url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | simp_rw [not_forall] at h | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : Β¬β β¦aβ aβ : ββ¦, f aβ = f aβ β aβ = aβ
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : β x x_1, β (_ : f x = f x_1), Β¬x = x_1
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : Β¬β β¦aβ aβ : ββ¦, f aβ = f aβ β aβ = aβ
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | rcases h with β¨a, b, h, h0β© | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : β x x_1, β (_ : f x = f x_1), Β¬x = x_1
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : β x x_1, β (_ : f x = f x_1), Β¬x = x_1
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | refine β¨a.succ, b.succ, Nat.succ_ne_succ.mpr h0, Ξ» k h1 β¦ ?_β© | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
β’ Even (Ξ© ((a.succ + k) * (b.succ + k))) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | have X (c : β) : c.succ + k β 0 := c.succ_add k βΈ (c + k).succ_ne_zero | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
β’ Even (Ξ© ((a.succ + k) * (b.succ + k))) | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
X : β (c : β), c.succ + k β 0
β’ Even (Ξ© ((a.succ + k) * (b.succ + k))) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
β’ Even (Ξ© ((a.succ + k) * (b.succ + k)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | rw [Even_iff_bodd, cardFactors_mul (X _) (X _),
Nat.bodd_add, xor_eq_false_iff_eq] | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
X : β (c : β), c.succ + k β 0
β’ Even (Ξ© ((a.succ + k) * (b.succ + k))) | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
X : β (c : β), c.succ + k β 0
β’ (Ξ© (a.succ + k)).bodd = (Ξ© (b.succ + k)).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
X : β (c : β), c.succ + k β 0
β’ Even (Ξ© ((a.succ + k) * (b.succ + k)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | exact congr_fun h β¨k, h1β© | case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
X : β (c : β), c.succ + k β 0
β’ (Ξ© (a.succ + k)).bodd = (Ξ© (b.succ + k)).bodd | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
a b : β
h : f a = f b
h0 : Β¬a = b
k : β
h1 : k < N
X : β (c : β), c.succ + k β 0
β’ (Ξ© (a.succ + k)).bodd = (Ξ© (b.succ + k)).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.exists_lt_omega_bodd_ne_succ | [63, 1] | [71, 10] | by_contra h | a : β
β’ β b, a β€ b β§ (Ξ© b).bodd β (Ξ© b.succ).bodd | a : β
h : Β¬β b, a β€ b β§ (Ξ© b).bodd β (Ξ© b.succ).bodd
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
β’ β b, a β€ b β§ (Ξ© b).bodd β (Ξ© b.succ).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.exists_lt_omega_bodd_ne_succ | [63, 1] | [71, 10] | rw [not_exists] at h | a : β
h : Β¬β b, a β€ b β§ (Ξ© b).bodd β (Ξ© b.succ).bodd
β’ False | a : β
h : β (x : β), Β¬(a β€ x β§ (Ξ© x).bodd β (Ξ© x.succ).bodd)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
h : Β¬β b, a β€ b β§ (Ξ© b).bodd β (Ξ© b.succ).bodd
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.exists_lt_omega_bodd_ne_succ | [63, 1] | [71, 10] | simp_rw [not_and, not_not] at h | a : β
h : β (x : β), Β¬(a β€ x β§ (Ξ© x).bodd β (Ξ© x.succ).bodd)
β’ False | a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
h : β (x : β), Β¬(a β€ x β§ (Ξ© x).bodd β (Ξ© x.succ).bodd)
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.exists_lt_omega_bodd_ne_succ | [63, 1] | [71, 10] | rcases a.exists_infinite_primes with β¨p, h0, h1β© | a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
β’ False | case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.exists_lt_omega_bodd_ne_succ | [63, 1] | [71, 10] | apply absurd (eventually_const_of_map_succ_eq h
p (p * p) h0 (h0.trans (Nat.le_mul_self p))) | case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ False | case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ Β¬(Ξ© p).bodd = (Ξ© (p * p)).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.exists_lt_omega_bodd_ne_succ | [63, 1] | [71, 10] | rw [cardFactors_apply_prime h1, β sq, cardFactors_apply_prime_pow h1] | case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ Β¬(Ξ© p).bodd = (Ξ© (p * p)).bodd | case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ Β¬Nat.bodd 1 = Nat.bodd 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ Β¬(Ξ© p).bodd = (Ξ© (p * p)).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.exists_lt_omega_bodd_ne_succ | [63, 1] | [71, 10] | trivial | case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ Β¬Nat.bodd 1 = Nat.bodd 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a : β
h : β (x : β), a β€ x β (Ξ© x).bodd = (Ξ© x.succ).bodd
p : β
h0 : a β€ p
h1 : p.Prime
β’ Β¬Nat.bodd 1 = Nat.bodd 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | wlog h0 : a β€ b | a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
β’ a = b | case inr
a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
this : β {a b : β}, (β (k : β), Even (Ξ© ((a + k) * (b + k)))) β a β€ b β a = b
h0 : Β¬a β€ b
β’ a = b
a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
h0 : a β€ b
β’ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
β’ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | rw [le_iff_exists_add] at h0 | a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
h0 : a β€ b
β’ a = b | a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
h0 : β c, b = a + c
β’ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
h0 : a β€ b
β’ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | rcases h0 with β¨_ | c, rflβ© | a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
h0 : β c, b = a + c
β’ a = b | case intro.zero
b : β
h : β (k : β), Even (Ξ© ((b + k) * (b + k)))
β’ b = b
case intro.succ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
β’ a = a + (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
h0 : β c, b = a + c
β’ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | rfl | case intro.zero
b : β
h : β (k : β), Even (Ξ© ((b + k) * (b + k)))
β’ b = b
case intro.succ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
β’ a = a + (c + 1) | case intro.succ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
β’ a = a + (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.zero
b : β
h : β (k : β), Even (Ξ© ((b + k) * (b + k)))
β’ b = b
case intro.succ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
β’ a = a + (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | rcases exists_lt_omega_bodd_ne_succ a.succ with β¨b, h0, h1β© | case intro.succ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
β’ a = a + (c + 1) | case intro.succ.intro.intro
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
h1 : (Ξ© b).bodd β (Ξ© b.succ).bodd
β’ a = a + (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
β’ a = a + (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | revert h1 | case intro.succ.intro.intro
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
h1 : (Ξ© b).bodd β (Ξ© b.succ).bodd
β’ a = a + (c + 1) | case intro.succ.intro.intro
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
β’ (Ξ© b).bodd β (Ξ© b.succ).bodd β a = a + (c + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
h1 : (Ξ© b).bodd β (Ξ© b.succ).bodd
β’ a = a + (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | apply absurd | case intro.succ.intro.intro
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
β’ (Ξ© b).bodd β (Ξ© b.succ).bodd β a = a + (c + 1) | case intro.succ.intro.intro.hβ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
β’ (Ξ© b).bodd β (Ξ© b.succ).bodd β a = a + (c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | specialize h (a * c + (b - a) * c.succ) | case intro.succ.intro.intro.hβ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | case intro.succ.intro.intro.hβ
a c b : β
h0 : a.succ β€ b
h : Even (Ξ© ((a + (a * c + (b - a) * c.succ)) * (a + (c + 1) + (a * c + (b - a) * c.succ))))
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro.hβ
a c : β
h : β (k : β), Even (Ξ© ((a + k) * (a + (c + 1) + k)))
b : β
h0 : a.succ β€ b
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | rw [Even_iff_bodd, add_right_comm, β add_assoc, a.add_comm, β Nat.mul_succ,
β add_mul, Nat.add_sub_of_le (a.le_succ.trans h0), β Nat.succ_mul] at h | case intro.succ.intro.intro.hβ
a c b : β
h0 : a.succ β€ b
h : Even (Ξ© ((a + (a * c + (b - a) * c.succ)) * (a + (c + 1) + (a * c + (b - a) * c.succ))))
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | case intro.succ.intro.intro.hβ
a c b : β
h0 : a.succ β€ b
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro.hβ
a c b : β
h0 : a.succ β€ b
h : Even (Ξ© ((a + (a * c + (b - a) * c.succ)) * (a + (c + 1) + (a * c + (b - a) * c.succ))))
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | replace h0 := (Nat.zero_lt_of_lt h0).ne.symm | case intro.succ.intro.intro.hβ
a c b : β
h0 : a.succ β€ b
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro.hβ
a c b : β
h0 : a.succ β€ b
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | have h1 := b.succ_ne_zero | case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
h1 : b.succ β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | have h2 := c.succ_ne_zero | case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
h1 : b.succ β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
h1 : b.succ β 0
h2 : c.succ β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
h1 : b.succ β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | rwa [cardFactors_mul (Nat.mul_ne_zero h0 h2) (Nat.mul_ne_zero h1 h2),
cardFactors_mul h0 h2, cardFactors_mul h1 h2, Nat.bodd_add,
xor_eq_false_iff_eq, Nat.bodd_add, Nat.bodd_add, Bool.xor_right_inj] at h | case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
h1 : b.succ β 0
h2 : c.succ β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ.intro.intro.hβ
a c b : β
h : (Ξ© (b * c.succ * (b.succ * c.succ))).bodd = false
h0 : b β 0
h1 : b.succ β 0
h2 : c.succ β 0
β’ (Ξ© b).bodd = (Ξ© b.succ).bodd
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part2 | [74, 1] | [89, 78] | exact (this (Ξ» k β¦ Nat.mul_comm _ _ βΈ h k) (Nat.le_of_not_ge h0)).symm | case inr
a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
this : β {a b : β}, (β (k : β), Even (Ξ© ((a + k) * (b + k)))) β a β€ b β a = b
h0 : Β¬a β€ b
β’ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
a b : β
h : β (k : β), Even (Ξ© ((a + k) * (b + k)))
this : β {a b : β}, (β (k : β), Even (Ξ© ((a + k) * (b + k)))) β a β€ b β a = b
h0 : Β¬a β€ b
β’ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_is_good | [22, 1] | [24, 33] | rw [sub_one_mul, mul_sub_one, sub_sub, β add_sub_assoc x, sub_add_cancel] | R : Type u_1
instβ : NonAssocRing R
x y : R
β’ (fun x => x - 1) (x * y + 1) = (fun x => x - 1) x * (fun x => x - 1) y + (fun x => x - 1) (x + y) | R : Type u_1
instβ : NonAssocRing R
x y : R
β’ (fun x => x - 1) (x * y + 1) = x * y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : NonAssocRing R
x y : R
β’ (fun x => x - 1) (x * y + 1) = (fun x => x - 1) x * (fun x => x - 1) y + (fun x => x - 1) (x + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_is_good | [22, 1] | [24, 33] | exact add_sub_cancel_right _ _ | R : Type u_1
instβ : NonAssocRing R
x y : R
β’ (fun x => x - 1) (x * y + 1) = x * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : NonAssocRing R
x y : R
β’ (fun x => x - 1) (x * y + 1) = x * y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | suffices β x y, f (x + y) = f x + f y + 1 by
have h0 (x y) : f (x + y + 1) = f (x + 1) + f (y + 1) := by
rw [h, h, h, this, add_assoc, add_add_add_comm]
have h1 (x y) : f (x * y + 1) = f (x + 1) * f (y + 1) := by
rw [hf.is_good, h, h, this, add_assoc, β add_assoc,
β mul_add_one (f x), β add_one_mul (f x)]
exact β¨β¨β¨β¨Ξ» x β¦ f (x + 1), (h 1).trans <| by rw [hf.map_one, zero_add]β©, h1β©,
((h 0).trans hf.map_zero_add_one), h0β©,
Ξ» x β¦ congrArg f (sub_add_cancel x 1).symmβ© | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
β’ β Ο, β (x : R), f x = Ο (x - 1) | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
β’ β (x y : R), f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
β’ β Ο, β (x : R), f x = Ο (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h0 (x y) : f (x * y) + 1 = f x * f y + f (x + y) :=
(h _).symm.trans (hf.is_good x y) | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
β’ β (x y : R), f (x + y) = f x + f y + 1 | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
β’ β (x y : R), f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
β’ β (x y : R), f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h1 (x y) : f (x * y) = f x * f y + f (x + y) - 1 :=
eq_sub_of_add_eq (h0 x y) | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
β’ β (x y : R), f (x + y) = f x + f y + 1 | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
β’ β (x y : R), f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
β’ β (x y : R), f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h2 (x y) : f ((x + 1) * y) = f (x * y) + (f y + 1) := by
rw [h1, h, add_right_comm, add_one_mul (f x), h, add_add_add_comm,
β h0, add_right_comm, add_sub_cancel_right] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
β’ β (x y : R), f (x + y) = f x + f y + 1 | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
β’ β (x y : R), f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
β’ β (x y : R), f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | intro x y | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
β’ β (x y : R), f (x + y) = f x + f y + 1 | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
β’ f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
β’ β (x y : R), f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h3 : f (x * ((x + 1) * y)) = f ((x + 1) * (x * y)) := by
rw [add_one_mul x, add_one_mul x, mul_add] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
β’ f (x + y) = f x + f y + 1 | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
h3 : f (x * ((x + 1) * y)) = f ((x + 1) * (x * y))
β’ f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
β’ f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rwa [h2, h1, h2, mul_add, β add_assoc, add_right_comm _ _ 1, h0, β h,
sub_eq_iff_eq_add, add_assoc, add_assoc, add_assoc, add_right_inj,
β add_left_inj 1, add_assoc, β h, add_right_comm, β mul_one_add (x + 1),
h2, add_left_comm, mul_one_add x, add_assoc, add_right_inj, add_comm 1,
h, mul_add_one (f x), h0, add_assoc, add_assoc (_ * _), add_right_inj,
β add_assoc, add_left_inj, β add_assoc, eq_comm] at h3 | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
h3 : f (x * ((x + 1) * y)) = f ((x + 1) * (x * y))
β’ f (x + y) = f x + f y + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
h3 : f (x * ((x + 1) * y)) = f ((x + 1) * (x * y))
β’ f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h0 (x y) : f (x + y + 1) = f (x + 1) + f (y + 1) := by
rw [h, h, h, this, add_assoc, add_add_add_comm] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
β’ β Ο, β (x : R), f x = Ο (x - 1) | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
β’ β Ο, β (x : R), f x = Ο (x - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
β’ β Ο, β (x : R), f x = Ο (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h1 (x y) : f (x * y + 1) = f (x + 1) * f (y + 1) := by
rw [hf.is_good, h, h, this, add_assoc, β add_assoc,
β mul_add_one (f x), β add_one_mul (f x)] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
β’ β Ο, β (x : R), f x = Ο (x - 1) | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : β (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
β’ β Ο, β (x : R), f x = Ο (x - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
β’ β Ο, β (x : R), f x = Ο (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | exact β¨β¨β¨β¨Ξ» x β¦ f (x + 1), (h 1).trans <| by rw [hf.map_one, zero_add]β©, h1β©,
((h 0).trans hf.map_zero_add_one), h0β©,
Ξ» x β¦ congrArg f (sub_add_cancel x 1).symmβ© | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : β (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
β’ β Ο, β (x : R), f x = Ο (x - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : β (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
β’ β Ο, β (x : R), f x = Ο (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [h, h, h, this, add_assoc, add_add_add_comm] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
x y : R
β’ f (x + y + 1) = f (x + 1) + f (y + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
x y : R
β’ f (x + y + 1) = f (x + 1) + f (y + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [hf.is_good, h, h, this, add_assoc, β add_assoc,
β mul_add_one (f x), β add_one_mul (f x)] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
x y : R
β’ f (x * y + 1) = f (x + 1) * f (y + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
x y : R
β’ f (x * y + 1) = f (x + 1) * f (y + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [hf.map_one, zero_add] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : β (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
β’ f 1 + 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
this : β (x y : R), f (x + y) = f x + f y + 1
h0 : β (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : β (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
β’ f 1 + 1 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [h1, h, add_right_comm, add_one_mul (f x), h, add_add_add_comm,
β h0, add_right_comm, add_sub_cancel_right] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
x y : R
β’ f ((x + 1) * y) = f (x * y) + (f y + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
x y : R
β’ f ((x + 1) * y) = f (x * y) + (f y + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [add_one_mul x, add_one_mul x, mul_add] | R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
β’ f (x * ((x + 1) * y)) = f ((x + 1) * (x * y)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
instβΒΉ : NonAssocRing R
S : Type u_1
instβ : NonAssocRing S
f : R β S
hf : NontrivialGood f
h : β (x : R), f (x + 1) = f x + 1
h0 : β (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : β (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : β (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
β’ f (x * ((x + 1) * y)) = f ((x + 1) * (x * y))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rw [add_zero, Ico_self] | a : β
β’ β x β Ico a (a + 0), (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0 | a : β
β’ β x β β
, (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
β’ β x β Ico a (a + 0), (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rfl | a : β
β’ β x β β
, (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
β’ β x β β
, (-1) ^ x = bif Nat.bodd 0 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rw [Nat.Ico_succ_singleton, sum_singleton] | a : β
β’ β x β Ico a (a + 1), (-1) ^ x = bif Nat.bodd 1 then (-1) ^ a else 0 | a : β
β’ (-1) ^ a = bif Nat.bodd 1 then (-1) ^ a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
β’ β x β Ico a (a + 1), (-1) ^ x = bif Nat.bodd 1 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rfl | a : β
β’ (-1) ^ a = bif Nat.bodd 1 then (-1) ^ a else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
β’ (-1) ^ a = bif Nat.bodd 1 then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | have h : a β€ a + n := a.le_add_right n | a n : β
β’ β x β Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0 | a n : β
h : a β€ a + n
β’ β x β Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a n : β
β’ β x β Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.sum_neg_one_pow_Ico | [37, 1] | [46, 74] | rw [β add_assoc, sum_Ico_succ_top (h.trans (a + n).le_succ),
pow_succ', neg_one_mul, sum_Ico_succ_top h, sum_neg_one_pow_Ico a n,
add_neg_cancel_right, Nat.bodd_add, Nat.bodd_two, Bool.xor_false] | a n : β
h : a β€ a + n
β’ β x β Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a n : β
h : a β€ a + n
β’ β x β Ico a (a + (n + 2)), (-1) ^ x = bif (n + 2).bodd then (-1) ^ a else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | rw [weight, latticeRect, sum_product] | q : (β Γ β) Γ β Γ β
β’ weight (latticeRect q) = bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | q : (β Γ β) Γ β Γ β
β’ β x β Ico q.1.1 (q.1.1 + q.2.1), β y β Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ (x, y).1 * (-1) ^ (x, y).2 =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (β Γ β) Γ β Γ β
β’ weight (latticeRect q) = bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | simp_rw [β mul_sum] | q : (β Γ β) Γ β Γ β
β’ β x β Ico q.1.1 (q.1.1 + q.2.1), β y β Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ (x, y).1 * (-1) ^ (x, y).2 =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | q : (β Γ β) Γ β Γ β
β’ β x β Ico q.1.1 (q.1.1 + q.2.1), (-1) ^ x * β i β Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ i =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (β Γ β) Γ β Γ β
β’ β x β Ico q.1.1 (q.1.1 + q.2.1), β y β Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ (x, y).1 * (-1) ^ (x, y).2 =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | rw [β sum_mul, sum_neg_one_pow_Ico, sum_neg_one_pow_Ico] | q : (β Γ β) Γ β Γ β
β’ β x β Ico q.1.1 (q.1.1 + q.2.1), (-1) ^ x * β i β Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ i =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | q : (β Γ β) Γ β Γ β
β’ ((bif q.2.1.bodd then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (β Γ β) Γ β Γ β
β’ β x β Ico q.1.1 (q.1.1 + q.2.1), (-1) ^ x * β i β Ico q.1.2 (q.1.2 + q.2.2), (-1) ^ i =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | cases q.2.1.bodd | q : (β Γ β) Γ β Γ β
β’ ((bif q.2.1.bodd then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | case false
q : (β Γ β) Γ β Γ β
β’ ((bif false then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif false && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
case true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
q : (β Γ β) Γ β Γ β
β’ ((bif q.2.1.bodd then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | exact zero_mul _ | case false
q : (β Γ β) Γ β Γ β
β’ ((bif false then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif false && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
case true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | case true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case false
q : (β Γ β) Γ β Γ β
β’ ((bif false then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif false && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
case true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | cases q.2.2.bodd | case true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0 | case true.false
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif false then (-1) ^ q.1.2 else 0) =
bif true && false then (-1) ^ (q.1.1 + q.1.2) else 0
case true.true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif true then (-1) ^ q.1.2 else 0) =
bif true && true then (-1) ^ (q.1.1 + q.1.2) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif q.2.2.bodd then (-1) ^ q.1.2 else 0) =
bif true && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight | [55, 1] | [61, 62] | exacts [mul_zero _, (pow_add _ _ _).symm] | case true.false
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif false then (-1) ^ q.1.2 else 0) =
bif true && false then (-1) ^ (q.1.1 + q.1.2) else 0
case true.true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif true then (-1) ^ q.1.2 else 0) =
bif true && true then (-1) ^ (q.1.1 + q.1.2) else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case true.false
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif false then (-1) ^ q.1.2 else 0) =
bif true && false then (-1) ^ (q.1.1 + q.1.2) else 0
case true.true
q : (β Γ β) Γ β Γ β
β’ ((bif true then (-1) ^ q.1.1 else 0) * bif true then (-1) ^ q.1.2 else 0) =
bif true && true then (-1) ^ (q.1.1 + q.1.2) else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | rw [latticeRect_weight, β Bool.and_eq_true] | q : (β Γ β) Γ β Γ β
β’ 0 < weight (latticeRect q) β (q.2.1.bodd = true β§ q.2.2.bodd = true) β§ (q.1.1 + q.1.2).bodd = false | q : (β Γ β) Γ β Γ β
β’ (0 < bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0) β
(q.2.1.bodd && q.2.2.bodd) = true β§ (q.1.1 + q.1.2).bodd = false | Please generate a tactic in lean4 to solve the state.
STATE:
q : (β Γ β) Γ β Γ β
β’ 0 < weight (latticeRect q) β (q.2.1.bodd = true β§ q.2.2.bodd = true) β§ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | cases q.2.1.bodd && q.2.2.bodd | q : (β Γ β) Γ β Γ β
β’ (0 < bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0) β
(q.2.1.bodd && q.2.2.bodd) = true β§ (q.1.1 + q.1.2).bodd = false | case false
q : (β Γ β) Γ β Γ β
β’ (0 < bif false then (-1) ^ (q.1.1 + q.1.2) else 0) β false = true β§ (q.1.1 + q.1.2).bodd = false
case true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ (q.1.1 + q.1.2) else 0) β true = true β§ (q.1.1 + q.1.2).bodd = false | Please generate a tactic in lean4 to solve the state.
STATE:
q : (β Γ β) Γ β Γ β
β’ (0 < bif q.2.1.bodd && q.2.2.bodd then (-1) ^ (q.1.1 + q.1.2) else 0) β
(q.2.1.bodd && q.2.2.bodd) = true β§ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | exact Ξ» h β¦ absurd h (le_refl 0).not_lt | case false
q : (β Γ β) Γ β Γ β
β’ (0 < bif false then (-1) ^ (q.1.1 + q.1.2) else 0) β false = true β§ (q.1.1 + q.1.2).bodd = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case false
q : (β Γ β) Γ β Γ β
β’ (0 < bif false then (-1) ^ (q.1.1 + q.1.2) else 0) β false = true β§ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | rw [neg_one_pow_eq_pow_mod_two (R := β€), Nat.mod_two_of_bodd] | case true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ (q.1.1 + q.1.2) else 0) β true = true β§ (q.1.1 + q.1.2).bodd = false | case true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif (q.1.1 + q.1.2).bodd then 1 else 0 else 0) β true = true β§ (q.1.1 + q.1.2).bodd = false | Please generate a tactic in lean4 to solve the state.
STATE:
case true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ (q.1.1 + q.1.2) else 0) β true = true β§ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | cases (q.1.1 + q.1.2).bodd | case true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif (q.1.1 + q.1.2).bodd then 1 else 0 else 0) β true = true β§ (q.1.1 + q.1.2).bodd = false | case true.false
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif false then 1 else 0 else 0) β true = true β§ false = false
case true.true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif true then 1 else 0 else 0) β true = true β§ true = false | Please generate a tactic in lean4 to solve the state.
STATE:
case true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif (q.1.1 + q.1.2).bodd then 1 else 0 else 0) β true = true β§ (q.1.1 + q.1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.latticeRect_weight_pos_imp | [63, 1] | [72, 69] | exacts [Ξ» _ β¦ β¨rfl, rflβ©, Ξ» h β¦ absurd neg_one_lt_zero h.not_lt] | case true.false
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif false then 1 else 0 else 0) β true = true β§ false = false
case true.true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif true then 1 else 0 else 0) β true = true β§ true = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case true.false
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif false then 1 else 0 else 0) β true = true β§ false = false
case true.true
q : (β Γ β) Γ β Γ β
β’ (0 < bif true then (-1) ^ bif true then 1 else 0 else 0) β true = true β§ true = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | suffices β i β I, 0 < weight (latticeRect (Q i))
from this.elim Ξ» i h3 β¦ β¨i, h3.1, latticeRect_weight_pos_imp h3.2β© | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect β Q) h
β’ β i β I, ((Q i).2.1.bodd = true β§ (Q i).2.2.bodd = true) β§ ((Q i).1.1 + (Q i).1.2).bodd = false | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect β Q) h
β’ β i β I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect β Q) h
β’ β i β I, ((Q i).2.1.bodd = true β§ (Q i).2.2.bodd = true) β§ ((Q i).1.1 + (Q i).1.2).bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | apply_fun weight at h1 | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect β Q) h
β’ β i β I, 0 < weight (latticeRect (Q i)) | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : weight (latticeRect ((0, 0), m, n)) = weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : latticeRect ((0, 0), m, n) = I.disjiUnion (latticeRect β Q) h
β’ β i β I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | rw [latticeRect_weight] at h1 | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : weight (latticeRect ((0, 0), m, n)) = weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i)) | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 :
(bif ((0, 0), m, n).2.1.bodd && ((0, 0), m, n).2.2.bodd then (-1) ^ (((0, 0), m, n).1.1 + ((0, 0), m, n).1.2)
else 0) =
weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : weight (latticeRect ((0, 0), m, n)) = weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | simp only at h1 | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 :
(bif ((0, 0), m, n).2.1.bodd && ((0, 0), m, n).2.2.bodd then (-1) ^ (((0, 0), m, n).1.1 + ((0, 0), m, n).1.2)
else 0) =
weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i)) | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif m.bodd && n.bodd then (-1) ^ (0 + 0) else 0) = weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 :
(bif ((0, 0), m, n).2.1.bodd && ((0, 0), m, n).2.2.bodd then (-1) ^ (((0, 0), m, n).1.1 + ((0, 0), m, n).1.2)
else 0) =
weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | rw [disjiUnion_weight_eq, Function.comp, h0.1, h0.2, add_zero] at h1 | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif m.bodd && n.bodd then (-1) ^ (0 + 0) else 0) = weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i)) | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ β i β I, 0 < weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif m.bodd && n.bodd then (-1) ^ (0 + 0) else 0) = weight (I.disjiUnion (latticeRect β Q) h)
β’ β i β I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | apply exists_lt_of_sum_lt | ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ β i β I, 0 < weight (latticeRect (Q i)) | case Hlt
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ β i β I, 0 < β i β I, weight (latticeRect (Q i)) | Please generate a tactic in lean4 to solve the state.
STATE:
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ β i β I, 0 < weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | rw [sum_const_zero, β h1] | case Hlt
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ β i β I, 0 < β i β I, weight (latticeRect (Q i)) | case Hlt
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ 0 < bif true && true then (-1) ^ 0 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case Hlt
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ β i β I, 0 < β i β I, weight (latticeRect (Q i))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/C1/C1.lean | IMOSL.IMO2017C1.final_solution | [77, 1] | [90, 24] | exact Int.zero_lt_one | case Hlt
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ 0 < bif true && true then (-1) ^ 0 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Hlt
ΞΉ : Type u_1
Q : ΞΉ β (β Γ β) Γ β Γ β
m n : β
I : Finset ΞΉ
h : (βI).PairwiseDisjoint (latticeRect β Q)
h0 : m.bodd = true β§ n.bodd = true
h1 : (bif true && true then (-1) ^ 0 else 0) = β i β I, weight ((fun x => latticeRect (Q x)) i)
β’ 0 < bif true && true then (-1) ^ 0 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.dvd_sq_iff_dvd_sq_of_dvd_add | [33, 1] | [39, 47] | have h0 {u v w : β} (h0 : u β£ v + w) : u β£ v ^ 2 β u β£ v * w := by
apply dvd_iff_of_dvd_add
rw [sq, β Nat.mul_add]
exact h0.mul_left v | a b c : β
h : c β£ a + b
β’ c β£ a ^ 2 β c β£ b ^ 2 | a b c : β
h : c β£ a + b
h0 : β {u v w : β}, u β£ v + w β (u β£ v ^ 2 β u β£ v * w)
β’ c β£ a ^ 2 β c β£ b ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
h : c β£ a + b
β’ c β£ a ^ 2 β c β£ b ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.dvd_sq_iff_dvd_sq_of_dvd_add | [33, 1] | [39, 47] | rw [h0 h, mul_comm, β h0 (a.add_comm b βΈ h)] | a b c : β
h : c β£ a + b
h0 : β {u v w : β}, u β£ v + w β (u β£ v ^ 2 β u β£ v * w)
β’ c β£ a ^ 2 β c β£ b ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
h : c β£ a + b
h0 : β {u v w : β}, u β£ v + w β (u β£ v ^ 2 β u β£ v * w)
β’ c β£ a ^ 2 β c β£ b ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.dvd_sq_iff_dvd_sq_of_dvd_add | [33, 1] | [39, 47] | apply dvd_iff_of_dvd_add | a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v ^ 2 β u β£ v * w | case h
a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v ^ 2 + v * w | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v ^ 2 β u β£ v * w
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.dvd_sq_iff_dvd_sq_of_dvd_add | [33, 1] | [39, 47] | rw [sq, β Nat.mul_add] | case h
a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v ^ 2 + v * w | case h
a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v * (v + w) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v ^ 2 + v * w
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.dvd_sq_iff_dvd_sq_of_dvd_add | [33, 1] | [39, 47] | exact h0.mul_left v | case h
a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v * (v + w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b c : β
h : c β£ a + b
u v w : β
h0 : u β£ v + w
β’ u β£ v * (v + w)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | rw [Nat.dvd_prime_pow h] at h1 | a p : β
h : p.Prime
h0 : a < p
h1 : p + a β£ p ^ 2
β’ a = 0 | a p : β
h : p.Prime
h0 : a < p
h1 : β k β€ 2, p + a = p ^ k
β’ a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a p : β
h : p.Prime
h0 : a < p
h1 : p + a β£ p ^ 2
β’ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | rcases h1 with β¨k, h1, h2β© | a p : β
h : p.Prime
h0 : a < p
h1 : β k β€ 2, p + a = p ^ k
β’ a = 0 | case intro.intro
a p : β
h : p.Prime
h0 : a < p
k : β
h1 : k β€ 2
h2 : p + a = p ^ k
β’ a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a p : β
h : p.Prime
h0 : a < p
h1 : β k β€ 2, p + a = p ^ k
β’ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | rw [Nat.le_add_one_iff, Nat.le_add_one_iff, zero_add, le_zero_iff] at h1 | case intro.intro
a p : β
h : p.Prime
h0 : a < p
k : β
h1 : k β€ 2
h2 : p + a = p ^ k
β’ a = 0 | case intro.intro
a p : β
h : p.Prime
h0 : a < p
k : β
h1 : (k = 0 β¨ k = 1) β¨ k = 1 + 1
h2 : p + a = p ^ k
β’ a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a p : β
h : p.Prime
h0 : a < p
k : β
h1 : k β€ 2
h2 : p + a = p ^ k
β’ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | rcases h1 with (rfl | rfl) | rfl | case intro.intro
a p : β
h : p.Prime
h0 : a < p
k : β
h1 : (k = 0 β¨ k = 1) β¨ k = 1 + 1
h2 : p + a = p ^ k
β’ a = 0 | case intro.intro.inl.inl
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ 0
β’ a = 0
case intro.intro.inl.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ 1
β’ a = 0
case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a p : β
h : p.Prime
h0 : a < p
k : β
h1 : (k = 0 β¨ k = 1) β¨ k = 1 + 1
h2 : p + a = p ^ k
β’ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | exact absurd h2 (h.one_lt.trans_le le_self_add).ne.symm | case intro.intro.inl.inl
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ 0
β’ a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl.inl
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ 0
β’ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | rwa [pow_one, add_right_eq_self] at h2 | case intro.intro.inl.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ 1
β’ a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ 1
β’ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | refine absurd ((add_lt_add_left h0 p).trans_le ?_) h2.not_lt | case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ a = 0 | case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ p + p β€ p ^ (1 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | rw [sq, β two_mul] | case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ p + p β€ p ^ (1 + 1) | case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ 2 * p β€ p * p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ p + p β€ p ^ (1 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.eq_zero_of_prime_add_dvd_sq | [41, 1] | [50, 62] | exact Nat.mul_le_mul_right p h.two_le | case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ 2 * p β€ p * p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr
a p : β
h : p.Prime
h0 : a < p
h2 : p + a = p ^ (1 + 1)
β’ 2 * p β€ p * p
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.linear_is_good | [61, 1] | [62, 67] | rw [sq, add_mul, Nat.mul_assoc, mul_left_comm] | C k a b : β
xβ : C < a + b
β’ a ^ 2 + b * (fun x => k * x) a = (a + (fun x => k * x) b) * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C k a b : β
xβ : C < a + b
β’ a ^ 2 + b * (fun x => k * x) a = (a + (fun x => k * x) b) * a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | have h0 {n} (h0 : C < n) : f n β€ n * f 1 := by
rw [β Nat.succ_le_succ_iff, Nat.succ_eq_one_add, Nat.succ_eq_one_add]
exact Nat.le_of_dvd (Nat.add_pos_left Nat.one_pos _)
(h 1 n <| Nat.lt_one_add_iff.mpr h0.le) | C : β
f : β β β
h : good C f
β’ β k, f = fun x => k * x | C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
β’ β k, f = fun x => k * x | Please generate a tactic in lean4 to solve the state.
STATE:
C : β
f : β β β
h : good C f
β’ β k, f = fun x => k * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | have h1 {p} (h1 : C < p) (h2 : p.Prime) : β k β€ f 1, f p = k * p := by
suffices p β£ f p ^ 2 by
rcases h2.dvd_of_dvd_pow this with β¨k, h3β©
exact β¨k, Nat.le_of_mul_le_mul_left (h3.ge.trans (h0 h1)) h2.pos,
h3.trans (p.mul_comm k)β©
rcases exists_gt (C + f p) with β¨n, h3β©
replace h3 := Nat.exists_eq_add_of_le <|
h3.le.trans (Nat.le_mul_of_pos_right _ h2.pos)
rcases h3 with β¨a, h3β©; rw [add_right_comm] at h3
specialize h (C + a) p <|
(C.le_add_right a).trans_lt (Nat.lt_add_of_pos_right h2.pos)
replace h3 : p β£ C + a + f p := β¨n, h3.symm.trans (n.mul_comm p)β©
rw [β dvd_sq_iff_dvd_sq_of_dvd_add h3, dvd_iff_of_dvd_add (h3.trans h)]
exact p.dvd_mul_right _ | C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
β’ β k, f = fun x => k * x | C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
β’ β k, f = fun x => k * x | Please generate a tactic in lean4 to solve the state.
STATE:
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
β’ β k, f = fun x => k * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rcases C.succ.exists_infinite_primes with β¨p, (h3 : C < p), h4β© | C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
β’ β k, f = fun x => k * x | case intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
β’ β k, f = fun x => k * x | Please generate a tactic in lean4 to solve the state.
STATE:
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
β’ β k, f = fun x => k * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rcases h1 h3 h4 with β¨k, -, h5β© | case intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
β’ β k, f = fun x => k * x | case intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
β’ β k, f = fun x => k * x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
β’ β k, f = fun x => k * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | refine β¨k, funext Ξ» n β¦ ?_β© | case intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
β’ β k, f = fun x => k * x | case intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n : β
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
β’ β k, f = fun x => k * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rcases h2 p with β¨Bp, hpβ© | case intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n : β
β’ f n = k * n | case intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n : β
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rcases h2 n with β¨Bn, hnβ© | case intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
β’ f n = k * n | case intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rcases (max Bp Bn).succ.exists_infinite_primes with β¨q, h6, h7β© | case intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
β’ f n = k * n | case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : (max Bp Bn).succ β€ q
h7 : q.Prime
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rw [Nat.succ_le_iff, max_lt_iff] at h6 | case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : (max Bp Bn).succ β€ q
h7 : q.Prime
β’ f n = k * n | case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : (max Bp Bn).succ β€ q
h7 : q.Prime
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | specialize hp q h7 h6.1 | case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
β’ f n = k * n | case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
hp : β k, f q = k * q β§ f p = k * p
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp : β
hp : β (p_1 : β), p_1.Prime β Bp < p_1 β β k, f p_1 = k * p_1 β§ f p = k * p
Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rcases hp with β¨kp, h8, h9β© | case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
hp : β k, f q = k * q β§ f p = k * p
β’ f n = k * n | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : f p = kp * p
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
hp : β k, f q = k * q β§ f p = k * p
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rw [h5, Nat.mul_left_inj h4.ne_zero] at h9 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : f p = kp * p
β’ f n = k * n | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : k = kp
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : f p = kp * p
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | specialize hn q h7 h6.2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : k = kp
β’ f n = k * n | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : k = kp
hn : β k, f q = k * q β§ f n = k * n
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn : β
hn : β (p : β), p.Prime β Bn < p β β k, f p = k * p β§ f n = k * n
q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : k = kp
β’ f n = k * n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2019/N4/N4.lean | IMOSL.IMO2019N4.good_is_linear | [64, 1] | [125, 24] | rcases hn with β¨kn, h10, h11β© | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : k = kp
hn : β k, f q = k * q β§ f n = k * n
β’ f n = k * n | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : k = kp
kn : β
h10 : f q = kn * q
h11 : f n = kn * n
β’ f n = k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
C : β
f : β β β
h : good C f
h0 : β {n : β}, C < n β f n β€ n * f 1
h1 : β {p : β}, C < p β p.Prime β β k β€ f 1, f p = k * p
h2 : β (x : β), β B, β (p : β), p.Prime β B < p β β k, f p = k * p β§ f x = k * x
p : β
h3 : C < p
h4 : p.Prime
k : β
h5 : f p = k * p
n Bp Bn q : β
h6 : Bp < q β§ Bn < q
h7 : q.Prime
kp : β
h8 : f q = kp * q
h9 : k = kp
hn : β k, f q = k * q β§ f n = k * n
β’ f n = k * n
TACTIC:
|
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