : c (n + 1) = c D ⊒ c n - b n ≀ c (n + 1) - b D
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr G : Type u_1 D : β„• inst✝ : LinearOrderedAddCommGroup G b c : β„• β†’ G h : βˆ€ (n : β„•), D ≀ n β†’ b (n + 1) ≀ max (b n) (c n - b n) h0 : βˆ€ (n : β„•), D ≀ n β†’ c (n + 1) ≀ max (c n) (2 β€’ b n) h1 : Monotone b h2 : Monotone c n✝ : β„• h3 : max (c D) (2 β€’ b D) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) n : β„• h4 : D ≀ n h5 : max (c n) (2 β€’ b n) ≀ max (2 β€’ b D) (2 β€’ (c D - b D)) h6 : b n < c n - b n h7 : c (n + 1) = c D ⊒ c n - b n ≀ c (n + 1) - b D TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2010/A1/A1.lean
IMOSL.IMO2010A1.Int_good_iff_MonoidGood
[40, 1]
[41, 67]
rw [Int.floor_int, smul_eq_mul, id_def]
R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R f : β„€ β†’ R m n : β„€ ⊒ f (⌊mβŒ‹ β€’ n) = f m * β†‘βŒŠf nβŒ‹ ↔ f (m * n) = f m * β†‘βŒŠf nβŒ‹
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 inst✝¹ : LinearOrderedRing R inst✝ : FloorRing R f : β„€ β†’ R m n : β„€ ⊒ f (⌊mβŒ‹ β€’ n) = f m * β†‘βŒŠf nβŒ‹ ↔ f (m * n) = f m * β†‘βŒŠf nβŒ‹ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.d_zero
[29, 1]
[30, 72]
rw [d, sum_range_one, Int.Nat.cast_ofNat_Int, Int.zero_mul, sub_zero]
z : β„• β†’ β„€ ⊒ d z 0 = z 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ ⊒ d z 0 = z 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.d_succ
[32, 1]
[35, 52]
rw [d, sum_range_succ, Int.natCast_add, Int.natCast_one, add_one_mul (Ξ± := β„€), add_sub_add_right_eq_sub]
z : β„• β†’ β„€ n : β„• ⊒ d z (n + 1) = (range (n + 1)).sum z - ↑n * z (n + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ n : β„• ⊒ d z (n + 1) = (range (n + 1)).sum z - ↑n * z (n + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.d_one
[37, 1]
[38, 64]
rw [d_succ, sum_range_one, Nat.cast_zero, zero_mul, sub_zero]
z : β„• β†’ β„€ ⊒ d z 1 = z 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ ⊒ d z 1 = z 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.main_lemma
[42, 1]
[45, 79]
rw [d_succ, d, sub_sub, ← mul_add_one (Ξ± := β„€)]
z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ d z (n + 1) ≀ d z n - ↑n
z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ (range (n + 1)).sum z - ↑n * z (n + 1) ≀ (range (n + 1)).sum z - ↑n * (z n + 1)
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ d z (n + 1) ≀ d z n - ↑n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.main_lemma
[42, 1]
[45, 79]
exact Int.sub_le_sub_left (Int.mul_le_mul_of_nonneg_left (h n.lt_succ_self) (Int.ofNat_zero_le n)) _
z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ (range (n + 1)).sum z - ↑n * z (n + 1) ≀ (range (n + 1)).sum z - ↑n * (z n + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ (range (n + 1)).sum z - ↑n * z (n + 1) ≀ (range (n + 1)).sum z - ↑n * (z n + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.binom_bound
[47, 1]
[53, 76]
rw [Nat.choose, Nat.choose_one_right, Nat.cast_add, ← sub_sub, sub_right_comm]
z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ d z (n + 1) ≀ z 0 - ↑((n + 1).choose 2)
z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ d z (n + 1) ≀ z 0 - ↑(n.choose (1 + 1)) - ↑n
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ d z (n + 1) ≀ z 0 - ↑((n + 1).choose 2) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.binom_bound
[47, 1]
[53, 76]
exact Int.le_sub_right_of_add_le <| (Int.add_le_of_le_sub_right (main_lemma h n)).trans (binom_bound n)
z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ d z (n + 1) ≀ z 0 - ↑(n.choose (1 + 1)) - ↑n
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ h : StrictMono z n : β„• ⊒ d z (n + 1) ≀ z 0 - ↑(n.choose (1 + 1)) - ↑n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.final_solution_part2
[103, 1]
[107, 37]
rw [eq_greatestDPos_iff h h0, d_succ]
z : β„• β†’ β„€ h : StrictMono z h0 : 0 < z 0 N : β„• ⊒ N = greatestDPos h ↔ ↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≀ ↑N * z (N + 1)
z : β„• β†’ β„€ h : StrictMono z h0 : 0 < z 0 N : β„• ⊒ 0 < d z N ∧ (range (N + 1)).sum z - ↑N * z (N + 1) ≀ 0 ↔ ↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≀ ↑N * z (N + 1)
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ h : StrictMono z h0 : 0 < z 0 N : β„• ⊒ N = greatestDPos h ↔ ↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≀ ↑N * z (N + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A1/A1.lean
IMOSL.IMO2014A1.final_solution_part2
[103, 1]
[107, 37]
exact and_congr sub_pos sub_nonpos
z : β„• β†’ β„€ h : StrictMono z h0 : 0 < z 0 N : β„• ⊒ 0 < d z N ∧ (range (N + 1)).sum z - ↑N * z (N + 1) ≀ 0 ↔ ↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≀ ↑N * z (N + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„• β†’ β„€ h : StrictMono z h0 : 0 < z 0 N : β„• ⊒ 0 < d z N ∧ (range (N + 1)).sum z - ↑N * z (N + 1) ≀ 0 ↔ ↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≀ ↑N * z (N + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2008/A3/A3.lean
IMOSL.IMO2008A3.final_solution_part_2_general
[79, 1]
[94, 42]
rwa [prod_lex_lt_iff, Nat.succ_lt_succ_iff, Nat.succ_inj', ← prod_lex_lt_iff]
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p < q ⊒ (p.1.succ, p.2) < (q.1.succ, q.2)
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p < q ⊒ (p.1.succ, p.2) < (q.1.succ, q.2) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2008/A3/A3.lean
IMOSL.IMO2008A3.final_solution_part_2_general
[79, 1]
[94, 42]
rw [prod_lex_lt_iff] at h2 ⊒
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p < q ⊒ (p.1, Ο†^[3 ^ p.1] p.2) < (q.1, Ο†^[3 ^ q.1] q.2)
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2 ⊒ (p.1, Ο†^[3 ^ p.1] p.2).1 < (q.1, Ο†^[3 ^ q.1] q.2).1 ∨ (p.1, Ο†^[3 ^ p.1] p.2).1 = (q.1, Ο†^[3 ^ q.1] q.2).1 ∧ (p.1, Ο†^[3 ^ p.1] p.2).2 < (q.1, Ο†^[3 ^ q.1] q.2).2
Please generate a tactic in lean4 to solve the state. STATE: Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p < q ⊒ (p.1, Ο†^[3 ^ p.1] p.2) < (q.1, Ο†^[3 ^ q.1] q.2) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2008/A3/A3.lean
IMOSL.IMO2008A3.final_solution_part_2_general
[79, 1]
[94, 42]
refine h2.imp_right Ξ» h3 ↦ ⟨h3.1, (h.iterate _ h3.2).trans_eq ?_⟩
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2 ⊒ (p.1, Ο†^[3 ^ p.1] p.2).1 < (q.1, Ο†^[3 ^ q.1] q.2).1 ∨ (p.1, Ο†^[3 ^ p.1] p.2).1 = (q.1, Ο†^[3 ^ q.1] q.2).1 ∧ (p.1, Ο†^[3 ^ p.1] p.2).2 < (q.1, Ο†^[3 ^ q.1] q.2).2
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2 h3 : p.1 = q.1 ∧ p.2 < q.2 ⊒ Ο†^[3 ^ p.1] q.2 = (q.1, Ο†^[3 ^ q.1] q.2).2
Please generate a tactic in lean4 to solve the state. STATE: Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2 ⊒ (p.1, Ο†^[3 ^ p.1] p.2).1 < (q.1, Ο†^[3 ^ q.1] q.2).1 ∨ (p.1, Ο†^[3 ^ p.1] p.2).1 = (q.1, Ο†^[3 ^ q.1] q.2).1 ∧ (p.1, Ο†^[3 ^ p.1] p.2).2 < (q.1, Ο†^[3 ^ q.1] q.2).2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2008/A3/A3.lean
IMOSL.IMO2008A3.final_solution_part_2_general
[79, 1]
[94, 42]
rw [← h3.1]
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2 h3 : p.1 = q.1 ∧ p.2 < q.2 ⊒ Ο†^[3 ^ p.1] q.2 = (q.1, Ο†^[3 ^ q.1] q.2).2
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p q : Lex (β„• Γ— Ξ²) h2 : p.1 < q.1 ∨ p.1 = q.1 ∧ p.2 < q.2 h3 : p.1 = q.1 ∧ p.2 < q.2 ⊒ Ο†^[3 ^ p.1] q.2 = (q.1, Ο†^[3 ^ q.1] q.2).2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2008/A3/A3.lean
IMOSL.IMO2008A3.final_solution_part_2_general
[79, 1]
[94, 42]
refine prod_lex_lt_iff.mpr <| Or.inr <| ⟨rfl, ?_⟩
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).2) < ((p.1.succ, p.2).1, Ο†^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2)
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).2).2 < ((p.1.succ, p.2).1, Ο†^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2).2
Please generate a tactic in lean4 to solve the state. STATE: Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).2) < ((p.1.succ, p.2).1, Ο†^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2008/A3/A3.lean
IMOSL.IMO2008A3.final_solution_part_2_general
[79, 1]
[94, 42]
rw [← Function.iterate_add_apply, ← two_mul, pow_succ']
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).2).2 < ((p.1.succ, p.2).1, Ο†^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2).2
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[2 * 3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] p.2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[2 * 3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] p.2).2).2 < ((p.1.succ, p.2).1, Ο†^[3 * 3 ^ p.1] (p.1.succ, p.2).2).2
Please generate a tactic in lean4 to solve the state. STATE: Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] (p.1, Ο†^[3 ^ p.1] p.2).2).2).2 < ((p.1.succ, p.2).1, Ο†^[3 ^ (p.1.succ, p.2).1] (p.1.succ, p.2).2).2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2008/A3/A3.lean
IMOSL.IMO2008A3.final_solution_part_2_general
[79, 1]
[94, 42]
exact h.strictMono_iterate_of_lt_map (h0 p.2) (Nat.mul_lt_mul_of_pos_right (Nat.lt_succ_self 2) (pow_pos (Nat.succ_pos 2) _))
Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[2 * 3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] p.2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[2 * 3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] p.2).2).2 < ((p.1.succ, p.2).1, Ο†^[3 * 3 ^ p.1] (p.1.succ, p.2).2).2
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ² : Type u_1 Ο† : Ξ² β†’ Ξ² inst✝ : Preorder Ξ² h : StrictMono Ο† h0 : βˆ€ (x : Ξ²), x < Ο† x p : Lex (β„• Γ— Ξ²) ⊒ (((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[2 * 3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] p.2).1.succ, ((p.1, Ο†^[3 ^ p.1] p.2).1, Ο†^[2 * 3 ^ (p.1, Ο†^[3 ^ p.1] p.2).1] p.2).2).2 < ((p.1.succ, p.2).1, Ο†^[3 * 3 ^ p.1] (p.1.succ, p.2).2).2 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_nat_add
[131, 1]
[133, 80]
unfold XpowMul
m n : β„• P : 𝔽₂X ⊒ XpowMul (m + n) P = XpowMul n (XpowMul m P)
m n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun k => k + (m + n)) P.toFinset } = { toFinset := Finset.image (fun k => k + n) { toFinset := Finset.image (fun k => k + m) P.toFinset }.toFinset }
Please generate a tactic in lean4 to solve the state. STATE: m n : β„• P : 𝔽₂X ⊒ XpowMul (m + n) P = XpowMul n (XpowMul m P) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_nat_add
[131, 1]
[133, 80]
rw [𝔽₂X.ext_iff, eq_comm, Finset.image_image, comp_add_right]
m n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun k => k + (m + n)) P.toFinset } = { toFinset := Finset.image (fun k => k + n) { toFinset := Finset.image (fun k => k + m) P.toFinset }.toFinset }
no goals
Please generate a tactic in lean4 to solve the state. STATE: m n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun k => k + (m + n)) P.toFinset } = { toFinset := Finset.image (fun k => k + n) { toFinset := Finset.image (fun k => k + m) P.toFinset }.toFinset } TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_sum
[139, 1]
[142, 71]
rw [Finset.sum_insert h, Finset.sum_insert h, XpowMul_𝔽₂X_add, h0]
ΞΉ : Type u_1 n : β„• inst✝ : DecidableEq ΞΉ f : ΞΉ β†’ 𝔽₂X S✝ : Finset ΞΉ i : ΞΉ S : Finset ΞΉ h : i βˆ‰ S h0 : XpowMul n (S.sum f) = βˆ‘ i ∈ S, XpowMul n (f i) ⊒ XpowMul n ((insert i S).sum f) = βˆ‘ i ∈ insert i S, XpowMul n (f i)
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 n : β„• inst✝ : DecidableEq ΞΉ f : ΞΉ β†’ 𝔽₂X S✝ : Finset ΞΉ i : ΞΉ S : Finset ΞΉ h : i βˆ‰ S h0 : XpowMul n (S.sum f) = βˆ‘ i ∈ S, XpowMul n (f i) ⊒ XpowMul n ((insert i S).sum f) = βˆ‘ i ∈ insert i S, XpowMul n (f i) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.sum_Xpow_eq_ofFinset
[144, 1]
[147, 55]
rw [Finset.sum_insert h, h0]
i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ (insert i S).sum Xpow = ofFinset (insert i S)
i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ Xpow i + ofFinset S = ofFinset (insert i S)
Please generate a tactic in lean4 to solve the state. STATE: i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ (insert i S).sum Xpow = ofFinset (insert i S) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.sum_Xpow_eq_ofFinset
[144, 1]
[147, 55]
exact 𝔽₂X.ext _ _ (symmDiff_singleton_eq_insert h)
i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ Xpow i + ofFinset S = ofFinset (insert i S)
no goals
Please generate a tactic in lean4 to solve the state. STATE: i : β„• S : Finset β„• h : i βˆ‰ S h0 : S.sum Xpow = ofFinset S ⊒ Xpow i + ofFinset S = ofFinset (insert i S) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.Xpow_add
[188, 1]
[189, 53]
rw [← XpowMul_eq_Xpow_mul, XpowMul_Xpow, add_comm]
k m : β„• ⊒ Xpow (k + m) = Xpow k * Xpow m
no goals
Please generate a tactic in lean4 to solve the state. STATE: k m : β„• ⊒ Xpow (k + m) = Xpow k * Xpow m TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_eq_mul_Xpow
[191, 1]
[194, 59]
rw [XpowMul_Xpow, Xpow_add]
n k : β„• ⊒ XpowMul n (Xpow k) = Xpow k * Xpow n
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„• ⊒ XpowMul n (Xpow k) = Xpow k * Xpow n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.XpowMul_eq_mul_Xpow
[191, 1]
[194, 59]
rw [XpowMul_𝔽₂X_add, 𝔽₂X.add_mul, h, h0]
n : β„• P✝ Q✝ : 𝔽₂X h : XpowMul n P✝ = P✝ * Xpow n h0 : XpowMul n Q✝ = Q✝ * Xpow n ⊒ XpowMul n (P✝ + Q✝) = (P✝ + Q✝) * Xpow n
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P✝ Q✝ : 𝔽₂X h : XpowMul n P✝ = P✝ * Xpow n h0 : XpowMul n Q✝ = Q✝ * Xpow n ⊒ XpowMul n (P✝ + Q✝) = (P✝ + Q✝) * Xpow n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_comm
[196, 1]
[199, 55]
rw [← XpowMul_eq_Xpow_mul, XpowMul_eq_mul_Xpow]
P : 𝔽₂X n : β„• ⊒ P * Xpow n = Xpow n * P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• ⊒ P * Xpow n = Xpow n * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_comm
[196, 1]
[199, 55]
rw [𝔽₂X.add_mul, 𝔽₂X.mul_add, h, h0]
P P✝ Q✝ : 𝔽₂X h : P * P✝ = P✝ * P h0 : P * Q✝ = Q✝ * P ⊒ P * (P✝ + Q✝) = (P✝ + Q✝) * P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P P✝ Q✝ : 𝔽₂X h : P * P✝ = P✝ * P h0 : P * Q✝ = Q✝ * P ⊒ P * (P✝ + Q✝) = (P✝ + Q✝) * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_XpowMul_left
[201, 1]
[205, 72]
rw [← XpowMul_eq_mul_Xpow, ← XpowMul_eq_mul_Xpow, ← XpowMul_nat_add, ← XpowMul_nat_add, Nat.add_comm]
n : β„• P : 𝔽₂X k : β„• ⊒ XpowMul n P * Xpow k = XpowMul n (P * Xpow k)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P : 𝔽₂X k : β„• ⊒ XpowMul n P * Xpow k = XpowMul n (P * Xpow k) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_XpowMul_left
[201, 1]
[205, 72]
rw [𝔽₂X.mul_add, 𝔽₂X.mul_add, h, h0, XpowMul_𝔽₂X_add]
n : β„• P P✝ Q✝ : 𝔽₂X h : XpowMul n P * P✝ = XpowMul n (P * P✝) h0 : XpowMul n P * Q✝ = XpowMul n (P * Q✝) ⊒ XpowMul n P * (P✝ + Q✝) = XpowMul n (P * (P✝ + Q✝))
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P P✝ Q✝ : 𝔽₂X h : XpowMul n P * P✝ = XpowMul n (P * P✝) h0 : XpowMul n P * Q✝ = XpowMul n (P * Q✝) ⊒ XpowMul n P * (P✝ + Q✝) = XpowMul n (P * (P✝ + Q✝)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_XpowMul_right
[207, 1]
[208, 48]
rw [P.mul_comm, mul_XpowMul_left, Q.mul_comm]
n : β„• P Q : 𝔽₂X ⊒ P * XpowMul n Q = XpowMul n (P * Q)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P Q : 𝔽₂X ⊒ P * XpowMul n Q = XpowMul n (P * Q) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_assoc
[210, 1]
[213, 68]
rw [← XpowMul_eq_mul_Xpow, ← XpowMul_eq_mul_Xpow, mul_XpowMul_right]
P Q : 𝔽₂X n : β„• ⊒ P * Q * Xpow n = P * (Q * Xpow n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: P Q : 𝔽₂X n : β„• ⊒ P * Q * Xpow n = P * (Q * Xpow n) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.mul_assoc
[210, 1]
[213, 68]
rw [𝔽₂X.mul_add, 𝔽₂X.mul_add, 𝔽₂X.mul_add, h, h0]
P Q P✝ Q✝ : 𝔽₂X h : P * Q * P✝ = P * (Q * P✝) h0 : P * Q * Q✝ = P * (Q * Q✝) ⊒ P * Q * (P✝ + Q✝) = P * (Q * (P✝ + Q✝))
no goals
Please generate a tactic in lean4 to solve the state. STATE: P Q P✝ Q✝ : 𝔽₂X h : P * Q * P✝ = P * (Q * P✝) h0 : P * Q * Q✝ = P * (Q * Q✝) ⊒ P * Q * (P✝ + Q✝) = P * (Q * (P✝ + Q✝)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.square_eq_mul_self
[235, 1]
[238, 73]
rw [square_add, square_Xpow, 𝔽₂X.add_mul, 𝔽₂X.mul_add, ← Xpow_add, Nat.two_mul, 𝔽₂X.mul_add, ← h, ← P.mul_comm, CharTwo.add_add_add_cancel_middle]
n : β„• P : 𝔽₂X h : P.square = P * P ⊒ (Xpow n + P).square = (Xpow n + P) * (Xpow n + P)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P : 𝔽₂X h : P.square = P * P ⊒ (Xpow n + P).square = (Xpow n + P) * (Xpow n + P) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.square_XpowMul
[240, 1]
[243, 82]
unfold square XpowMul
n : β„• P : 𝔽₂X ⊒ (XpowMul n P).square = XpowMul (2 * n) P.square
n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun n => 2 * n) { toFinset := Finset.image (fun k => k + n) P.toFinset }.toFinset } = { toFinset := Finset.image (fun k => k + 2 * n) { toFinset := Finset.image (fun n => 2 * n) P.toFinset }.toFinset }
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P : 𝔽₂X ⊒ (XpowMul n P).square = XpowMul (2 * n) P.square TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.square_XpowMul
[240, 1]
[243, 82]
rw [Finset.image_image, Finset.image_image]
n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun n => 2 * n) { toFinset := Finset.image (fun k => k + n) P.toFinset }.toFinset } = { toFinset := Finset.image (fun k => k + 2 * n) { toFinset := Finset.image (fun n => 2 * n) P.toFinset }.toFinset }
n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image ((fun n => 2 * n) ∘ fun k => k + n) P.toFinset } = { toFinset := Finset.image ((fun k => k + 2 * n) ∘ fun n => 2 * n) P.toFinset }
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image (fun n => 2 * n) { toFinset := Finset.image (fun k => k + n) P.toFinset }.toFinset } = { toFinset := Finset.image (fun k => k + 2 * n) { toFinset := Finset.image (fun n => 2 * n) P.toFinset }.toFinset } TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.square_XpowMul
[240, 1]
[243, 82]
exact 𝔽₂X.ext _ _ (congrArg P.toFinset.image <| funext Ξ» n ↦ Nat.mul_add 2 _ _)
n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image ((fun n => 2 * n) ∘ fun k => k + n) P.toFinset } = { toFinset := Finset.image ((fun k => k + 2 * n) ∘ fun n => 2 * n) P.toFinset }
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P : 𝔽₂X ⊒ { toFinset := Finset.image ((fun n => 2 * n) ∘ fun k => k + n) P.toFinset } = { toFinset := Finset.image ((fun k => k + 2 * n) ∘ fun n => 2 * n) P.toFinset } TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.square_mul
[245, 1]
[249, 77]
rw [← XpowMul_eq_mul_Xpow, square_XpowMul, square_Xpow, XpowMul_eq_mul_Xpow]
P : 𝔽₂X n : β„• ⊒ (P * Xpow n).square = P.square * (Xpow n).square
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• ⊒ (P * Xpow n).square = P.square * (Xpow n).square TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.square_mul
[245, 1]
[249, 77]
rw [P.mul_add, square_add, h, h0, square_add, 𝔽₂X.mul_add]
P P✝ Q✝ : 𝔽₂X h : (P * P✝).square = P.square * P✝.square h0 : (P * Q✝).square = P.square * Q✝.square ⊒ (P * (P✝ + Q✝)).square = P.square * (P✝ + Q✝).square
no goals
Please generate a tactic in lean4 to solve the state. STATE: P P✝ Q✝ : 𝔽₂X h : (P * P✝).square = P.square * P✝.square h0 : (P * Q✝).square = P.square * Q✝.square ⊒ (P * (P✝ + Q✝)).square = P.square * (P✝ + Q✝).square TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_of_ne_zero
[266, 1]
[269, 28]
rw [𝔽₂X.natPow, if_neg h]
n : β„• P : 𝔽₂X h : n β‰  0 ⊒ P.natPow n = if n % 2 = 0 then P.square.natPow (n / 2) else P.square.natPow (n / 2) * P
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• P : 𝔽₂X h : n β‰  0 ⊒ P.natPow n = if n % 2 = 0 then P.square.natPow (n / 2) else P.square.natPow (n / 2) * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul
[271, 1]
[277, 62]
rcases Decidable.eq_or_ne n 0 with rfl | h
P : 𝔽₂X n : β„• ⊒ P.natPow (2 * n) = P.square.natPow n
case inl P : 𝔽₂X ⊒ P.natPow (2 * 0) = P.square.natPow 0 case inr P : 𝔽₂X n : β„• h : n β‰  0 ⊒ P.natPow (2 * n) = P.square.natPow n
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• ⊒ P.natPow (2 * n) = P.square.natPow n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul
[271, 1]
[277, 62]
rw [Nat.mul_zero, 𝔽₂X.natPow_zero, 𝔽₂X.natPow_zero]
case inl P : 𝔽₂X ⊒ P.natPow (2 * 0) = P.square.natPow 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl P : 𝔽₂X ⊒ P.natPow (2 * 0) = P.square.natPow 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul
[271, 1]
[277, 62]
have h0 : 0 < 2 := Nat.two_pos
case inr P : 𝔽₂X n : β„• h : n β‰  0 ⊒ P.natPow (2 * n) = P.square.natPow n
case inr P : 𝔽₂X n : β„• h : n β‰  0 h0 : 0 < 2 ⊒ P.natPow (2 * n) = P.square.natPow n
Please generate a tactic in lean4 to solve the state. STATE: case inr P : 𝔽₂X n : β„• h : n β‰  0 ⊒ P.natPow (2 * n) = P.square.natPow n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul
[271, 1]
[277, 62]
rw [P.natPow_of_ne_zero (Nat.mul_ne_zero h0.ne.symm h), if_pos (Nat.mul_mod_right _ _), Nat.mul_div_right _ h0]
case inr P : 𝔽₂X n : β„• h : n β‰  0 h0 : 0 < 2 ⊒ P.natPow (2 * n) = P.square.natPow n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr P : 𝔽₂X n : β„• h : n β‰  0 h0 : 0 < 2 ⊒ P.natPow (2 * n) = P.square.natPow n TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul_add_one
[279, 1]
[283, 51]
rw [P.natPow_of_ne_zero (2 * n).add_one_ne_zero, Nat.mul_add_mod, Nat.mul_add_div Nat.two_pos, if_neg Nat.one_ne_zero, Nat.div_eq_of_lt Nat.one_lt_two, Nat.add_zero]
P : 𝔽₂X n : β„• ⊒ P.natPow (2 * n + 1) = P.square.natPow n * P
no goals
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• ⊒ P.natPow (2 * n + 1) = P.square.natPow n * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
rw [← n.div_add_mod 2]
P : 𝔽₂X n : β„• ⊒ P.natPow n.succ = P.natPow n * P
P : 𝔽₂X n : β„• ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• ⊒ P.natPow n.succ = P.natPow n * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
rcases n.mod_two_eq_zero_or_one with h0 | h0
P : 𝔽₂X n : β„• ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
case inl P : 𝔽₂X n : β„• h0 : n % 2 = 0 ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P case inr P : 𝔽₂X n : β„• h0 : n % 2 = 1 ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
rw [h0, Nat.add_zero, 𝔽₂X.natPow_two_mul_add_one, 𝔽₂X.natPow_two_mul]
case inl P : 𝔽₂X n : β„• h0 : n % 2 = 0 ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl P : 𝔽₂X n : β„• h0 : n % 2 = 0 ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
rw [h0, Nat.succ_eq_add_one, ← Nat.mul_succ 2, 𝔽₂X.natPow_two_mul, 𝔽₂X.natPow_two_mul_add_one, 𝔽₂X.mul_assoc, ← P.square_eq_mul_self]
case inr P : 𝔽₂X n : β„• h0 : n % 2 = 1 ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
case inr P : 𝔽₂X n : β„• h0 : n % 2 = 1 ⊒ P.square.natPow (n / 2).succ = P.square.natPow (n / 2) * P.square
Please generate a tactic in lean4 to solve the state. STATE: case inr P : 𝔽₂X n : β„• h0 : n % 2 = 1 ⊒ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
exact (square P).natPow_succ (n / 2)
case inr P : 𝔽₂X n : β„• h0 : n % 2 = 1 ⊒ P.square.natPow (n / 2).succ = P.square.natPow (n / 2) * P.square
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr P : 𝔽₂X n : β„• h0 : n % 2 = 1 ⊒ P.square.natPow (n / 2).succ = P.square.natPow (n / 2) * P.square TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
apply Nat.bitwise_rec_lemma
P : 𝔽₂X n : β„• a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : n % 2 = 1 ⊒ (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 ⟨P.square, n / 2⟩ ⟨P, n⟩
case hNe P : 𝔽₂X n : β„• a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : n % 2 = 1 ⊒ n β‰  0
Please generate a tactic in lean4 to solve the state. STATE: P : 𝔽₂X n : β„• a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : n % 2 = 1 ⊒ (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 ⟨P.square, n / 2⟩ ⟨P, n⟩ TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
rintro rfl
case hNe P : 𝔽₂X n : β„• a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : n % 2 = 1 ⊒ n β‰  0
case hNe P : 𝔽₂X a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, 0⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : 0 % 2 = 1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case hNe P : 𝔽₂X n : β„• a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : n % 2 = 1 ⊒ n β‰  0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/Extra/F2X/Defs.lean
IMOSL.IMO2012A5.𝔽₂X.natPow_succ
[291, 1]
[299, 63]
exact absurd h0.symm Nat.one_ne_zero
case hNe P : 𝔽₂X a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, 0⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : 0 % 2 = 1 ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hNe P : 𝔽₂X a✝ : βˆ€ (y : (_ : 𝔽₂X) Γ—' β„•), (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, 0⟩ β†’ y.1.natPow y.2.succ = y.1.natPow y.2 * y.1 h0 : 0 % 2 = 1 ⊒ False TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean
IMOSL.IMO2012A5.zero_is_good
[21, 1]
[22, 37]
rw [add_zero, zero_mul]
R : Type u_1 S : Type u_2 inst✝¹ : NonAssocSemiring R inst✝ : NonAssocSemiring S x✝¹ x✝ : R ⊒ (fun x => 0) (x✝¹ * x✝ + 1) = (fun x => 0) x✝¹ * (fun x => 0) x✝ + (fun x => 0) (x✝¹ + x✝)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_1 S : Type u_2 inst✝¹ : NonAssocSemiring R inst✝ : NonAssocSemiring S x✝¹ x✝ : R ⊒ (fun x => 0) (x✝¹ * x✝ + 1) = (fun x => 0) x✝¹ * (fun x => 0) x✝ + (fun x => 0) (x✝¹ + x✝) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean
IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero
[24, 1]
[34, 89]
rw [zero_add, ← h, one_mul]
R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f ⊒ 0 + f (1 + 1) = f 1 * f 1 + f (1 + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f ⊒ 0 + f (1 + 1) = f 1 * f 1 + f (1 + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean
IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero
[24, 1]
[34, 89]
have h1 := h x 0
R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 x : R ⊒ 0 = f x * (f 0 + 1)
R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 x : R h1 : f (x * 0 + 1) = f x * f 0 + f (x + 0) ⊒ 0 = f x * (f 0 + 1)
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 x : R ⊒ 0 = f x * (f 0 + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean
IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero
[24, 1]
[34, 89]
rwa [mul_zero, zero_add, h0, add_zero, ← mul_add_one (f x)] at h1
R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 x : R h1 : f (x * 0 + 1) = f x * f 0 + f (x + 0) ⊒ 0 = f x * (f 0 + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 x : R h1 : f (x * 0 + 1) = f x * f 0 + f (x + 0) ⊒ 0 = f x * (f 0 + 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean
IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero
[24, 1]
[34, 89]
specialize h1 x
R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 h1 : βˆ€ (x : R), 0 = f x * (f 0 + 1) h2 : f 0 = 0 x : R ⊒ f x = 0
R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 h2 : f 0 = 0 x : R h1 : 0 = f x * (f 0 + 1) ⊒ f x = 0
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 h1 : βˆ€ (x : R), 0 = f x * (f 0 + 1) h2 : f 0 = 0 x : R ⊒ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean
IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero
[24, 1]
[34, 89]
rwa [h2, zero_add, mul_one, eq_comm] at h1
R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 h2 : f 0 = 0 x : R h1 : 0 = f x * (f 0 + 1) ⊒ f x = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: R : Type u_2 S : Type u_1 inst✝³ : NonAssocSemiring R inst✝² : NonAssocSemiring S inst✝¹ : IsCancelAdd S inst✝ : NoZeroDivisors S f : R β†’ S h : good f h0 : f 1 = 0 h2 : f 0 = 0 x : R h1 : 0 = f x * (f 0 + 1) ⊒ f x = 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.exists_ne_pow_eq
[24, 1]
[31, 22]
obtain ⟨m, -, n, -, h, h0⟩ := Set.infinite_univ.exists_ne_map_eq_of_mapsTo h0 (Set.toFinite _)
k : β„€ h : k β‰  0 b : β„€ h0 : Set.MapsTo (fun x => b ^ x % k) Set.univ ↑(Finset.Ico 0 |k|) ⊒ βˆƒ m n, m β‰  n ∧ b ^ m % k = b ^ n % k
case intro.intro.intro.intro.intro k : β„€ h✝ : k β‰  0 b : β„€ h0✝ : Set.MapsTo (fun x => b ^ x % k) Set.univ ↑(Finset.Ico 0 |k|) m n : β„• h : m β‰  n h0 : b ^ m % k = b ^ n % k ⊒ βˆƒ m n, m β‰  n ∧ b ^ m % k = b ^ n % k
Please generate a tactic in lean4 to solve the state. STATE: k : β„€ h : k β‰  0 b : β„€ h0 : Set.MapsTo (fun x => b ^ x % k) Set.univ ↑(Finset.Ico 0 |k|) ⊒ βˆƒ m n, m β‰  n ∧ b ^ m % k = b ^ n % k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.exists_ne_pow_eq
[24, 1]
[31, 22]
exact ⟨m, n, h, h0⟩
case intro.intro.intro.intro.intro k : β„€ h✝ : k β‰  0 b : β„€ h0✝ : Set.MapsTo (fun x => b ^ x % k) Set.univ ↑(Finset.Ico 0 |k|) m n : β„• h : m β‰  n h0 : b ^ m % k = b ^ n % k ⊒ βˆƒ m n, m β‰  n ∧ b ^ m % k = b ^ n % k
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro k : β„€ h✝ : k β‰  0 b : β„€ h0✝ : Set.MapsTo (fun x => b ^ x % k) Set.univ ↑(Finset.Ico 0 |k|) m n : β„• h : m β‰  n h0 : b ^ m % k = b ^ n % k ⊒ βˆƒ m n, m β‰  n ∧ b ^ m % k = b ^ n % k TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.exists_ne_pow_eq
[24, 1]
[31, 22]
rw [Finset.coe_Ico, Set.mem_Ico]
k : β„€ h : k β‰  0 b : β„€ x : β„• x✝ : x ∈ Set.univ ⊒ (fun x => b ^ x % k) x ∈ ↑(Finset.Ico 0 |k|)
k : β„€ h : k β‰  0 b : β„€ x : β„• x✝ : x ∈ Set.univ ⊒ 0 ≀ (fun x => b ^ x % k) x ∧ (fun x => b ^ x % k) x < |k|
Please generate a tactic in lean4 to solve the state. STATE: k : β„€ h : k β‰  0 b : β„€ x : β„• x✝ : x ∈ Set.univ ⊒ (fun x => b ^ x % k) x ∈ ↑(Finset.Ico 0 |k|) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.exists_ne_pow_eq
[24, 1]
[31, 22]
exact ⟨(b ^ x).emod_nonneg h, (b ^ x).emod_lt h⟩
k : β„€ h : k β‰  0 b : β„€ x : β„• x✝ : x ∈ Set.univ ⊒ 0 ≀ (fun x => b ^ x % k) x ∧ (fun x => b ^ x % k) x < |k|
no goals
Please generate a tactic in lean4 to solve the state. STATE: k : β„€ h : k β‰  0 b : β„€ x : β„• x✝ : x ∈ Set.univ ⊒ 0 ≀ (fun x => b ^ x % k) x ∧ (fun x => b ^ x % k) x < |k| TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.linear_good'
[44, 1]
[46, 74]
rw [add_sub_add_right_eq_sub, mul_add, add_sub_cancel_left, add_one_mul (Ξ± := β„€), add_sub_add_right_eq_sub, ← mul_sub, add_sub_cancel_right, ← mul_add]
k m x y : β„€ ⊒ (fun x => k * x + m) (y + (fun x => k * x + m) x) - (fun x => k * x + m) y = (fun x => k * x + m) ((k + 1) * x) - (fun x => k * x + m) x + k * m
no goals
Please generate a tactic in lean4 to solve the state. STATE: k m x y : β„€ ⊒ (fun x => k * x + m) (y + (fun x => k * x + m) x) - (fun x => k * x + m) y = (fun x => k * x + m) ((k + 1) * x) - (fun x => k * x + m) x + k * m TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.linear_good
[48, 1]
[50, 43]
nth_rw 1 [← sub_add_cancel b 1, ← Int.mul_ediv_cancel' h]
b c : β„€ h : b - 1 ∣ c ⊒ good b c fun x => (b - 1) * x + c / (b - 1)
b c : β„€ h : b - 1 ∣ c ⊒ good (b - 1 + 1) ((b - 1) * (c / (b - 1))) fun x => (b - 1) * x + c / (b - 1)
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ h : b - 1 ∣ c ⊒ good b c fun x => (b - 1) * x + c / (b - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.linear_good
[48, 1]
[50, 43]
exact linear_good' (b - 1) (c / (b - 1))
b c : β„€ h : b - 1 ∣ c ⊒ good (b - 1 + 1) ((b - 1) * (c / (b - 1))) fun x => (b - 1) * x + c / (b - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ h : b - 1 ∣ c ⊒ good (b - 1 + 1) ((b - 1) * (c / (b - 1))) fun x => (b - 1) * x + c / (b - 1) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_map_zero_add
[58, 1]
[59, 63]
rw [← sub_eq_iff_eq_add, h, mul_zero, sub_self, zero_add]
b c : β„€ f : β„€ β†’ β„€ h : good b c f y : β„€ ⊒ f (y + f 0) = c + f y
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 y : β„€ ⊒ f (y + f 0) = c + f y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_mul_map_zero_add
[61, 1]
[65, 33]
have h0 n : f (y + f 0 * (n + 1)) = c + f (y + f 0 * n) := by rw [mul_add_one (Ξ± := β„€), ← add_assoc, map_map_zero_add h]
b c : β„€ f : β„€ β†’ β„€ h : good b c f y k : β„€ ⊒ f (y + f 0 * k) = c * k + f y
b c : β„€ f : β„€ β†’ β„€ h : good b c f y k : β„€ h0 : βˆ€ (n : β„€), f (y + f 0 * (n + 1)) = c + f (y + f 0 * n) ⊒ f (y + f 0 * k) = c * k + f y
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ f : β„€ β†’ β„€ h : good b c f y k : β„€ ⊒ f (y + f 0 * k) = c * k + f y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_mul_map_zero_add
[61, 1]
[65, 33]
replace h0 := Extra.IntIntLinearSolverAlt (f := Ξ» n ↦ f (y + f 0 * n)) h0 k
b c : β„€ f : β„€ β†’ β„€ h : good b c f y k : β„€ h0 : βˆ€ (n : β„€), f (y + f 0 * (n + 1)) = c + f (y + f 0 * n) ⊒ f (y + f 0 * k) = c * k + f y
b c : β„€ f : β„€ β†’ β„€ h : good b c f y k : β„€ h0 : f (y + f 0 * k) = c * k + f (y + f 0 * 0) ⊒ f (y + f 0 * k) = c * k + f y
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ f : β„€ β†’ β„€ h : good b c f y k : β„€ h0 : βˆ€ (n : β„€), f (y + f 0 * (n + 1)) = c + f (y + f 0 * n) ⊒ f (y + f 0 * k) = c * k + f y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_mul_map_zero_add
[61, 1]
[65, 33]
rwa [mul_zero, add_zero] at h0
b c : β„€ f : β„€ β†’ β„€ h : good b c f y k : β„€ h0 : f (y + f 0 * k) = c * k + f (y + f 0 * 0) ⊒ f (y + f 0 * k) = c * k + f y
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 y k : β„€ h0 : f (y + f 0 * k) = c * k + f (y + f 0 * 0) ⊒ f (y + f 0 * k) = c * k + f y TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_mul_map_zero_add
[61, 1]
[65, 33]
rw [mul_add_one (Ξ± := β„€), ← add_assoc, map_map_zero_add h]
b c : β„€ f : β„€ β†’ β„€ h : good b c f y k n : β„€ ⊒ f (y + f 0 * (n + 1)) = c + f (y + f 0 * n)
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 y k n : β„€ ⊒ f (y + f 0 * (n + 1)) = c + f (y + f 0 * n) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_b_pow_mul_eq_of_map_eq
[67, 1]
[74, 45]
rwa [pow_zero, one_mul, one_mul]
b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y ⊒ f (b ^ 0 * x) = f (b ^ 0 * y)
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 x y : β„€ h0 : f x = f y ⊒ f (b ^ 0 * x) = f (b ^ 0 * y) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_b_pow_mul_eq_of_map_eq
[67, 1]
[74, 45]
rw [pow_succ', mul_assoc, mul_assoc]
b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y k : β„• ⊒ f (b ^ (k + 1) * x) = f (b ^ (k + 1) * y)
b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y k : β„• ⊒ f (b * (b ^ k * x)) = f (b * (b ^ k * y))
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y k : β„• ⊒ f (b ^ (k + 1) * x) = f (b ^ (k + 1) * y) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_b_pow_mul_eq_of_map_eq
[67, 1]
[74, 45]
have h1 := h (b ^ k * y) 0
b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y k : β„• ⊒ f (b * (b ^ k * x)) = f (b * (b ^ k * y))
b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y k : β„• h1 : f (0 + f (b ^ k * y)) - f 0 = f (b * (b ^ k * y)) - f (b ^ k * y) + c ⊒ f (b * (b ^ k * x)) = f (b * (b ^ k * y))
Please generate a tactic in lean4 to solve the state. STATE: b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y k : β„• ⊒ f (b * (b ^ k * x)) = f (b * (b ^ k * y)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_b_pow_mul_eq_of_map_eq
[67, 1]
[74, 45]
rwa [← map_b_pow_mul_eq_of_map_eq h0 k, h, add_left_inj, sub_left_inj] at h1
b c : β„€ f : β„€ β†’ β„€ h : good b c f x y : β„€ h0 : f x = f y k : β„• h1 : f (0 + f (b ^ k * y)) - f 0 = f (b * (b ^ k * y)) - f (b ^ k * y) + c ⊒ f (b * (b ^ k * x)) = f (b * (b ^ k * y))
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 x y : β„€ h0 : f x = f y k : β„• h1 : f (0 + f (b ^ k * y)) - f 0 = f (b * (b ^ k * y)) - f (b ^ k * y) + c ⊒ f (b * (b ^ k * x)) = f (b * (b ^ k * y)) TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_is_linear
[78, 1]
[103, 43]
suffices f.Injective by intro n have h2 := eq_add_of_sub_eq' (h 0 (b * n)) rw [mul_zero, sub_self, zero_add, ← sub_left_inj (a := f n), add_sub_right_comm, ← h n n, sub_left_inj] at h2 rw [sub_one_mul, ← add_sub_right_comm, this h2, add_sub_cancel_left]
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 ⊒ βˆ€ (n : β„€), f n = (b - 1) * n + f 0
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 ⊒ Function.Injective f
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 ⊒ βˆ€ (n : β„€), f n = (b - 1) * n + f 0 TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_is_linear
[78, 1]
[103, 43]
have h2 : f 0 β‰  0 := Ξ» h2 ↦ by have h3 := map_map_zero_add h 0 rw [zero_add, h2, h2, add_zero] at h3 exact h1 h3.symm
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 ⊒ Function.Injective f
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 ⊒ Function.Injective f
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 ⊒ Function.Injective f TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_is_linear
[78, 1]
[103, 43]
obtain ⟨m, n, h3, h4⟩ := exists_ne_pow_eq h2 b
b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 ⊒ Function.Injective f
case intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n h4 : b ^ m % f 0 = b ^ n % f 0 ⊒ Function.Injective f
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 h2 : f 0 β‰  0 ⊒ Function.Injective f TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_is_linear
[78, 1]
[103, 43]
rw [Int.emod_eq_emod_iff_emod_sub_eq_zero, ← Int.dvd_iff_emod_eq_zero] at h4
case intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n h4 : b ^ m % f 0 = b ^ n % f 0 ⊒ Function.Injective f
case intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n h4 : f 0 ∣ b ^ m - b ^ n ⊒ Function.Injective f
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n h4 : b ^ m % f 0 = b ^ n % f 0 ⊒ Function.Injective f TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_is_linear
[78, 1]
[103, 43]
rcases h4 with ⟨N, h4⟩
case intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n h4 : f 0 ∣ b ^ m - b ^ n ⊒ Function.Injective f
case intro.intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n N : β„€ h4 : b ^ m - b ^ n = f 0 * N ⊒ Function.Injective f
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n h4 : f 0 ∣ b ^ m - b ^ n ⊒ Function.Injective f TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_is_linear
[78, 1]
[103, 43]
intro x y h5
case intro.intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n N : β„€ h4 : b ^ m - b ^ n = f 0 * N ⊒ Function.Injective f
case intro.intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n N : β„€ h4 : b ^ m - b ^ n = f 0 * N x y : β„€ h5 : f x = f y ⊒ x = y
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n N : β„€ h4 : b ^ m - b ^ n = f 0 * N ⊒ Function.Injective f TACTIC:
https://github.com/mortarsanjaya/IMOSLLean4.git
be127d301e366822fbeeeda49d9fd5b998fb4eb5
IMOSLLean4/IMO2014/A4/A4.lean
IMOSL.IMO2014A4.map_is_linear
[78, 1]
[103, 43]
apply map_b_pow_mul_eq_of_map_eq h at h5
case intro.intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n N : β„€ h4 : b ^ m - b ^ n = f 0 * N x y : β„€ h5 : f x = f y ⊒ x = y
case intro.intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n N : β„€ h4 : b ^ m - b ^ n = f 0 * N x y : β„€ h5 : βˆ€ (k : β„•), f (b ^ k * x) = f (b ^ k * y) ⊒ x = y
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro b c : β„€ f : β„€ β†’ β„€ h : good b c f h0 : 1 < b.natAbs h1 : c β‰  0 h2 : f 0 β‰  0 m n : β„• h3 : m β‰  n N : β„€ h4 : b ^ m - b ^ n = f 0 * N x y : β„€ h5 : f x = f y ⊒ x = y TACTIC: