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/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases exists_iter_add_large_eq h a k with ⟨N, h4⟩ | case intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
⊢ ∃ N, 0 < minimalPeriod f (f^[N] 0) | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ ∃ N, 0 < minimalPeriod f (f^[N] 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
⊢ ∃ N, 0 < minimalPeriod f (f^[N] 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | refine ⟨N + 1 + M, IsPeriodicPt.minimalPeriod_pos h1 ?_⟩ | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ ∃ N, 0 < minimalPeriod f (f^[N] 0) | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[N + 1 + M] 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ ∃ N, 0 < minimalPeriod f (f^[N] 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [iterate_add_apply, ← h3, Commute.iterate_iterate_self] | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[N + 1 + M] 0) | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[K] (f^[N + 1] a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[N + 1 + M] 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | refine IsPeriodicPt.apply_iterate ?_ _ | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[K] (f^[N + 1] a)) | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[N + 1] a) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[K] (f^[N + 1] a))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [IsPeriodicPt, IsFixedPt, ← iterate_add_apply,
iterate_succ_apply, ← h2, Commute.iterate_self, ← h4,
add_left_comm, ← add_assoc, iterate_succ_apply'] | case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[N + 1] a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
K M : ℕ
h3 : f^[K] a = f^[M] 0
N : ℕ
h4 : f^[N + k] a = f^[N] (a + ↑k)
⊢ IsPeriodicPt f k (f^[N + 1] a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases this with ⟨N, h1⟩ | f : ℤ → ℤ
h : good f
h0 : ¬Injective f
this : ∃ N, 0 < minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
this : ∃ N, 0 < minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | let k := f.minimalPeriod (f^[N] 0) | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | let F := λ n ↦ |f^[n] 0| | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | refine ⟨Extra.seqMax F (N + k) + 1, λ n ↦ Int.lt_add_one_of_le <|
(n.le_total (N + k)).elim (Extra.le_seqMax_of_le F) (λ h2 ↦ ?_)⟩ | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
n : ℕ
h2 : N + k ≤ n
⊢ |f^[n] 0| ≤ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
⊢ ∃ M, ∀ (n : ℕ), |f^[n] 0| < M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [le_iff_exists_add] at h2 | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
n : ℕ
h2 : N + k ≤ n
⊢ |f^[n] 0| ≤ Extra.seqMax F (N + k) | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
n : ℕ
h2 : ∃ c, n = N + k + c
⊢ |f^[n] 0| ≤ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
n : ℕ
h2 : N + k ≤ n
⊢ |f^[n] 0| ≤ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases h2 with ⟨c, rfl⟩ | case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
n : ℕ
h2 : ∃ c, n = N + k + c
⊢ |f^[n] 0| ≤ Extra.seqMax F (N + k) | case intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
c : ℕ
⊢ |f^[N + k + c] 0| ≤ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
n : ℕ
h2 : ∃ c, n = N + k + c
⊢ |f^[n] 0| ≤ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [add_rotate, iterate_add_apply, ← iterate_mod_minimalPeriod_eq,
Nat.add_mod_left, ← iterate_add_apply, add_comm] | case intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
c : ℕ
⊢ |f^[N + k + c] 0| ≤ Extra.seqMax F (N + k) | case intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
c : ℕ
⊢ |f^[N + c % k] 0| ≤ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
c : ℕ
⊢ |f^[N + k + c] 0| ≤ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exact Extra.le_seqMax_of_le F (add_le_add_left (c.mod_lt h1).le N) | case intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
c : ℕ
⊢ |f^[N + c % k] 0| ≤ Extra.seqMax F (N + k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
N : ℕ
h1 : 0 < minimalPeriod f (f^[N] 0)
k : ℕ := minimalPeriod f (f^[N] 0)
F : ℕ → ℤ := fun n => |f^[n] 0|
c : ℕ
⊢ |f^[N + c % k] 0| ≤ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | suffices ∃ a b, a < b ∧ f a = f b by
rcases this with ⟨a, b, h1, h2⟩
apply sub_pos_of_lt at h1
refine ⟨a, (b - a).natAbs, Int.natAbs_pos.mpr h1.ne.symm, ?_⟩
rw [Int.natCast_natAbs, abs_of_pos h1, add_sub_cancel, h2] | f : ℤ → ℤ
h : good f
h0 : ¬Injective f
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a | f : ℤ → ℤ
h : good f
h0 : ¬Injective f
⊢ ∃ a b, a < b ∧ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | simp_rw [Injective, not_forall] at h0 | f : ℤ → ℤ
h : good f
h0 : ¬Injective f
⊢ ∃ a b, a < b ∧ f a = f b | f : ℤ → ℤ
h : good f
h0 : ∃ x x_1, ∃ (_ : f x = f x_1), ¬x = x_1
⊢ ∃ a b, a < b ∧ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
⊢ ∃ a b, a < b ∧ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases h0 with ⟨a, b, h0, h1⟩ | f : ℤ → ℤ
h : good f
h0 : ∃ x x_1, ∃ (_ : f x = f x_1), ¬x = x_1
⊢ ∃ a b, a < b ∧ f a = f b | case intro.intro.intro
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
⊢ ∃ a b, a < b ∧ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
h0 : ∃ x x_1, ∃ (_ : f x = f x_1), ¬x = x_1
⊢ ∃ a b, a < b ∧ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases ne_iff_lt_or_gt.mp h1 with h2 | h2 | case intro.intro.intro
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
⊢ ∃ a b, a < b ∧ f a = f b | case intro.intro.intro.inl
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
h2 : a < b
⊢ ∃ a b, a < b ∧ f a = f b
case intro.intro.intro.inr
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
h2 : a > b
⊢ ∃ a b, a < b ∧ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
⊢ ∃ a b, a < b ∧ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exacts [⟨a, b, h2, h0⟩, ⟨b, a, h2, h0.symm⟩] | case intro.intro.intro.inl
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
h2 : a < b
⊢ ∃ a b, a < b ∧ f a = f b
case intro.intro.intro.inr
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
h2 : a > b
⊢ ∃ a b, a < b ∧ f a = f b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
h2 : a < b
⊢ ∃ a b, a < b ∧ f a = f b
case intro.intro.intro.inr
f : ℤ → ℤ
h : good f
a b : ℤ
h0 : f a = f b
h1 : ¬a = b
h2 : a > b
⊢ ∃ a b, a < b ∧ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases this with ⟨a, b, h1, h2⟩ | f : ℤ → ℤ
h : good f
h0 : ¬Injective f
this : ∃ a b, a < b ∧ f a = f b
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a | case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h1 : a < b
h2 : f a = f b
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
this : ∃ a b, a < b ∧ f a = f b
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | apply sub_pos_of_lt at h1 | case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h1 : a < b
h2 : f a = f b
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a | case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h2 : f a = f b
h1 : 0 < b - a
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h1 : a < b
h2 : f a = f b
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | refine ⟨a, (b - a).natAbs, Int.natAbs_pos.mpr h1.ne.symm, ?_⟩ | case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h2 : f a = f b
h1 : 0 < b - a
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a | case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h2 : f a = f b
h1 : 0 < b - a
⊢ f (a + ↑(b - a).natAbs) = f a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h2 : f a = f b
h1 : 0 < b - a
⊢ ∃ a k, 0 < k ∧ f (a + ↑k) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [Int.natCast_natAbs, abs_of_pos h1, add_sub_cancel, h2] | case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h2 : f a = f b
h1 : 0 < b - a
⊢ f (a + ↑(b - a).natAbs) = f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a b : ℤ
h2 : f a = f b
h1 : 0 < b - a
⊢ f (a + ↑(b - a).natAbs) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases le_total a 0 with h3 | h3 | f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
⊢ ∃ K M, f^[K] a = f^[M] 0 | case inl
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
⊢ ∃ K M, f^[K] a = f^[M] 0
case inr
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
⊢ ∃ K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
⊢ ∃ K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases exists_iter_add_large_eq h a a.natAbs with ⟨N, h4⟩ | case inl
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
⊢ ∃ K M, f^[K] a = f^[M] 0 | case inl.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
N : ℕ
h4 : f^[N + a.natAbs] a = f^[N] (a + ↑a.natAbs)
⊢ ∃ K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
⊢ ∃ K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [Int.natCast_natAbs, abs_of_nonpos h3, add_neg_self] at h4 | case inl.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
N : ℕ
h4 : f^[N + a.natAbs] a = f^[N] (a + ↑a.natAbs)
⊢ ∃ K M, f^[K] a = f^[M] 0 | case inl.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
N : ℕ
h4 : f^[N + a.natAbs] a = f^[N] 0
⊢ ∃ K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
N : ℕ
h4 : f^[N + a.natAbs] a = f^[N] (a + ↑a.natAbs)
⊢ ∃ K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exact ⟨_, _, h4⟩ | case inl.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
N : ℕ
h4 : f^[N + a.natAbs] a = f^[N] 0
⊢ ∃ K M, f^[K] a = f^[M] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : a ≤ 0
N : ℕ
h4 : f^[N + a.natAbs] a = f^[N] 0
⊢ ∃ K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases exists_iter_add_large_eq h 0 a.natAbs with ⟨N, h4⟩ | case inr
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
⊢ ∃ K M, f^[K] a = f^[M] 0 | case inr.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
N : ℕ
h4 : f^[N + a.natAbs] 0 = f^[N] (0 + ↑a.natAbs)
⊢ ∃ K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
⊢ ∃ K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [Int.natCast_natAbs, abs_of_nonneg h3, zero_add] at h4 | case inr.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
N : ℕ
h4 : f^[N + a.natAbs] 0 = f^[N] (0 + ↑a.natAbs)
⊢ ∃ K M, f^[K] a = f^[M] 0 | case inr.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
N : ℕ
h4 : f^[N + a.natAbs] 0 = f^[N] a
⊢ ∃ K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
N : ℕ
h4 : f^[N + a.natAbs] 0 = f^[N] (0 + ↑a.natAbs)
⊢ ∃ K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exact ⟨_, _, h4.symm⟩ | case inr.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
N : ℕ
h4 : f^[N + a.natAbs] 0 = f^[N] a
⊢ ∃ K M, f^[K] a = f^[M] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
f : ℤ → ℤ
h : good f
h0 : ¬Injective f
a : ℤ
k : ℕ
h1 : 0 < k
h2 : f (a + ↑k) = f a
h3 : 0 ≤ a
N : ℕ
h4 : f^[N + a.natAbs] 0 = f^[N] a
⊢ ∃ K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rcases h0 with ⟨M, h0⟩ | f : ℤ → ℤ
h : good f
h0 : ∃ M, ∀ (n : ℕ), |f^[n] 0| < M
⊢ f = 0 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
h0 : ∃ M, ∀ (n : ℕ), |f^[n] 0| < M
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h1 (a : ℤ) : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) := by
have h := h a (-a); rwa [a.natAbs_neg, add_neg_self,
← two_mul, neg_mul, ← sub_eq_add_neg, ← mul_sub] at h | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
⊢ f = 0 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | replace h0 (a : ℤ) (h2 : M ≤ a) : f.IsPeriodicPt (2 * a.natAbs ^ 2) 0 :=
Int.eq_zero_of_abs_lt_dvd ⟨_, h1 a⟩ ((h0 _).trans_le h2) | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
⊢ f = 0 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | replace h0 : f.IsPeriodicPt 2 0 := by
have h2 := (h0 _ M.le_refl).gcd (h0 _ M.lt_succ.le)
have h3 : IsCoprime M (M + 1) :=
⟨-1, 1, by rw [one_mul, neg_one_mul, neg_add_cancel_left]⟩
rwa [Nat.gcd_mul_left, ← Int.natAbs_pow, ← Int.natAbs_pow,
← Int.gcd_eq_natAbs, Int.gcd_eq_one_iff_coprime.mpr h3.pow, mul_one] at h2 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
⊢ f = 0 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | replace h1 (a : ℤ) (h3 : a ≠ 0) : f a = f (-a) := by
specialize h1 a
rw [h0.mul_const, zero_eq_mul] at h1
exact eq_of_sub_eq_zero (h1.resolve_left h3) | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
⊢ f = 0 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | suffices h2 : ∀ a : ℤ, a ≠ 0 → f a = 0 by
have h3 : f (f 2) = 1 * f 1 + 1 * f 1 := h 1 1
rw [h2 1 one_ne_zero, h2 _ two_ne_zero] at h3
exact funext λ a ↦ (ne_or_eq a 0).elim (h2 a) (λ h4 ↦ h4.symm ▸ h3) | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
⊢ f = 0 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
⊢ ∀ (a : ℤ), a ≠ 0 → f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | intro a h2 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
⊢ ∀ (a : ℤ), a ≠ 0 → f a = 0 | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
⊢ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
⊢ ∀ (a : ℤ), a ≠ 0 → f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | obtain ⟨n, h3⟩ : ∃ n : ℕ, a.natAbs ^ 2 = n.succ :=
Nat.exists_eq_succ_of_ne_zero (pow_ne_zero 2 <| Int.natAbs_ne_zero.mpr h2) | case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
⊢ f a = 0 | case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
⊢ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
⊢ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h4 := map_iterate_sq h (-a) | case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
⊢ f a = 0 | case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
h4 : f^[(-a).natAbs ^ 2] (-a) = -a * f (-a)
⊢ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
⊢ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [Int.natAbs_neg, h3, iterate_succ_apply, ← h1 a h2, ← iterate_succ_apply,
← h3, map_iterate_sq h, neg_mul, eq_neg_self_iff, mul_eq_zero] at h4 | case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
h4 : f^[(-a).natAbs ^ 2] (-a) = -a * f (-a)
⊢ f a = 0 | case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
h4 : a = 0 ∨ f a = 0
⊢ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
h4 : f^[(-a).natAbs ^ 2] (-a) = -a * f (-a)
⊢ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | exact h4.resolve_left h2 | case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
h4 : a = 0 ∨ f a = 0
⊢ f a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
a : ℤ
h2 : a ≠ 0
n : ℕ
h3 : a.natAbs ^ 2 = n.succ
h4 : a = 0 ∨ f a = 0
⊢ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h := h a (-a) | f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
a : ℤ
⊢ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) | f : ℤ → ℤ
h✝ : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
a : ℤ
h : f^[a.natAbs ^ 2 + (-a).natAbs ^ 2] (a + -a) = a * f a + -a * f (-a)
⊢ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
a : ℤ
⊢ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rwa [a.natAbs_neg, add_neg_self,
← two_mul, neg_mul, ← sub_eq_add_neg, ← mul_sub] at h | f : ℤ → ℤ
h✝ : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
a : ℤ
h : f^[a.natAbs ^ 2 + (-a).natAbs ^ 2] (a + -a) = a * f a + -a * f (-a)
⊢ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h✝ : good f
M : ℤ
h0 : ∀ (n : ℕ), |f^[n] 0| < M
a : ℤ
h : f^[a.natAbs ^ 2 + (-a).natAbs ^ 2] (a + -a) = a * f a + -a * f (-a)
⊢ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h2 := (h0 _ M.le_refl).gcd (h0 _ M.lt_succ.le) | f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
⊢ IsPeriodicPt f 2 0 | f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
⊢ IsPeriodicPt f 2 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
⊢ IsPeriodicPt f 2 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h3 : IsCoprime M (M + 1) :=
⟨-1, 1, by rw [one_mul, neg_one_mul, neg_add_cancel_left]⟩ | f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
⊢ IsPeriodicPt f 2 0 | f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
h3 : IsCoprime M (M + 1)
⊢ IsPeriodicPt f 2 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
⊢ IsPeriodicPt f 2 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rwa [Nat.gcd_mul_left, ← Int.natAbs_pow, ← Int.natAbs_pow,
← Int.gcd_eq_natAbs, Int.gcd_eq_one_iff_coprime.mpr h3.pow, mul_one] at h2 | f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
h3 : IsCoprime M (M + 1)
⊢ IsPeriodicPt f 2 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
h3 : IsCoprime M (M + 1)
⊢ IsPeriodicPt f 2 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [one_mul, neg_one_mul, neg_add_cancel_left] | f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
⊢ -1 * M + 1 * (M + 1) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : ∀ (a : ℤ), M ≤ a → IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
⊢ -1 * M + 1 * (M + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | specialize h1 a | f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
⊢ f a = f (-a) | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
h1 : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
⊢ f a = f (-a) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h1 : ∀ (a : ℤ), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
⊢ f a = f (-a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [h0.mul_const, zero_eq_mul] at h1 | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
h1 : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
⊢ f a = f (-a) | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
h1 : a = 0 ∨ f a - f (-a) = 0
⊢ f a = f (-a) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
h1 : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
⊢ f a = f (-a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | exact eq_of_sub_eq_zero (h1.resolve_left h3) | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
h1 : a = 0 ∨ f a - f (-a) = 0
⊢ f a = f (-a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
a : ℤ
h3 : a ≠ 0
h1 : a = 0 ∨ f a - f (-a) = 0
⊢ f a = f (-a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h3 : f (f 2) = 1 * f 1 + 1 * f 1 := h 1 1 | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
⊢ f = 0 | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
h3 : f (f 2) = 1 * f 1 + 1 * f 1
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [h2 1 one_ne_zero, h2 _ two_ne_zero] at h3 | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
h3 : f (f 2) = 1 * f 1 + 1 * f 1
⊢ f = 0 | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
h3 : f 0 = 1 * 0 + 1 * 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
h3 : f (f 2) = 1 * f 1 + 1 * f 1
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | exact funext λ a ↦ (ne_or_eq a 0).elim (h2 a) (λ h4 ↦ h4.symm ▸ h3) | f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
h3 : f 0 = 1 * 0 + 1 * 0
⊢ f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : good f
M : ℤ
h0 : IsPeriodicPt f 2 0
h1 : ∀ (a : ℤ), a ≠ 0 → f a = f (-a)
h2 : ∀ (a : ℤ), a ≠ 0 → f a = 0
h3 : f 0 = 1 * 0 + 1 * 0
⊢ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.castMonoidHom_is_MonoidGood | [33, 1] | [34, 65] | rw [Int.floor_intCast, ← Int.cast_mul, ← φ.map_mul] | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℤ
m n : M
⊢ (fun x => ↑(φ x)) (m * n) = (fun x => ↑(φ x)) m * ↑⌊(fun x => ↑(φ x)) n⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℤ
m n : M
⊢ (fun x => ↑(φ x)) (m * n) = (fun x => ↑(φ x)) m * ↑⌊(fun x => ↑(φ x)) n⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | change (1 + ε) * _ = (1 + ε) * _ * ⌊(1 + ε) * _⌋ | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (fun x => (1 + ε) * ↑(φ x)) (m * n) = (fun x => (1 + ε) * ↑(φ x)) m * ↑⌊(fun x => (1 + ε) * ↑(φ x)) n⌋ | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ (m * n)) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (fun x => (1 + ε) * ↑(φ x)) (m * n) = (fun x => (1 + ε) * ↑(φ x)) m * ↑⌊(fun x => (1 + ε) * ↑(φ x)) n⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | rw [φ.map_mul, Nat.cast_mul, ← mul_assoc] | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ (m * n)) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ m) * ↑(φ n) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ (m * n)) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | apply congrArg | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ m) * ↑(φ n) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ ↑(φ n) = ↑⌊(1 + ε) * ↑(φ n)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ m) * ↑(φ n) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | rw [one_add_mul ε, Int.floor_nat_add, Int.cast_add, Int.cast_natCast,
← nsmul_eq_mul', self_eq_add_right, Int.cast_eq_zero, Int.floor_eq_zero_iff] | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ ↑(φ n) = ↑⌊(1 + ε) * ↑(φ n)⌋ | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ φ n • ε ∈ Set.Ico 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ ↑(φ n) = ↑⌊(1 + ε) * ↑(φ n)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | exact ⟨nsmul_nonneg h _, abs_eq_self.mpr h ▸ h0 (φ n)⟩ | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ φ n • ε ∈ Set.Ico 0 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ φ n • ε ∈ Set.Ico 0 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | simp only [h] | R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (fun n => if n ∈ A then C else 0) (m * n) =
(fun n => if n ∈ A then C else 0) m * ↑⌊(fun n => if n ∈ A then C else 0) n⌋ | R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (fun n => if n ∈ A then C else 0) (m * n) =
(fun n => if n ∈ A then C else 0) m * ↑⌊(fun n => if n ∈ A then C else 0) n⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | by_cases h1 : n ∈ A | R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | rw [if_pos h1, h0, Int.cast_one, mul_one] | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = if m ∈ A then C else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | exact if_congr (and_iff_left h1) rfl rfl | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = if m ∈ A then C else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = if m ∈ A then C else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | rw [if_neg h1, Int.floor_zero, Int.cast_zero, mul_zero] | case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | exact if_neg λ h2 ↦ h1 h2.2 | case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.map_eq_map_one_mul_floor | [66, 1] | [67, 21] | rw [← hf, one_mul] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
x : M
⊢ f x = f 1 * ↑⌊f x⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
x : M
⊢ f x = f 1 * ↑⌊f x⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | have h := map_eq_map_one_mul_floor hf 1 | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = f 1 * ↑⌊f 1⌋
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
⊢ f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | rw [← sub_eq_zero, ← mul_one_sub, mul_eq_zero] at h | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = f 1 * ↑⌊f 1⌋
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = f 1 * ↑⌊f 1⌋
⊢ f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | revert h | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
⊢ f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0 → f = 0 ∨ ⌊f 1⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0
⊢ f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | refine Or.imp (λ h ↦ funext λ n ↦ ?_)
(λ h ↦ Int.cast_eq_one.mp (eq_of_sub_eq_zero h).symm) | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
⊢ f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0 → f = 0 ∨ ⌊f 1⌋ = 1 | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0
n : M
⊢ f n = 0 n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
⊢ f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0 → f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | rw [map_eq_map_one_mul_floor hf, h, zero_mul] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0
n : M
⊢ f n = 0 n | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0
n : M
⊢ 0 = 0 n | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0
n : M
⊢ f n = 0 n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | rfl | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0
n : M
⊢ 0 = 0 n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f 1 = 0
n : M
⊢ 0 = 0 n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.fract_eq_eps_mul_floor | [80, 1] | [81, 89] | rw [Int.fract, Int.fract, h, Int.cast_one, sub_one_mul, ← map_eq_map_one_mul_floor hf] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
⊢ Int.fract (f x) = Int.fract (f 1) * ↑⌊f x⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
⊢ Int.fract (f x) = Int.fract (f 1) * ↑⌊f x⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | have h0 : f 1 ≠ 0 := λ h0 ↦ Int.zero_ne_one <| by rw [← h, h0, Int.floor_zero] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 ≠ 0
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | have h1 := hf x y | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 ≠ 0
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 ≠ 0
h1 : f (x * y) = f x * ↑⌊f y⌋
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 ≠ 0
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | rwa [map_eq_map_one_mul_floor hf, map_eq_map_one_mul_floor hf x, mul_assoc, ← sub_eq_zero,
← mul_sub, mul_eq_zero, or_iff_right h0, ← Int.cast_mul, sub_eq_zero, Int.cast_inj] at h1 | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 ≠ 0
h1 : f (x * y) = f x * ↑⌊f y⌋
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 ≠ 0
h1 : f (x * y) = f x * ↑⌊f y⌋
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | rw [← h, h0, Int.floor_zero] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 = 0
⊢ 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x y : M
h0 : f 1 = 0
⊢ 0 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | rcases floor_unbounded_of_one_lt h0 N with ⟨y, h1⟩ | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋ | case intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | use x * y | case intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋ | case h
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ↑(N + 1) < ⌊f (x * y)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | rw [floor_map_mul hf h, Nat.cast_succ, ← one_mul ((N : ℤ) + 1)] | case h
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ↑(N + 1) < ⌊f (x * y)⌋ | case h
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ 1 * (↑N + 1) < ⌊f x⌋ * ⌊f y⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ↑(N + 1) < ⌊f (x * y)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | exact mul_lt_mul_of_nonneg_of_pos h0 h1 Int.one_nonneg (N.cast_nonneg.trans_lt h1) | case h
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ 1 * (↑N + 1) < ⌊f x⌋ * ⌊f y⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ 1 * (↑N + 1) < ⌊f x⌋ * ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rw [map_eq_map_one_mul_floor hf,
← Int.natAbs_of_nonneg (h1 x), Int.cast_natCast] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f x = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x) | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f 1 * ↑⌊f x⌋.natAbs = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f x = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rfl | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f 1 * ↑⌊f x⌋.natAbs = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f 1 * ↑⌊f x⌋.natAbs = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | refine ⟨{x : M | ⌊f x⌋ ≠ 0}, λ x y ↦ ?_, λ x ↦ ?_⟩ | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
⊢ ∃ A, ∃ (_ : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A), ∀ (x : M), f x = if x ∈ A then f 1 else 0 | case refine_1
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x y : M
⊢ x * y ∈ {x | ⌊f x⌋ ≠ 0} ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0}
case refine_2
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
⊢ ∃ A, ∃ (_ : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A), ∀ (x : M), f x = if x ∈ A then f 1 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | by_cases h3 : ⌊f x⌋ = 0 | case refine_2
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0
case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | suffices ⌊f x⌋ = 1 by rw [if_pos (by rwa [Set.mem_setOf_eq]),
map_eq_map_one_mul_floor hf, this, Int.cast_one, mul_one] | case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | specialize h1 x | case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
⊢ ⌊f x⌋ = 1 | case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 0 ≤ ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
⊢ ⌊f x⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rw [le_iff_eq_or_lt, eq_comm, or_iff_right h3,
Int.lt_iff_add_one_le, zero_add, le_iff_eq_or_lt] at h1 | case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 0 ≤ ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋ ∨ 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 0 ≤ ⌊f x⌋
⊢ ⌊f x⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rcases h1 with h1 | h1 | case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋ ∨ 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg.inl
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋
⊢ ⌊f x⌋ = 1
case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋ ∨ 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | exact h1.symm | case neg.inl
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋
⊢ ⌊f x⌋ = 1
case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋
⊢ ⌊f x⌋ = 1
case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | refine h2.elim λ N ↦ ?_ | case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
⊢ N • Int.fract (f 1) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rcases floor_unbounded_of_one_lt hf h h1 N with ⟨y, h4⟩ | case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
⊢ N • Int.fract (f 1) < 1 | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ N • Int.fract (f 1) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
⊢ N • Int.fract (f 1) < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rw [nsmul_eq_mul', ← Int.cast_natCast] | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ N • Int.fract (f 1) < 1 | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑↑N < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ N • Int.fract (f 1) < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | apply (mul_lt_mul_of_pos_left (Int.cast_lt.mpr h4) h0).trans | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑↑N < 1 | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑⌊f y⌋ < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑↑N < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rw [← fract_eq_eps_mul_floor hf h] | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑⌊f y⌋ < 1 | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f y) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑⌊f y⌋ < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | exact Int.fract_lt_one _ | case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f y) < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f y) < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rw [Set.mem_setOf_eq, floor_map_mul hf h, mul_ne_zero_iff] | case refine_1
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x y : M
⊢ x * y ∈ {x | ⌊f x⌋ ≠ 0} ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0} | case refine_1
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x y : M
⊢ ⌊f x⌋ ≠ 0 ∧ ⌊f y⌋ ≠ 0 ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x y : M
⊢ x * y ∈ {x | ⌊f x⌋ ≠ 0} ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rfl | case refine_1
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x y : M
⊢ ⌊f x⌋ ≠ 0 ∧ ⌊f y⌋ ≠ 0 ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x y : M
⊢ ⌊f x⌋ ≠ 0 ∧ ⌊f y⌋ ≠ 0 ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rw [map_eq_map_one_mul_floor hf, h3, Int.cast_zero, mul_zero] | case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ 0 = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | refine (if_neg ?_).symm | case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ 0 = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ x ∉ {x | ⌊f x⌋ ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ 0 = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rwa [Set.mem_setOf_eq, not_not] | case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ x ∉ {x | ⌊f x⌋ ≠ 0} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ⌊f x⌋ = 0
⊢ x ∉ {x | ⌊f x⌋ ≠ 0}
TACTIC:
|
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