url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_singleton_sum_eq | [48, 1] | [52, 61] | rw [symmDiff_singleton_eq_erase h, ← S.add_sum_erase f h, CharTwo.add_add_cancel_left] | case pos
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
h : i ∈ S
⊢ (symmDiff {i} S).sum f = f i + S.sum f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
h : i ∈ S
⊢ (symmDiff {i} S).sum f = f i + S.sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_singleton_sum_eq | [48, 1] | [52, 61] | rw [symmDiff_singleton_eq_insert h, Finset.sum_insert h] | case neg
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
h : i ∉ S
⊢ (symmDiff {i} S).sum f = f i + S.sum f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
h : i ∉ S
⊢ (symmDiff {i} S).sum f = f i + S.sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_sum_eq | [54, 1] | [60, 53] | revert T | M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S T : Finset ι
⊢ (symmDiff S T).sum f = S.sum f + T.sum f | M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ ∀ (T : Finset ι), (symmDiff S T).sum f = S.sum f + T.sum f | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S T : Finset ι
⊢ (symmDiff S T).sum f = S.sum f + T.sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_sum_eq | [54, 1] | [60, 53] | apply Finset.induction | M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ ∀ (T : Finset ι), (symmDiff S T).sum f = S.sum f + T.sum f | case empty
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ (symmDiff S ∅).sum f = S.sum f + ∅.sum f
case insert
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
a ∉ s → (symmDiff S s).sum f = S.sum f + s.sum f → (symmDiff S (insert a s)).sum f = S.sum f + (insert a s).sum f | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ ∀ (T : Finset ι), (symmDiff S T).sum f = S.sum f + T.sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_sum_eq | [54, 1] | [60, 53] | have h : symmDiff S ∅ = S := by exact symmDiff_bot _ | case empty
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ (symmDiff S ∅).sum f = S.sum f + ∅.sum f | case empty
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
h : symmDiff S ∅ = S
⊢ (symmDiff S ∅).sum f = S.sum f + ∅.sum f | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ (symmDiff S ∅).sum f = S.sum f + ∅.sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_sum_eq | [54, 1] | [60, 53] | rw [Finset.sum_empty, add_zero, h] | case empty
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
h : symmDiff S ∅ = S
⊢ (symmDiff S ∅).sum f = S.sum f + ∅.sum f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
h : symmDiff S ∅ = S
⊢ (symmDiff S ∅).sum f = S.sum f + ∅.sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_sum_eq | [54, 1] | [60, 53] | exact symmDiff_bot _ | M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ symmDiff S ∅ = S | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ symmDiff S ∅ = S
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_sum_eq | [54, 1] | [60, 53] | intro i T h h0 | case insert
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
a ∉ s → (symmDiff S s).sum f = S.sum f + s.sum f → (symmDiff S (insert a s)).sum f = S.sum f + (insert a s).sum f | case insert
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
i : ι
T : Finset ι
h : i ∉ T
h0 : (symmDiff S T).sum f = S.sum f + T.sum f
⊢ (symmDiff S (insert i T)).sum f = S.sum f + (insert i T).sum f | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
a ∉ s → (symmDiff S s).sum f = S.sum f + s.sum f → (symmDiff S (insert a s)).sum f = S.sum f + (insert a s).sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_sum_eq | [54, 1] | [60, 53] | rw [Finset.sum_insert h, add_left_comm, ← h0, ← symmDiff_singleton_eq_insert h,
symmDiff_left_comm, symmDiff_singleton_sum_eq] | case insert
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
i : ι
T : Finset ι
h : i ∉ T
h0 : (symmDiff S T).sum f = S.sum f + T.sum f
⊢ (symmDiff S (insert i T)).sum f = S.sum f + (insert i T).sum f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
S : Finset ι
i : ι
T : Finset ι
h : i ∉ T
h0 : (symmDiff S T).sum f = S.sum f + T.sum f
⊢ (symmDiff S (insert i T)).sum f = S.sum f + (insert i T).sum f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_natAbs_le | [39, 1] | [48, 81] | rw [← Nat.cast_le (α := ℤ), Int.cast_natAbs, Int.cast_natAbs,
Int.cast_abs, Int.cast_id, Int.cast_abs, Int.cast_id] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ ⌊f r⌋.natAbs ≤ ⌊r⌋.natAbs | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ |⌊f r⌋| ≤ |⌊r⌋| | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ ⌊f r⌋.natAbs ≤ ⌊r⌋.natAbs
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_natAbs_le | [39, 1] | [48, 81] | rcases le_total 0 ⌊r⌋ with h | h | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ |⌊f r⌋| ≤ |⌊r⌋| | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : 0 ≤ ⌊r⌋
⊢ |⌊f r⌋| ≤ |⌊r⌋|
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ⌊r⌋ ≤ 0
⊢ |⌊f r⌋| ≤ |⌊r⌋| | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ |⌊f r⌋| ≤ |⌊r⌋|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_natAbs_le | [39, 1] | [48, 81] | rw [abs_eq_self.mpr h, abs_eq_self.mpr (floor_f_nonneg h), ← Int.cast_le (R := R)] | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : 0 ≤ ⌊r⌋
⊢ |⌊f r⌋| ≤ |⌊r⌋| | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : 0 ≤ ⌊r⌋
⊢ ↑⌊f r⌋ ≤ ↑⌊r⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : 0 ≤ ⌊r⌋
⊢ |⌊f r⌋| ≤ |⌊r⌋|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_natAbs_le | [39, 1] | [48, 81] | exact (Int.floor_le _).trans <| mul_le_of_le_one_right
(Int.cast_nonneg.mpr h) (Int.fract_lt_one r).le | case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : 0 ≤ ⌊r⌋
⊢ ↑⌊f r⌋ ≤ ↑⌊r⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : 0 ≤ ⌊r⌋
⊢ ↑⌊f r⌋ ≤ ↑⌊r⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_natAbs_le | [39, 1] | [48, 81] | rw [abs_eq_neg_self.mpr h, abs_eq_neg_self.mpr (floor_f_nonpos h),
neg_le_neg_iff, Int.le_floor] | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ⌊r⌋ ≤ 0
⊢ |⌊f r⌋| ≤ |⌊r⌋| | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ⌊r⌋ ≤ 0
⊢ ↑⌊r⌋ ≤ f r | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ⌊r⌋ ≤ 0
⊢ |⌊f r⌋| ≤ |⌊r⌋|
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_natAbs_le | [39, 1] | [48, 81] | exact le_mul_of_le_one_right (Int.cast_nonpos.mpr h) (Int.fract_lt_one r).le | case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ⌊r⌋ ≤ 0
⊢ ↑⌊r⌋ ≤ f r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ⌊r⌋ ≤ 0
⊢ ↑⌊r⌋ ≤ f r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_natAbs_eventually_const | [54, 1] | [61, 22] | have ha : Antitone (λ k ↦ ⌊f^[k] r⌋.natAbs) := λ k m h0 ↦ by
rcases Nat.exists_eq_add_of_le h0 with ⟨c, rfl⟩; simp only
rw [Nat.add_comm, f.iterate_add_apply]
exact floor_f_iter_natAbs_le _ c | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => C | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
ha : Antitone fun k => ⌊f^[k] r⌋.natAbs
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => 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, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_natAbs_eventually_const | [54, 1] | [61, 22] | rcases NatSeq_antitone_imp_const ha with ⟨C, N, ha⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
ha : Antitone fun k => ⌊f^[k] r⌋.natAbs
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => C | case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
ha✝ : Antitone fun k => ⌊f^[k] r⌋.natAbs
C N : ℕ
ha : ∀ (n : ℕ), ⌊f^[n + N] r⌋.natAbs = C
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => 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
ha : Antitone fun k => ⌊f^[k] r⌋.natAbs
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_natAbs_eventually_const | [54, 1] | [61, 22] | exact ⟨C, N, 0, ha⟩ | case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
ha✝ : Antitone fun k => ⌊f^[k] r⌋.natAbs
C N : ℕ
ha : ∀ (n : ℕ), ⌊f^[n + N] r⌋.natAbs = C
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
ha✝ : Antitone fun k => ⌊f^[k] r⌋.natAbs
C N : ℕ
ha : ∀ (n : ℕ), ⌊f^[n + N] r⌋.natAbs = C
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋.natAbs) fun x => C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_natAbs_eventually_const | [54, 1] | [61, 22] | rcases Nat.exists_eq_add_of_le h0 with ⟨c, rfl⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k m : ℕ
h0 : k ≤ m
⊢ (fun k => ⌊f^[k] r⌋.natAbs) m ≤ (fun k => ⌊f^[k] r⌋.natAbs) k | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k c : ℕ
h0 : k ≤ k + c
⊢ (fun k => ⌊f^[k] r⌋.natAbs) (k + c) ≤ (fun k => ⌊f^[k] r⌋.natAbs) k | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k m : ℕ
h0 : k ≤ m
⊢ (fun k => ⌊f^[k] r⌋.natAbs) m ≤ (fun k => ⌊f^[k] r⌋.natAbs) k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_natAbs_eventually_const | [54, 1] | [61, 22] | simp only | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k c : ℕ
h0 : k ≤ k + c
⊢ (fun k => ⌊f^[k] r⌋.natAbs) (k + c) ≤ (fun k => ⌊f^[k] r⌋.natAbs) k | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k c : ℕ
h0 : k ≤ k + c
⊢ ⌊f^[k + c] r⌋.natAbs ≤ ⌊f^[k] r⌋.natAbs | 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
k c : ℕ
h0 : k ≤ k + c
⊢ (fun k => ⌊f^[k] r⌋.natAbs) (k + c) ≤ (fun k => ⌊f^[k] r⌋.natAbs) k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_natAbs_eventually_const | [54, 1] | [61, 22] | rw [Nat.add_comm, f.iterate_add_apply] | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k c : ℕ
h0 : k ≤ k + c
⊢ ⌊f^[k + c] r⌋.natAbs ≤ ⌊f^[k] r⌋.natAbs | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k c : ℕ
h0 : k ≤ k + c
⊢ ⌊f^[c] (f^[k] r)⌋.natAbs ≤ ⌊f^[k] r⌋.natAbs | 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
k c : ℕ
h0 : k ≤ k + c
⊢ ⌊f^[k + c] r⌋.natAbs ≤ ⌊f^[k] r⌋.natAbs
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_natAbs_eventually_const | [54, 1] | [61, 22] | exact floor_f_iter_natAbs_le _ c | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
k c : ℕ
h0 : k ≤ k + c
⊢ ⌊f^[c] (f^[k] r)⌋.natAbs ≤ ⌊f^[k] r⌋.natAbs | no goals | 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
k c : ℕ
h0 : k ≤ k + c
⊢ ⌊f^[c] (f^[k] r)⌋.natAbs ≤ ⌊f^[k] r⌋.natAbs
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | rcases floor_f_iter_natAbs_eventually_const r with ⟨C, N, K, h⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C | case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋.natAbs) (k + N) = (fun x => C) (k + K)
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑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, EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | refine ⟨C, N, K, λ n ↦ (Int.natAbs_eq_iff.mp (h n)).elim (λ h0 ↦ h0.trans ?_) id⟩ | case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋.natAbs) (k + N) = (fun x => C) (k + K)
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C | case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋.natAbs) (k + N) = (fun x => C) (k + K)
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑((fun x => C) (n + K))
⊢ ↑((fun x => C) (n + K)) = (fun x => -↑C) (n + K) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋.natAbs) (k + N) = (fun x => C) (k + K)
⊢ ∃ C, EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | simp only at h h0 ⊢ | case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋.natAbs) (k + N) = (fun x => C) (k + K)
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑((fun x => C) (n + K))
⊢ ↑((fun x => C) (n + K)) = (fun x => -↑C) (n + K) | case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
⊢ ↑C = -↑C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋.natAbs) (k + N) = (fun x => C) (k + K)
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑((fun x => C) (n + K))
⊢ ↑((fun x => C) (n + K)) = (fun x => -↑C) (n + K)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | rcases C.eq_zero_or_pos with rfl | h1 | case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
⊢ ↑C = -↑C | case intro.intro.intro.inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N K n : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = 0
h0 : ⌊f^[n + N] r⌋ = ↑0
⊢ ↑0 = -↑0
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
⊢ ↑C = -↑C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
⊢ ↑C = -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | rfl | case intro.intro.intro.inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N K n : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = 0
h0 : ⌊f^[n + N] r⌋ = ↑0
⊢ ↑0 = -↑0
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
⊢ ↑C = -↑C | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
⊢ ↑C = -↑C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N K n : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = 0
h0 : ⌊f^[n + N] r⌋ = ↑0
⊢ ↑0 = -↑0
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
⊢ ↑C = -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | specialize h (n + 1) | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
⊢ ↑C = -↑C | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f^[n + 1 + N] r⌋.natAbs = C
⊢ ↑C = -↑C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋.natAbs = C
n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
⊢ ↑C = -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | rw [Nat.add_right_comm, f.iterate_succ_apply'] at h | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f^[n + 1 + N] r⌋.natAbs = C
⊢ ↑C = -↑C | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f (f^[n + N] r)⌋.natAbs = C
⊢ ↑C = -↑C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f^[n + 1 + N] r⌋.natAbs = C
⊢ ↑C = -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | replace h1 : 0 < ⌊f^[n + N] r⌋ := by rwa [h0, Nat.cast_pos] | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f (f^[n + N] r)⌋.natAbs = C
⊢ ↑C = -↑C | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h : ⌊f (f^[n + N] r)⌋.natAbs = C
h1 : 0 < ⌊f^[n + N] r⌋
⊢ ↑C = -↑C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f (f^[n + N] r)⌋.natAbs = C
⊢ ↑C = -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | rw [← h, Int.natCast_natAbs, abs_eq_self.mpr (floor_f_nonneg h1.le)] at h0 | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h : ⌊f (f^[n + N] r)⌋.natAbs = C
h1 : 0 < ⌊f^[n + N] r⌋
⊢ ↑C = -↑C | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ⌊f (f^[n + N] r)⌋
h : ⌊f (f^[n + N] r)⌋.natAbs = C
h1 : 0 < ⌊f^[n + N] r⌋
⊢ ↑C = -↑C | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h : ⌊f (f^[n + N] r)⌋.natAbs = C
h1 : 0 < ⌊f^[n + N] r⌋
⊢ ↑C = -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | exact absurd h0.symm (floor_f_lt_of_floor_pos h1).ne | case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ⌊f (f^[n + N] r)⌋
h : ⌊f (f^[n + N] r)⌋.natAbs = C
h1 : 0 < ⌊f^[n + N] r⌋
⊢ ↑C = -↑C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ⌊f (f^[n + N] r)⌋
h : ⌊f (f^[n + N] r)⌋.natAbs = C
h1 : 0 < ⌊f^[n + N] r⌋
⊢ ↑C = -↑C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.floor_f_iter_eventually_const | [66, 1] | [74, 55] | rwa [h0, Nat.cast_pos] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C N K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f (f^[n + N] r)⌋.natAbs = C
⊢ 0 < ⌊f^[n + N] r⌋ | 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 K n : ℕ
h0 : ⌊f^[n + N] r⌋ = ↑C
h1 : C > 0
h : ⌊f (f^[n + N] r)⌋.natAbs = C
⊢ 0 < ⌊f^[n + N] r⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_zero | [76, 1] | [80, 77] | rw [EventuallyEqual.const_right_iff] at h ⊢ | 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 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ∃ N, ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
⊢ ∃ N, ∀ (k : ℕ), f^[k + N] r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
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.case_floor_eventually_zero | [76, 1] | [80, 77] | rcases h with ⟨N, h⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ∃ N, ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
⊢ ∃ N, ∀ (k : ℕ), f^[k + N] r = 0 | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
⊢ ∃ N, ∀ (k : ℕ), f^[k + N] r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ∃ N, ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
⊢ ∃ N, ∀ (k : ℕ), f^[k + N] r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_zero | [76, 1] | [80, 77] | refine ⟨N + 1, λ k ↦ ?_⟩ | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
⊢ ∃ N, ∀ (k : ℕ), f^[k + N] r = 0 | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
k : ℕ
⊢ f^[k + (N + 1)] r = 0 | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
⊢ ∃ N, ∀ (k : ℕ), f^[k + N] r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_zero | [76, 1] | [80, 77] | rw [← Nat.add_assoc, f.iterate_succ_apply', f, h, Int.cast_zero, zero_mul] | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
k : ℕ
⊢ f^[k + (N + 1)] r = 0 | no goals | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = 0
k : ℕ
⊢ f^[k + (N + 1)] r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rw [EventuallyEqual.const_right_iff] at h | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -1
⊢ ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1]) | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ∃ N, ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -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:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -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.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rcases h with ⟨N, h⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ∃ N, ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1]) | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -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:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
h : ∃ N, ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -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.case_floor_eventually_neg_one | [82, 1] | [105, 27] | refine ⟨-f^[N] r, ?_, ?_⟩ | case intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ ∃ s, (0 < s ∧ s < 1) ∧ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [-s, s - 1]) | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1
case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [- -f^[N] r, -f^[N] r - 1]) | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -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.case_floor_eventually_neg_one | [82, 1] | [105, 27] | have h0 := h 0 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : ⌊f^[0 + N] r⌋ = -1
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1 | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rw [Nat.zero_add, Int.floor_eq_iff, Int.cast_neg, Int.cast_one, and_comm] at h0 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : ⌊f^[0 + N] r⌋ = -1
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : f^[N] r < -1 + 1 ∧ -1 ≤ f^[N] r
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1 | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : ⌊f^[0 + N] r⌋ = -1
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | revert h0 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : f^[N] r < -1 + 1 ∧ -1 ≤ f^[N] r
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ f^[N] r < -1 + 1 ∧ -1 ≤ f^[N] r → 0 < -f^[N] r ∧ -f^[N] r < 1 | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : f^[N] r < -1 + 1 ∧ -1 ≤ f^[N] r
⊢ 0 < -f^[N] r ∧ -f^[N] r < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | refine And.imp (λ h0 ↦ neg_pos_of_neg (h0.trans_eq <| neg_add_self 1))
(λ h0 ↦ neg_lt_of_neg_lt (h0.lt_or_eq.resolve_right λ h1 ↦ ?_)) | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ f^[N] r < -1 + 1 ∧ -1 ≤ f^[N] r → 0 < -f^[N] r ∧ -f^[N] r < 1 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : -1 ≤ f^[N] r
h1 : -1 = f^[N] r
⊢ False | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ f^[N] r < -1 + 1 ∧ -1 ≤ f^[N] r → 0 < -f^[N] r ∧ -f^[N] r < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | replace h0 : Int.fract (-1 : R) = 0 := Int.fract_neg_eq_zero.mpr Int.fract_one | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : -1 ≤ f^[N] r
h1 : -1 = f^[N] r
⊢ False | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
⊢ False | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : -1 ≤ f^[N] r
h1 : -1 = f^[N] r
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | specialize h 1 | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
⊢ False | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
h : ⌊f^[1 + N] r⌋ = -1
⊢ False | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rw [f.iterate_add_apply, f.iterate_one,
← h1, f, h0, mul_zero, Int.floor_zero, zero_eq_neg] at h | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
h : ⌊f^[1 + N] r⌋ = -1
⊢ False | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
h : 1 = 0
⊢ False | 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
N : ℕ
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
h : ⌊f^[1 + N] r⌋ = -1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | exact one_ne_zero h | case intro.refine_1
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
h : 1 = 0
⊢ False | 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
N : ℕ
h1 : -1 = f^[N] r
h0 : Int.fract (-1) = 0
h : 1 = 0
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | have h0 (k : ℕ) : f^[(k + 1) + N] r = -f^[k + N] r - 1 := by
rw [Nat.add_right_comm, f.iterate_succ_apply', f, Int.fract, h,
Int.cast_neg, Int.cast_one, neg_one_mul, neg_sub', neg_neg] | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [- -f^[N] r, -f^[N] r - 1]) | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
⊢ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [- -f^[N] r, -f^[N] r - 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
⊢ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [- -f^[N] r, -f^[N] r - 1])
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | refine ⟨N, 0, λ k ↦ ?_⟩ | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
⊢ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [- -f^[N] r, -f^[N] r - 1]) | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ (fun x => f^[x] r) (k + N) = NatSeq_ofList [- -f^[N] r, -f^[N] r - 1] (k + 0) | 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
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
⊢ EventuallyEqual (fun x => f^[x] r) (NatSeq_ofList [- -f^[N] r, -f^[N] r - 1])
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | change f^[k + N] r = [_, _].getI (k % 2) | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ (fun x => f^[x] r) (k + N) = NatSeq_ofList [- -f^[N] r, -f^[N] r - 1] (k + 0) | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ f^[k + N] r = [- -f^[N] r, -f^[N] r - 1].getI (k % 2) | 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
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ (fun x => f^[x] r) (k + N) = NatSeq_ofList [- -f^[N] r, -f^[N] r - 1] (k + 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rw [neg_neg, ← N.zero_add, ← h0, zero_add 1, ← add_assoc, k.add_zero, ← h.map_mod_nat] | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ f^[k + N] r = [- -f^[N] r, -f^[N] r - 1].getI (k % 2) | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2) | 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
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ f^[k + N] r = [- -f^[N] r, -f^[N] r - 1].getI (k % 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | obtain h1 | h1 : k % 2 = 0 ∨ k % 2 = 1 := Nat.mod_two_eq_zero_or_one k | case intro.refine_2
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2) | case intro.refine_2.inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 0
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2)
case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2) | 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
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | all_goals rw [h1]; rfl | case intro.refine_2.inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 0
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2)
case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.inl
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 0
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2)
case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rw [Nat.add_right_comm, f.iterate_succ_apply', f, Int.fract, h,
Int.cast_neg, Int.cast_one, neg_one_mul, neg_sub', neg_neg] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
k : ℕ
⊢ f^[k + 1 + N] r = -f^[k + N] r - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
k : ℕ
⊢ f^[k + 1 + N] r = -f^[k + N] r - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rw [h0, neg_sub, sub_neg_eq_add, add_sub_cancel_left] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
k : ℕ
⊢ -f^[k + 1 + N] r - 1 = (fun x => f^[x + N] r) k | 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
N : ℕ
h : ∀ (k : ℕ), ⌊f^[k + N] r⌋ = -1
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
k : ℕ
⊢ -f^[k + 1 + N] r - 1 = (fun x => f^[x + N] r) k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rw [h1] | case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2) | case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[1 + N] r = [f^[0 + N] r, f^[1 + N] r].getI 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[k % 2 + N] r = [f^[0 + N] r, f^[1 + N] r].getI (k % 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.case_floor_eventually_neg_one | [82, 1] | [105, 27] | rfl | case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[1 + N] r = [f^[0 + N] r, f^[1 + N] r].getI 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.inr
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
N : ℕ
h0 : ∀ (k : ℕ), f^[k + 1 + N] r = -f^[k + N] r - 1
h : Function.Periodic (fun x => f^[x + N] r) 2
k : ℕ
h1 : k % 2 = 1
⊢ f^[1 + N] r = [f^[0 + N] r, f^[1 + N] r].getI 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.f_alt_formula | [107, 1] | [110, 79] | rw [sq, f, Int.fract, mul_sub, mul_sub, sub_sub, ← add_one_mul (α := R), sub_add_cancel,
mul_sub, sub_left_inj, ← mul_assoc, sub_one_mul, ← mul_sub_one, mul_assoc] | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
⊢ (↑⌊r⌋ - 1) * f r - ↑⌊r⌋ ^ 2 = ↑⌊r⌋ * ((↑⌊r⌋ - 1) * r - ↑⌊r⌋ ^ 2) | 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
⊢ (↑⌊r⌋ - 1) * f r - ↑⌊r⌋ ^ 2 = ↑⌊r⌋ * ((↑⌊r⌋ - 1) * r - ↑⌊r⌋ ^ 2)
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] | rcases h with ⟨N, _, h⟩ | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑C
⊢ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋) (k + N) = (fun x => -↑C) (k + w✝)
⊢ ∃ 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
C : ℕ
hC : 1 < C
h : EventuallyEqual (fun x => ⌊f^[x] r⌋) fun x => -↑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.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | simp only at h | case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋) (k + N) = (fun x => -↑C) (k + w✝)
⊢ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.intro
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, 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.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (fun x => ⌊f^[x] r⌋) (k + N) = (fun x => -↑C) (k + w✝)
⊢ ∃ 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.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | have h0 (k) : (C + 1) * f^[(k + 1) + N] r + C ^ 2
= -C * ((C + 1) * f^[k + N] r + C ^ 2) := by
have h0 := f_alt_formula (f^[k + N] r)
rw [h, Int.cast_neg, Int.cast_natCast, neg_sq, ← neg_add', neg_mul, ← neg_add',
neg_mul, neg_inj, neg_mul, ← neg_add', mul_neg, ← neg_mul] at h0
rwa [Nat.add_right_comm, f.iterate_succ_apply'] | case intro.intro
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, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.intro
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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ ∃ 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.intro
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, 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.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | replace h0 : ∀ k, (C + 1) * f^[k + N] r + C ^ 2
= (-C) ^ k * ((C + 1) * f^[N] r + C ^ 2) :=
Nat.rec (by rw [Nat.zero_add, pow_zero, one_mul])
(λ k h1 ↦ by rw [h0, h1, ← mul_assoc, ← pow_succ']) | case intro.intro
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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.intro
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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ((↑C + 1) * f^[N] r + ↑C ^ 2)
⊢ ∃ 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.intro
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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ ∃ 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.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | set ε := (C + 1) * f^[N] r + C ^ 2 | case intro.intro
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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ((↑C + 1) * f^[N] r + ↑C ^ 2)
⊢ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.intro
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) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ ∃ 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.intro
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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ((↑C + 1) * f^[N] r + ↑C ^ 2)
⊢ ∃ 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.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | refine ⟨ε, ?_, N, 0, λ k ↦ eq_neg_add_of_add_eq <| (add_comm _ _).trans (h0 k)⟩ | case intro.intro
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) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ ∃ ε, Infinitesimal ε ∧ EventuallyEqual (fun x => (↑C + 1) * f^[x] r) fun x => -↑C ^ 2 + (-↑C) ^ x * ε | case intro.intro
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) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ Infinitesimal ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
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) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ ∃ 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.case_floor_eventually_neg_of_one_lt | [112, 1] | [143, 46] | replace h0 (k) : |(C + 1) * Int.fract (f^[k + N] r) - C| = C ^ k • |ε| := by
specialize h0 k
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
rw [sub_eq_add_neg, h0, abs_mul, abs_pow, abs_neg,
Nat.abs_cast, nsmul_eq_mul, Nat.cast_pow] | case intro.intro
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) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ Infinitesimal ε | case intro.intro
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|
⊢ Infinitesimal ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
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) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
⊢ Infinitesimal 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] | replace h (s : R) : |(C + 1) * Int.fract s - C| < ((C + 1 + C : ℕ) : R) := by
apply (abs_sub _ _).trans_lt
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)]
refine mul_lt_of_lt_one_right (Nat.cast_pos.mpr C.succ_pos) (Int.fract_lt_one _) | case intro.intro
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|
⊢ Infinitesimal ε | case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
h : ∀ (s : R), |(↑C + 1) * Int.fract s - ↑C| < ↑(C + 1 + C)
⊢ Infinitesimal ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
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|
⊢ Infinitesimal 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] | exact Infinitesimal.iff_nsmul_Nat_bdd.mpr ⟨C + 1 + C, λ k ↦
(nsmul_le_nsmul_left (abs_nonneg ε) (Nat.lt_pow_self hC k).le).trans_lt <|
(h0 _).symm.trans_lt (h (f^[k + N] r))⟩ | case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
h : ∀ (s : R), |(↑C + 1) * Int.fract s - ↑C| < ↑(C + 1 + C)
⊢ Infinitesimal ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
ε : R := (↑C + 1) * f^[N] r + ↑C ^ 2
h0 : ∀ (k : ℕ), |(↑C + 1) * Int.fract (f^[k + N] r) - ↑C| = C ^ k • |ε|
h : ∀ (s : R), |(↑C + 1) * Int.fract s - ↑C| < ↑(C + 1 + C)
⊢ Infinitesimal 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] | have h0 := f_alt_formula (f^[k + N] r) | 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
k : ℕ
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2) | 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
k : ℕ
h0 :
(↑⌊f^[k + N] r⌋ - 1) * f (f^[k + N] r) - ↑⌊f^[k + N] r⌋ ^ 2 =
↑⌊f^[k + N] r⌋ * ((↑⌊f^[k + N] r⌋ - 1) * f^[k + N] r - ↑⌊f^[k + N] r⌋ ^ 2)
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2) | 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
k : ℕ
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
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 [h, Int.cast_neg, Int.cast_natCast, neg_sq, ← neg_add', neg_mul, ← neg_add',
neg_mul, neg_inj, neg_mul, ← neg_add', mul_neg, ← neg_mul] 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
k : ℕ
h0 :
(↑⌊f^[k + N] r⌋ - 1) * f (f^[k + N] r) - ↑⌊f^[k + N] r⌋ ^ 2 =
↑⌊f^[k + N] r⌋ * ((↑⌊f^[k + N] r⌋ - 1) * f^[k + N] r - ↑⌊f^[k + N] r⌋ ^ 2)
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2) | 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
k : ℕ
h0 : (↑C + 1) * f (f^[k + N] r) + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2) | 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
k : ℕ
h0 :
(↑⌊f^[k + N] r⌋ - 1) * f (f^[k + N] r) - ↑⌊f^[k + N] r⌋ ^ 2 =
↑⌊f^[k + N] r⌋ * ((↑⌊f^[k + N] r⌋ - 1) * f^[k + N] r - ↑⌊f^[k + N] r⌋ ^ 2)
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
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] | rwa [Nat.add_right_comm, f.iterate_succ_apply'] | 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
k : ℕ
h0 : (↑C + 1) * f (f^[k + N] r) + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2) | 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
k : ℕ
h0 : (↑C + 1) * f (f^[k + N] r) + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
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.zero_add, pow_zero, one_mul] | 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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ (↑C + 1) * f^[Nat.zero + N] r + ↑C ^ 2 = (-↑C) ^ Nat.zero * ((↑C + 1) * f^[N] r + ↑C ^ 2) | 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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
⊢ (↑C + 1) * f^[Nat.zero + N] r + ↑C ^ 2 = (-↑C) ^ Nat.zero * ((↑C + 1) * f^[N] r + ↑C ^ 2)
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 [h0, h1, ← mul_assoc, ← pow_succ'] | 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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
k : ℕ
h1 : (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ((↑C + 1) * f^[N] r + ↑C ^ 2)
⊢ (↑C + 1) * f^[k.succ + N] r + ↑C ^ 2 = (-↑C) ^ k.succ * ((↑C + 1) * f^[N] r + ↑C ^ 2) | 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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + 1 + N] r + ↑C ^ 2 = -↑C * ((↑C + 1) * f^[k + N] r + ↑C ^ 2)
k : ℕ
h1 : (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ((↑C + 1) * f^[N] r + ↑C ^ 2)
⊢ (↑C + 1) * f^[k.succ + N] r + ↑C ^ 2 = (-↑C) ^ k.succ * ((↑C + 1) * f^[N] r + ↑C ^ 2)
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] | specialize h0 k | 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) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
k : ℕ
⊢ |(↑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) * f^[k + N] r + ↑C ^ 2 = (-↑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
h0 : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r + ↑C ^ 2 = (-↑C) ^ k * ε
k : ℕ
⊢ |(↑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 [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:
|
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