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/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rw [mem_symmDiff, mem_Icc] at h2 | case flip.refine_2.refine_1
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : j ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j ∉ X.board
h2 : i + n ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
⊢ False | case flip.refine_2.refine_1
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : j ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j ∉ X.board
h2 : i + n ∈ X.board ∧ ¬(i ≤ i + n ∧ i + n ≤ i + n) ∨ (i ≤ i + n ∧ i + n ≤ i + n) ∧ i + n ∉ X.board
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_1
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : j ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j ∉ X.board
h2 : i + n ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | exact h2.elim (λ h2 ↦ h2.2 ⟨le_self_add, (i + n).le_refl⟩) (λ h2 ↦ h2.2 h) | case flip.refine_2.refine_1
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : j ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j ∉ X.board
h2 : i + n ∈ X.board ∧ ¬(i ≤ i + n ∧ i + n ≤ i + n) ∨ (i ≤ i + n ∧ i + n ≤ i + n) ∧ i + n ∉ X.board
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_1
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : j ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j ∉ X.board
h2 : i + n ∈ X.board ∧ ¬(i ≤ i + n ∧ i + n ≤ i + n) ∨ (i ≤ i + n ∧ i + n ≤ i + n) ∧ i + n ∉ X.board
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rw [mem_symmDiff, or_iff_right (λ h2 ↦ h1 h2.1), mem_Icc] at h0 | case flip.refine_2.refine_2
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : j ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j ∉ X.board
⊢ j ≤ i + n | case flip.refine_2.refine_2
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : (i ≤ j ∧ j ≤ i + n) ∧ j ∉ X.board
h1 : j ∉ X.board
⊢ j ≤ i + n | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_2
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : j ∈ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j ∉ X.board
⊢ j ≤ i + n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | exact h0.1.2 | case flip.refine_2.refine_2
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : (i ≤ j ∧ j ≤ i + n) ∧ j ∉ X.board
h1 : j ∉ X.board
⊢ j ≤ i + n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_2
n : ℕ
X : GameState n
i : ℕ
h : i + n ∈ X.board
j : ℕ
h0 : (i ≤ j ∧ j ≤ i + n) ∧ j ∉ X.board
h1 : j ∉ X.board
⊢ j ≤ i + n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.isReachable_sum_two_pow_add_numMoves | [155, 1] | [162, 60] | rw [ValidMove_numMoves h0, ← add_assoc, Nat.succ_le_iff] | n : ℕ
X Y : GameState n
h : X.IsReachable Y
a✝ c✝ : GameState n
h0 : a✝.ValidMove c✝
x✝ : ReflTransGen ValidMove c✝ Y
h1 : ∑ i ∈ Y.board, 2 ^ i + Y.numMoves ≤ ∑ i ∈ c✝.board, 2 ^ i + c✝.numMoves
⊢ ∑ i ∈ c✝.board, 2 ^ i + c✝.numMoves ≤ ∑ i ∈ a✝.board, 2 ^ i + a✝.numMoves | n : ℕ
X Y : GameState n
h : X.IsReachable Y
a✝ c✝ : GameState n
h0 : a✝.ValidMove c✝
x✝ : ReflTransGen ValidMove c✝ Y
h1 : ∑ i ∈ Y.board, 2 ^ i + Y.numMoves ≤ ∑ i ∈ c✝.board, 2 ^ i + c✝.numMoves
⊢ ∑ i ∈ c✝.board, 2 ^ i + a✝.numMoves < ∑ i ∈ a✝.board, 2 ^ i + a✝.numMoves | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
X Y : GameState n
h : X.IsReachable Y
a✝ c✝ : GameState n
h0 : a✝.ValidMove c✝
x✝ : ReflTransGen ValidMove c✝ Y
h1 : ∑ i ∈ Y.board, 2 ^ i + Y.numMoves ≤ ∑ i ∈ c✝.board, 2 ^ i + c✝.numMoves
⊢ ∑ i ∈ c✝.board, 2 ^ i + c✝.numMoves ≤ ∑ i ∈ a✝.board, 2 ^ i + a✝.numMoves
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.isReachable_sum_two_pow_add_numMoves | [155, 1] | [162, 60] | exact Nat.add_lt_add_right (ValidMove_sum_two_pow h0) _ | n : ℕ
X Y : GameState n
h : X.IsReachable Y
a✝ c✝ : GameState n
h0 : a✝.ValidMove c✝
x✝ : ReflTransGen ValidMove c✝ Y
h1 : ∑ i ∈ Y.board, 2 ^ i + Y.numMoves ≤ ∑ i ∈ c✝.board, 2 ^ i + c✝.numMoves
⊢ ∑ i ∈ c✝.board, 2 ^ i + a✝.numMoves < ∑ i ∈ a✝.board, 2 ^ i + a✝.numMoves | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
X Y : GameState n
h : X.IsReachable Y
a✝ c✝ : GameState n
h0 : a✝.ValidMove c✝
x✝ : ReflTransGen ValidMove c✝ Y
h1 : ∑ i ∈ Y.board, 2 ^ i + Y.numMoves ≤ ∑ i ∈ c✝.board, 2 ^ i + c✝.numMoves
⊢ ∑ i ∈ c✝.board, 2 ^ i + a✝.numMoves < ∑ i ∈ a✝.board, 2 ^ i + a✝.numMoves
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_central_card_mod_two | [178, 1] | [183, 50] | rw [centralCards, filter_symmDiff, ← centralCards,
symmDiff_card_mod_two, Iic_filter_dvd_card] | n : ℕ
X Y : GameState n
h : X.ValidMove Y
i : ℕ
x✝ : i + n ∈ X.board
⊢ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.centralCards.card % 2 =
(X.centralCards.card + 1) % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
X Y : GameState n
h : X.ValidMove Y
i : ℕ
x✝ : i + n ∈ X.board
⊢ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.centralCards.card % 2 =
(X.centralCards.card + 1) % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.isReachable_central_card_add_numMoves_mod_two | [185, 1] | [192, 58] | rw [ValidMove_numMoves h0, Nat.add_mod, ValidMove_central_card_mod_two h0,
← Nat.add_mod, add_add_add_comm, Nat.add_mod_right] | n : ℕ
X Y : GameState n
h : X.IsReachable Y
a✝ c✝ : GameState n
h0 : a✝.ValidMove c✝
x✝ : ReflTransGen ValidMove c✝ Y
h1 : (Y.centralCards.card + Y.numMoves) % 2 = (c✝.centralCards.card + c✝.numMoves) % 2
⊢ (c✝.centralCards.card + c✝.numMoves) % 2 = (a✝.centralCards.card + a✝.numMoves) % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
X Y : GameState n
h : X.IsReachable Y
a✝ c✝ : GameState n
h0 : a✝.ValidMove c✝
x✝ : ReflTransGen ValidMove c✝ Y
h1 : (Y.centralCards.card + Y.numMoves) % 2 = (c✝.centralCards.card + c✝.numMoves) % 2
⊢ (c✝.centralCards.card + c✝.numMoves) % 2 = (a✝.centralCards.card + a✝.numMoves) % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.numMoves_mod_two_eq_div_of_ends | [205, 1] | [210, 60] | have h1 := isReachable_central_card_add_numMoves_mod_two h | M n : ℕ
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
⊢ X.numMoves % 2 = M / n.succ % 2 | M n : ℕ
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
h1 : (X.centralCards.card + X.numMoves) % 2 = ((init M n).centralCards.card + (init M n).numMoves) % 2
⊢ X.numMoves % 2 = M / n.succ % 2 | Please generate a tactic in lean4 to solve the state.
STATE:
M n : ℕ
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
⊢ X.numMoves % 2 = M / n.succ % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.numMoves_mod_two_eq_div_of_ends | [205, 1] | [210, 60] | rwa [filter_central_init_card, numMoves_init, Nat.add_zero,
filter_central_ends h0, card_empty, Nat.zero_add] at h1 | M n : ℕ
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
h1 : (X.centralCards.card + X.numMoves) % 2 = ((init M n).centralCards.card + (init M n).numMoves) % 2
⊢ X.numMoves % 2 = M / n.succ % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M n : ℕ
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
h1 : (X.centralCards.card + X.numMoves) % 2 = ((init M n).centralCards.card + (init M n).numMoves) % 2
⊢ X.numMoves % 2 = M / n.succ % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | have h0 : f (c + 1) = 0 := by cases h with
| Left h => rw [h, hf.map_one, mul_zero]
| Right h => rw [add_comm, h, hf.map_one, zero_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (-c) = f c | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
⊢ f (-c) = f c | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (-c) = f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | have h1 := hf.is_good (c + 1) (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
⊢ f (-c) = f c | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
h1 : f ((c + 1) * -1 + 1) = f (c + 1) * f (-1) + f (c + 1 + -1)
⊢ f (-c) = f c | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
⊢ f (-c) = f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | rwa [h0, zero_mul, zero_add, add_neg_cancel_right,
mul_neg_one, neg_add_rev, neg_add_cancel_comm] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
h1 : f ((c + 1) * -1 + 1) = f (c + 1) * f (-1) + f (c + 1 + -1)
⊢ f (-c) = f c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (c + 1) = 0
h1 : f ((c + 1) * -1 + 1) = f (c + 1) * f (-1) + f (c + 1 + -1)
⊢ f (-c) = f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | cases h with
| Left h => rw [h, hf.map_one, mul_zero]
| Right h => rw [add_comm, h, hf.map_one, zero_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
⊢ f (c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | rw [h, hf.map_one, mul_zero] | case Left
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f (c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Left
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f (c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_neg | [39, 1] | [45, 57] | rw [add_comm, h, hf.map_one, zero_mul] | case Right
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f (c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Right
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f (c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | cases h with
| Left h => specialize h (-c); rwa [add_neg_self, h0, neg_mul] at h
| Right h => specialize h (-c); rwa [neg_add_self, h0, mul_neg] at h | R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
⊢ f 0 = -(f c * f c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | specialize h (-c) | case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f 0 = -(f c * f c) | case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (c + -c) = -f c * f (-c)
⊢ f 0 = -(f c * f c) | Please generate a tactic in lean4 to solve the state.
STATE:
case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (c + x) = -f c * f x
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | rwa [add_neg_self, h0, neg_mul] at h | case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (c + -c) = -f c * f (-c)
⊢ f 0 = -(f c * f c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Left
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (c + -c) = -f c * f (-c)
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | specialize h (-c) | case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f 0 = -(f c * f c) | case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (-c + c) = f (-c) * -f c
⊢ f 0 = -(f c * f c) | Please generate a tactic in lean4 to solve the state.
STATE:
case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : ∀ (x : R), f (x + c) = f x * -f c
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | rwa [neg_add_self, h0, mul_neg] at h | case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (-c + c) = f (-c) * -f c
⊢ f 0 = -(f c * f c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Right
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h0 : f (-c) = f c
h : f (-c + c) = f (-c) * -f c
⊢ f 0 = -(f c * f c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_mul_self_eq_one | [47, 1] | [52, 47] | rwa [hf.map_zero, neg_inj, eq_comm] at h1 | R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
h1 : f 0 = -(f c * f c)
⊢ f c * f c = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.3631
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : f (-c) = f c
h1 : f 0 = -(f c * f c)
⊢ f c * f c = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | cases map_eq_one_or_neg_one hf h with
| inl h => rw [h]; exact Commute.neg_one_left (f x)
| inr h => rw [h, neg_neg]; exact Commute.one_left (f x) | R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ Commute (-f c) (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ Commute (-f c) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | rw [h] | case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-f c) (f x) | case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-1) (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-f c) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | exact Commute.neg_one_left (f x) | case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-1) (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = 1
⊢ Commute (-1) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | rw [h, neg_neg] | case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute (-f c) (f x) | case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute 1 (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute (-f c) (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.map_commute | [57, 9] | [60, 59] | exact Commute.one_left (f x) | case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute 1 (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type ?u.6428
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h✝ : QuasiPeriodic f c
x : R
h : f c = -1
⊢ Commute 1 (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.imp_left | [62, 1] | [63, 63] | rw [add_comm, h0, map_commute hf h] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (x + c) = f x * -f c
x : R
⊢ f (c + x) = -f c * f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (x + c) = f x * -f c
x : R
⊢ f (c + x) = -f c * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.imp_right | [65, 1] | [66, 65] | rw [add_comm, h0, map_commute hf h] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (c + x) = -f c * f x
x : R
⊢ f (x + c) = f x * -f c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
h0 : ∀ (x : R), f (c + x) = -f c * f x
x : R
⊢ f (x + c) = f x * -f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.iff_left2 | [74, 1] | [76, 63] | rw [neg_mul, hf.is_good, eq_neg_iff_add_eq_zero, add_comm] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (c + x) = -f c * f x ↔ f (c * x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (c + x) = -f c * f x ↔ f (c * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.iff_right2 | [78, 1] | [80, 63] | rw [mul_neg, hf.is_good, eq_neg_iff_add_eq_zero, add_comm] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (x + c) = f x * -f c ↔ f (x * c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
x : R
⊢ f (x + c) = f x * -f c ↔ f (x * c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | rw [iff_right2 hf] at h ⊢ | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (d * c) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d : R
⊢ ∀ (x : R), f (x * (d * c) + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (d * c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | intro x | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d : R
⊢ ∀ (x : R), f (x * (d * c) + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * (d * c) + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d : R
⊢ ∀ (x : R), f (x * (d * c) + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | rw [← mul_assoc] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * (d * c) + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * d * c + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * (d * c) + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_left | [91, 1] | [93, 45] | exact h (x * d) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * d * c + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (x * c + 1) = 0
d x : R
⊢ f (x * d * c + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | rw [iff_left2 hf] at h ⊢ | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (c * d) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d : R
⊢ ∀ (x : R), f (c * d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : QuasiPeriodic f c
d : R
⊢ QuasiPeriodic f (c * d)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | intro x | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d : R
⊢ ∀ (x : R), f (c * d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d : R
⊢ ∀ (x : R), f (c * d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | rw [mul_assoc] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * (d * x) + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.mul_right | [95, 1] | [97, 43] | exact h (d * x) | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * (d * x) + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
c : R
h : ∀ (x : R), f (c * x + 1) = 0
d x : R
⊢ f (c * (d * x) + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_eq_zero_iff | [112, 1] | [114, 79] | rw [add_zero, (iff_right hf.toNontrivialGood).mp h, h0, neg_neg, mul_one] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0✝ : c ≠ 0
h0 : f c = -1
x : R
⊢ f (x + c) = f (x + 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0✝ : c ≠ 0
h0 : f c = -1
x : R
⊢ f (x + c) = f (x + 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_QuasiPeriodic_eq | [120, 1] | [127, 60] | have h3 (c) := (iff_right hf.toNontrivialGood (c := c)).mp | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
⊢ f (x + d) = f (x + c) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
h3 : ∀ (c : R), QuasiPeriodic f c → ∀ (x : R), f (x + c) = f x * -f c
⊢ f (x + d) = f (x + c) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
⊢ f (x + d) = f (x + c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_QuasiPeriodic_eq | [120, 1] | [127, 60] | rw [h3 c h, h3 d h1, h2, reduced_map_eq_one hf h h0] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
h3 : ∀ (c : R), QuasiPeriodic f c → ∀ (x : R), f (x + c) = f x * -f c
⊢ f (x + d) = f (x + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : QuasiPeriodic f d
h2 : f d = 1
x : R
h3 : ∀ (c : R), QuasiPeriodic f c → ∀ (x : R), f (x + c) = f x * -f c
⊢ f (x + d) = f (x + c)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [iff_left2 hf.toNontrivialGood] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ QuasiPeriodic f d | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ ∀ (x : R), f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ QuasiPeriodic f d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | intro x | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ ∀ (x : R), f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
⊢ ∀ (x : R), f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | have h2 := reduced_map_eq_one hf h h0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
⊢ f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | have h3 : f (d * (c + x) + 1) = -f (d * x + 1) := by
rw [iff_left hf.toNontrivialGood] at h
rw [hf.is_good, h, add_left_comm, h, h2, neg_one_mul,
neg_one_mul, mul_neg, ← neg_add, hf.is_good] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : f (d * (c + x) + 1) = -f (d * x + 1)
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [mul_add, h1, zero_add, eq_neg_iff_add_eq_zero, ← two_mul, mul_eq_zero] at h3 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : f (d * (c + x) + 1) = -f (d * x + 1)
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : 2 = 0 ∨ f (d * x + 1) = 0
⊢ f (d * x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : f (d * (c + x) + 1) = -f (d * x + 1)
⊢ f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | refine h3.resolve_left λ h3 ↦ h0 ?_ | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : 2 = 0 ∨ f (d * x + 1) = 0
⊢ f (d * x + 1) = 0 | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3✝ : 2 = 0 ∨ f (d * x + 1) = 0
h3 : 2 = 0
⊢ c = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3 : 2 = 0 ∨ f (d * x + 1) = 0
⊢ f (d * x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [reduced_eq_zero_iff hf h, h2, eq_neg_iff_add_eq_zero, one_add_one_eq_two, h3] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3✝ : 2 = 0 ∨ f (d * x + 1) = 0
h3 : 2 = 0
⊢ c = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3✝ : 2 = 0 ∨ f (d * x + 1) = 0
h3 : 2 = 0
⊢ c = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [iff_left hf.toNontrivialGood] at h | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [hf.is_good, h, add_left_comm, h, h2, neg_one_mul,
neg_one_mul, mul_neg, ← neg_add, hf.is_good] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_QuasiPeriod_equiv_cases | [148, 1] | [152, 68] | rwa [sub_one_mul, sub_eq_zero] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : ∀ (d : R), d * c = 0 → QuasiPeriodic f d := fun d => reduced_mul_left_eq_zero_imp hf h h0
h2 : d * c = c
⊢ (d - 1) * c = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : ∀ (d : R), d * c = 0 → QuasiPeriodic f d := fun d => reduced_mul_left_eq_zero_imp hf h h0
h2 : d * c = c
⊢ (d - 1) * c = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/IntLinearSolver.lean | IMOSL.Extra.IntLinearSolver | [21, 1] | [27, 68] | rw [zero_zsmul, zero_add] | G : Type u_1
g : G
inst✝ : AddGroup G
f : ℤ → G
h : ∀ (n : ℤ), f (n + 1) = g + f n
n : ℤ
⊢ f 0 = 0 • g + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
g : G
inst✝ : AddGroup G
f : ℤ → G
h : ∀ (n : ℤ), f (n + 1) = g + f n
n : ℤ
⊢ f 0 = 0 • g + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/IntLinearSolver.lean | IMOSL.Extra.IntLinearSolver | [21, 1] | [27, 68] | rw [h, h1, ← add_assoc, add_comm, one_add_zsmul] | G : Type u_1
g : G
inst✝ : AddGroup G
f : ℤ → G
h : ∀ (n : ℤ), f (n + 1) = g + f n
n k : ℤ
x✝ : 0 ≤ k
h1 : f k = k • g + f 0
⊢ f (k + 1) = (k + 1) • g + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
g : G
inst✝ : AddGroup G
f : ℤ → G
h : ∀ (n : ℤ), f (n + 1) = g + f n
n k : ℤ
x✝ : 0 ≤ k
h1 : f k = k • g + f 0
⊢ f (k + 1) = (k + 1) • g + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/IntLinearSolver.lean | IMOSL.Extra.IntLinearSolver | [21, 1] | [27, 68] | rwa [← add_right_inj g, ← h, sub_add_cancel,
← add_assoc, ← one_add_zsmul, add_sub_cancel] | G : Type u_1
g : G
inst✝ : AddGroup G
f : ℤ → G
h : ∀ (n : ℤ), f (n + 1) = g + f n
n k : ℤ
x✝ : k ≤ 0
h1 : f k = k • g + f 0
⊢ f (k - 1) = (k - 1) • g + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
g : G
inst✝ : AddGroup G
f : ℤ → G
h : ∀ (n : ℤ), f (n + 1) = g + f n
n k : ℤ
x✝ : k ≤ 0
h1 : f k = k • g + f 0
⊢ f (k - 1) = (k - 1) • g + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/IntLinearSolver.lean | IMOSL.Extra.IntIntLinearSolverAlt | [29, 1] | [31, 48] | rw [IntLinearSolver h, smul_eq_mul, mul_comm] | g : ℤ
f : ℤ → ℤ
h : ∀ (n : ℤ), f (n + 1) = g + f n
n : ℤ
⊢ f n = g * n + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
g : ℤ
f : ℤ → ℤ
h : ∀ (n : ℤ), f (n + 1) = g + f n
n : ℤ
⊢ f n = g * n + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.surjective_iff | [45, 1] | [47, 55] | rw [← Set.range_iff_surjective, ← compl_inj_iff,
← h.rangeCompl_spec, Set.compl_univ, coe_eq_empty] | α : Type u_1
f : α → α
h : FinChainFn f
⊢ Surjective f ↔ h.rangeCompl = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : α → α
h : FinChainFn f
⊢ Surjective f ↔ h.rangeCompl = ∅
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | wlog h3 : m < n | α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
⊢ f^[m] a ≠ f^[n] b | case inr
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
this :
∀ {α : Type u_1} {f : α → α} (h : FinChainFn f) {m n : ℕ} {a b : α},
m ≠ n → a ∈ h.rangeCompl → b ∈ h.rangeCompl → m < n → f^[m] a ≠ f^[n] b
h3 : ¬m < n
⊢ f^[m] a ≠ f^[n] b
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
⊢ f^[m] a ≠ f^[n] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | exact (this h h0.symm h2 h1 <| (le_of_not_lt h3).lt_of_ne h0.symm).symm | case inr
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
this :
∀ {α : Type u_1} {f : α → α} (h : FinChainFn f) {m n : ℕ} {a b : α},
m ≠ n → a ∈ h.rangeCompl → b ∈ h.rangeCompl → m < n → f^[m] a ≠ f^[n] b
h3 : ¬m < n
⊢ f^[m] a ≠ f^[n] b
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
this :
∀ {α : Type u_1} {f : α → α} (h : FinChainFn f) {m n : ℕ} {a b : α},
m ≠ n → a ∈ h.rangeCompl → b ∈ h.rangeCompl → m < n → f^[m] a ≠ f^[n] b
h3 : ¬m < n
⊢ f^[m] a ≠ f^[n] b
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rcases Nat.exists_eq_add_of_le h3.le with ⟨k, rfl⟩ | α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : m < m + k
⊢ f^[m] a ≠ f^[m + k] b | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m n : ℕ
a b : α
h0 : m ≠ n
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
h3 : m < n
⊢ f^[m] a ≠ f^[n] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [Nat.lt_add_right_iff_pos] at h3 | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : m < m + k
⊢ f^[m] a ≠ f^[m + k] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ f^[m] a ≠ f^[m + k] b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : m < m + k
⊢ f^[m] a ≠ f^[m + k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [f.iterate_add_apply m k b, (h.injective.iterate m).ne_iff] | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ f^[m] a ≠ f^[m + k] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ a ≠ f^[k] b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ f^[m] a ≠ f^[m + k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rintro rfl | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ a ≠ f^[k] b | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : f^[k] b ∈ h.rangeCompl
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
a b : α
h1 : a ∈ h.rangeCompl
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
⊢ a ≠ f^[k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [mem_rangeCompl_iff, Set.mem_range] at h1 | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : f^[k] b ∈ h.rangeCompl
⊢ False | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : f^[k] b ∈ h.rangeCompl
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | refine h1 ⟨f^[k.pred] b, ?_⟩ | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ False | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ f (f^[k.pred] b) = f^[k] b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_apply_ne_of_mem_rangeCompl_iter_ne | [49, 1] | [59, 59] | rw [← f.iterate_succ_apply', Nat.succ_pred_eq_of_pos h3] | case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ f (f^[k.pred] b) = f^[k] b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
f✝ : α✝ → α✝
h✝ : FinChainFn f✝
α : Type u_1
f : α → α
h : FinChainFn f
m : ℕ
b : α
h2 : b ∈ h.rangeCompl
k : ℕ
h0 : m ≠ m + k
h3 : 0 < k
h1 : ¬∃ y, f y = f^[k] b
⊢ f (f^[k.pred] b) = f^[k] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_spec | [88, 1] | [92, 72] | rw [exactIterRange, coe_image, h.rangeCompl_spec, iterate_succ,
Set.range_comp _ f, Set.range_diff_image (h.injective.iterate n)] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
a : α
⊢ a ∈ ↑(h.exactIterRange n) ↔ a ∈ Set.range f^[n] \ Set.range f^[n + 1] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
a : α
⊢ a ∈ ↑(h.exactIterRange n) ↔ a ∈ Set.range f^[n] \ Set.range f^[n + 1]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | rw [disjoint_iff_ne] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ Disjoint (h.exactIterRange m) (h.exactIterRange n) | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ ∀ a ∈ h.exactIterRange m, ∀ b ∈ h.exactIterRange n, a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ Disjoint (h.exactIterRange m) (h.exactIterRange n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | intro a h1 b h2 | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ ∀ a ∈ h.exactIterRange m, ∀ b ∈ h.exactIterRange n, a ≠ b | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : a ∈ h.exactIterRange m
b : α
h2 : b ∈ h.exactIterRange n
⊢ a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
⊢ ∀ a ∈ h.exactIterRange m, ∀ b ∈ h.exactIterRange n, a ≠ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | rw [h.mem_exactIterRange_iff] at h1 h2 | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : a ∈ h.exactIterRange m
b : α
h2 : b ∈ h.exactIterRange n
⊢ a ≠ b | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : ∃ b ∈ h.rangeCompl, f^[m] b = a
b : α
h2 : ∃ b_1 ∈ h.rangeCompl, f^[n] b_1 = b
⊢ a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : a ∈ h.exactIterRange m
b : α
h2 : b ∈ h.exactIterRange n
⊢ a ≠ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | rcases h1 with ⟨a, h1, rfl⟩ | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : ∃ b ∈ h.rangeCompl, f^[m] b = a
b : α
h2 : ∃ b_1 ∈ h.rangeCompl, f^[n] b_1 = b
⊢ a ≠ b | case intro.intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
b : α
h2 : ∃ b_1 ∈ h.rangeCompl, f^[n] b_1 = b
a : α
h1 : a ∈ h.rangeCompl
⊢ f^[m] a ≠ b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : ∃ b ∈ h.rangeCompl, f^[m] b = a
b : α
h2 : ∃ b_1 ∈ h.rangeCompl, f^[n] b_1 = b
⊢ a ≠ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | rcases h2 with ⟨b, h2, rfl⟩ | case intro.intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
b : α
h2 : ∃ b_1 ∈ h.rangeCompl, f^[n] b_1 = b
a : α
h1 : a ∈ h.rangeCompl
⊢ f^[m] a ≠ b | case intro.intro.intro.intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : a ∈ h.rangeCompl
b : α
h2 : b ∈ h.rangeCompl
⊢ f^[m] a ≠ f^[n] b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
b : α
h2 : ∃ b_1 ∈ h.rangeCompl, f^[n] b_1 = b
a : α
h1 : a ∈ h.rangeCompl
⊢ f^[m] a ≠ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.exactIterRange_disjoint_of_ne | [97, 1] | [104, 59] | exact h.iter_apply_ne_of_mem_rangeCompl_iter_ne h0 h1 h2 | case intro.intro.intro.intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : a ∈ h.rangeCompl
b : α
h2 : b ∈ h.rangeCompl
⊢ f^[m] a ≠ f^[n] b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
m n : ℕ
h0 : m ≠ n
a : α
h1 : a ∈ h.rangeCompl
b : α
h2 : b ∈ h.rangeCompl
⊢ f^[m] a ≠ f^[n] b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_spec | [128, 1] | [139, 23] | rw [iterRangeCompl_zero, coe_empty,
iterate_zero, Set.range_id, Set.compl_univ] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
⊢ ↑(h.iterRangeCompl 0) = (Set.range f^[0])ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
⊢ ↑(h.iterRangeCompl 0) = (Set.range f^[0])ᶜ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_spec | [128, 1] | [139, 23] | rw [h.iterRangeCompl_succ, coe_union, iterRangeCompl_spec n,
h.exactIterRange_spec, Set.diff_eq, Set.inter_union_distrib_right,
Set.union_compl_self, Set.univ_inter, ← Set.compl_inter] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
⊢ ↑(h.iterRangeCompl (n + 1)) = (Set.range f^[n + 1])ᶜ | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
⊢ (Set.range f^[n + 1] ∩ Set.range f^[n])ᶜ = (Set.range f^[n + 1])ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
⊢ ↑(h.iterRangeCompl (n + 1)) = (Set.range f^[n + 1])ᶜ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_spec | [128, 1] | [139, 23] | refine congr_arg _ (Set.inter_eq_left.mpr λ x h1 ↦ ?_) | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
⊢ (Set.range f^[n + 1] ∩ Set.range f^[n])ᶜ = (Set.range f^[n + 1])ᶜ | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
x : α
h1 : x ∈ Set.range f^[n + 1]
⊢ x ∈ Set.range f^[n] | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
⊢ (Set.range f^[n + 1] ∩ Set.range f^[n])ᶜ = (Set.range f^[n + 1])ᶜ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_spec | [128, 1] | [139, 23] | rw [Set.mem_range] at h1 ⊢ | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
x : α
h1 : x ∈ Set.range f^[n + 1]
⊢ x ∈ Set.range f^[n] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
x : α
h1 : ∃ y, f^[n + 1] y = x
⊢ ∃ y, f^[n] y = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
x : α
h1 : x ∈ Set.range f^[n + 1]
⊢ x ∈ Set.range f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_spec | [128, 1] | [139, 23] | rcases h1 with ⟨y, rfl⟩ | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
x : α
h1 : ∃ y, f^[n + 1] y = x
⊢ ∃ y, f^[n] y = x | case intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
y : α
⊢ ∃ y_1, f^[n] y_1 = f^[n + 1] y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
x : α
h1 : ∃ y, f^[n + 1] y = x
⊢ ∃ y, f^[n] y = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_spec | [128, 1] | [139, 23] | exact ⟨f y, rfl⟩ | case intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
y : α
⊢ ∃ y_1, f^[n] y_1 = f^[n + 1] y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
y : α
⊢ ∃ y_1, f^[n] y_1 = f^[n + 1] y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_card | [162, 1] | [169, 55] | have h0 := card_union_of_disjoint
(h.iterRangeCompl_disjoint_exactIterRange n.le_refl) | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
⊢ (h.iterRangeCompl (n + 1)).card = (n + 1) * h.rangeCompl.card | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
h0 : (h.exactIterRange n ∪ h.iterRangeCompl n).card = (h.exactIterRange n).card + (h.iterRangeCompl n).card
⊢ (h.iterRangeCompl (n + 1)).card = (n + 1) * h.rangeCompl.card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
⊢ (h.iterRangeCompl (n + 1)).card = (n + 1) * h.rangeCompl.card
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iterRangeCompl_card | [162, 1] | [169, 55] | rw [h.iterRangeCompl_succ, h0, h.exactIterRange_card,
iterRangeCompl_card n, Nat.succ_mul, add_comm] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
h0 : (h.exactIterRange n ∪ h.iterRangeCompl n).card = (h.exactIterRange n).card + (h.iterRangeCompl n).card
⊢ (h.iterRangeCompl (n + 1)).card = (n + 1) * h.rangeCompl.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
n : ℕ
h0 : (h.exactIterRange n ∪ h.iterRangeCompl n).card = (h.exactIterRange n).card + (h.iterRangeCompl n).card
⊢ (h.iterRangeCompl (n + 1)).card = (n + 1) * h.rangeCompl.card
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_range_of_rangeCompl_singleton | [171, 1] | [175, 78] | rw [h.iterRangeCompl_succ, exactIterRange, h0, image_singleton,
iter_range_of_rangeCompl_singleton h0 n, range_succ, image_insert] | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
a : α
h0 : h.rangeCompl = {a}
n : ℕ
⊢ h.iterRangeCompl (n + 1) = image (fun k => f^[k] a) (range (n + 1)) | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
a : α
h0 : h.rangeCompl = {a}
n : ℕ
⊢ {f^[n] a} ∪ image (fun k => f^[k] a) (range n) = insert (f^[n] a) (image (fun k => f^[k] a) (range n)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
a : α
h0 : h.rangeCompl = {a}
n : ℕ
⊢ h.iterRangeCompl (n + 1) = image (fun k => f^[k] a) (range (n + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/FinChainFn.lean | IMOSL.IMO2013A5.FinChainFn.iter_range_of_rangeCompl_singleton | [171, 1] | [175, 78] | rfl | α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
a : α
h0 : h.rangeCompl = {a}
n : ℕ
⊢ {f^[n] a} ∪ image (fun k => f^[k] a) (range n) = insert (f^[n] a) (image (fun k => f^[k] a) (range n)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : α → α
h : FinChainFn f
a : α
h0 : h.rangeCompl = {a}
n : ℕ
⊢ {f^[n] a} ∪ image (fun k => f^[k] a) (range n) = insert (f^[n] a) (image (fun k => f^[k] a) (range n))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Ring/Hom.lean | IMOSL.Extra.CharTwo.ofRingHom | [21, 1] | [24, 47] | rw [← one_add_one_eq_two, ← φ.map_one,
← φ.map_add, add_self_eq_zero, φ.map_zero] | R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CharTwo R
inst✝ : NonAssocSemiring S
φ : R →+* S
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : CharTwo R
inst✝ : NonAssocSemiring S
φ : R →+* S
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.abs_le_max_seqMax | [43, 1] | [49, 49] | rw [le_max_iff] | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
⊢ |a n| ≤ max (Extra.seqMax (-a) n) (2 • Extra.seqMax a n) | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
⊢ |a n| ≤ Extra.seqMax (-a) n ∨ |a n| ≤ 2 • Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
⊢ |a n| ≤ max (Extra.seqMax (-a) n) (2 • Extra.seqMax a n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.abs_le_max_seqMax | [43, 1] | [49, 49] | refine (le_total (a n) 0).imp (λ h ↦ ?_) (λ h ↦ ?_) | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
⊢ |a n| ≤ Extra.seqMax (-a) n ∨ |a n| ≤ 2 • Extra.seqMax a n | case refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : a n ≤ 0
⊢ |a n| ≤ Extra.seqMax (-a) n
case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
⊢ |a n| ≤ 2 • Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
⊢ |a n| ≤ Extra.seqMax (-a) n ∨ |a n| ≤ 2 • Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.abs_le_max_seqMax | [43, 1] | [49, 49] | rw [abs_of_nonpos h] | case refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : a n ≤ 0
⊢ |a n| ≤ Extra.seqMax (-a) n | case refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : a n ≤ 0
⊢ -a n ≤ Extra.seqMax (-a) n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : a n ≤ 0
⊢ |a n| ≤ Extra.seqMax (-a) n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.abs_le_max_seqMax | [43, 1] | [49, 49] | exact Extra.le_seqMax_self (-a) n | case refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : a n ≤ 0
⊢ -a n ≤ Extra.seqMax (-a) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : a n ≤ 0
⊢ -a n ≤ Extra.seqMax (-a) n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.abs_le_max_seqMax | [43, 1] | [49, 49] | rw [abs_of_nonneg h, two_nsmul] | case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
⊢ |a n| ≤ 2 • Extra.seqMax a n | case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
⊢ a n ≤ Extra.seqMax a n + Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
⊢ |a n| ≤ 2 • Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.abs_le_max_seqMax | [43, 1] | [49, 49] | have h0 := Extra.le_seqMax_self a n | case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
⊢ a n ≤ Extra.seqMax a n + Extra.seqMax a n | case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
h0 : a n ≤ Extra.seqMax a n
⊢ a n ≤ Extra.seqMax a n + Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
⊢ a n ≤ Extra.seqMax a n + Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.abs_le_max_seqMax | [43, 1] | [49, 49] | exact le_add_of_le_of_nonneg h0 (h.trans h0) | case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
h0 : a n ≤ Extra.seqMax a n
⊢ a n ≤ Extra.seqMax a n + Extra.seqMax a n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
a : ℕ → G
n : ℕ
h : 0 ≤ a n
h0 : a n ≤ Extra.seqMax a n
⊢ a n ≤ Extra.seqMax a n + Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good1_bdd_above | [51, 1] | [57, 78] | rcases Extra.exists_map_eq_seqMax a n with ⟨i, h2, h1⟩ | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n | case intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
i : ℕ
h2 : i ≤ n
h1 : a i = Extra.seqMax a n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good1_bdd_above | [51, 1] | [57, 78] | rw [le_iff_exists_add] at h2 | case intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
i : ℕ
h2 : i ≤ n
h1 : a i = Extra.seqMax a n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n | case intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
i : ℕ
h2 : ∃ c, n = i + c
h1 : a i = Extra.seqMax a n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
i : ℕ
h2 : i ≤ n
h1 : a i = Extra.seqMax a n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good1_bdd_above | [51, 1] | [57, 78] | rcases h2 with ⟨j, rfl⟩ | case intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
i : ℕ
h2 : ∃ c, n = i + c
h1 : a i = Extra.seqMax a n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ a (i + j + 1) ≤ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good1 D a
h0 : D ≤ n
i : ℕ
h2 : ∃ c, n = i + c
h1 : a i = Extra.seqMax a n
⊢ a (n + 1) ≤ Extra.seqMax (-a) n - Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good1_bdd_above | [51, 1] | [57, 78] | apply (h i j h0).trans | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ a (i + j + 1) ≤ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j) | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ -(a i + a j) ≤ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ a (i + j + 1) ≤ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good1_bdd_above | [51, 1] | [57, 78] | rw [← h1, neg_add_rev, ← sub_eq_add_neg] | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ -(a i + a j) ≤ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j) | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ -a j - a i ≤ Extra.seqMax (-a) (i + j) - a i | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ -(a i + a j) ≤ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good1_bdd_above | [51, 1] | [57, 78] | exact sub_le_sub_right (Extra.le_seqMax_of_le (-a) (j.le_add_left i)) (a i) | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ -a j - a i ≤ Extra.seqMax (-a) (i + j) - a i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good1 D a
i j : ℕ
h0 : D ≤ i + j
h1 : a i = Extra.seqMax a (i + j)
⊢ -a j - a i ≤ Extra.seqMax (-a) (i + j) - a i
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good2_bdd_below | [64, 1] | [69, 48] | rcases h n h0 with ⟨i, j, rfl, h0⟩ | G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good2 D a
h0 : D ≤ n
⊢ -a (n + 1) ≤ 2 • Extra.seqMax a n | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good2 D a
i j : ℕ
h0✝ : D ≤ i + j
h0 : a (i + j + 1) = -(a i + a j)
⊢ -a (i + j + 1) ≤ 2 • Extra.seqMax a (i + j) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D n : ℕ
a : ℕ → G
h : good2 D a
h0 : D ≤ n
⊢ -a (n + 1) ≤ 2 • Extra.seqMax a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good2_bdd_below | [64, 1] | [69, 48] | rw [h0, neg_neg, two_nsmul] | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good2 D a
i j : ℕ
h0✝ : D ≤ i + j
h0 : a (i + j + 1) = -(a i + a j)
⊢ -a (i + j + 1) ≤ 2 • Extra.seqMax a (i + j) | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good2 D a
i j : ℕ
h0✝ : D ≤ i + j
h0 : a (i + j + 1) = -(a i + a j)
⊢ a i + a j ≤ Extra.seqMax a (i + j) + Extra.seqMax a (i + j) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good2 D a
i j : ℕ
h0✝ : D ≤ i + j
h0 : a (i + j + 1) = -(a i + a j)
⊢ -a (i + j + 1) ≤ 2 • Extra.seqMax a (i + j)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.good2_bdd_below | [64, 1] | [69, 48] | exact add_le_add (Extra.le_seqMax_of_le a (i.le_add_right j))
(Extra.le_seqMax_of_le a (j.le_add_left i)) | case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good2 D a
i j : ℕ
h0✝ : D ≤ i + j
h0 : a (i + j + 1) = -(a i + a j)
⊢ a i + a j ≤ Extra.seqMax a (i + j) + Extra.seqMax a (i + j) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
G : Type u_1
inst✝ : LinearOrderedAddCommGroup G
D : ℕ
a : ℕ → G
h : good2 D a
i j : ℕ
h0✝ : D ≤ i + j
h0 : a (i + j + 1) = -(a i + a j)
⊢ a i + a j ≤ Extra.seqMax a (i + j) + Extra.seqMax a (i + j)
TACTIC:
|
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