url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.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:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases exists_iter_add_large_eq h a k with β¨N, h4β© | case intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
β’ β N, 0 < minimalPeriod f (f^[N] 0) | case intro.intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ β N, 0 < minimalPeriod f (f^[N] 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
K M : β
h3 : f^[K] a = f^[M] 0
β’ β N, 0 < minimalPeriod f (f^[N] 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | refine β¨N + 1 + M, IsPeriodicPt.minimalPeriod_pos h1 ?_β© | case intro.intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ β N, 0 < minimalPeriod f (f^[N] 0) | case intro.intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[N + 1 + M] 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ β N, 0 < minimalPeriod f (f^[N] 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [iterate_add_apply, β h3, Commute.iterate_iterate_self] | case intro.intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[N + 1 + M] 0) | case intro.intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[K] (f^[N + 1] a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[N + 1 + M] 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | refine IsPeriodicPt.apply_iterate ?_ _ | case intro.intro.intro.intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[K] (f^[N + 1] a)) | case intro.intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[N + 1] a) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[K] (f^[N + 1] a))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [IsPeriodicPt, IsFixedPt, β iterate_add_apply,
iterate_succ_apply, β h2, Commute.iterate_self, β h4,
add_left_comm, β add_assoc, iterate_succ_apply'] | case intro.intro.intro.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
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[N + 1] a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
K M : β
h3 : f^[K] a = f^[M] 0
N : β
h4 : f^[N + k] a = f^[N] (a + βk)
β’ IsPeriodicPt f k (f^[N + 1] a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases this with β¨N, h1β© | f : β€ β β€
h : good f
h0 : Β¬Injective f
this : β N, 0 < minimalPeriod f (f^[N] 0)
β’ β M, β (n : β), |f^[n] 0| < M | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
β’ β M, β (n : β), |f^[n] 0| < M | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : Β¬Injective f
this : β N, 0 < minimalPeriod f (f^[N] 0)
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] | let k := f.minimalPeriod (f^[N] 0) | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
β’ β M, β (n : β), |f^[n] 0| < M | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
β’ β M, β (n : β), |f^[n] 0| < M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
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] | let F := Ξ» n β¦ |f^[n] 0| | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
β’ β M, β (n : β), |f^[n] 0| < M | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
β’ β M, β (n : β), |f^[n] 0| < M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
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] | 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© | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
β’ β M, β (n : β), |f^[n] 0| < M | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
n : β
h2 : N + k β€ n
β’ |f^[n] 0| β€ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
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] | rw [le_iff_exists_add] at h2 | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
n : β
h2 : N + k β€ n
β’ |f^[n] 0| β€ Extra.seqMax F (N + k) | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
n : β
h2 : β c, n = N + k + c
β’ |f^[n] 0| β€ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
n : β
h2 : N + k β€ n
β’ |f^[n] 0| β€ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases h2 with β¨c, rflβ© | case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
n : β
h2 : β c, n = N + k + c
β’ |f^[n] 0| β€ Extra.seqMax F (N + k) | case intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
c : β
β’ |f^[N + k + c] 0| β€ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
n : β
h2 : β c, n = N + k + c
β’ |f^[n] 0| β€ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [add_rotate, iterate_add_apply, β iterate_mod_minimalPeriod_eq,
Nat.add_mod_left, β iterate_add_apply, add_comm] | case intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
c : β
β’ |f^[N + k + c] 0| β€ Extra.seqMax F (N + k) | case intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
c : β
β’ |f^[N + c % k] 0| β€ Extra.seqMax F (N + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
c : β
β’ |f^[N + k + c] 0| β€ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exact Extra.le_seqMax_of_le F (add_le_add_left (c.mod_lt h1).le N) | case intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
c : β
β’ |f^[N + c % k] 0| β€ Extra.seqMax F (N + k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
N : β
h1 : 0 < minimalPeriod f (f^[N] 0)
k : β := minimalPeriod f (f^[N] 0)
F : β β β€ := fun n => |f^[n] 0|
c : β
β’ |f^[N + c % k] 0| β€ Extra.seqMax F (N + k)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | suffices 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] | f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β a k, 0 < k β§ f (a + βk) = f a | f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β a b, a < b β§ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β a k, 0 < k β§ f (a + βk) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | simp_rw [Injective, not_forall] at h0 | f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β a b, a < b β§ f a = f b | f : β€ β β€
h : good f
h0 : β x x_1, β (_ : f x = f x_1), Β¬x = x_1
β’ β a b, a < b β§ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : Β¬Injective f
β’ β a b, a < b β§ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases h0 with β¨a, b, h0, h1β© | f : β€ β β€
h : good f
h0 : β x x_1, β (_ : f x = f x_1), Β¬x = x_1
β’ β a b, a < b β§ f a = f b | case intro.intro.intro
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
β’ β a b, a < b β§ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : β x x_1, β (_ : f x = f x_1), Β¬x = x_1
β’ β a b, a < b β§ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases ne_iff_lt_or_gt.mp h1 with h2 | h2 | case intro.intro.intro
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
β’ β a b, a < b β§ f a = f b | case intro.intro.intro.inl
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
h2 : a < b
β’ β a b, a < b β§ f a = f b
case intro.intro.intro.inr
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
h2 : a > b
β’ β a b, a < b β§ f a = f b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
β’ β a b, a < b β§ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exacts [β¨a, b, h2, h0β©, β¨b, a, h2, h0.symmβ©] | case intro.intro.intro.inl
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
h2 : a < b
β’ β a b, a < b β§ f a = f b
case intro.intro.intro.inr
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
h2 : a > b
β’ β a b, a < b β§ f a = f b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
h2 : a < b
β’ β a b, a < b β§ f a = f b
case intro.intro.intro.inr
f : β€ β β€
h : good f
a b : β€
h0 : f a = f b
h1 : Β¬a = b
h2 : a > b
β’ β a b, a < b β§ f a = f b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases this with β¨a, b, h1, h2β© | f : β€ β β€
h : good f
h0 : Β¬Injective f
this : β a b, a < b β§ f a = f b
β’ β a k, 0 < k β§ f (a + βk) = f a | case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h1 : a < b
h2 : f a = f b
β’ β a k, 0 < k β§ f (a + βk) = f a | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : Β¬Injective f
this : β a b, a < b β§ f a = f b
β’ β a k, 0 < k β§ f (a + βk) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | apply sub_pos_of_lt at h1 | case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h1 : a < b
h2 : f a = f b
β’ β a k, 0 < k β§ f (a + βk) = f a | case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h2 : f a = f b
h1 : 0 < b - a
β’ β a k, 0 < k β§ f (a + βk) = f a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h1 : a < b
h2 : f a = f b
β’ β a k, 0 < k β§ f (a + βk) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | refine β¨a, (b - a).natAbs, Int.natAbs_pos.mpr h1.ne.symm, ?_β© | case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h2 : f a = f b
h1 : 0 < b - a
β’ β a k, 0 < k β§ f (a + βk) = f a | case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h2 : f a = f b
h1 : 0 < b - a
β’ f (a + β(b - a).natAbs) = f a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h2 : f a = f b
h1 : 0 < b - a
β’ β a k, 0 < k β§ f (a + βk) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [Int.natCast_natAbs, abs_of_pos h1, add_sub_cancel, h2] | case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h2 : f a = f b
h1 : 0 < b - a
β’ f (a + β(b - a).natAbs) = f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a b : β€
h2 : f a = f b
h1 : 0 < b - a
β’ f (a + β(b - a).natAbs) = f a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases le_total a 0 with h3 | h3 | f : 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 K M, f^[K] a = f^[M] 0 | case inl
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : a β€ 0
β’ β K M, f^[K] a = f^[M] 0
case inr
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : 0 β€ a
β’ β K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
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 K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases exists_iter_add_large_eq h a a.natAbs with β¨N, h4β© | case inl
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : a β€ 0
β’ β K M, f^[K] a = f^[M] 0 | case inl.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
h3 : a β€ 0
N : β
h4 : f^[N + a.natAbs] a = f^[N] (a + βa.natAbs)
β’ β K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : a β€ 0
β’ β K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [Int.natCast_natAbs, abs_of_nonpos h3, add_neg_self] at h4 | case inl.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : a β€ 0
N : β
h4 : f^[N + a.natAbs] a = f^[N] (a + βa.natAbs)
β’ β K M, f^[K] a = f^[M] 0 | case inl.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
h3 : a β€ 0
N : β
h4 : f^[N + a.natAbs] a = f^[N] 0
β’ β K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.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
h3 : a β€ 0
N : β
h4 : f^[N + a.natAbs] a = f^[N] (a + βa.natAbs)
β’ β K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exact β¨_, _, h4β© | case inl.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
h3 : a β€ 0
N : β
h4 : f^[N + a.natAbs] a = f^[N] 0
β’ β K M, f^[K] a = f^[M] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : a β€ 0
N : β
h4 : f^[N + a.natAbs] a = f^[N] 0
β’ β K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rcases exists_iter_add_large_eq h 0 a.natAbs with β¨N, h4β© | case inr
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : 0 β€ a
β’ β K M, f^[K] a = f^[M] 0 | case inr.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
h3 : 0 β€ a
N : β
h4 : f^[N + a.natAbs] 0 = f^[N] (0 + βa.natAbs)
β’ β K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : 0 β€ a
β’ β K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | rw [Int.natCast_natAbs, abs_of_nonneg h3, zero_add] at h4 | case inr.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : 0 β€ a
N : β
h4 : f^[N + a.natAbs] 0 = f^[N] (0 + βa.natAbs)
β’ β K M, f^[K] a = f^[M] 0 | case inr.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
h3 : 0 β€ a
N : β
h4 : f^[N + a.natAbs] 0 = f^[N] a
β’ β K M, f^[K] a = f^[M] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.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
h3 : 0 β€ a
N : β
h4 : f^[N + a.natAbs] 0 = f^[N] (0 + βa.natAbs)
β’ β K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.orbit_zero_bdd_of_not_injective | [90, 1] | [130, 53] | exact β¨_, _, h4.symmβ© | case inr.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
h3 : 0 β€ a
N : β
h4 : f^[N + a.natAbs] 0 = f^[N] a
β’ β K M, f^[K] a = f^[M] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
f : β€ β β€
h : good f
h0 : Β¬Injective f
a : β€
k : β
h1 : 0 < k
h2 : f (a + βk) = f a
h3 : 0 β€ a
N : β
h4 : f^[N + a.natAbs] 0 = f^[N] a
β’ β K M, f^[K] a = f^[M] 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rcases h0 with β¨M, h0β© | f : β€ β β€
h : good f
h0 : β M, β (n : β), |f^[n] 0| < M
β’ f = 0 | case intro
f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
h0 : β M, β (n : β), |f^[n] 0| < M
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h1 (a : β€) : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) := by
have h := h a (-a); rwa [a.natAbs_neg, add_neg_self,
β two_mul, neg_mul, β sub_eq_add_neg, β mul_sub] at h | case intro
f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
β’ f = 0 | case intro
f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | replace h0 (a : β€) (h2 : M β€ a) : f.IsPeriodicPt (2 * a.natAbs ^ 2) 0 :=
Int.eq_zero_of_abs_lt_dvd β¨_, h1 aβ© ((h0 _).trans_le h2) | case intro
f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
β’ f = 0 | case intro
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | replace h0 : f.IsPeriodicPt 2 0 := by
have h2 := (h0 _ M.le_refl).gcd (h0 _ M.lt_succ.le)
have h3 : IsCoprime M (M + 1) :=
β¨-1, 1, by rw [one_mul, neg_one_mul, neg_add_cancel_left]β©
rwa [Nat.gcd_mul_left, β Int.natAbs_pow, β Int.natAbs_pow,
β Int.gcd_eq_natAbs, Int.gcd_eq_one_iff_coprime.mpr h3.pow, mul_one] at h2 | case intro
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
β’ f = 0 | case intro
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | replace h1 (a : β€) (h3 : a β 0) : f a = f (-a) := by
specialize h1 a
rw [h0.mul_const, zero_eq_mul] at h1
exact eq_of_sub_eq_zero (h1.resolve_left h3) | case intro
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
β’ f = 0 | case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | suffices h2 : β a : β€, a β 0 β f a = 0 by
have h3 : f (f 2) = 1 * f 1 + 1 * f 1 := h 1 1
rw [h2 1 one_ne_zero, h2 _ two_ne_zero] at h3
exact funext Ξ» a β¦ (ne_or_eq a 0).elim (h2 a) (Ξ» h4 β¦ h4.symm βΈ h3) | case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
β’ f = 0 | case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
β’ β (a : β€), a β 0 β f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | intro a h2 | case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
β’ β (a : β€), a β 0 β f a = 0 | case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
β’ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
β’ β (a : β€), a β 0 β f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | obtain β¨n, h3β© : β n : β, a.natAbs ^ 2 = n.succ :=
Nat.exists_eq_succ_of_ne_zero (pow_ne_zero 2 <| Int.natAbs_ne_zero.mpr h2) | case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
β’ f a = 0 | case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
β’ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
β’ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h4 := map_iterate_sq h (-a) | case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
β’ f a = 0 | case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
h4 : f^[(-a).natAbs ^ 2] (-a) = -a * f (-a)
β’ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
β’ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [Int.natAbs_neg, h3, iterate_succ_apply, β h1 a h2, β iterate_succ_apply,
β h3, map_iterate_sq h, neg_mul, eq_neg_self_iff, mul_eq_zero] at h4 | case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
h4 : f^[(-a).natAbs ^ 2] (-a) = -a * f (-a)
β’ f a = 0 | case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
h4 : a = 0 β¨ f a = 0
β’ f a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
h4 : f^[(-a).natAbs ^ 2] (-a) = -a * f (-a)
β’ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | exact h4.resolve_left h2 | case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
h4 : a = 0 β¨ f a = 0
β’ f a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
a : β€
h2 : a β 0
n : β
h3 : a.natAbs ^ 2 = n.succ
h4 : a = 0 β¨ f a = 0
β’ f a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h := h a (-a) | f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
a : β€
β’ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) | f : β€ β β€
hβ : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
a : β€
h : f^[a.natAbs ^ 2 + (-a).natAbs ^ 2] (a + -a) = a * f a + -a * f (-a)
β’ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
a : β€
β’ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rwa [a.natAbs_neg, add_neg_self,
β two_mul, neg_mul, β sub_eq_add_neg, β mul_sub] at h | f : β€ β β€
hβ : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
a : β€
h : f^[a.natAbs ^ 2 + (-a).natAbs ^ 2] (a + -a) = a * f a + -a * f (-a)
β’ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
hβ : good f
M : β€
h0 : β (n : β), |f^[n] 0| < M
a : β€
h : f^[a.natAbs ^ 2 + (-a).natAbs ^ 2] (a + -a) = a * f a + -a * f (-a)
β’ f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h2 := (h0 _ M.le_refl).gcd (h0 _ M.lt_succ.le) | f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
β’ IsPeriodicPt f 2 0 | f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
β’ IsPeriodicPt f 2 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
β’ IsPeriodicPt f 2 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h3 : IsCoprime M (M + 1) :=
β¨-1, 1, by rw [one_mul, neg_one_mul, neg_add_cancel_left]β© | f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
β’ IsPeriodicPt f 2 0 | f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
h3 : IsCoprime M (M + 1)
β’ IsPeriodicPt f 2 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
β’ IsPeriodicPt f 2 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rwa [Nat.gcd_mul_left, β Int.natAbs_pow, β Int.natAbs_pow,
β Int.gcd_eq_natAbs, Int.gcd_eq_one_iff_coprime.mpr h3.pow, mul_one] at h2 | f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
h3 : IsCoprime M (M + 1)
β’ IsPeriodicPt f 2 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
h3 : IsCoprime M (M + 1)
β’ IsPeriodicPt f 2 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [one_mul, neg_one_mul, neg_add_cancel_left] | f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
β’ -1 * M + 1 * (M + 1) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : β (a : β€), M β€ a β IsPeriodicPt f (2 * a.natAbs ^ 2) 0
h2 : IsPeriodicPt f ((2 * M.natAbs ^ 2).gcd (2 * (M + 1).natAbs ^ 2)) 0
β’ -1 * M + 1 * (M + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | specialize h1 a | f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
β’ f a = f (-a) | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
h1 : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
β’ f a = f (-a) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h1 : β (a : β€), f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
β’ f a = f (-a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [h0.mul_const, zero_eq_mul] at h1 | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
h1 : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
β’ f a = f (-a) | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
h1 : a = 0 β¨ f a - f (-a) = 0
β’ f a = f (-a) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
h1 : f^[2 * a.natAbs ^ 2] 0 = a * (f a - f (-a))
β’ f a = f (-a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | exact eq_of_sub_eq_zero (h1.resolve_left h3) | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
h1 : a = 0 β¨ f a - f (-a) = 0
β’ f a = f (-a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
a : β€
h3 : a β 0
h1 : a = 0 β¨ f a - f (-a) = 0
β’ f a = f (-a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | have h3 : f (f 2) = 1 * f 1 + 1 * f 1 := h 1 1 | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
β’ f = 0 | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
h3 : f (f 2) = 1 * f 1 + 1 * f 1
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | rw [h2 1 one_ne_zero, h2 _ two_ne_zero] at h3 | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
h3 : f (f 2) = 1 * f 1 + 1 * f 1
β’ f = 0 | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
h3 : f 0 = 1 * 0 + 1 * 0
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
h3 : f (f 2) = 1 * f 1 + 1 * f 1
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2020/A6/A6.lean | IMOSL.IMO2020A6.eq_zero_of_not_injective | [132, 1] | [162, 27] | exact funext Ξ» a β¦ (ne_or_eq a 0).elim (h2 a) (Ξ» h4 β¦ h4.symm βΈ h3) | f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
h3 : f 0 = 1 * 0 + 1 * 0
β’ f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β€ β β€
h : good f
M : β€
h0 : IsPeriodicPt f 2 0
h1 : β (a : β€), a β 0 β f a = f (-a)
h2 : β (a : β€), a β 0 β f a = 0
h3 : f 0 = 1 * 0 + 1 * 0
β’ f = 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
β’ |a n| β€ max (Extra.seqMax (-a) n) (2 β’ Extra.seqMax a n) | G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
β’ |a n| β€ Extra.seqMax (-a) n β¨ |a n| β€ 2 β’ Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
β’ |a n| β€ max (Extra.seqMax (-a) n) (2 β’ 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 (Ξ» h β¦ ?_) (Ξ» h β¦ ?_) | G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
β’ |a n| β€ Extra.seqMax (-a) n β¨ |a n| β€ 2 β’ Extra.seqMax a n | case refine_1
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : a n β€ 0
β’ |a n| β€ Extra.seqMax (-a) n
case refine_2
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
β’ |a n| β€ 2 β’ Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
β’ |a n| β€ Extra.seqMax (-a) n β¨ |a n| β€ 2 β’ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : a n β€ 0
β’ |a n| β€ Extra.seqMax (-a) n | case refine_1
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : a n β€ 0
β’ -a n β€ Extra.seqMax (-a) n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : a n β€ 0
β’ |a n| β€ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : a n β€ 0
β’ -a n β€ Extra.seqMax (-a) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : a n β€ 0
β’ -a n β€ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
β’ |a n| β€ 2 β’ Extra.seqMax a n | case refine_2
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
β’ a n β€ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
β’ |a n| β€ 2 β’ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
β’ a n β€ Extra.seqMax a n + Extra.seqMax a n | case refine_2
G : Type u_1
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
h0 : a n β€ Extra.seqMax a n
β’ a n β€ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
β’ a n β€ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
h0 : a n β€ Extra.seqMax a n
β’ a n β€ 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
instβ : LinearOrderedAddCommGroup G
a : β β G
n : β
h : 0 β€ a n
h0 : a n β€ Extra.seqMax a n
β’ a n β€ 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 β¨i, h2, h1β© | G : Type u_1
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
β’ a (n + 1) β€ Extra.seqMax (-a) n - Extra.seqMax a n | case intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
i : β
h2 : i β€ n
h1 : a i = Extra.seqMax a n
β’ a (n + 1) β€ Extra.seqMax (-a) n - Extra.seqMax a n | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
β’ a (n + 1) β€ 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
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
i : β
h2 : i β€ n
h1 : a i = Extra.seqMax a n
β’ a (n + 1) β€ Extra.seqMax (-a) n - Extra.seqMax a n | case intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
i : β
h2 : β c, n = i + c
h1 : a i = Extra.seqMax a n
β’ a (n + 1) β€ 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
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
i : β
h2 : i β€ n
h1 : a i = Extra.seqMax a n
β’ a (n + 1) β€ 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 β¨j, rflβ© | case intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
i : β
h2 : β c, n = i + c
h1 : a i = Extra.seqMax a n
β’ a (n + 1) β€ Extra.seqMax (-a) n - Extra.seqMax a n | case intro.intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ a (i + j + 1) β€ 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
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good1 D a
h0 : D β€ n
i : β
h2 : β c, n = i + c
h1 : a i = Extra.seqMax a n
β’ a (n + 1) β€ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ a (i + j + 1) β€ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j) | case intro.intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ -(a i + a j) β€ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ a (i + j + 1) β€ 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 [β h1, neg_add_rev, β sub_eq_add_neg] | case intro.intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ -(a i + a j) β€ Extra.seqMax (-a) (i + j) - Extra.seqMax a (i + j) | case intro.intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ -a j - a i β€ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ -(a i + a j) β€ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ -a j - a i β€ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good1 D a
i j : β
h0 : D β€ i + j
h1 : a i = Extra.seqMax a (i + j)
β’ -a j - a i β€ 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 β¨i, j, rfl, h0β© | G : Type u_1
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good2 D a
h0 : D β€ n
β’ -a (n + 1) β€ 2 β’ Extra.seqMax a n | case intro.intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good2 D a
i j : β
h0β : D β€ i + j
h0 : a (i + j + 1) = -(a i + a j)
β’ -a (i + j + 1) β€ 2 β’ Extra.seqMax a (i + j) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D n : β
a : β β G
h : good2 D a
h0 : D β€ n
β’ -a (n + 1) β€ 2 β’ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good2 D a
i j : β
h0β : D β€ i + j
h0 : a (i + j + 1) = -(a i + a j)
β’ -a (i + j + 1) β€ 2 β’ Extra.seqMax a (i + j) | case intro.intro.intro
G : Type u_1
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good2 D a
i j : β
h0β : D β€ i + j
h0 : a (i + j + 1) = -(a i + a j)
β’ a i + a j β€ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good2 D a
i j : β
h0β : D β€ i + j
h0 : a (i + j + 1) = -(a i + a j)
β’ -a (i + j + 1) β€ 2 β’ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good2 D a
i j : β
h0β : D β€ i + j
h0 : a (i + j + 1) = -(a i + a j)
β’ a i + a j β€ 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
instβ : LinearOrderedAddCommGroup G
D : β
a : β β G
h : good2 D a
i j : β
h0β : D β€ i + j
h0 : a (i + j + 1) = -(a i + a j)
β’ a i + a j β€ 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.c_bdd | [87, 1] | [92, 69] | suffices b K = b (K + 1) from
β¨this.symm, (h0 K h2).trans_eq (this βΈ max_eq_right h3)β© | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K β€ 2 β’ b K
β’ b (K + 1) = b K β§ c (K + 1) β€ 2 β’ b (K + 1) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K β€ 2 β’ b K
β’ b K = b (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K β€ 2 β’ b K
β’ b (K + 1) = b K β§ c (K + 1) β€ 2 β’ b (K + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.c_bdd | [87, 1] | [92, 69] | rw [two_nsmul, β sub_le_iff_le_add] at h3 | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K β€ 2 β’ b K
β’ b K = b (K + 1) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K - b K β€ b K
β’ b K = b (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K β€ 2 β’ b K
β’ b K = b (K + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.c_bdd | [87, 1] | [92, 69] | exact (h1 K.le_succ).antisymm ((h K h2).trans_eq (max_eq_left h3)) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K - b K β€ b K
β’ b K = b (K + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : D β€ K
h3 : c K - b K β€ b K
β’ b K = b (K + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd | [94, 1] | [100, 67] | have X {M : β} : D β€ M β 2 β’ b M < c M β c (M + 1) = c M := Ξ» h3 h4 β¦
((h0 M h3).trans_eq (max_eq_left_of_lt h4)).antisymm (h2 M.le_succ) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
β’ 2 β’ b K < c K β c (K + 1) = c D | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
β’ 2 β’ b K < c K β c (K + 1) = c D | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
β’ 2 β’ b K < c K β c (K + 1) = c D
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd | [94, 1] | [100, 67] | refine Nat.le_induction (X D.le_refl) (Ξ» n h4 h5 h6 β¦ ?_) K h3 | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
β’ 2 β’ b K < c K β c (K + 1) = c D | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
n : β
h4 : D β€ n
h5 : 2 β’ b n < c n β c (n + 1) = c D
h6 : 2 β’ b (n + 1) < c (n + 1)
β’ c (n + 1 + 1) = c D | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
β’ 2 β’ b K < c K β c (K + 1) = c D
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd | [94, 1] | [100, 67] | rw [X (Nat.le_step h4) h6] | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
n : β
h4 : D β€ n
h5 : 2 β’ b n < c n β c (n + 1) = c D
h6 : 2 β’ b (n + 1) < c (n + 1)
β’ c (n + 1 + 1) = c D | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
n : β
h4 : D β€ n
h5 : 2 β’ b n < c n β c (n + 1) = c D
h6 : 2 β’ b (n + 1) < c (n + 1)
β’ c n.succ = c D | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
n : β
h4 : D β€ n
h5 : 2 β’ b n < c n β c (n + 1) = c D
h6 : 2 β’ b (n + 1) < c (n + 1)
β’ c (n + 1 + 1) = c D
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.c_succ_eq_D_of_b_bdd | [94, 1] | [100, 67] | exact h5 (lt_of_not_le Ξ» h7 β¦ h6.not_le (c_bdd h h0 h1 h4 h7).2) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
n : β
h4 : D β€ n
h5 : 2 β’ b n < c n β c (n + 1) = c D
h6 : 2 β’ b (n + 1) < c (n + 1)
β’ c n.succ = c D | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
K : β
h2 : Monotone c
h3 : D β€ K
X : β {M : β}, D β€ M β 2 β’ b M < c M β c (M + 1) = c M
n : β
h4 : D β€ n
h5 : 2 β’ b n < c n β c (n + 1) = c D
h6 : 2 β’ b (n + 1) < c (n + 1)
β’ c n.succ = c D
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | have h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D)) := by
refine max_le (le_max_of_two_nsmul_le_add ?_) (le_max_left _ _)
rw [β nsmul_add, add_sub_cancel] | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
β’ max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D)) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
β’ max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
β’ max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | refine (le_total n D).elim (Ξ» h4 β¦ h3.trans' (max_le_max (h2 h4) (nsmul_le_nsmul_right (h1 h4) 2)))
(Nat.le_induction h3 (Ξ» n h4 h5 β¦ ?_) n) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
β’ max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D)) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
nβ : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
n : β
h4 : D β€ n
h5 : max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D))
β’ max (c (n + 1)) (2 β’ b (n + 1)) β€ max (2 β’ b D) (2 β’ (c D - b D)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
β’ max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | rcases le_or_lt (c n) (2 β’ b n) with h6 | h6 | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
nβ : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
n : β
h4 : D β€ n
h5 : max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D))
β’ max (c (n + 1)) (2 β’ b (n + 1)) β€ max (2 β’ b D) (2 β’ (c D - b D)) | case inl
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
nβ : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
n : β
h4 : D β€ n
h5 : max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D))
h6 : c n β€ 2 β’ b n
β’ max (c (n + 1)) (2 β’ b (n + 1)) β€ max (2 β’ b D) (2 β’ (c D - b D))
case inr
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
nβ : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
n : β
h4 : D β€ n
h5 : max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D))
h6 : 2 β’ b n < c n
β’ max (c (n + 1)) (2 β’ b (n + 1)) β€ max (2 β’ b D) (2 β’ (c D - b D)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
nβ : β
h3 : max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
n : β
h4 : D β€ n
h5 : max (c n) (2 β’ b n) β€ max (2 β’ b D) (2 β’ (c D - b D))
β’ max (c (n + 1)) (2 β’ b (n + 1)) β€ max (2 β’ b D) (2 β’ (c D - b D))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A4/A4.lean | IMOSL.IMO2017A4.max_two_nsmul_b_and_c_bdd | [102, 1] | [120, 44] | refine max_le (le_max_of_two_nsmul_le_add ?_) (le_max_left _ _) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
β’ max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D)) | G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
β’ 2 β’ c D β€ 2 β’ b D + 2 β’ (c D - b D) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
D : β
instβ : LinearOrderedAddCommGroup G
b c : β β G
h : β (n : β), D β€ n β b (n + 1) β€ max (b n) (c n - b n)
h0 : β (n : β), D β€ n β c (n + 1) β€ max (c n) (2 β’ b n)
h1 : Monotone b
h2 : Monotone c
n : β
β’ max (c D) (2 β’ b D) β€ max (2 β’ b D) (2 β’ (c D - b D))
TACTIC:
|
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