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/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact Nat.not_dvd_of_pos_of_lt n.succ_pos <|
lt_add_of_pos_right _ (n.succ + c).succ_pos | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ Β¬n + 1 + (n + 1 + (c + 1)) β£ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ Β¬n + 1 + (n + 1 + (c + 1)) β£ n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | intro n h0 | k c : β
h : good k c
β’ good (k + 1) (2 * c + 1) | k c : β
h : good k c
n : β
h0 : 0 < n
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(n + (2 * c + 1)) / βn = (map (fun m => β(m + 1) / βm) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
k c : β
h : good k c
β’ good (k + 1) (2 * c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rcases n.even_or_odd' with β¨t, rfl | rflβ© | k c : β
h : good k c
n : β
h0 : 0 < n
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(n + (2 * c + 1)) / βn = (map (fun m => β(m + 1) / βm) S).prod | case intro.inl
k c : β
h : good k c
t : β
h0 : 0 < 2 * t
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod
case intro.inr
k c : β
h : good k c
t : β
h0 : 0 < 2 * t + 1
β’ β S,
card S = k + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
k c : β
h : good k c
n : β
h0 : 0 < n
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(n + (2 * c + 1)) / βn = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | replace h0 := pos_of_mul_pos_right h0 (Nat.zero_le 2) | case intro.inl
k c : β
h : good k c
t : β
h0 : 0 < 2 * t
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod | case intro.inl
k c : β
h : good k c
t : β
h0 : 0 < t
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
k c : β
h : good k c
t : β
h0 : 0 < 2 * t
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rcases h t h0 with β¨T, rfl, h1, h2β© | case intro.inl
k c : β
h : good k c
t : β
h0 : 0 < t
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
β’ β S,
card S = card T + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
k c : β
h : good k c
t : β
h0 : 0 < t
β’ β S, card S = k + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | have X := t.add_pos_left h0 c | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
β’ β S,
card S = card T + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β S,
card S = card T + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
β’ β S,
card S = card T + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | refine β¨(2 * (t + c)) ::β T, card_cons _ T,
forall_mem_cons.mpr β¨mul_pos (Nat.succ_pos 1) X, h1β©, ?_β© | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β S,
card S = card T + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) (2 * (t + c) ::β T)).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β S,
card S = card T + 1 β§ (β m β S, 0 < m) β§ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rw [map_cons, prod_cons, β h2, β add_assoc, β mul_add,
div_mul_div_comm, Nat.cast_mul, Nat.cast_mul, mul_right_comm] | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) (2 * (t + c) ::β T)).prod | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β(2 * (t + c) + 1) / (β2 * βt) = β(2 * (t + c) + 1) * β(t + c) / (β2 * βt * β(t + c)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β(2 * t + (2 * c + 1)) / β(2 * t) = (map (fun m => β(m + 1) / βm) (2 * (t + c) ::β T)).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | exact (mul_div_mul_right _ _ <| Nat.cast_ne_zero.mpr X.ne.symm).symm | case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β(2 * (t + c) + 1) / (β2 * βt) = β(2 * (t + c) + 1) * β(t + c) / (β2 * βt * β(t + c)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : β
h0 : 0 < t
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + c) / βt = (map (fun m => β(m + 1) / βm) T).prod
X : 0 < t + c
β’ β(2 * (t + c) + 1) / (β2 * βt) = β(2 * (t + c) + 1) * β(t + c) / (β2 * βt * β(t + c))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | have X := t.succ_pos | case intro.inr
k c : β
h : good k c
t : β
h0 : 0 < 2 * t + 1
β’ β S,
card S = k + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod | case intro.inr
k c : β
h : good k c
t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
β’ β S,
card S = k + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
k c : β
h : good k c
t : β
h0 : 0 < 2 * t + 1
β’ β S,
card S = k + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rcases h (t + 1) X with β¨T, rfl, h1, h2β© | case intro.inr
k c : β
h : good k c
t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
β’ β S,
card S = k + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod | case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β S,
card S = card T + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
k c : β
h : good k c
t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
β’ β S,
card S = k + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | refine β¨(2 * t + 1) ::β T, card_cons _ T,
forall_mem_cons.mpr β¨(2 * t).succ_pos, h1β©, ?_β© | case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β S,
card S = card T + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod | case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) ((2 * t + 1) ::β T)).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β S,
card S = card T + 1 β§
(β m β S, 0 < m) β§ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rw [map_cons, prod_cons, β h2, add_add_add_comm, add_right_comm,
add_assoc (2 * t) 1, β mul_add_one (Ξ± := β), β mul_add, div_mul_div_comm,
Nat.cast_mul, Nat.cast_mul, mul_right_comm] | case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) ((2 * t + 1) ::β T)).prod | case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β2 * β(t + 1 + c) / β(2 * t + 1) = β2 * β(t + 1 + c) * β(t + 1) / (β(2 * t + 1) * β(t + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β(2 * t + 1 + (2 * c + 1)) / β(2 * t + 1) = (map (fun m => β(m + 1) / βm) ((2 * t + 1) ::β T)).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | exact (mul_div_mul_right _ _ <| Nat.cast_ne_zero.mpr X.ne.symm).symm | case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β2 * β(t + 1 + c) / β(2 * t + 1) = β2 * β(t + 1 + c) * β(t + 1) / (β(2 * t + 1) * β(t + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr.intro.intro.intro
c t : β
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset β
h : good (card T) c
h1 : β m β T, 0 < m
h2 : β(t + 1 + c) / β(t + 1) = (map (fun m => β(m + 1) / βm) T).prod
β’ β2 * β(t + 1 + c) / β(2 * t + 1) = β2 * β(t + 1 + c) * β(t + 1) / (β(2 * t + 1) * β(t + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | rw [pow_zero, Nat.sub_self, add_zero] | n : β
h : 0 < n
β’ β(n + (2 ^ 0 - 1)) / βn = (map (fun m => β(m + 1) / βm) 0).prod | n : β
h : 0 < n
β’ βn / βn = (map (fun m => β(m + 1) / βm) 0).prod | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
h : 0 < n
β’ β(n + (2 ^ 0 - 1)) / βn = (map (fun m => β(m + 1) / βm) 0).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | exact div_self (Nat.cast_ne_zero.mpr h.ne.symm) | n : β
h : 0 < n
β’ βn / βn = (map (fun m => β(m + 1) / βm) 0).prod | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
h : 0 < n
β’ βn / βn = (map (fun m => β(m + 1) / βm) 0).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | have h := good_two_mul_add_one (final_solution k) | k : β
β’ good (k + 1) (2 ^ (k + 1) - 1) | k : β
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
β’ good (k + 1) (2 ^ (k + 1) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
k : β
β’ good (k + 1) (2 ^ (k + 1) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | have h0 := k.one_le_pow 2 (Nat.succ_pos 1) | k : β
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
β’ good (k + 1) (2 ^ (k + 1) - 1) | k : β
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
h0 : 1 β€ 2 ^ k
β’ good (k + 1) (2 ^ (k + 1) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
k : β
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
β’ good (k + 1) (2 ^ (k + 1) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | rwa [two_mul, add_assoc, Nat.sub_add_cancel h0, add_comm _ (2 ^ k),
β Nat.add_sub_assoc h0, β two_mul, β pow_succ'] at h | k : β
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
h0 : 1 β€ 2 ^ k
β’ good (k + 1) (2 ^ (k + 1) - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : β
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
h0 : 1 β€ 2 ^ k
β’ good (k + 1) (2 ^ (k + 1) - 1)
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/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | refine Nat.rec (Ξ» _ β¦ h.not_dvd_one) (Ξ» r h0 h1 β¦ ?_) | p : β
h : p.Prime
k : β
β’ β r < p, Β¬p β£ (p * k + r).descFactorial r | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ (p * k + r.succ).descFactorial r.succ | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
h : p.Prime
k : β
β’ β r < p, Β¬p β£ (p * k + r).descFactorial r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | rw [Nat.add_succ, Nat.succ_descFactorial_succ] | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ (p * k + r.succ).descFactorial r.succ | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ (p * k + r + 1) * (p * k + r).descFactorial r | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ (p * k + r.succ).descFactorial r.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | refine h.not_dvd_mul ?_ (h0 <| r.lt_succ_self.trans h1) | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ (p * k + r + 1) * (p * k + r).descFactorial r | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ p * k + r + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ (p * k + r + 1) * (p * k + r).descFactorial r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | rw [add_assoc, Nat.dvd_add_right β¨k, rflβ©] | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ p * k + r + 1 | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ r + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ p * k + r + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_not_dvd_descFactorial | [20, 1] | [26, 47] | exact Nat.not_dvd_of_pos_of_lt r.succ_pos h1 | p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ r + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
h : p.Prime
k r : β
h0 : r < p β Β¬p β£ (p * k + r).descFactorial r
h1 : r.succ < p
β’ Β¬p β£ r + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_binom_not_dvd | [28, 1] | [35, 59] | apply prime_not_dvd_descFactorial h k.pred p.pred (p.pred_lt_self h.pos) | k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ False | k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ p β£ (p * k.pred + p.pred).descFactorial p.pred | Please generate a tactic in lean4 to solve the state.
STATE:
k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_binom_not_dvd | [28, 1] | [35, 59] | rw [β Nat.mul_dvd_mul_iff_left (p * k.pred + p.pred).succ_pos,
β Nat.succ_descFactorial_succ, β Nat.succ_eq_add_one, β Nat.add_succ,
β Nat.succ_eq_add_one, Nat.succ_pred h.ne_zero, β p.mul_succ,
Nat.succ_pred h0, Nat.descFactorial_eq_factorial_mul_choose, mul_comm] | k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ p β£ (p * k.pred + p.pred).descFactorial p.pred | k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ p * (p * k) β£ p.factorial * (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ p β£ (p * k.pred + p.pred).descFactorial p.pred
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.prime_binom_not_dvd | [28, 1] | [35, 59] | exact mul_dvd_mul (Nat.dvd_factorial h.pos p.le_refl) h1 | k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ p * (p * k) β£ p.factorial * (p * k).choose p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k p : β
h : p.Prime
h0 : k β 0
h1 : p * k β£ (p * k).choose p
β’ p * (p * k) β£ p.factorial * (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | refine β¨Ξ» h0 β¦ ?_, Ξ» h0 n h1 β¦ ?_β© | m : β
h : 1 < m
β’ (β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)) β m.Prime | case refine_1
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
β’ m.Prime
case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : 2 * n β€ m
β’ n β£ n.choose (m - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
m : β
h : 1 < m
β’ (β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)) β m.Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | by_cases h1 : 2 β£ m | case refine_1
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
β’ m.Prime
case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : 2 * n β€ m
β’ n β£ n.choose (m - 2 * n) | case pos
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
h1 : 2 β£ m
β’ m.Prime
case neg
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
h1 : Β¬2 β£ m
β’ m.Prime
case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : 2 * n β€ m
β’ n β£ n.choose (m - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
β’ m.Prime
case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : 2 * n β€ m
β’ n β£ n.choose (m - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h1 with β¨k, rflβ© | case pos
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
h1 : 2 β£ m
β’ m.Prime | case pos.intro
k : β
h : 1 < 2 * k
h0 : β (n : β), 2 * n β€ 2 * k β n β£ n.choose (2 * k - 2 * n)
β’ (2 * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
h1 : 2 β£ m
β’ m.Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | specialize h0 k (2 * k).le_refl | case pos.intro
k : β
h : 1 < 2 * k
h0 : β (n : β), 2 * n β€ 2 * k β n β£ n.choose (2 * k - 2 * n)
β’ (2 * k).Prime | case pos.intro
k : β
h : 1 < 2 * k
h0 : k β£ k.choose (2 * k - 2 * k)
β’ (2 * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : β
h : 1 < 2 * k
h0 : β (n : β), 2 * n β€ 2 * k β n β£ n.choose (2 * k - 2 * n)
β’ (2 * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.sub_self, k.choose_zero_right, Nat.dvd_one] at h0 | case pos.intro
k : β
h : 1 < 2 * k
h0 : k β£ k.choose (2 * k - 2 * k)
β’ (2 * k).Prime | case pos.intro
k : β
h : 1 < 2 * k
h0 : k = 1
β’ (2 * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : β
h : 1 < 2 * k
h0 : k β£ k.choose (2 * k - 2 * k)
β’ (2 * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [h0, mul_one] | case pos.intro
k : β
h : 1 < 2 * k
h0 : k = 1
β’ (2 * k).Prime | case pos.intro
k : β
h : 1 < 2 * k
h0 : k = 1
β’ Nat.Prime 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : β
h : 1 < 2 * k
h0 : k = 1
β’ (2 * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact Nat.prime_two | case pos.intro
k : β
h : 1 < 2 * k
h0 : k = 1
β’ Nat.Prime 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
k : β
h : 1 < 2 * k
h0 : k = 1
β’ Nat.Prime 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | obtain β¨p, h2, k, rflβ© : β p : β, p.Prime β§ p β£ m :=
β¨m.minFac, Nat.minFac_prime h.ne.symm, m.minFac_dvdβ© | case neg
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
h1 : Β¬2 β£ m
β’ m.Prime | case neg.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * k
h0 : β (n : β), 2 * n β€ p * k β n β£ n.choose (p * k - 2 * n)
h1 : Β¬2 β£ p * k
β’ (p * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
m : β
h : 1 < m
h0 : β (n : β), 2 * n β€ m β n β£ n.choose (m - 2 * n)
h1 : Β¬2 β£ m
β’ m.Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.two_dvd_ne_zero, β Nat.odd_iff, Nat.odd_mul] at h1 | case neg.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * k
h0 : β (n : β), 2 * n β€ p * k β n β£ n.choose (p * k - 2 * n)
h1 : Β¬2 β£ p * k
β’ (p * k).Prime | case neg.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * k
h0 : β (n : β), 2 * n β€ p * k β n β£ n.choose (p * k - 2 * n)
h1 : Odd p β§ Odd k
β’ (p * k).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * k
h0 : β (n : β), 2 * n β€ p * k β n β£ n.choose (p * k - 2 * n)
h1 : Β¬2 β£ p * k
β’ (p * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h1 with β¨-, k, rflβ© | case neg.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * k
h0 : β (n : β), 2 * n β€ p * k β n β£ n.choose (p * k - 2 * n)
h1 : Odd p β§ Odd k
β’ (p * k).Prime | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
β’ (p * (2 * k + 1)).Prime | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * k
h0 : β (n : β), 2 * n β€ p * k β n β£ n.choose (p * k - 2 * n)
h1 : Odd p β§ Odd k
β’ (p * k).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | suffices k = 0 by rwa [this, mul_zero, zero_add, mul_one] | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
β’ (p * (2 * k + 1)).Prime | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
β’ k = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
β’ (p * (2 * k + 1)).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | by_contra h1 | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
β’ k = 0 | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
h1 : Β¬k = 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
β’ k = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | apply prime_binom_not_dvd h2 h1 | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
h1 : Β¬k = 0
β’ False | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
h1 : Β¬k = 0
β’ p * k β£ (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
h1 : Β¬k = 0
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | specialize h0 (p * k) | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
h1 : Β¬k = 0
β’ p * k β£ (p * k).choose p | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h1 : Β¬k = 0
h0 : 2 * (p * k) β€ p * (2 * k + 1) β p * k β£ (p * k).choose (p * (2 * k + 1) - 2 * (p * k))
β’ p * k β£ (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
h1 : Β¬k = 0
β’ p * k β£ (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [mul_add_one (Ξ± := β), mul_left_comm, Nat.add_sub_cancel_left] at h0 | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h1 : Β¬k = 0
h0 : 2 * (p * k) β€ p * (2 * k + 1) β p * k β£ (p * k).choose (p * (2 * k + 1) - 2 * (p * k))
β’ p * k β£ (p * k).choose p | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h1 : Β¬k = 0
h0 : p * (2 * k) β€ p * (2 * k) + p β p * k β£ (p * k).choose p
β’ p * k β£ (p * k).choose p | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h1 : Β¬k = 0
h0 : 2 * (p * k) β€ p * (2 * k + 1) β p * k β£ (p * k).choose (p * (2 * k + 1) - 2 * (p * k))
β’ p * k β£ (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact h0 (Nat.le_add_right (p * (2 * k)) p) | case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h1 : Β¬k = 0
h0 : p * (2 * k) β€ p * (2 * k) + p β p * k β£ (p * k).choose p
β’ p * k β£ (p * k).choose p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h1 : Β¬k = 0
h0 : p * (2 * k) β€ p * (2 * k) + p β p * k β£ (p * k).choose p
β’ p * k β£ (p * k).choose p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rwa [this, mul_zero, zero_add, mul_one] | p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
this : k = 0
β’ (p * (2 * k + 1)).Prime | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
h2 : p.Prime
k : β
h : 1 < p * (2 * k + 1)
h0 : β (n : β), 2 * n β€ p * (2 * k + 1) β n β£ n.choose (p * (2 * k + 1) - 2 * n)
this : k = 0
β’ (p * (2 * k + 1)).Prime
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [le_iff_exists_add] at h1 | case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : 2 * n β€ m
β’ n β£ n.choose (m - 2 * n) | case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : β c, m = 2 * n + c
β’ n β£ n.choose (m - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : 2 * n β€ m
β’ n β£ n.choose (m - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h1 with β¨c, rflβ© | case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : β c, m = 2 * n + c
β’ n β£ n.choose (m - 2 * n) | case refine_2.intro
n c : β
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
β’ n β£ n.choose (2 * n + c - 2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
m : β
h : 1 < m
h0 : m.Prime
n : β
h1 : β c, m = 2 * n + c
β’ n β£ n.choose (m - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.add_sub_cancel_left] | case refine_2.intro
n c : β
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
β’ n β£ n.choose (2 * n + c - 2 * n) | case refine_2.intro
n c : β
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
β’ n β£ n.choose c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
n c : β
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
β’ n β£ n.choose (2 * n + c - 2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases c with _ | c | case refine_2.intro
n c : β
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
β’ n β£ n.choose c | case refine_2.intro.zero
n : β
h : 1 < 2 * n + 0
h0 : (2 * n + 0).Prime
β’ n β£ n.choose 0
case refine_2.intro.succ
n c : β
h : 1 < 2 * n + (c + 1)
h0 : (2 * n + (c + 1)).Prime
β’ n β£ n.choose (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
n c : β
h : 1 < 2 * n + c
h0 : (2 * n + c).Prime
β’ n β£ n.choose c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases n with _ | n | case refine_2.intro.succ
n c : β
h : 1 < 2 * n + (c + 1)
h0 : (2 * n + (c + 1)).Prime
β’ n β£ n.choose (c + 1) | case refine_2.intro.succ.zero
c : β
h : 1 < 2 * 0 + (c + 1)
h0 : (2 * 0 + (c + 1)).Prime
β’ 0 β£ Nat.choose 0 (c + 1)
case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
β’ n + 1 β£ (n + 1).choose (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ
n c : β
h : 1 < 2 * n + (c + 1)
h0 : (2 * n + (c + 1)).Prime
β’ n β£ n.choose (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [Nat.add_zero, Nat.prime_mul_iff] at h0 | case refine_2.intro.zero
n : β
h : 1 < 2 * n + 0
h0 : (2 * n + 0).Prime
β’ n β£ n.choose 0 | case refine_2.intro.zero
n : β
h : 1 < 2 * n + 0
h0 : Nat.Prime 2 β§ n = 1 β¨ n.Prime β§ 2 = 1
β’ n β£ n.choose 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.zero
n : β
h : 1 < 2 * n + 0
h0 : (2 * n + 0).Prime
β’ n β£ n.choose 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rcases h0 with β¨-, rflβ© | β¨-, h1β© | case refine_2.intro.zero
n : β
h : 1 < 2 * n + 0
h0 : Nat.Prime 2 β§ n = 1 β¨ n.Prime β§ 2 = 1
β’ n β£ n.choose 0 | case refine_2.intro.zero.inl.intro
h : 1 < 2 * 1 + 0
β’ 1 β£ Nat.choose 1 0
case refine_2.intro.zero.inr.intro
n : β
h : 1 < 2 * n + 0
h1 : 2 = 1
β’ n β£ n.choose 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.zero
n : β
h : 1 < 2 * n + 0
h0 : Nat.Prime 2 β§ n = 1 β¨ n.Prime β§ 2 = 1
β’ n β£ n.choose 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exacts [Nat.dvd_refl 1, absurd h1 (Nat.succ_ne_self 1)] | case refine_2.intro.zero.inl.intro
h : 1 < 2 * 1 + 0
β’ 1 β£ Nat.choose 1 0
case refine_2.intro.zero.inr.intro
n : β
h : 1 < 2 * n + 0
h1 : 2 = 1
β’ n β£ n.choose 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.zero.inl.intro
h : 1 < 2 * 1 + 0
β’ 1 β£ Nat.choose 1 0
case refine_2.intro.zero.inr.intro
n : β
h : 1 < 2 * n + 0
h1 : 2 = 1
β’ n β£ n.choose 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact Nat.dvd_refl 0 | case refine_2.intro.succ.zero
c : β
h : 1 < 2 * 0 + (c + 1)
h0 : (2 * 0 + (c + 1)).Prime
β’ 0 β£ Nat.choose 0 (c + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.zero
c : β
h : 1 < 2 * 0 + (c + 1)
h0 : (2 * 0 + (c + 1)).Prime
β’ 0 β£ Nat.choose 0 (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | refine (Nat.Coprime.dvd_mul_right ?_).mp
β¨_, (n.succ_mul_choose_eq c).symmβ© | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
β’ n + 1 β£ (n + 1).choose (c + 1) | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
β’ (n + 1).Coprime c.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
β’ n + 1 β£ (n + 1).choose (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [two_mul, add_assoc] at h0 | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
β’ (n + 1).Coprime c.succ | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ (n + 1).Coprime c.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (2 * (n + 1) + (c + 1)).Prime
β’ (n + 1).Coprime c.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | rw [β Nat.coprime_self_add_right, β Nat.coprime_self_add_right,
Nat.coprime_comm, h0.coprime_iff_not_dvd] | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ (n + 1).Coprime c.succ | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ Β¬n + 1 + (n + 1 + (c + 1)) β£ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ (n + 1).Coprime c.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N3/N3.lean | IMOSL.IMO2012N3.final_solution | [38, 1] | [77, 52] | exact Nat.not_dvd_of_pos_of_lt n.succ_pos <|
lt_add_of_pos_right _ (n.succ + c).succ_pos | case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ Β¬n + 1 + (n + 1 + (c + 1)) β£ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.succ.succ
c n : β
h : 1 < 2 * (n + 1) + (c + 1)
h0 : (n + 1 + (n + 1 + (c + 1))).Prime
β’ Β¬n + 1 + (n + 1 + (c + 1)) β£ n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.and_and_or_not_iff | [36, 1] | [39, 48] | rw [and_assoc, and_or_left, and_not_self_iff,
or_false_iff, and_assoc, and_comm (b := Q)] | P Q R : Prop
β’ (P β§ Q) β§ (R β¨ Β¬Q) β (P β§ R) β§ Q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P Q R : Prop
β’ (P β§ Q) β§ (R β¨ Β¬Q) β (P β§ R) β§ Q
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.Iic_filter_dvd_card | [41, 1] | [47, 45] | let h := (k + (n + 1)).card_multiples (n + 1) | k n : β
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1 | k n : β
h : (filter (fun e => n + 1 β£ e + 1) (range (k + (n + 1)))).card = (k + (n + 1)) / (n + 1) :=
Nat.card_multiples (k + (n + 1)) (n + 1)
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
k n : β
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.Iic_filter_dvd_card | [41, 1] | [47, 45] | rwa [range_eq_Ico, β Ico_union_Ico_eq_Ico k.zero_le le_self_add, filter_union,
card_union_of_disjoint (disjoint_filter_filter <| Ico_disjoint_Ico_consecutive 0 k _),
β range_eq_Ico, Nat.card_multiples, k.add_div_right n.succ_pos,
Nat.succ_eq_add_one, add_right_inj] at h | k n : β
h : (filter (fun e => n + 1 β£ e + 1) (range (k + (n + 1)))).card = (k + (n + 1)) / (n + 1) :=
Nat.card_multiples (k + (n + 1)) (n + 1)
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k n : β
h : (filter (fun e => n + 1 β£ e + 1) (range (k + (n + 1)))).card = (k + (n + 1)) / (n + 1) :=
Nat.card_multiples (k + (n + 1)) (n + 1)
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.symmDiff_card_add_two_mul_inter_card | [61, 1] | [64, 62] | rw [two_mul, β add_assoc, β card_union_of_disjoint (disjoint_symmDiff_inter A B),
symmDiff_union_inter_eq_union, card_union_add_card_inter] | Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card + 2 * (A β© B).card = A.card + B.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card + 2 * (A β© B).card = A.card + B.card
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.symmDiff_card_mod_two | [66, 1] | [67, 73] | rw [β symmDiff_card_add_two_mul_inter_card, Nat.add_mul_mod_self_left] | Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card % 2 = (A.card + B.card) % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card % 2 = (A.card + B.card) % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.filter_symmDiff | [74, 1] | [78, 60] | rw [mem_filter, mem_symmDiff, mem_symmDiff,
mem_filter, mem_filter, not_and_or, not_and_or,
and_and_or_not_iff, and_and_or_not_iff, β or_and_right] | Ξ± : Type u_1
instβΒΉ : DecidableEq Ξ±
A B : Finset Ξ±
p : Ξ± β Prop
instβ : DecidablePred p
x : Ξ±
β’ x β filter p (symmDiff A B) β x β symmDiff (filter p A) (filter p B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβΒΉ : DecidableEq Ξ±
A B : Finset Ξ±
p : Ξ± β Prop
instβ : DecidablePred p
x : Ξ±
β’ x β filter p (symmDiff A B) β x β symmDiff (filter p A) (filter p B)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.Ends_iff | [126, 1] | [132, 47] | rw [mem_range, β not_le] | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ i β range n | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ Β¬n β€ i | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ i β range n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.Ends_iff | [126, 1] | [132, 47] | refine Ξ» h1 β¦ h _ <| ValidMove.flip (i - n) ?_ | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ Β¬n β€ i | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
h1 : n β€ i
β’ i - n + n β X.board | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ Β¬n β€ i
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.Ends_iff | [126, 1] | [132, 47] | rwa [Nat.sub_add_cancel h1] | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
h1 : n β€ i
β’ i - n + n β X.board | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
h1 : n β€ i
β’ i - n + n β X.board
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rcases h with β¨i, hβ© | n : β
X Y : GameState n
h : X.ValidMove Y
β’ { ofColex := Y.board } < { ofColex := X.board } | case flip
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { ofColex := { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board } <
{ ofColex := X.board } | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X Y : GameState n
h : X.ValidMove Y
β’ { ofColex := Y.board } < { ofColex := X.board }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | refine Colex.toColex_lt_toColex.mpr β¨?_, Ξ» j h0 h1 β¦ ?_β© | case flip
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { ofColex := { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board } <
{ ofColex := X.board } | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β X.board
case flip.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ β b β X.board, b β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β§ j β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case flip
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { ofColex := { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board } <
{ ofColex := X.board }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rw [Ne, symmDiff_eq_left] | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β X.board | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ Β¬Icc i (i + n) = β₯ | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β X.board
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | exact ne_empty_of_mem (left_mem_Icc.mpr le_self_add) | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ Β¬Icc i (i + n) = β₯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ Β¬Icc i (i + n) = β₯
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | refine β¨i + n, h, Ξ» h2 β¦ ?_, ?_β© | case flip.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ β b β X.board, b β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β§ j β€ b | case flip.refine_2.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
h2 : i + n β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
β’ False
case flip.refine_2.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ j β€ i + n | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ β b β X.board, b β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β§ j β€ b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rw [mem_symmDiff, mem_Icc] at h2 | case flip.refine_2.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
h2 : i + n β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
β’ False | case flip.refine_2.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
h2 : i + n β X.board β§ Β¬(i β€ i + n β§ i + n β€ i + n) β¨ (i β€ i + n β§ i + n β€ i + n) β§ i + n β X.board
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
h2 : i + n β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | exact h2.elim (Ξ» h2 β¦ h2.2 β¨le_self_add, (i + n).le_reflβ©) (Ξ» h2 β¦ h2.2 h) | case flip.refine_2.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
h2 : i + n β X.board β§ Β¬(i β€ i + n β§ i + n β€ i + n) β¨ (i β€ i + n β§ i + n β€ i + n) β§ i + n β X.board
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
h2 : i + n β X.board β§ Β¬(i β€ i + n β§ i + n β€ i + n) β¨ (i β€ i + n β§ i + n β€ i + n) β§ i + n β X.board
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rw [mem_symmDiff, or_iff_right (Ξ» h2 β¦ h1 h2.1), mem_Icc] at h0 | case flip.refine_2.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ j β€ i + n | case flip.refine_2.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : (i β€ j β§ j β€ i + n) β§ j β X.board
h1 : j β X.board
β’ j β€ i + n | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ j β€ i + n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | exact h0.1.2 | case flip.refine_2.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : (i β€ j β§ j β€ i + n) β§ j β X.board
h1 : j β X.board
β’ j β€ i + n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : (i β€ j β§ j β€ i + n) β§ j β X.board
h1 : j β X.board
β’ j β€ i + n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.isReachable_sum_two_pow_add_numMoves | [155, 1] | [162, 60] | rw [ValidMove_numMoves h0, β add_assoc, Nat.succ_le_iff] | n : β
X Y : GameState n
h : X.IsReachable Y
aβ cβ : GameState n
h0 : aβ.ValidMove cβ
xβ : ReflTransGen ValidMove cβ Y
h1 : β i β Y.board, 2 ^ i + Y.numMoves β€ β i β cβ.board, 2 ^ i + cβ.numMoves
β’ β i β cβ.board, 2 ^ i + cβ.numMoves β€ β i β aβ.board, 2 ^ i + aβ.numMoves | n : β
X Y : GameState n
h : X.IsReachable Y
aβ cβ : GameState n
h0 : aβ.ValidMove cβ
xβ : ReflTransGen ValidMove cβ Y
h1 : β i β Y.board, 2 ^ i + Y.numMoves β€ β i β cβ.board, 2 ^ i + cβ.numMoves
β’ β i β cβ.board, 2 ^ i + aβ.numMoves < β i β aβ.board, 2 ^ i + aβ.numMoves | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X Y : GameState n
h : X.IsReachable Y
aβ cβ : GameState n
h0 : aβ.ValidMove cβ
xβ : ReflTransGen ValidMove cβ Y
h1 : β i β Y.board, 2 ^ i + Y.numMoves β€ β i β cβ.board, 2 ^ i + cβ.numMoves
β’ β i β cβ.board, 2 ^ i + cβ.numMoves β€ β i β aβ.board, 2 ^ i + aβ.numMoves
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.isReachable_sum_two_pow_add_numMoves | [155, 1] | [162, 60] | exact Nat.add_lt_add_right (ValidMove_sum_two_pow h0) _ | n : β
X Y : GameState n
h : X.IsReachable Y
aβ cβ : GameState n
h0 : aβ.ValidMove cβ
xβ : ReflTransGen ValidMove cβ Y
h1 : β i β Y.board, 2 ^ i + Y.numMoves β€ β i β cβ.board, 2 ^ i + cβ.numMoves
β’ β i β cβ.board, 2 ^ i + aβ.numMoves < β i β aβ.board, 2 ^ i + aβ.numMoves | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X Y : GameState n
h : X.IsReachable Y
aβ cβ : GameState n
h0 : aβ.ValidMove cβ
xβ : ReflTransGen ValidMove cβ Y
h1 : β i β Y.board, 2 ^ i + Y.numMoves β€ β i β cβ.board, 2 ^ i + cβ.numMoves
β’ β i β cβ.board, 2 ^ i + aβ.numMoves < β i β aβ.board, 2 ^ i + aβ.numMoves
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_central_card_mod_two | [178, 1] | [183, 50] | rw [centralCards, filter_symmDiff, β centralCards,
symmDiff_card_mod_two, Iic_filter_dvd_card] | n : β
X Y : GameState n
h : X.ValidMove Y
i : β
xβ : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.centralCards.card % 2 =
(X.centralCards.card + 1) % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X Y : GameState n
h : X.ValidMove Y
i : β
xβ : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.centralCards.card % 2 =
(X.centralCards.card + 1) % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.isReachable_central_card_add_numMoves_mod_two | [185, 1] | [192, 58] | rw [ValidMove_numMoves h0, Nat.add_mod, ValidMove_central_card_mod_two h0,
β Nat.add_mod, add_add_add_comm, Nat.add_mod_right] | n : β
X Y : GameState n
h : X.IsReachable Y
aβ cβ : GameState n
h0 : aβ.ValidMove cβ
xβ : ReflTransGen ValidMove cβ Y
h1 : (Y.centralCards.card + Y.numMoves) % 2 = (cβ.centralCards.card + cβ.numMoves) % 2
β’ (cβ.centralCards.card + cβ.numMoves) % 2 = (aβ.centralCards.card + aβ.numMoves) % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X Y : GameState n
h : X.IsReachable Y
aβ cβ : GameState n
h0 : aβ.ValidMove cβ
xβ : ReflTransGen ValidMove cβ Y
h1 : (Y.centralCards.card + Y.numMoves) % 2 = (cβ.centralCards.card + cβ.numMoves) % 2
β’ (cβ.centralCards.card + cβ.numMoves) % 2 = (aβ.centralCards.card + aβ.numMoves) % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.numMoves_mod_two_eq_div_of_ends | [205, 1] | [210, 60] | have h1 := isReachable_central_card_add_numMoves_mod_two h | M n : β
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
β’ X.numMoves % 2 = M / n.succ % 2 | M n : β
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
h1 : (X.centralCards.card + X.numMoves) % 2 = ((init M n).centralCards.card + (init M n).numMoves) % 2
β’ X.numMoves % 2 = M / n.succ % 2 | Please generate a tactic in lean4 to solve the state.
STATE:
M n : β
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
β’ X.numMoves % 2 = M / n.succ % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.numMoves_mod_two_eq_div_of_ends | [205, 1] | [210, 60] | rwa [filter_central_init_card, numMoves_init, Nat.add_zero,
filter_central_ends h0, card_empty, Nat.zero_add] at h1 | M n : β
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
h1 : (X.centralCards.card + X.numMoves) % 2 = ((init M n).centralCards.card + (init M n).numMoves) % 2
β’ X.numMoves % 2 = M / n.succ % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M n : β
X : GameState n
h : (init M n).IsReachable X
h0 : X.Ends
h1 : (X.centralCards.card + X.numMoves) % 2 = ((init M n).centralCards.card + (init M n).numMoves) % 2
β’ X.numMoves % 2 = M / n.succ % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.and_and_or_not_iff | [36, 1] | [39, 48] | rw [and_assoc, and_or_left, and_not_self_iff,
or_false_iff, and_assoc, and_comm (b := Q)] | P Q R : Prop
β’ (P β§ Q) β§ (R β¨ Β¬Q) β (P β§ R) β§ Q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P Q R : Prop
β’ (P β§ Q) β§ (R β¨ Β¬Q) β (P β§ R) β§ Q
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.Iic_filter_dvd_card | [41, 1] | [47, 45] | let h := (k + (n + 1)).card_multiples (n + 1) | k n : β
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1 | k n : β
h : (filter (fun e => n + 1 β£ e + 1) (range (k + (n + 1)))).card = (k + (n + 1)) / (n + 1) :=
Nat.card_multiples (k + (n + 1)) (n + 1)
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
k n : β
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.Iic_filter_dvd_card | [41, 1] | [47, 45] | rwa [range_eq_Ico, β Ico_union_Ico_eq_Ico k.zero_le le_self_add, filter_union,
card_union_of_disjoint (disjoint_filter_filter <| Ico_disjoint_Ico_consecutive 0 k _),
β range_eq_Ico, Nat.card_multiples, k.add_div_right n.succ_pos,
Nat.succ_eq_add_one, add_right_inj] at h | k n : β
h : (filter (fun e => n + 1 β£ e + 1) (range (k + (n + 1)))).card = (k + (n + 1)) / (n + 1) :=
Nat.card_multiples (k + (n + 1)) (n + 1)
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k n : β
h : (filter (fun e => n + 1 β£ e + 1) (range (k + (n + 1)))).card = (k + (n + 1)) / (n + 1) :=
Nat.card_multiples (k + (n + 1)) (n + 1)
β’ (filter (fun i => n + 1 β£ i + 1) (Icc k (k + n))).card = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.symmDiff_card_add_two_mul_inter_card | [61, 1] | [64, 62] | rw [two_mul, β add_assoc, β card_union_of_disjoint (disjoint_symmDiff_inter A B),
symmDiff_union_inter_eq_union, card_union_add_card_inter] | Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card + 2 * (A β© B).card = A.card + B.card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card + 2 * (A β© B).card = A.card + B.card
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.symmDiff_card_mod_two | [66, 1] | [67, 73] | rw [β symmDiff_card_add_two_mul_inter_card, Nat.add_mul_mod_self_left] | Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card % 2 = (A.card + B.card) % 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβ : DecidableEq Ξ±
A B : Finset Ξ±
β’ (symmDiff A B).card % 2 = (A.card + B.card) % 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.filter_symmDiff | [74, 1] | [78, 60] | rw [mem_filter, mem_symmDiff, mem_symmDiff,
mem_filter, mem_filter, not_and_or, not_and_or,
and_and_or_not_iff, and_and_or_not_iff, β or_and_right] | Ξ± : Type u_1
instβΒΉ : DecidableEq Ξ±
A B : Finset Ξ±
p : Ξ± β Prop
instβ : DecidablePred p
x : Ξ±
β’ x β filter p (symmDiff A B) β x β symmDiff (filter p A) (filter p B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
instβΒΉ : DecidableEq Ξ±
A B : Finset Ξ±
p : Ξ± β Prop
instβ : DecidablePred p
x : Ξ±
β’ x β filter p (symmDiff A B) β x β symmDiff (filter p A) (filter p B)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.Ends_iff | [126, 1] | [132, 47] | rw [mem_range, β not_le] | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ i β range n | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ Β¬n β€ i | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ i β range n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.Ends_iff | [126, 1] | [132, 47] | refine Ξ» h1 β¦ h _ <| ValidMove.flip (i - n) ?_ | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ Β¬n β€ i | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
h1 : n β€ i
β’ i - n + n β X.board | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
β’ Β¬n β€ i
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.Ends_iff | [126, 1] | [132, 47] | rwa [Nat.sub_add_cancel h1] | n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
h1 : n β€ i
β’ i - n + n β X.board | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X : GameState n
h : X.Ends
i : β
h0 : i β X.board
h1 : n β€ i
β’ i - n + n β X.board
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rcases h with β¨i, hβ© | n : β
X Y : GameState n
h : X.ValidMove Y
β’ { ofColex := Y.board } < { ofColex := X.board } | case flip
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { ofColex := { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board } <
{ ofColex := X.board } | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
X Y : GameState n
h : X.ValidMove Y
β’ { ofColex := Y.board } < { ofColex := X.board }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | refine Colex.toColex_lt_toColex.mpr β¨?_, Ξ» j h0 h1 β¦ ?_β© | case flip
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { ofColex := { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board } <
{ ofColex := X.board } | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β X.board
case flip.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ β b β X.board, b β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β§ j β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case flip
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { ofColex := { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board } <
{ ofColex := X.board }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | rw [Ne, symmDiff_eq_left] | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β X.board | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ Β¬Icc i (i + n) = β₯ | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β X.board
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | exact ne_empty_of_mem (left_mem_Icc.mpr le_self_add) | case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ Β¬Icc i (i + n) = β₯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
β’ Β¬Icc i (i + n) = β₯
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/C1/C1.lean | IMOSL.IMO2009C1.GameState.ValidMove_Colex | [138, 1] | [148, 19] | refine β¨i + n, h, Ξ» h2 β¦ ?_, ?_β© | case flip.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ β b β X.board, b β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β§ j β€ b | case flip.refine_2.refine_1
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
h2 : i + n β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
β’ False
case flip.refine_2.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ j β€ i + n | Please generate a tactic in lean4 to solve the state.
STATE:
case flip.refine_2
n : β
X : GameState n
i : β
h : i + n β X.board
j : β
h0 : j β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board
h1 : j β X.board
β’ β b β X.board, b β { board := symmDiff X.board (Icc i (i + n)), numMoves := X.numMoves.succ }.board β§ j β€ b
TACTIC:
|
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