url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | rw [sq, ← Int.fract_add_floor (f^[k + N] r), h, Int.cast_neg, Int.cast_natCast, mul_add,
mul_neg, add_one_mul (C : R) C, add_assoc, neg_add_rev, neg_add_cancel_right] at h0 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
k : ℕ
h0 : (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε| | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
k : ℕ
h0 : (↑C + 1) * Int.fract (f^[k + N] r) + -↑C = (-↑C) ^ k * ε
⊢ |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε| | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
k : ℕ
h0 : (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | rw [sub_eq_add_neg, h0, abs_mul, abs_pow, abs_neg,
Nat.abs_cast, nsmul_eq_mul, Nat.cast_pow] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
k : ℕ
h0 : (↑C + 1) * Int.fract (f^[k + N] r) + -↑C = (-↑C) ^ k * ε
⊢ |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
k : ℕ
h0 : (↑C + 1) * Int.fract (f^[k + N] r) + -↑C = (-↑C) ^ k * ε
⊢ |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | apply (abs_sub _ _).trans_lt | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ |(↑C + 1) * Int.fract s - ↑C| < ↑(C + 1 + C) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ |(↑C + 1) * Int.fract s| + |↑C| < ↑(C + 1 + C) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ |(↑C + 1) * Int.fract s - ↑C| < ↑(C + 1 + C)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | rw [Nat.abs_cast, Nat.cast_add, add_lt_add_iff_right, ← Nat.cast_succ,
abs_mul, Nat.abs_cast, abs_eq_self.mpr (Int.fract_nonneg s)] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ |(↑C + 1) * Int.fract s| + |↑C| < ↑(C + 1 + C) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ ↑C.succ * Int.fract s < ↑(C + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ |(↑C + 1) * Int.fract s| + |↑C| < ↑(C + 1 + C)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | refine mul_lt_of_lt_one_right (Nat.cast_pos.mpr C.succ_pos) (Int.fract_lt_one _) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ ↑C.succ * Int.fract s < ↑(C + 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
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -↑C
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
s : R
⊢ ↑C.succ * Int.fract s < ↑(C + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | rcases floor_f_iter_eventually_const r with ⟨C, h⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ (EventuallyEqual (fun x => f^[x] r) fun x => 0) ∨
(∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])) ∨
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
⊢ (EventuallyEqual (fun x => f^[x] r) fun x => 0) ∨
(∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])) ∨
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ (EventuallyEqual (fun x => f^[x] r) fun x => 0) ∨
(∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])) ∨
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | refine C.eq_zero_or_pos.imp ?_ (λ h0 ↦ (h0 : 1 ≤ C).eq_or_lt.imp ?_ ?_) | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
⊢ (EventuallyEqual (fun x => f^[x] r) fun x => 0) ∨
(∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])) ∨
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
⊢ C = 0 → EventuallyEqual (fun x => f^[x] r) fun x => 0
case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0 : 1 ≤ C
⊢ 1 = C → ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])
case intro.refine_3
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0 : 1 ≤ C
⊢ 1 < C →
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
⊢ (EventuallyEqual (fun x => f^[x] r) fun x => 0) ∨
(∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])) ∨
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | rintro rfl | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
⊢ C = 0 → EventuallyEqual (fun x => f^[x] r) fun x => 0 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑0
⊢ EventuallyEqual (fun x => f^[x] r) fun x => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
⊢ C = 0 → EventuallyEqual (fun x => f^[x] r) fun x => 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | exact case_floor_eventually_zero h | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑0
⊢ EventuallyEqual (fun x => f^[x] r) fun x => 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑0
⊢ EventuallyEqual (fun x => f^[x] r) fun x => 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | rintro rfl | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0 : 1 ≤ C
⊢ 1 = C → ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1]) | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑1
h0 : 1 ≤ 1
⊢ ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1]) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0 : 1 ≤ C
⊢ 1 = C → ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | exact case_floor_eventually_neg_one h | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑1
h0 : 1 ≤ 1
⊢ ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1]) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑1
h0 : 1 ≤ 1
⊢ ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1])
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | intro h0 | case intro.refine_3
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0 : 1 ≤ C
⊢ 1 < C →
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.refine_3
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0✝ : 1 ≤ C
h0 : 1 < C
⊢ ∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_3
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0 : 1 ≤ C
⊢ 1 < C →
∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution_general | [151, 1] | [163, 70] | exact ⟨C, h0, case_floor_eventually_neg_of_one_lt h0 h⟩ | case intro.refine_3
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0✝ : 1 ≤ C
h0 : 1 < C
⊢ ∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_3
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
h0✝ : 1 ≤ C
h0 : 1 < C
⊢ ∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.Archimedean_f_iter_classification | [165, 1] | [172, 21] | rintro ⟨C, h, ε, h0, h1⟩ | R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
⊢ (∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε) →
∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 | case intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
h : 1 < C
ε : R
h0 : Infinitesimal ε
h1 : EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε
⊢ ∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
⊢ (∃ C, 1 < C ∧ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε) →
∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.Archimedean_f_iter_classification | [165, 1] | [172, 21] | simp only [h0.zero_of_Archimedean, mul_zero, add_zero] at h1 | case intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
h : 1 < C
ε : R
h0 : Infinitesimal ε
h1 : EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε
⊢ ∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 | case intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
h : 1 < C
ε : R
h0 : Infinitesimal ε
h1 : EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2
⊢ ∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
h : 1 < C
ε : R
h0 : Infinitesimal ε
h1 : EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε
⊢ ∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.Archimedean_f_iter_classification | [165, 1] | [172, 21] | exact ⟨C, h, h1⟩ | case intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
h : 1 < C
ε : R
h0 : Infinitesimal ε
h1 : EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2
⊢ ∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
h : 1 < C
ε : R
h0 : Infinitesimal ε
h1 : EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2
⊢ ∃ C, 1 < C ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | rcases Archimedean_f_iter_classification r with
⟨N, _, h⟩ | ⟨s, -, N, M, h⟩ | ⟨C, hC, N, _, h⟩ | R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r | case inl.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => f^[x] r) (k + N) = (fun x => 0) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
case inr.inl.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), (fun x => f^[x] r) (k + N) = NatSeq_ofList [-s, s - 1] (k + M)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => (↑C + 1) * f^[x] r) (k + N) = (fun x => -↑C ^ 2) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | all_goals
simp only at h; refine ⟨N, λ k h0 ↦ ?_⟩
rcases Nat.exists_eq_add_of_le' h0 with ⟨k, rfl⟩ | case inl.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => f^[x] r) (k + N) = (fun x => 0) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
case inr.inl.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), (fun x => f^[x] r) (k + N) = NatSeq_ofList [-s, s - 1] (k + M)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => (↑C + 1) * f^[x] r) (k + N) = (fun x => -↑C ^ 2) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r | case inl.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
N w✝ : ℕ
h : ∀ (k : ℕ), f^[k + N] r = 0
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r
case inr.inl.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), f^[k + N] r = NatSeq_ofList [-s, s - 1] (k + M)
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r
case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => f^[x] r) (k + N) = (fun x => 0) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
case inr.inl.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), (fun x => f^[x] r) (k + N) = NatSeq_ofList [-s, s - 1] (k + M)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => (↑C + 1) * f^[x] r) (k + N) = (fun x => -↑C ^ 2) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | simp only at h | case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => (↑C + 1) * f^[x] r) (k + N) = (fun x => -↑C ^ 2) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r | case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => (↑C + 1) * f^[x] r) (k + N) = (fun x => -↑C ^ 2) (k + w✝)
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | refine ⟨N, λ k h0 ↦ ?_⟩ | case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r | case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : N ≤ k
⊢ f^[k + 2] r = f^[k] r | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
⊢ ∃ N, ∀ (k : ℕ), N ≤ k → f^[k + 2] r = f^[k] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | rcases Nat.exists_eq_add_of_le' h0 with ⟨k, rfl⟩ | case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : N ≤ k
⊢ f^[k + 2] r = f^[k] r | case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : N ≤ k
⊢ f^[k + 2] r = f^[k] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | rw [h, Nat.add_right_comm, h] | case inl.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
N w✝ : ℕ
h : ∀ (k : ℕ), f^[k + N] r = 0
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
N w✝ : ℕ
h : ∀ (k : ℕ), f^[k + N] r = 0
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | rw [h, Nat.add_right_comm, h, Nat.add_right_comm] | case inr.inl.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), f^[k + N] r = NatSeq_ofList [-s, s - 1] (k + M)
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r | case inr.inl.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), f^[k + N] r = NatSeq_ofList [-s, s - 1] (k + M)
k : ℕ
h0 : N ≤ k + N
⊢ NatSeq_ofList [-s, s - 1] (k + M + 2) = NatSeq_ofList [-s, s - 1] (k + M) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), f^[k + N] r = NatSeq_ofList [-s, s - 1] (k + M)
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | exact NatSeq_ofList_periodic [-s, s - 1] _ | case inr.inl.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), f^[k + N] r = NatSeq_ofList [-s, s - 1] (k + M)
k : ℕ
h0 : N ≤ k + N
⊢ NatSeq_ofList [-s, s - 1] (k + M + 2) = NatSeq_ofList [-s, s - 1] (k + M) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r s : R
N M : ℕ
h : ∀ (k : ℕ), f^[k + N] r = NatSeq_ofList [-s, s - 1] (k + M)
k : ℕ
h0 : N ≤ k + N
⊢ NatSeq_ofList [-s, s - 1] (k + M + 2) = NatSeq_ofList [-s, s - 1] (k + M)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | replace h0 := (h (k + 2)).trans (h k).symm | case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r | case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : (↑C + 1) * f^[k + 2 + N] r = (↑C + 1) * f^[k + N] r
⊢ f^[k + N + 2] r = f^[k + N] r | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : N ≤ k + N
⊢ f^[k + N + 2] r = f^[k + N] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | rw [Nat.add_right_comm, mul_eq_mul_left_iff] at h0 | case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : (↑C + 1) * f^[k + 2 + N] r = (↑C + 1) * f^[k + N] r
⊢ f^[k + N + 2] r = f^[k + N] r | case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : f^[k + N + 2] r = f^[k + N] r ∨ ↑C + 1 = 0
⊢ f^[k + N + 2] r = f^[k + N] r | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : (↑C + 1) * f^[k + 2 + N] r = (↑C + 1) * f^[k + N] r
⊢ f^[k + N + 2] r = f^[k + N] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | exact h0.resolve_right (add_pos (Nat.cast_pos.mpr (pos_of_gt hC)) one_pos).ne.symm | case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : f^[k + N + 2] r = f^[k + N] r ∨ ↑C + 1 = 0
⊢ f^[k + N + 2] r = f^[k + N] r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : f^[k + N + 2] r = f^[k + N] r ∨ ↑C + 1 = 0
⊢ f^[k + N + 2] r = f^[k + N] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.Semiring_of_two_eq_zero | [21, 1] | [22, 41] | rw [← two_mul, h, zero_mul] | R : Type u_1
inst✝ : NonAssocSemiring R
h : 2 = 0
x : R
⊢ x + x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocSemiring R
h : 2 = 0
x : R
⊢ x + x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.two_eq_zero | [29, 1] | [30, 46] | rw [← one_add_one_eq_two, add_self_eq_zero] | R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : CharTwo R
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : CharTwo R
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.add_mul_self_of_Commute | [32, 1] | [34, 71] | rw [add_mul, mul_add, mul_add, ← add_assoc, h, add_add_cancel_right] | R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : CharTwo R
x y : R
h : x * y = y * x
⊢ (x + y) * (x + y) = x * x + y * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : CharTwo R
x y : R
h : x * y = y * x
⊢ (x + y) * (x + y) = x * x + y * y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.add_one_mul_self | [36, 1] | [37, 77] | rw [add_mul_self_of_Commute ((mul_one x).trans (one_mul x).symm), one_mul] | R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : CharTwo R
x : R
⊢ (x + 1) * (x + 1) = x * x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : CharTwo R
x : R
⊢ (x + 1) * (x + 1) = x * x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.mul_self_eq_one_iff | [39, 1] | [40, 86] | rw [← add_eq_zero_iff_eq, ← add_one_mul_self, mul_self_eq_zero, add_eq_zero_iff_eq] | R : Type u_1
inst✝² : NonAssocSemiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x : R
⊢ x * x = 1 ↔ x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : NonAssocSemiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x : R
⊢ x * x = 1 ↔ x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.add_sq_of_Commute | [54, 1] | [55, 45] | rw [sq, sq, sq, add_mul_self_of_Commute h] | R : Type u_1
inst✝¹ : Semiring R
inst✝ : CharTwo R
x y : R
h : x * y = y * x
⊢ (x + y) ^ 2 = x ^ 2 + y ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : CharTwo R
x y : R
h : x * y = y * x
⊢ (x + y) ^ 2 = x ^ 2 + y ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.add_one_sq | [57, 1] | [58, 32] | rw [sq, sq, add_one_mul_self] | R : Type u_1
inst✝¹ : Semiring R
inst✝ : CharTwo R
x : R
⊢ (x + 1) ^ 2 = x ^ 2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : CharTwo R
x : R
⊢ (x + 1) ^ 2 = x ^ 2 + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.sq_eq_one_iff | [60, 1] | [61, 31] | rw [sq, mul_self_eq_one_iff] | R : Type u_1
inst✝² : Semiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x : R
⊢ x ^ 2 = 1 ↔ x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : Semiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x : R
⊢ x ^ 2 = 1 ↔ x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.mul_self_eq_iff | [81, 1] | [82, 82] | rw [← add_eq_zero_iff_eq, ← add_mul_self, mul_self_eq_zero, add_eq_zero_iff_eq] | R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x y : R
⊢ x * x = y * y ↔ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x y : R
⊢ x * x = y * y ↔ x = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Basic.lean | IMOSL.Extra.CharTwo.sq_eq_iff | [84, 1] | [85, 31] | rw [sq, sq, mul_self_eq_iff] | R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x y : R
⊢ x ^ 2 = y ^ 2 ↔ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CharTwo R
inst✝ : NoZeroDivisors R
x y : R
⊢ x ^ 2 = y ^ 2 ↔ x = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.iterate_add_mul_eq | [31, 1] | [35, 59] | rw [k.mul_succ, ← Nat.add_assoc, f.iterate_add,
iterate_add_mul_eq h t, ← f.iterate_add, h] | n k : ℕ
S : Type u_1
f : S → S
h : f^[n + k] = f^[n]
t : ℕ
⊢ f^[n + k * (t + 1)] = f^[n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
S : Type u_1
f : S → S
h : f^[n + k] = f^[n]
t : ℕ
⊢ f^[n + k * (t + 1)] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | apply (Set.range_comp_subset_range f f).antisymm | S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f^[2] = Set.range f | S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f^[2] = Set.range f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | rcases Nat.exists_eq_add_of_le h with ⟨k, rfl⟩ | S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f ⊆ Set.range (f ∘ f)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | nth_rw 1 [← h0, f.iterate_add] | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range (f^[2] ∘ f^[k]) ⊆ Set.range (f ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range f ⊆ Set.range (f ∘ f)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | exact Set.range_comp_subset_range _ _ | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range (f^[2] ∘ f^[k]) ⊆ Set.range (f ∘ f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range (f^[2] ∘ f^[k]) ⊆ Set.range (f ∘ f)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_lt_iterate_eq | [46, 1] | [49, 73] | obtain ⟨a, b, h, h0⟩ : ∃ a b : ℕ, a ≠ b ∧ f^[a] = f^[b] :=
Finite.exists_ne_map_eq_of_infinite _ | S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ a b, a < b ∧ f^[a] = f^[b] | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a ≠ b
h0 : f^[a] = f^[b]
⊢ ∃ a b, a < b ∧ f^[a] = f^[b] | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ a b, a < b ∧ f^[a] = f^[b]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_lt_iterate_eq | [46, 1] | [49, 73] | exact h.lt_or_lt.elim (λ h ↦ ⟨a, b, h, h0⟩) (λ h ↦ ⟨b, a, h, h0.symm⟩) | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a ≠ b
h0 : f^[a] = f^[b]
⊢ ∃ a b, a < b ∧ f^[a] = f^[b] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a ≠ b
h0 : f^[a] = f^[b]
⊢ ∃ a b, a < b ∧ f^[a] = f^[b]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | obtain ⟨a, b, h, h0⟩ := exists_lt_iterate_eq f | S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a < b
h0 : f^[a] = f^[b]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rcases Nat.exists_eq_add_of_le h.le with ⟨c, rfl⟩ | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a < b
h0 : f^[a] = f^[b]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : a < a + c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a < b
h0 : f^[a] = f^[b]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rw [Nat.lt_add_right_iff_pos] at h | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : a < a + c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : a < a + c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | refine ⟨c * a.succ, Nat.mul_pos h a.succ_pos, ?_⟩ | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[2 * (c * a.succ)] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rw [Nat.two_mul] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[2 * (c * a.succ)] = f^[c * a.succ] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c * a.succ] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[2 * (c * a.succ)] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | refine iterate_add_mul_eq ?_ _ | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c * a.succ] = f^[c * a.succ] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c * a.succ] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | obtain ⟨k, h1⟩ := Nat.exists_eq_add_of_le
(Nat.le_mul_of_pos_left a.succ h) | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c] = f^[c * a.succ] | case intro.intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
k : ℕ
h1 : c * a.succ = a.succ + k
⊢ f^[c * a.succ + c] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rw [h1, Nat.add_right_comm, f.iterate_add, Nat.succ_add,
f.iterate_succ, ← h0, ← f.iterate_succ, f.iterate_add] | case intro.intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
k : ℕ
h1 : c * a.succ = a.succ + k
⊢ f^[c * a.succ + c] = f^[c * a.succ] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
k : ℕ
h1 : c * a.succ = a.succ + k
⊢ f^[c * a.succ + c] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | rcases exists_iterate_idempotent f with ⟨n, h0, h1⟩ | S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
⊢ Set.range f^[2] = Set.range f | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ Set.range f^[2] = Set.range f | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
⊢ Set.range f^[2] = Set.range f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | refine range_iter_two_eq_of_exists_iter_eq_self
(Nat.lt_add_of_pos_left h0) (h _ ?_) | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ Set.range f^[2] = Set.range f | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1] ∘ f ∘ f^[n + 1] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ Set.range f^[2] = Set.range f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | rw [← f.iterate_succ', ← f.iterate_add] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1] ∘ f ∘ f^[n + 1] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1 + (n + 1).succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1] ∘ f ∘ f^[n + 1]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | change (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1 + (n + 1).succ] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1 + (n + 1).succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | rw [Nat.succ_add, ← Nat.two_mul, f.iterate_succ' (2 * n), h1] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/EventuallyEqual.lean | IMOSL.Extra.EventuallyEqual.symm | [26, 1] | [27, 66] | rcases h with ⟨N_a, N_b, h⟩ | α✝ : Sort u_1
a b : Nat → α✝
h : EventuallyEqual a b
⊢ EventuallyEqual b a | case intro.intro
α✝ : Sort u_1
a b : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
⊢ EventuallyEqual b a | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Sort u_1
a b : Nat → α✝
h : EventuallyEqual a b
⊢ EventuallyEqual b a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/EventuallyEqual.lean | IMOSL.Extra.EventuallyEqual.symm | [26, 1] | [27, 66] | exact ⟨N_b, N_a, λ k ↦ (h k).symm⟩ | case intro.intro
α✝ : Sort u_1
a b : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
⊢ EventuallyEqual b a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Sort u_1
a b : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
⊢ EventuallyEqual b a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/EventuallyEqual.lean | IMOSL.Extra.EventuallyEqual.trans | [29, 1] | [34, 85] | rcases h with ⟨N_a, N_b, h⟩ | α✝ : Sort u_1
a b c : Nat → α✝
h : EventuallyEqual a b
h0 : EventuallyEqual b c
⊢ EventuallyEqual a c | case intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
h0 : EventuallyEqual b c
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
⊢ EventuallyEqual a c | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Sort u_1
a b c : Nat → α✝
h : EventuallyEqual a b
h0 : EventuallyEqual b c
⊢ EventuallyEqual a c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/EventuallyEqual.lean | IMOSL.Extra.EventuallyEqual.trans | [29, 1] | [34, 85] | rcases h0 with ⟨K_b, K_c, h0⟩ | case intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
h0 : EventuallyEqual b c
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
⊢ EventuallyEqual a c | case intro.intro.intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
K_b K_c : Nat
h0 : ∀ (k : Nat), b (k + K_b) = c (k + K_c)
⊢ EventuallyEqual a c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
h0 : EventuallyEqual b c
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
⊢ EventuallyEqual a c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/EventuallyEqual.lean | IMOSL.Extra.EventuallyEqual.trans | [29, 1] | [34, 85] | refine ⟨K_b + N_a, K_c + N_b, λ k ↦ ?_⟩ | case intro.intro.intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
K_b K_c : Nat
h0 : ∀ (k : Nat), b (k + K_b) = c (k + K_c)
⊢ EventuallyEqual a c | case intro.intro.intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
K_b K_c : Nat
h0 : ∀ (k : Nat), b (k + K_b) = c (k + K_c)
k : Nat
⊢ a (k + (K_b + N_a)) = c (k + (K_c + N_b)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
K_b K_c : Nat
h0 : ∀ (k : Nat), b (k + K_b) = c (k + K_c)
⊢ EventuallyEqual a c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/EventuallyEqual.lean | IMOSL.Extra.EventuallyEqual.trans | [29, 1] | [34, 85] | rw [← Nat.add_assoc, h, Nat.add_right_comm, h0, Nat.add_right_comm, Nat.add_assoc] | case intro.intro.intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
K_b K_c : Nat
h0 : ∀ (k : Nat), b (k + K_b) = c (k + K_c)
k : Nat
⊢ a (k + (K_b + N_a)) = c (k + (K_c + N_b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α✝ : Sort u_1
a b c : Nat → α✝
N_a N_b : Nat
h : ∀ (k : Nat), a (k + N_a) = b (k + N_b)
K_b K_c : Nat
h0 : ∀ (k : Nat), b (k + K_b) = c (k + K_c)
k : Nat
⊢ a (k + (K_b + N_a)) = c (k + (K_c + N_b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | refine Nat.rec (λ _ ↦ h.not_dvd_one) (λ r h0 h1 ↦ ?_) | p : ℕ
h : p.Prime
k : ℕ
⊢ ∀ r < p, ¬p ∣ (p * k + r).descFactorial r | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ (p * k + r.succ).descFactorial r.succ | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
h : p.Prime
k : ℕ
⊢ ∀ r < p, ¬p ∣ (p * k + r).descFactorial r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | rw [Nat.add_succ, Nat.succ_descFactorial_succ] | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ (p * k + r.succ).descFactorial r.succ | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ (p * k + r + 1) * (p * k + r).descFactorial r | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ (p * k + r.succ).descFactorial r.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | refine h.not_dvd_mul ?_ (h0 <| r.lt_succ_self.trans h1) | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ (p * k + r + 1) * (p * k + r).descFactorial r | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ p * k + r + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ (p * k + r + 1) * (p * k + r).descFactorial r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | rw [add_assoc, Nat.dvd_add_right ⟨k, rfl⟩] | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ p * k + r + 1 | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ r + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ p * k + r + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | exact Nat.not_dvd_of_pos_of_lt r.succ_pos h1 | p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ r + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
h : p.Prime
k r : ℕ
h0 : r < p → ¬p ∣ (p * k + r).descFactorial r
h1 : r.succ < p
⊢ ¬p ∣ r + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_binom_not_dvd | [28, 1] | [35, 59] | apply prime_not_dvd_descFactorial h k.pred p.pred (p.pred_lt_self h.pos) | k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ False | k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ p ∣ (p * k.pred + p.pred).descFactorial p.pred | Please generate a tactic in lean4 to solve the state.
STATE:
k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_binom_not_dvd | [28, 1] | [35, 59] | rw [← Nat.mul_dvd_mul_iff_left (p * k.pred + p.pred).succ_pos,
← Nat.succ_descFactorial_succ, ← Nat.succ_eq_add_one, ← Nat.add_succ,
← Nat.succ_eq_add_one, Nat.succ_pred h.ne_zero, ← p.mul_succ,
Nat.succ_pred h0, Nat.descFactorial_eq_factorial_mul_choose, mul_comm] | k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ p ∣ (p * k.pred + p.pred).descFactorial p.pred | k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ p * (p * k) ∣ p.factorial * (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ p ∣ (p * k.pred + p.pred).descFactorial p.pred
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_binom_not_dvd | [28, 1] | [35, 59] | exact mul_dvd_mul (Nat.dvd_factorial h.pos p.le_refl) h1 | k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ p * (p * k) ∣ p.factorial * (p * k).choose p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k p : ℕ
h : p.Prime
h0 : k ≠ 0
h1 : p * k ∣ (p * k).choose p
⊢ p * (p * k) ∣ p.factorial * (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | refine ⟨λ h0 ↦ ?_, λ h0 n h1 ↦ ?_⟩ | m : ℕ
h : 1 < m
⊢ (∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)) ↔ m.Prime | case refine_1
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
⊢ m.Prime
case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : 2 * n ≤ m
⊢ n ∣ n.choose (m - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
h : 1 < m
⊢ (∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)) ↔ m.Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | by_cases h1 : 2 ∣ m | case refine_1
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
⊢ m.Prime
case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : 2 * n ≤ m
⊢ n ∣ n.choose (m - 2 * n) | case pos
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
h1 : 2 ∣ m
⊢ m.Prime
case neg
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
h1 : ¬2 ∣ m
⊢ m.Prime
case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : 2 * n ≤ m
⊢ n ∣ n.choose (m - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
⊢ m.Prime
case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : 2 * n ≤ m
⊢ n ∣ n.choose (m - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h1 with ⟨k, rfl⟩ | case pos
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
h1 : 2 ∣ m
⊢ m.Prime | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : ∀ (n : ℕ), 2 * n ≤ 2 * k → n ∣ n.choose (2 * k - 2 * n)
⊢ (2 * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
h1 : 2 ∣ m
⊢ m.Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | specialize h0 k (2 * k).le_refl | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : ∀ (n : ℕ), 2 * n ≤ 2 * k → n ∣ n.choose (2 * k - 2 * n)
⊢ (2 * k).Prime | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k ∣ k.choose (2 * k - 2 * k)
⊢ (2 * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : ∀ (n : ℕ), 2 * n ≤ 2 * k → n ∣ n.choose (2 * k - 2 * n)
⊢ (2 * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.sub_self, k.choose_zero_right, Nat.dvd_one] at h0 | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k ∣ k.choose (2 * k - 2 * k)
⊢ (2 * k).Prime | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k = 1
⊢ (2 * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k ∣ k.choose (2 * k - 2 * k)
⊢ (2 * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [h0, mul_one] | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k = 1
⊢ (2 * k).Prime | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k = 1
⊢ Nat.Prime 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k = 1
⊢ (2 * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact Nat.prime_two | case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k = 1
⊢ Nat.Prime 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : ℕ
h : 1 < 2 * k
h0 : k = 1
⊢ Nat.Prime 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | obtain ⟨p, h2, k, rfl⟩ : ∃ p : ℕ, p.Prime ∧ p ∣ m :=
⟨m.minFac, Nat.minFac_prime h.ne.symm, m.minFac_dvd⟩ | case neg
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
h1 : ¬2 ∣ m
⊢ m.Prime | case neg.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * k
h0 : ∀ (n : ℕ), 2 * n ≤ p * k → n ∣ n.choose (p * k - 2 * n)
h1 : ¬2 ∣ p * k
⊢ (p * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
m : ℕ
h : 1 < m
h0 : ∀ (n : ℕ), 2 * n ≤ m → n ∣ n.choose (m - 2 * n)
h1 : ¬2 ∣ m
⊢ m.Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.two_dvd_ne_zero, ← Nat.odd_iff, Nat.odd_mul] at h1 | case neg.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * k
h0 : ∀ (n : ℕ), 2 * n ≤ p * k → n ∣ n.choose (p * k - 2 * n)
h1 : ¬2 ∣ p * k
⊢ (p * k).Prime | case neg.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * k
h0 : ∀ (n : ℕ), 2 * n ≤ p * k → n ∣ n.choose (p * k - 2 * n)
h1 : Odd p ∧ Odd k
⊢ (p * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * k
h0 : ∀ (n : ℕ), 2 * n ≤ p * k → n ∣ n.choose (p * k - 2 * n)
h1 : ¬2 ∣ p * k
⊢ (p * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h1 with ⟨-, k, rfl⟩ | case neg.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * k
h0 : ∀ (n : ℕ), 2 * n ≤ p * k → n ∣ n.choose (p * k - 2 * n)
h1 : Odd p ∧ Odd k
⊢ (p * k).Prime | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
⊢ (p * (2 * k + 1)).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * k
h0 : ∀ (n : ℕ), 2 * n ≤ p * k → n ∣ n.choose (p * k - 2 * n)
h1 : Odd p ∧ Odd k
⊢ (p * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | suffices k = 0 by rwa [this, mul_zero, zero_add, mul_one] | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
⊢ (p * (2 * k + 1)).Prime | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
⊢ k = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
⊢ (p * (2 * k + 1)).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | by_contra h1 | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
⊢ k = 0 | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
h1 : ¬k = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
⊢ k = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | apply prime_binom_not_dvd h2 h1 | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
h1 : ¬k = 0
⊢ False | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
h1 : ¬k = 0
⊢ p * k ∣ (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
h1 : ¬k = 0
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | specialize h0 (p * k) | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
h1 : ¬k = 0
⊢ p * k ∣ (p * k).choose p | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h1 : ¬k = 0
h0 : 2 * (p * k) ≤ p * (2 * k + 1) → p * k ∣ (p * k).choose (p * (2 * k + 1) - 2 * (p * k))
⊢ p * k ∣ (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
h1 : ¬k = 0
⊢ p * k ∣ (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [mul_add_one (α := ℕ), mul_left_comm, Nat.add_sub_cancel_left] at h0 | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h1 : ¬k = 0
h0 : 2 * (p * k) ≤ p * (2 * k + 1) → p * k ∣ (p * k).choose (p * (2 * k + 1) - 2 * (p * k))
⊢ p * k ∣ (p * k).choose p | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h1 : ¬k = 0
h0 : p * (2 * k) ≤ p * (2 * k) + p → p * k ∣ (p * k).choose p
⊢ p * k ∣ (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h1 : ¬k = 0
h0 : 2 * (p * k) ≤ p * (2 * k + 1) → p * k ∣ (p * k).choose (p * (2 * k + 1) - 2 * (p * k))
⊢ p * k ∣ (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact h0 (Nat.le_add_right (p * (2 * k)) p) | case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h1 : ¬k = 0
h0 : p * (2 * k) ≤ p * (2 * k) + p → p * k ∣ (p * k).choose p
⊢ p * k ∣ (p * k).choose p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h1 : ¬k = 0
h0 : p * (2 * k) ≤ p * (2 * k) + p → p * k ∣ (p * k).choose p
⊢ p * k ∣ (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rwa [this, mul_zero, zero_add, mul_one] | p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
this : k = 0
⊢ (p * (2 * k + 1)).Prime | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
h2 : p.Prime
k : ℕ
h : 1 < p * (2 * k + 1)
h0 : ∀ (n : ℕ), 2 * n ≤ p * (2 * k + 1) → n ∣ n.choose (p * (2 * k + 1) - 2 * n)
this : k = 0
⊢ (p * (2 * k + 1)).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [le_iff_exists_add] at h1 | case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : 2 * n ≤ m
⊢ n ∣ n.choose (m - 2 * n) | case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : ∃ c, m = 2 * n + c
⊢ n ∣ n.choose (m - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : 2 * n ≤ m
⊢ n ∣ n.choose (m - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h1 with ⟨c, rfl⟩ | case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : ∃ c, m = 2 * n + c
⊢ n ∣ n.choose (m - 2 * n) | case refine_2.intro
n c : ℕ
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
⊢ n ∣ n.choose (2 * n + c - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
m : ℕ
h : 1 < m
h0 : m.Prime
n : ℕ
h1 : ∃ c, m = 2 * n + c
⊢ n ∣ n.choose (m - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.add_sub_cancel_left] | case refine_2.intro
n c : ℕ
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
⊢ n ∣ n.choose (2 * n + c - 2 * n) | case refine_2.intro
n c : ℕ
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
⊢ n ∣ n.choose c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
n c : ℕ
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
⊢ n ∣ n.choose (2 * n + c - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases c with _ | c | case refine_2.intro
n c : ℕ
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
⊢ n ∣ n.choose c | case refine_2.intro.zero
n : ℕ
h : 1 < 2 * n + 0
h0 : (2 * n + 0).Prime
⊢ n ∣ n.choose 0
case refine_2.intro.succ
n c : ℕ
h : 1 < 2 * n + (c + 1)
h0 : (2 * n + (c + 1)).Prime
⊢ n ∣ n.choose (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
n c : ℕ
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
⊢ n ∣ n.choose c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases n with _ | n | case refine_2.intro.succ
n c : ℕ
h : 1 < 2 * n + (c + 1)
h0 : (2 * n + (c + 1)).Prime
⊢ n ∣ n.choose (c + 1) | case refine_2.intro.succ.zero
c : ℕ
h : 1 < 2 * 0 + (c + 1)
h0 : (2 * 0 + (c + 1)).Prime
⊢ 0 ∣ Nat.choose 0 (c + 1)
case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
⊢ n + 1 ∣ (n + 1).choose (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ
n c : ℕ
h : 1 < 2 * n + (c + 1)
h0 : (2 * n + (c + 1)).Prime
⊢ n ∣ n.choose (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.add_zero, Nat.prime_mul_iff] at h0 | case refine_2.intro.zero
n : ℕ
h : 1 < 2 * n + 0
h0 : (2 * n + 0).Prime
⊢ n ∣ n.choose 0 | case refine_2.intro.zero
n : ℕ
h : 1 < 2 * n + 0
h0 : Nat.Prime 2 ∧ n = 1 ∨ n.Prime ∧ 2 = 1
⊢ n ∣ n.choose 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.zero
n : ℕ
h : 1 < 2 * n + 0
h0 : (2 * n + 0).Prime
⊢ n ∣ n.choose 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h0 with ⟨-, rfl⟩ | ⟨-, h1⟩ | case refine_2.intro.zero
n : ℕ
h : 1 < 2 * n + 0
h0 : Nat.Prime 2 ∧ n = 1 ∨ n.Prime ∧ 2 = 1
⊢ n ∣ n.choose 0 | case refine_2.intro.zero.inl.intro
h : 1 < 2 * 1 + 0
⊢ 1 ∣ Nat.choose 1 0
case refine_2.intro.zero.inr.intro
n : ℕ
h : 1 < 2 * n + 0
h1 : 2 = 1
⊢ n ∣ n.choose 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.zero
n : ℕ
h : 1 < 2 * n + 0
h0 : Nat.Prime 2 ∧ n = 1 ∨ n.Prime ∧ 2 = 1
⊢ n ∣ n.choose 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exacts [Nat.dvd_refl 1, absurd h1 (Nat.succ_ne_self 1)] | case refine_2.intro.zero.inl.intro
h : 1 < 2 * 1 + 0
⊢ 1 ∣ Nat.choose 1 0
case refine_2.intro.zero.inr.intro
n : ℕ
h : 1 < 2 * n + 0
h1 : 2 = 1
⊢ n ∣ n.choose 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.zero.inl.intro
h : 1 < 2 * 1 + 0
⊢ 1 ∣ Nat.choose 1 0
case refine_2.intro.zero.inr.intro
n : ℕ
h : 1 < 2 * n + 0
h1 : 2 = 1
⊢ n ∣ n.choose 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact Nat.dvd_refl 0 | case refine_2.intro.succ.zero
c : ℕ
h : 1 < 2 * 0 + (c + 1)
h0 : (2 * 0 + (c + 1)).Prime
⊢ 0 ∣ Nat.choose 0 (c + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.zero
c : ℕ
h : 1 < 2 * 0 + (c + 1)
h0 : (2 * 0 + (c + 1)).Prime
⊢ 0 ∣ Nat.choose 0 (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | refine (Nat.Coprime.dvd_mul_right ?_).mp
⟨_, (n.succ_mul_choose_eq c).symm⟩ | case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
⊢ n + 1 ∣ (n + 1).choose (c + 1) | case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
⊢ (n + 1).Coprime c.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
⊢ n + 1 ∣ (n + 1).choose (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [two_mul, add_assoc] at h0 | case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
⊢ (n + 1).Coprime c.succ | case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
⊢ (n + 1).Coprime c.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
⊢ (n + 1).Coprime c.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [← Nat.coprime_self_add_right, ← Nat.coprime_self_add_right,
Nat.coprime_comm, h0.coprime_iff_not_dvd] | case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
⊢ (n + 1).Coprime c.succ | case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
⊢ ¬n + 1 + (n + 1 + (c + 1)) ∣ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : ℕ
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
⊢ (n + 1).Coprime c.succ
TACTIC:
|
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