url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.f_self_lt_f_succ_of_divisors_card | [77, 1] | [84, 58] | rw [sum_const, card_range, smul_eq_mul, mul_comm] | n : β
h : n β 0
h0 : β k < n, k.succ.divisors.card < n.succ.divisors.card
β’ β x β range n, n.succ.divisors.card = n.succ.divisors.card * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
h : n β 0
h0 : β k < n, k.succ.divisors.card < n.succ.divisors.card
β’ β x β range n, n.succ.divisors.card = n.succ.divisors.card * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part1 | [87, 1] | [98, 83] | obtain β¨n, h0, h1β© : β n : β, N < n β§
β k : β, k < n β k.succ.divisors.card < n.succ.divisors.card := by
obtain β¨K, h0β© : β K : β, β k : β, k β€ N β k.succ.divisors.card β€ K :=
β¨Extra.seqMax (Ξ» n β¦ n.succ.divisors.card) N,
Ξ» _ β¦ Extra.le_seqMax_of_le (Ξ» n β¦ n.succ.divisors.card)β©
have h1 := exists_lt_card_divisor_succ K
refine β¨Nat.find h1,
(Nat.lt_find_iff h1 _).mpr Ξ» k h2 β¦ (h0 k h2).not_lt,
Ξ» k h2 β¦ (le_of_not_lt (Nat.find_min h1 h2)).trans_lt (Nat.find_spec h1)β© | N : β
β’ β b β {n | f n < f n.succ}, N < b | case intro.intro
N n : β
h0 : N < n
h1 : β k < n, k.succ.divisors.card < n.succ.divisors.card
β’ β b β {n | f n < f n.succ}, N < b | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ β b β {n | f n < f n.succ}, N < b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part1 | [87, 1] | [98, 83] | exact β¨n, f_self_lt_f_succ_of_divisors_card (Nat.not_eq_zero_of_lt h0) h1, h0β© | case intro.intro
N n : β
h0 : N < n
h1 : β k < n, k.succ.divisors.card < n.succ.divisors.card
β’ β b β {n | f n < f n.succ}, N < b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
N n : β
h0 : N < n
h1 : β k < n, k.succ.divisors.card < n.succ.divisors.card
β’ β b β {n | f n < f n.succ}, N < b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part1 | [87, 1] | [98, 83] | obtain β¨K, h0β© : β K : β, β k : β, k β€ N β k.succ.divisors.card β€ K :=
β¨Extra.seqMax (Ξ» n β¦ n.succ.divisors.card) N,
Ξ» _ β¦ Extra.le_seqMax_of_le (Ξ» n β¦ n.succ.divisors.card)β© | N : β
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card | case intro
N K : β
h0 : β k β€ N, k.succ.divisors.card β€ K
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part1 | [87, 1] | [98, 83] | have h1 := exists_lt_card_divisor_succ K | case intro
N K : β
h0 : β k β€ N, k.succ.divisors.card β€ K
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card | case intro
N K : β
h0 : β k β€ N, k.succ.divisors.card β€ K
h1 : β n, K < n.succ.divisors.card
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
N K : β
h0 : β k β€ N, k.succ.divisors.card β€ K
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part1 | [87, 1] | [98, 83] | refine β¨Nat.find h1,
(Nat.lt_find_iff h1 _).mpr Ξ» k h2 β¦ (h0 k h2).not_lt,
Ξ» k h2 β¦ (le_of_not_lt (Nat.find_min h1 h2)).trans_lt (Nat.find_spec h1)β© | case intro
N K : β
h0 : β k β€ N, k.succ.divisors.card β€ K
h1 : β n, K < n.succ.divisors.card
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
N K : β
h0 : β k β€ N, k.succ.divisors.card β€ K
h1 : β n, K < n.succ.divisors.card
β’ β n, N < n β§ β k < n, k.succ.divisors.card < n.succ.divisors.card
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.f_succ_lt_self_of_succ_prime_large | [100, 1] | [104, 25] | apply f_lt_f_of_g | n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ f n.succ < f n | case h
n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ g n.succ * n < g n * n.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ f n.succ < f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.f_succ_lt_self_of_succ_prime_large | [100, 1] | [104, 25] | rw [g_succ, card_divisors_prime h0, add_mul, Nat.mul_succ, add_lt_add_iff_left] | case h
n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ g n.succ * n < g n * n.succ | case h
n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ 2 * n < g n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ g n.succ * n < g n * n.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.f_succ_lt_self_of_succ_prime_large | [100, 1] | [104, 25] | exact two_mul_lt_g n h | case h
n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ 2 * n < g n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : β
h : 6 β€ n
h0 : n.succ.Prime
β’ 2 * n < g n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part2 | [107, 1] | [114, 59] | obtain β¨n, h, h0, h1β© : β n, 6 β€ n β§ n.succ.Prime β§ N < n := by
rcases (max 6 (N + 1) + 1).exists_infinite_primes with β¨_ | n, h, h0β©
exact absurd h0 Nat.not_prime_zero
rw [Nat.add_le_add_iff_right, max_le_iff] at h
exact β¨n, h.1, h0, h.2β© | N : β
β’ β b β {n | f n.succ < f n}, N < b | case intro.intro.intro
N n : β
h : 6 β€ n
h0 : n.succ.Prime
h1 : N < n
β’ β b β {n | f n.succ < f n}, N < b | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ β b β {n | f n.succ < f n}, N < b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part2 | [107, 1] | [114, 59] | exact β¨n, f_succ_lt_self_of_succ_prime_large h h0, h1β© | case intro.intro.intro
N n : β
h : 6 β€ n
h0 : n.succ.Prime
h1 : N < n
β’ β b β {n | f n.succ < f n}, N < b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
N n : β
h : 6 β€ n
h0 : n.succ.Prime
h1 : N < n
β’ β b β {n | f n.succ < f n}, N < b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part2 | [107, 1] | [114, 59] | rcases (max 6 (N + 1) + 1).exists_infinite_primes with β¨_ | n, h, h0β© | N : β
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n | case intro.zero.intro
N : β
h : max 6 (N + 1) + 1 β€ 0
h0 : Nat.Prime 0
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n
case intro.succ.intro
N n : β
h : max 6 (N + 1) + 1 β€ n + 1
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part2 | [107, 1] | [114, 59] | exact absurd h0 Nat.not_prime_zero | case intro.zero.intro
N : β
h : max 6 (N + 1) + 1 β€ 0
h0 : Nat.Prime 0
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n
case intro.succ.intro
N n : β
h : max 6 (N + 1) + 1 β€ n + 1
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n | case intro.succ.intro
N n : β
h : max 6 (N + 1) + 1 β€ n + 1
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.zero.intro
N : β
h : max 6 (N + 1) + 1 β€ 0
h0 : Nat.Prime 0
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n
case intro.succ.intro
N n : β
h : max 6 (N + 1) + 1 β€ n + 1
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part2 | [107, 1] | [114, 59] | rw [Nat.add_le_add_iff_right, max_le_iff] at h | case intro.succ.intro
N n : β
h : max 6 (N + 1) + 1 β€ n + 1
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n | case intro.succ.intro
N n : β
h : 6 β€ n β§ N + 1 β€ n
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro
N n : β
h : max 6 (N + 1) + 1 β€ n + 1
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/N3/N3.lean | IMOSL.IMO2006N3.final_solution_part2 | [107, 1] | [114, 59] | exact β¨n, h.1, h0, h.2β© | case intro.succ.intro
N n : β
h : 6 β€ n β§ N + 1 β€ n
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro
N n : β
h : 6 β€ n β§ N + 1 β€ n
h0 : (n + 1).Prime
β’ β n, 6 β€ n β§ n.succ.Prime β§ N < n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | have h1 n : N * (f (n + 1) - f n) = f N - f 0 := by
rw [mul_sub, sub_eq_iff_eq_add, β add_sub_right_comm, eq_sub_iff_add_eq',
β N.mul_zero, h0, zero_add, n.add_comm, β h0, mul_one] | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
β’ f = 0 β¨ β c, f = fun x => N * x + c | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), N * (f (n + 1) - f n) = f N - f 0
β’ f = 0 β¨ β c, f = fun x => N * x + c | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
β’ f = 0 β¨ β c, f = fun x => N * x + c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | replace h1 n : f (n + 1) = (f 1 - f 0) + f n :=
eq_add_of_sub_eq <| mul_left_cancelβ h <| by rw [h1, β h1 0, zero_add] | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), N * (f (n + 1) - f n) = f N - f 0
β’ f = 0 β¨ β c, f = fun x => N * x + c | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), f (n + 1) = f 1 - f 0 + f n
β’ f = 0 β¨ β c, f = fun x => N * x + c | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), N * (f (n + 1) - f n) = f N - f 0
β’ f = 0 β¨ β c, f = fun x => N * x + c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | generalize f 1 - f 0 = q at h1 | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), f (n + 1) = f 1 - f 0 + f n
β’ f = 0 β¨ β c, f = fun x => N * x + c | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f (n + 1) = q + f n
β’ f = 0 β¨ β c, f = fun x => N * x + c | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), f (n + 1) = f 1 - f 0 + f n
β’ f = 0 β¨ β c, f = fun x => N * x + c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | apply Extra.IntIntLinearSolverAlt at h1 | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f (n + 1) = q + f n
β’ f = 0 β¨ β c, f = fun x => N * x + c | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
β’ f = 0 β¨ β c, f = fun x => N * x + c | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f (n + 1) = q + f n
β’ f = 0 β¨ β c, f = fun x => N * x + c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | refine (em' (N = q)).imp (Ξ» h2 β¦ ?_) (Ξ» h2 β¦ β¨f 0, funext <| by rwa [h2]β©) | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
β’ f = 0 β¨ β c, f = fun x => N * x + c | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
β’ f = 0 β¨ β c, f = fun x => N * x + c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | have h3 := h0 0 0 | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
β’ f = 0 | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f (N * 0) + N * f 0 = f (f (0 + 0))
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [add_zero, N.mul_zero, h1 (f 0), add_comm, add_left_inj,
mul_eq_mul_right_iff, or_iff_right h2] at h3 | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f (N * 0) + N * f 0 = f (f (0 + 0))
β’ f = 0 | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f (N * 0) + N * f 0 = f (f (0 + 0))
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | specialize h0 0 1 | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
β’ f = 0 | N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : f (N * 0) + N * f 1 = f (f (0 + 1))
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [N.mul_zero, zero_add, h1 (f 1), add_comm, add_left_inj,
mul_eq_mul_right_iff, or_iff_right h2, h1, mul_one, h3, add_zero] at h0 | N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : f (N * 0) + N * f 1 = f (f (0 + 1))
β’ f = 0 | N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : q = 0
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : f (N * 0) + N * f 1 = f (f (0 + 1))
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | funext n | N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : q = 0
β’ f = 0 | case h
N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : q = 0
n : β€
β’ f n = 0 n | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : q = 0
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [h1, h0, h3, n.zero_mul, add_zero, Pi.zero_apply] | case h
N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : q = 0
n : β€
β’ f n = 0 n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : β€
h : N β 0
f : β€ β β€
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : Β¬N = q
h3 : f 0 = 0
h0 : q = 0
n : β€
β’ f n = 0 n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [mul_sub, sub_eq_iff_eq_add, β add_sub_right_comm, eq_sub_iff_add_eq',
β N.mul_zero, h0, zero_add, n.add_comm, β h0, mul_one] | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
n : β€
β’ N * (f (n + 1) - f n) = f N - f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
n : β€
β’ N * (f (n + 1) - f n) = f N - f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [h1, β h1 0, zero_add] | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), N * (f (n + 1) - f n) = f N - f 0
n : β€
β’ N * (f (n + 1) - f n) = N * (f 1 - f 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
h1 : β (n : β€), N * (f (n + 1) - f n) = f N - f 0
n : β€
β’ N * (f (n + 1) - f n) = N * (f 1 - f 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rwa [h2] | N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : N = q
β’ β (x : β€), f x = N * x + f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : β (a b : β€), f (N * a) + N * f b = f (f (a + b))
q : β€
h1 : β (n : β€), f n = q * n + f 0
h2 : N = q
β’ β (x : β€), f x = N * x + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rcases h0 with rfl | β¨c, rflβ© | N : β€
h : N β 0
f : β€ β β€
h0 : f = 0 β¨ β c, f = fun x => N * x + c
a b : β€
β’ f (N * a) + N * f b = f (f (a + b)) | case inl
N : β€
h : N β 0
a b : β€
β’ 0 (N * a) + N * 0 b = 0 (0 (a + b))
case inr.intro
N : β€
h : N β 0
a b c : β€
β’ (fun x => N * x + c) (N * a) + N * (fun x => N * x + c) b = (fun x => N * x + c) ((fun x => N * x + c) (a + b)) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β€
h : N β 0
f : β€ β β€
h0 : f = 0 β¨ β c, f = fun x => N * x + c
a b : β€
β’ f (N * a) + N * f b = f (f (a + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | exact (N * 0).zero_add.trans N.mul_zero | case inl
N : β€
h : N β 0
a b : β€
β’ 0 (N * a) + N * 0 b = 0 (0 (a + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
N : β€
h : N β 0
a b : β€
β’ 0 (N * a) + N * 0 b = 0 (0 (a + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/A1/A1.lean | IMOSL.IMO2019A1.final_solution | [23, 1] | [49, 62] | rw [add_right_comm, β mul_add, β add_assoc, β mul_add] | case inr.intro
N : β€
h : N β 0
a b c : β€
β’ (fun x => N * x + c) (N * a) + N * (fun x => N * x + c) b = (fun x => N * x + c) ((fun x => N * x + c) (a + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
N : β€
h : N β 0
a b c : β€
β’ (fun x => N * x + c) (N * a) + N * (fun x => N * x + c) b = (fun x => N * x + c) ((fun x => N * x + c) (a + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_comm | [96, 11] | [98, 71] | rw [mul_def, mul_def, mul_comm a, mul_comm b, add_comm (a * b')] | a b a' b' : β€
β’ { re := a, im := b } * { re := a', im := b' } = { re := a', im := b' } * { re := a, im := b } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' : β€
β’ { re := a, im := b } * { re := a', im := b' } = { re := a', im := b' } * { re := a, im := b }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_assoc | [100, 11] | [103, 39] | rw [mul_def, mul_def, mul_def, mul_def, β€Ο.mk.injEq] | a b a' b' a'' b'' : β€
β’ { re := a, im := b } * { re := a', im := b' } * { re := a'', im := b'' } =
{ re := a, im := b } * ({ re := a', im := b' } * { re := a'', im := b'' }) | a b a' b' a'' b'' : β€
β’ { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * a'' +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im β§
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * b'' +
a'' * { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im +
{ re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re * b +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : β€
β’ { re := a, im := b } * { re := a', im := b' } * { re := a'', im := b'' } =
{ re := a, im := b } * ({ re := a', im := b' } * { re := a'', im := b'' })
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_assoc | [100, 11] | [103, 39] | dsimp only | a b a' b' a'' b'' : β€
β’ { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * a'' +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im β§
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * b'' +
a'' * { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im +
{ re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re * b +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im | a b a' b' a'' b'' : β€
β’ (a * a' + b * b') * a'' + (a * b' + a' * b + b * b') * b'' =
a * (a' * a'' + b' * b'') + b * (a' * b'' + a'' * b' + b' * b'') β§
(a * a' + b * b') * b'' + a'' * (a * b' + a' * b + b * b') + (a * b' + a' * b + b * b') * b'' =
a * (a' * b'' + a'' * b' + b' * b'') + (a' * a'' + b' * b'') * b + b * (a' * b'' + a'' * b' + b' * b'') | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : β€
β’ { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * a'' +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im β§
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * b'' +
a'' * { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im +
{ re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re * b +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.im
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_assoc | [100, 11] | [103, 39] | constructor <;> ring | a b a' b' a'' b'' : β€
β’ (a * a' + b * b') * a'' + (a * b' + a' * b + b * b') * b'' =
a * (a' * a'' + b' * b'') + b * (a' * b'' + a'' * b' + b' * b'') β§
(a * a' + b * b') * b'' + a'' * (a * b' + a' * b + b * b') + (a * b' + a' * b + b * b') * b'' =
a * (a' * b'' + a'' * b' + b' * b'') + (a' * a'' + b' * b'') * b + b * (a' * b'' + a'' * b' + b' * b'') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : β€
β’ (a * a' + b * b') * a'' + (a * b' + a' * b + b * b') * b'' =
a * (a' * a'' + b' * b'') + b * (a' * b'' + a'' * b' + b' * b'') β§
(a * a' + b * b') * b'' + a'' * (a * b' + a' * b + b * b') + (a * b' + a' * b + b * b') * b'' =
a * (a' * b'' + a'' * b' + b' * b'') + (a' * a'' + b' * b'') * b + b * (a' * b'' + a'' * b' + b' * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.zero_mul | [105, 11] | [107, 46] | rw [mul_def, zero_def, zero_mul,
zero_mul, mul_zero, add_zero, add_zero] | a b : β€
β’ 0 * { re := a, im := b } = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β€
β’ 0 * { re := a, im := b } = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.one_mul | [112, 11] | [114, 56] | rw [mul_def, one_def, one_mul, one_mul,
zero_mul, mul_zero, add_zero, add_zero, add_zero] | a b : β€
β’ 1 * { re := a, im := b } = { re := a, im := b } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β€
β’ 1 * { re := a, im := b } = { re := a, im := b }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_add | [119, 11] | [124, 81] | rw [mul_def, mul_def, mul_def, add_def, add_def, β€Ο.mk.injEq] | a b a' b' a'' b'' : β€
β’ { re := a, im := b } * ({ re := a', im := b' } + { re := a'', im := b'' }) =
{ re := a, im := b } * { re := a', im := b' } + { re := a, im := b } * { re := a'', im := b'' } | a b a' b' a'' b'' : β€
β’ a * { re := a' + a'', im := b' + b'' }.re + b * { re := a' + a'', im := b' + b'' }.im =
a * a' + b * b' + (a * a'' + b * b'') β§
a * { re := a' + a'', im := b' + b'' }.im + { re := a' + a'', im := b' + b'' }.re * b +
b * { re := a' + a'', im := b' + b'' }.im =
a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : β€
β’ { re := a, im := b } * ({ re := a', im := b' } + { re := a'', im := b'' }) =
{ re := a, im := b } * { re := a', im := b' } + { re := a, im := b } * { re := a'', im := b'' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_add | [119, 11] | [124, 81] | dsimp only | a b a' b' a'' b'' : β€
β’ a * { re := a' + a'', im := b' + b'' }.re + b * { re := a' + a'', im := b' + b'' }.im =
a * a' + b * b' + (a * a'' + b * b'') β§
a * { re := a' + a'', im := b' + b'' }.im + { re := a' + a'', im := b' + b'' }.re * b +
b * { re := a' + a'', im := b' + b'' }.im =
a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | a b a' b' a'' b'' : β€
β’ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') β§
a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : β€
β’ a * { re := a' + a'', im := b' + b'' }.re + b * { re := a' + a'', im := b' + b'' }.im =
a * a' + b * b' + (a * a'' + b * b'') β§
a * { re := a' + a'', im := b' + b'' }.im + { re := a' + a'', im := b' + b'' }.re * b +
b * { re := a' + a'', im := b' + b'' }.im =
a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_add | [119, 11] | [124, 81] | constructor | a b a' b' a'' b'' : β€
β’ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') β§
a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | case left
a b a' b' a'' b'' : β€
β’ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'')
case right
a b a' b' a'' b'' : β€
β’ a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : β€
β’ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') β§
a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_add | [119, 11] | [124, 81] | rw [add_add_add_comm, β mul_add, β mul_add] | case left
a b a' b' a'' b'' : β€
β’ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
a b a' b' a'' b'' : β€
β’ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.mul_add | [119, 11] | [124, 81] | rw [add_add_add_comm, β mul_add, add_add_add_comm, β mul_add, β add_mul] | case right
a b a' b' a'' b'' : β€
β’ a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
a b a' b' a'' b'' : β€
β’ a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.add_mul | [126, 11] | [127, 55] | rw [β€Ο.mul_comm, β€Ο.mul_add, z.mul_comm, z.mul_comm] | x y z : β€Ο
β’ (x + y) * z = x * z + y * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : β€Ο
β’ (x + y) * z = x * z + y * z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_zero | [149, 1] | [150, 77] | change ((0 : β€) : R) + 0 β’ r = 0 | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ cast r 0 = 0 | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β0 + 0 β’ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ cast r 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_zero | [149, 1] | [150, 77] | rw [Int.cast_zero, zero_add, zero_zsmul] | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β0 + 0 β’ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β0 + 0 β’ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_one | [152, 1] | [153, 76] | change ((1 : β€) : R) + 0 β’ r = 1 | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ cast r 1 = 1 | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β1 + 0 β’ r = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ cast r 1 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_one | [152, 1] | [153, 76] | rw [Int.cast_one, zero_zsmul, add_zero] | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β1 + 0 β’ r = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β1 + 0 β’ r = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_Ο | [155, 1] | [156, 76] | change ((0 : β€) : R) + 1 β’ r = r | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ cast r Ο = r | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β0 + 1 β’ r = r | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ cast r Ο = r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_Ο | [155, 1] | [156, 76] | rw [Int.cast_zero, zero_add, one_zsmul] | R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β0 + 1 β’ r = r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddGroupWithOne R
r : R
β’ β0 + 1 β’ r = r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_add | [158, 1] | [162, 58] | change ((a + a' : β€) : R) + (b + b') β’ r = _ | R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ cast r ({ re := a, im := b } + { re := a', im := b' }) = cast r { re := a, im := b } + cast r { re := a', im := b' } | R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ β(a + a') + (b + b') β’ r = cast r { re := a, im := b } + cast r { re := a', im := b' } | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ cast r ({ re := a, im := b } + { re := a', im := b' }) = cast r { re := a, im := b } + cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_add | [158, 1] | [162, 58] | rw [Int.cast_add, add_zsmul, add_add_add_comm] | R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ β(a + a') + (b + b') β’ r = cast r { re := a, im := b } + cast r { re := a', im := b' } | R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ βa + b β’ r + (βa' + b' β’ r) = cast r { re := a, im := b } + cast r { re := a', im := b' } | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ β(a + a') + (b + b') β’ r = cast r { re := a, im := b } + cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_add | [158, 1] | [162, 58] | rfl | R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ βa + b β’ r + (βa' + b' β’ r) = cast r { re := a, im := b } + cast r { re := a', im := b' } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddCommGroupWithOne R
r : R
a b a' b' : β€
β’ βa + b β’ r + (βa' + b' β’ r) = cast r { re := a, im := b } + cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_mul | [166, 1] | [174, 76] | change β(a * a' + b * b') + (a * b' + a' * b + b * b') β’ r
= (βa + b β’ r) * (βa' + b' β’ r) | R : Type u_1
instβ : Ring R
r : R
h : r * r = r + 1
a b a' b' : β€
β’ cast r ({ re := a, im := b } * { re := a', im := b' }) = cast r { re := a, im := b } * cast r { re := a', im := b' } | R : Type u_1
instβ : Ring R
r : R
h : r * r = r + 1
a b a' b' : β€
β’ β(a * a' + b * b') + (a * b' + a' * b + b * b') β’ r = (βa + b β’ r) * (βa' + b' β’ r) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : Ring R
r : R
h : r * r = r + 1
a b a' b' : β€
β’ cast r ({ re := a, im := b } * { re := a', im := b' }) = cast r { re := a, im := b } * cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.β€Ο.cast_mul | [166, 1] | [174, 76] | rw [add_zsmul, Int.cast_add, add_add_add_comm, β zsmul_one (b * b'),
β zsmul_add, add_comm 1 r, β h, add_zsmul, β add_assoc, mul_zsmul,
zsmul_eq_mul, Int.cast_mul, β mul_add, add_mul, add_assoc, add_right_inj,
mul_add, β zsmul_eq_mul', mul_zsmul, add_right_inj, mul_zsmul, zsmul_eq_mul,
zsmul_eq_mul', zsmul_eq_mul, zsmul_eq_mul', mul_assoc, β mul_assoc] | R : Type u_1
instβ : Ring R
r : R
h : r * r = r + 1
a b a' b' : β€
β’ β(a * a' + b * b') + (a * b' + a' * b + b * b') β’ r = (βa + b β’ r) * (βa' + b' β’ r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : Ring R
r : R
h : r * r = r + 1
a b a' b' : β€
β’ β(a * a' + b * b') + (a * b' + a' * b + b * b') β’ r = (βa + b β’ r) * (βa' + b' β’ r)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_of_singleton | [32, 1] | [33, 62] | rw [NatSeq_ofList, List.length_singleton, Nat.mod_one] | Ξ± : Type u_1
instβ : Inhabited Ξ±
c : Ξ±
n : β
β’ NatSeq_ofList [c] n = c | Ξ± : Type u_1
instβ : Inhabited Ξ±
c : Ξ±
n : β
β’ [c].getI 0 = c | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : Inhabited Ξ±
c : Ξ±
n : β
β’ NatSeq_ofList [c] n = c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_of_singleton | [32, 1] | [33, 62] | rfl | Ξ± : Type u_1
instβ : Inhabited Ξ±
c : Ξ±
n : β
β’ [c].getI 0 = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : Inhabited Ξ±
c : Ξ±
n : β
β’ [c].getI 0 = c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | replace hN := List.length_pos.mp <| hN.trans_eq (List_ofNatSeq_length a N).symm | Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
hN : 0 < N
a : β β Ξ±
ha : Function.Periodic a N
n : β
β’ NatSeq_ofList (List_ofNatSeq a N) n = a n | Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ NatSeq_ofList (List_ofNatSeq a N) n = a n | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
hN : 0 < N
a : β β Ξ±
ha : Function.Periodic a N
n : β
β’ NatSeq_ofList (List_ofNatSeq a N) n = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | rw [NatSeq_nonempty_eq_get hN] | Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ NatSeq_ofList (List_ofNatSeq a N) n = a n | Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ (List_ofNatSeq a N).get β¨n % (List_ofNatSeq a N).length, β―β© = a n | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ NatSeq_ofList (List_ofNatSeq a N) n = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | unfold List_ofNatSeq | Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ (List_ofNatSeq a N).get β¨n % (List_ofNatSeq a N).length, β―β© = a n | Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ (List.map a (List.range N)).get β¨n % (List.map a (List.range N)).length, β―β© = a n | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ (List_ofNatSeq a N).get β¨n % (List_ofNatSeq a N).length, β―β© = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | rw [List.get_map, List.get_range, Fin.val_mk,
List.length_map, List.length_range, ha.map_mod_nat] | Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ (List.map a (List.range N)).get β¨n % (List.map a (List.range N)).length, β―β© = a n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : Inhabited Ξ±
N : β
a : β β Ξ±
ha : Function.Periodic a N
n : β
hN : List_ofNatSeq a N β []
β’ (List.map a (List.range N)).get β¨n % (List.map a (List.range N)).length, β―β© = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [add_inf] | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) (aβΒΉ + aβ β bβ) | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) ((aβΒΉ + aβ) β (aβΒΉ + bβ)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) (aβΒΉ + aβ β bβ)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofInf | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) ((aβΒΉ + aβ) β (aβΒΉ + bβ)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) ((aβΒΉ + aβ) β (aβΒΉ + bβ))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [add_sup] | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) (aβΒΉ + aβ β bβ) | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) ((aβΒΉ + aβ) β (aβΒΉ + bβ)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) (aβΒΉ + aβ β bβ)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofSup | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) ((aβΒΉ + aβ) β (aβΒΉ + bβ)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
haβ : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβΒΉ : G
ha : aβΒΉ β S
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβΒΉ + aβ) β
MetaClosure (fun x => x β S) (aβΒΉ + bβ) β MetaClosure (fun x => x β S) ((aβΒΉ + aβ) β (aβΒΉ + bβ))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [inf_add] | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) (aβ β bβ + b) | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) ((aβ + b) β (bβ + b)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) (aβ β bβ + b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofInf | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) ((aβ + b) β (bβ + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) ((aβ + b) β (bβ + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [sup_add] | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) (aβ β bβ + b) | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) ((aβ + b) β (bβ + b)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) (aβ β bβ + b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofSup | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) ((aβ + b) β (bβ + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x β S) a
hb : MetaClosure (fun x => x β S) b
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (aβ + b) β
MetaClosure (fun x => x β S) (bβ + b) β MetaClosure (fun x => x β S) ((aβ + b) β (bβ + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | rw [neg_inf] | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-(aβ β bβ)) | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-aβ β -bβ) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-(aβ β bβ))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | exact ofSup | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-aβ β -bβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-aβ β -bβ)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | rw [neg_sup] | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-(aβ β bβ)) | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-aβ β -bβ) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-(aβ β bβ))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | exact ofInf | G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-aβ β -bβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβΒ³ : Lattice G
instβΒ² : AddGroup G
instβΒΉ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
instβ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x β€ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x β S) a
aβ bβ : G
xβΒΉ : MetaClosure (fun x => x β S) aβ
xβ : MetaClosure (fun x => x β S) bβ
β’ MetaClosure (fun x => x β S) (-aβ) β MetaClosure (fun x => x β S) (-bβ) β MetaClosure (fun x => x β S) (-aβ β -bβ)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2022/A1/A1.lean | IMOSL.IMO2022A1.final_solution | [39, 1] | [52, 32] | have h2 (i : β) (h1 : 1 < a (i + 1)) (h2 : 1 < a (i + 2)) : False :=
(main_ineq h1.le h2 (h _) (h0 _)).asymm <| main_ineq2 (h i) h1 h2.le (h0 _) | R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
N : β
h1 : 2 β€ N
β’ a N β€ 1 | R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
N : β
h1 : 2 β€ N
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
β’ a N β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
N : β
h1 : 2 β€ N
β’ a N β€ 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2022/A1/A1.lean | IMOSL.IMO2022A1.final_solution | [39, 1] | [52, 32] | rcases Nat.exists_eq_add_of_le' h1 with β¨n, rflβ© | R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
N : β
h1 : 2 β€ N
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
β’ a N β€ 1 | case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1 : 2 β€ n + 2
β’ a (n + 2) β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
N : β
h1 : 2 β€ N
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
β’ a N β€ 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2022/A1/A1.lean | IMOSL.IMO2022A1.final_solution | [39, 1] | [52, 32] | refine le_of_not_lt Ξ» h1 β¦ (h0 (n + 1)).not_lt ?_ | case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1 : 2 β€ n + 2
β’ a (n + 2) β€ 1 | case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1β : 2 β€ n + 2
h1 : 1 < a (n + 2)
β’ a (n + 1) + a (n + 1 + 2) < a (n + 1 + 1) ^ 2 + a (n + 1) * a (n + 1 + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1 : 2 β€ n + 2
β’ a (n + 2) β€ 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2022/A1/A1.lean | IMOSL.IMO2022A1.final_solution | [39, 1] | [52, 32] | rw [β sub_lt_iff_lt_add, add_sub_assoc, β one_sub_mul] | case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1β : 2 β€ n + 2
h1 : 1 < a (n + 2)
β’ a (n + 1) + a (n + 1 + 2) < a (n + 1 + 1) ^ 2 + a (n + 1) * a (n + 1 + 2) | case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1β : 2 β€ n + 2
h1 : 1 < a (n + 2)
β’ a (n + 1) + (1 - a (n + 1)) * a (n + 1 + 2) < a (n + 1 + 1) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1β : 2 β€ n + 2
h1 : 1 < a (n + 2)
β’ a (n + 1) + a (n + 1 + 2) < a (n + 1 + 1) ^ 2 + a (n + 1) * a (n + 1 + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2022/A1/A1.lean | IMOSL.IMO2022A1.final_solution | [39, 1] | [52, 32] | exact (one_lt_pow h1 <| Nat.succ_ne_zero 1).trans_le' <|
add_le_of_le_sub_left <| mul_le_of_le_one_right
(sub_nonneg_of_le <| le_of_not_lt <| Ξ» h3 β¦ h2 _ h3 h1)
(le_of_not_lt <| h2 _ h1) | case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1β : 2 β€ n + 2
h1 : 1 < a (n + 2)
β’ a (n + 1) + (1 - a (n + 1)) * a (n + 1 + 2) < a (n + 1 + 1) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
instβ : LinearOrderedRing R
a : β β R
h : β (i : β), 0 β€ a i
h0 : β (i : β), a (i + 1) ^ 2 + a i * a (i + 2) β€ a i + a (i + 2)
h2 : β (i : β), 1 < a (i + 1) β 1 < a (i + 2) β False
n : β
h1β : 2 β€ n + 2
h1 : 1 < a (n + 2)
β’ a (n + 1) + (1 - a (n + 1)) * a (n + 1 + 2) < a (n + 1 + 1) ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.add_one_iterate | [32, 1] | [34, 81] | rw [iterate_succ_apply', add_one_iterate n a, add_assoc] | n : β
a : β€
β’ (fun x => x + 1)^[n + 1] a = a + β(n + 1) | n : β
a : β€
β’ a + (βn + 1) = a + β(n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
a : β€
β’ (fun x => x + 1)^[n + 1] a = a + β(n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.add_one_iterate | [32, 1] | [34, 81] | rfl | n : β
a : β€
β’ a + (βn + 1) = a + β(n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
a : β€
β’ a + (βn + 1) = a + β(n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_add_one | [36, 1] | [39, 62] | have h (c : β€) : c + ((c.natAbs ^ 2 : β) : β€) = c * (c + 1) := by
rw [Int.natCast_pow, Int.natAbs_sq, sq, β mul_one_add, add_comm] | a b : β€
β’ (fun x => x + 1)^[a.natAbs ^ 2 + b.natAbs ^ 2] (a + b) = a * (fun x => x + 1) a + b * (fun x => x + 1) b | a b : β€
h : β (c : β€), c + β(c.natAbs ^ 2) = c * (c + 1)
β’ (fun x => x + 1)^[a.natAbs ^ 2 + b.natAbs ^ 2] (a + b) = a * (fun x => x + 1) a + b * (fun x => x + 1) b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β€
β’ (fun x => x + 1)^[a.natAbs ^ 2 + b.natAbs ^ 2] (a + b) = a * (fun x => x + 1) a + b * (fun x => x + 1) b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_add_one | [36, 1] | [39, 62] | rw [add_one_iterate, Int.ofNat_add, add_add_add_comm, h, h] | a b : β€
h : β (c : β€), c + β(c.natAbs ^ 2) = c * (c + 1)
β’ (fun x => x + 1)^[a.natAbs ^ 2 + b.natAbs ^ 2] (a + b) = a * (fun x => x + 1) a + b * (fun x => x + 1) b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β€
h : β (c : β€), c + β(c.natAbs ^ 2) = c * (c + 1)
β’ (fun x => x + 1)^[a.natAbs ^ 2 + b.natAbs ^ 2] (a + b) = a * (fun x => x + 1) a + b * (fun x => x + 1) b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_add_one | [36, 1] | [39, 62] | rw [Int.natCast_pow, Int.natAbs_sq, sq, β mul_one_add, add_comm] | a b c : β€
β’ c + β(c.natAbs ^ 2) = c * (c + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β€
β’ c + β(c.natAbs ^ 2) = c * (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | rw [const_iterate, β Int.add_mul, Int.mul_zero] | a b : β€
β’ (fun x => 0)^[a.natAbs ^ 2 + b.natAbs ^ 2] (a + b) = a * (fun x => 0) a + b * (fun x => 0) b | a b : β€
β’ (bif (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 then a + b else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β€
β’ (fun x => 0)^[a.natAbs ^ 2 + b.natAbs ^ 2] (a + b) = a * (fun x => 0) a + b * (fun x => 0) b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | cases h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 | a b : β€
β’ (bif (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 then a + b else 0) = 0 | case false
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = false
β’ (bif false then a + b else 0) = 0
case true
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
β’ (bif true then a + b else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β€
β’ (bif (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 then a + b else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | rfl | case false
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = false
β’ (bif false then a + b else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case false
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = false
β’ (bif false then a + b else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | have h0 : β c : β€, c.natAbs ^ 2 = 0 β c = 0 :=
Ξ» c β¦ by rw [sq_eq_zero_iff, Int.natAbs_eq_zero] | case true
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
β’ (bif true then a + b else 0) = 0 | case true
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then a + b else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case true
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
β’ (bif true then a + b else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | rw [Nat.beq_eq, add_eq_zero, h0, h0] at h | case true
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then a + b else 0) = 0 | case true
a b : β€
h : a = 0 β§ b = 0
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then a + b else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case true
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then a + b else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | rw [h.1, h.2] | case true
a b : β€
h : a = 0 β§ b = 0
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then a + b else 0) = 0 | case true
a b : β€
h : a = 0 β§ b = 0
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then 0 + 0 else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case true
a b : β€
h : a = 0 β§ b = 0
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then a + b else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | rfl | case true
a b : β€
h : a = 0 β§ b = 0
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then 0 + 0 else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case true
a b : β€
h : a = 0 β§ b = 0
h0 : β (c : β€), c.natAbs ^ 2 = 0 β c = 0
β’ (bif true then 0 + 0 else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.good_zero | [46, 1] | [53, 23] | rw [sq_eq_zero_iff, Int.natAbs_eq_zero] | a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
c : β€
β’ c.natAbs ^ 2 = 0 β c = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β€
h : (a.natAbs ^ 2 + b.natAbs ^ 2).beq 0 = true
c : β€
β’ c.natAbs ^ 2 = 0 β c = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.map_iterate_sq | [65, 1] | [66, 59] | have h := h a 0 | f : β€ β β€
h : good f
a : β€
β’ f^[a.natAbs ^ 2] a = a * f a | f : β€ β β€
hβ : good f
a : β€
h : f^[a.natAbs ^ 2 + Int.natAbs 0 ^ 2] (a + 0) = a * f a + 0 * f 0
β’ f^[a.natAbs ^ 2] a = a * f a | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
a : β€
β’ f^[a.natAbs ^ 2] a = a * f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.map_iterate_sq | [65, 1] | [66, 59] | rwa [zero_mul, add_zero, add_zero] at h | f : β€ β β€
hβ : good f
a : β€
h : f^[a.natAbs ^ 2 + Int.natAbs 0 ^ 2] (a + 0) = a * f a + 0 * f 0
β’ f^[a.natAbs ^ 2] a = a * f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
hβ : good f
a : β€
h : f^[a.natAbs ^ 2 + Int.natAbs 0 ^ 2] (a + 0) = a * f a + 0 * f 0
β’ f^[a.natAbs ^ 2] a = a * f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.map_iterate_sq_add_one | [71, 1] | [75, 50] | have h0 := h (a + 1) (-1) | f : β€ β β€
h : good f
a : β€
β’ f^[(a + 1).natAbs ^ 2 + 1] a = f^[(a + 1).natAbs ^ 2] (a + 1) | f : β€ β β€
h : good f
a : β€
h0 : f^[(a + 1).natAbs ^ 2 + (-1).natAbs ^ 2] (a + 1 + -1) = (a + 1) * f (a + 1) + -1 * f (-1)
β’ f^[(a + 1).natAbs ^ 2 + 1] a = f^[(a + 1).natAbs ^ 2] (a + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
a : β€
β’ f^[(a + 1).natAbs ^ 2 + 1] a = f^[(a + 1).natAbs ^ 2] (a + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.map_iterate_sq_add_one | [71, 1] | [75, 50] | rwa [add_neg_cancel_right, map_neg_one h,
mul_zero, add_zero, β map_iterate_sq h] at h0 | f : β€ β β€
h : good f
a : β€
h0 : f^[(a + 1).natAbs ^ 2 + (-1).natAbs ^ 2] (a + 1 + -1) = (a + 1) * f (a + 1) + -1 * f (-1)
β’ f^[(a + 1).natAbs ^ 2 + 1] a = f^[(a + 1).natAbs ^ 2] (a + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
a : β€
h0 : f^[(a + 1).natAbs ^ 2 + (-1).natAbs ^ 2] (a + 1 + -1) = (a + 1) * f (a + 1) + -1 * f (-1)
β’ f^[(a + 1).natAbs ^ 2 + 1] a = f^[(a + 1).natAbs ^ 2] (a + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.exists_iter_add_large_eq | [80, 1] | [88, 67] | rcases exists_iter_add_large_eq a k with β¨N, h0β© | f : β€ β β€
h : good f
a : β€
k : β
β’ β N, f^[N + (k + 1)] a = f^[N] (a + β(k + 1)) | case intro
f : β€ β β€
h : good f
a : β€
k N : β
h0 : f^[N + k] a = f^[N] (a + βk)
β’ β N, f^[N + (k + 1)] a = f^[N] (a + β(k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
a : β€
k : β
β’ β N, f^[N + (k + 1)] a = f^[N] (a + β(k + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.exists_iter_add_large_eq | [80, 1] | [88, 67] | refine β¨N + (a + k + 1).natAbs ^ 2, ?_β© | case intro
f : β€ β β€
h : good f
a : β€
k N : β
h0 : f^[N + k] a = f^[N] (a + βk)
β’ β N, f^[N + (k + 1)] a = f^[N] (a + β(k + 1)) | case intro
f : β€ β β€
h : good f
a : β€
k N : β
h0 : f^[N + k] a = f^[N] (a + βk)
β’ f^[N + (a + βk + 1).natAbs ^ 2 + (k + 1)] a = f^[N + (a + βk + 1).natAbs ^ 2] (a + β(k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
a : β€
k N : β
h0 : f^[N + k] a = f^[N] (a + βk)
β’ β N, f^[N + (k + 1)] a = f^[N] (a + β(k + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.exists_iter_add_large_eq | [80, 1] | [88, 67] | rw [f.iterate_add_apply N, Nat.cast_succ, β add_assoc a,
β map_iterate_sq_add_one h, Commute.iterate_iterate_self, β h0,
β iterate_add_apply, add_comm _ (N + k), add_add_add_comm] | case intro
f : β€ β β€
h : good f
a : β€
k N : β
h0 : f^[N + k] a = f^[N] (a + βk)
β’ f^[N + (a + βk + 1).natAbs ^ 2 + (k + 1)] a = f^[N + (a + βk + 1).natAbs ^ 2] (a + β(k + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
a : β€
k N : β
h0 : f^[N + k] a = f^[N] (a + βk)
β’ f^[N + (a + βk + 1).natAbs ^ 2 + (k + 1)] a = f^[N + (a + βk + 1).natAbs ^ 2] (a + β(k + 1))
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 β N : β, 0 < f.minimalPeriod (f^[N] 0) by
rcases this with β¨N, h1β©
let k := f.minimalPeriod (f^[N] 0)
let F := Ξ» n β¦ |f^[n] 0|
refine β¨Extra.seqMax F (N + k) + 1, Ξ» n β¦ Int.lt_add_one_of_le <|
(n.le_total (N + k)).elim (Extra.le_seqMax_of_le F) (Ξ» h2 β¦ ?_)β©
rw [le_iff_exists_add] at h2; rcases h2 with β¨c, rflβ©
rw [add_rotate, iterate_add_apply, β iterate_mod_minimalPeriod_eq,
Nat.add_mod_left, β iterate_add_apply, add_comm]
exact Extra.le_seqMax_of_le F (add_le_add_left (c.mod_lt h1).le N) | f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β M, β (n : β), |f^[n] 0| < M | f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β N, 0 < minimalPeriod f (f^[N] 0) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β M, β (n : β), |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] | obtain β¨a, k, h1, h2β© : β (a : β€) (k : β), 0 < k β§ f (a + k) = f a := by
suffices β a b, a < b β§ f a = f b by
rcases this with β¨a, b, h1, h2β©
apply sub_pos_of_lt at h1
refine β¨a, (b - a).natAbs, Int.natAbs_pos.mpr h1.ne.symm, ?_β©
rw [Int.natCast_natAbs, abs_of_pos h1, add_sub_cancel, h2]
simp_rw [Injective, not_forall] at h0
rcases h0 with β¨a, b, h0, h1β©
rcases ne_iff_lt_or_gt.mp h1 with h2 | h2
exacts [β¨a, b, h2, h0β©, β¨b, a, h2, h0.symmβ©] | f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β N, 0 < minimalPeriod f (f^[N] 0) | case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
β’ β N, 0 < minimalPeriod f (f^[N] 0) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β N, 0 < minimalPeriod f (f^[N] 0)
TACTIC:
|
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