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/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | replace h0 (k : β) : (k.succ : F) / a k.succ β€ a k + k / a k := by
rw [add_div' _ _ _ (h k).ne.symm, div_le_div_iff (h _) (h k), β sq]
exact (div_le_iff' <| add_pos_of_pos_of_nonneg
(pow_pos (h k) 2) k.cast_nonneg).mp (h0 k) | F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ * a k / (a k ^ 2 + βk) β€ a k.succ
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a | F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ * a k / (a k ^ 2 + βk) β€ a k.succ
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | refine Nat.le_induction ?_ ?_ | F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β2 β€ (range 2).sum a
case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a β β(n + 1) β€ (range (n + 1)).sum a | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [add_div' _ _ _ (h k).ne.symm, div_le_div_iff (h _) (h k), β sq] | F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ * a k / (a k ^ 2 + βk) β€ a k.succ
k : β
β’ βk.succ / a k.succ β€ a k + βk / a k | F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ * a k / (a k ^ 2 + βk) β€ a k.succ
k : β
β’ βk.succ * a k β€ (a k ^ 2 + βk) * a k.succ | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ * a k / (a k ^ 2 + βk) β€ a k.succ
k : β
β’ βk.succ / a k.succ β€ a k + βk / a k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | exact (div_le_iff' <| add_pos_of_pos_of_nonneg
(pow_pos (h k) 2) k.cast_nonneg).mp (h0 k) | F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ * a k / (a k ^ 2 + βk) β€ a k.succ
k : β
β’ βk.succ * a k β€ (a k ^ 2 + βk) * a k.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ * a k / (a k ^ 2 + βk) β€ a k.succ
k : β
β’ βk.succ * a k β€ (a k ^ 2 + βk) * a k.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [Nat.cast_two, sum_range_succ, sum_range_one] | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β2 β€ (range 2).sum a | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ 2 β€ a 0 + a 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β2 β€ (range 2).sum a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | specialize h0 0 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ 2 β€ a 0 + a 1 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β(Nat.succ 0) / a (Nat.succ 0) β€ a 0 + β0 / a 0
β’ 2 β€ a 0 + a 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ 2 β€ a 0 + a 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [Nat.cast_one, Nat.cast_zero, zero_div, add_zero] at h0 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β(Nat.succ 0) / a (Nat.succ 0) β€ a 0 + β0 / a 0
β’ 2 β€ a 0 + a 1 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 β€ a 0 + a 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β(Nat.succ 0) / a (Nat.succ 0) β€ a 0 + β0 / a 0
β’ 2 β€ a 0 + a 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | apply (add_le_add_right h0 _).trans' | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 β€ a 0 + a 1 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 β€ 1 / a (Nat.succ 0) + a 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 β€ a 0 + a 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [div_add' _ _ _ (h 1).ne.symm, le_div_iff (h 1), β sq] | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 β€ 1 / a (Nat.succ 0) + a 1 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 * a 1 β€ 1 + a 1 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 β€ 1 / a (Nat.succ 0) + a 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | have h1 := two_mul_le_add_sq 1 (a 1) | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 * a 1 β€ 1 + a 1 ^ 2 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
h1 : 2 * 1 * a 1 β€ 1 ^ 2 + a 1 ^ 2
β’ 2 * a 1 β€ 1 + a 1 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
β’ 2 * a 1 β€ 1 + a 1 ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rwa [mul_one, one_pow] at h1 | case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
h1 : 2 * 1 * a 1 β€ 1 ^ 2 + a 1 ^ 2
β’ 2 * a 1 β€ 1 + a 1 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : 1 / a (Nat.succ 0) β€ a 0
h1 : 2 * 1 * a 1 β€ 1 ^ 2 + a 1 ^ 2
β’ 2 * a 1 β€ 1 + a 1 ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | intro n h1 h2 | case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a β β(n + 1) β€ (range (n + 1)).sum a | case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
β’ β(n + 1) β€ (range (n + 1)).sum a | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β (n : β), 2 β€ n β βn β€ (range n).sum a β β(n + 1) β€ (range (n + 1)).sum a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [sum_range_succ, Nat.cast_succ] | case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
β’ β(n + 1) β€ (range (n + 1)).sum a | case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
β’ βn + 1 β€ β x β range n, a x + a n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
β’ β(n + 1) β€ (range (n + 1)).sum a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rcases le_total 1 (a n) with h3 | h3 | case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
β’ βn + 1 β€ β x β range n, a x + a n | case refine_2.inl
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : 1 β€ a n
β’ βn + 1 β€ β x β range n, a x + a n
case refine_2.inr
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn + 1 β€ β x β range n, a x + a n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
β’ βn + 1 β€ β x β range n, a x + a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | refine le_trans ?_ (add_le_add_right (?_ : (n : F) / a n β€ _) _) | case refine_2.inr
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn + 1 β€ β x β range n, a x + a n | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn + 1 β€ βn / a n + a n
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn + 1 β€ β x β range n, a x + a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [div_add' _ _ _ (h n).ne.symm, le_div_iff (h n), add_one_mul (Ξ± := F),
β sub_le_iff_le_add, add_sub_assoc, β mul_one_sub,
β le_sub_iff_add_le', β mul_one_sub, β sub_nonneg, β sub_mul] | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn + 1 β€ βn / a n + a n
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 0 β€ (βn - a n) * (1 - a n)
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn + 1 β€ βn / a n + a n
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | refine mul_nonneg (sub_nonneg.mpr (h3.trans ?_)) (sub_nonneg.mpr h3) | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 0 β€ (βn - a n) * (1 - a n)
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 1 β€ βn
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 0 β€ (βn - a n) * (1 - a n)
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [Nat.one_le_cast] | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 1 β€ βn
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 1 β€ n
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 1 β€ βn
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | exact one_le_two.trans h1 | case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 1 β€ n
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_1
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ 1 β€ n
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | clear h1 h2 h3 | case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x | case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
β’ βn / a n β€ β x β range n, a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : a n β€ 1
β’ βn / a n β€ β x β range n, a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | induction' n with n n_ih | case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
β’ βn / a n β€ β x β range n, a x | case refine_2.inr.refine_2.zero
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β0 / a 0 β€ β x β range 0, a x
case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ β x β range (n + 1), a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_2
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
β’ βn / a n β€ β x β range n, a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [Nat.cast_zero, sum_range_zero, zero_div] | case refine_2.inr.refine_2.zero
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β0 / a 0 β€ β x β range 0, a x
case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ β x β range (n + 1), a x | case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ β x β range (n + 1), a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_2.zero
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
β’ β0 / a 0 β€ β x β range 0, a x
case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ β x β range (n + 1), a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | rw [sum_range_succ, add_comm _ (a n)] | case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ β x β range (n + 1), a x | case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ a n + β x β range n, a x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ β x β range (n + 1), a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | refine (h0 n).trans (add_le_add_left n_ih (a n)) | case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ a n + β x β range n, a x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.refine_2.succ
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
n_ih : βn / a n β€ β x β range n, a x
β’ β(n + 1) / a (n + 1) β€ a n + β x β range n, a x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A1/A1.lean | IMOSL.IMO2015A1.final_solution | [33, 1] | [65, 53] | exact add_le_add h2 h3 | case refine_2.inl
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : 1 β€ a n
β’ βn + 1 β€ β x β range n, a x + a n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl
F : Type u_1
instβ : LinearOrderedField F
a : β β F
h : β (k : β), 0 < a k
h0 : β (k : β), βk.succ / a k.succ β€ a k + βk / a k
n : β
h1 : 2 β€ n
h2 : βn β€ (range n).sum a
h3 : 1 β€ a n
β’ βn + 1 β€ β x β range n, a x + a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_infinite_map_eq_map_zero | [31, 1] | [40, 58] | obtain β¨n, h0, h1β© := h0 (b + (f 0 - f b).natAbs + 1) | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (k : β), β n, k β€ n β§ f n = f 0
b : β
β’ f b = f 0 | case intro.intro
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : b + (f 0 - f b).natAbs + 1 β€ n
h1 : f n = f 0
β’ f b = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (k : β), β n, k β€ n β§ f n = f 0
b : β
β’ f b = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_infinite_map_eq_map_zero | [31, 1] | [40, 58] | specialize h n b | case intro.intro
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : b + (f 0 - f b).natAbs + 1 β€ n
h1 : f n = f 0
β’ f b = f 0 | case intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : b + (f 0 - f b).natAbs + 1 β€ n
h1 : f n = f 0
h : βn - βb β£ f n - f b
β’ f b = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : b + (f 0 - f b).natAbs + 1 β€ n
h1 : f n = f 0
β’ f b = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_infinite_map_eq_map_zero | [31, 1] | [40, 58] | rw [Nat.succ_le_iff, lt_iff_exists_add] at h0 | case intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : b + (f 0 - f b).natAbs + 1 β€ n
h1 : f n = f 0
h : βn - βb β£ f n - f b
β’ f b = f 0 | case intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : β c > 0, n = b + (f 0 - f b).natAbs + c
h1 : f n = f 0
h : βn - βb β£ f n - f b
β’ f b = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : b + (f 0 - f b).natAbs + 1 β€ n
h1 : f n = f 0
h : βn - βb β£ f n - f b
β’ f b = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_infinite_map_eq_map_zero | [31, 1] | [40, 58] | rcases h0 with β¨C, h0, rflβ© | case intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : β c > 0, n = b + (f 0 - f b).natAbs + c
h1 : f n = f 0
h : βn - βb β£ f n - f b
β’ f b = f 0 | case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : β(b + (f 0 - f b).natAbs + C) - βb β£ f (b + (f 0 - f b).natAbs + C) - f b
β’ f b = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b n : β
h0 : β c > 0, n = b + (f 0 - f b).natAbs + c
h1 : f n = f 0
h : βn - βb β£ f n - f b
β’ f b = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_infinite_map_eq_map_zero | [31, 1] | [40, 58] | rw [h1, add_assoc, Nat.cast_add, add_sub_cancel_left, Int.natCast_dvd] at h | case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : β(b + (f 0 - f b).natAbs + C) - βb β£ f (b + (f 0 - f b).natAbs + C) - f b
β’ f b = f 0 | case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : (f 0 - f b).natAbs + C β£ (f 0 - f b).natAbs
β’ f b = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : β(b + (f 0 - f b).natAbs + C) - βb β£ f (b + (f 0 - f b).natAbs + C) - f b
β’ f b = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_infinite_map_eq_map_zero | [31, 1] | [40, 58] | have h2 := Nat.eq_zero_of_dvd_of_lt h (Nat.lt_add_of_pos_right h0) | case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : (f 0 - f b).natAbs + C β£ (f 0 - f b).natAbs
β’ f b = f 0 | case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : (f 0 - f b).natAbs + C β£ (f 0 - f b).natAbs
h2 : (f 0 - f b).natAbs = 0
β’ f b = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : (f 0 - f b).natAbs + C β£ (f 0 - f b).natAbs
β’ f b = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_infinite_map_eq_map_zero | [31, 1] | [40, 58] | rwa [Int.natAbs_eq_zero, sub_eq_zero, eq_comm] at h2 | case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : (f 0 - f b).natAbs + C β£ (f 0 - f b).natAbs
h2 : (f 0 - f b).natAbs = 0
β’ f b = f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : β β β€
h0β : β (k : β), β n, k β€ n β§ f n = f 0
b C : β
h0 : C > 0
h1 : f (b + (f 0 - f b).natAbs + C) = f 0
h : (f 0 - f b).natAbs + C β£ (f 0 - f b).natAbs
h2 : (f 0 - f b).natAbs = 0
β’ f b = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | have h1 : f 0 β 0 := Ξ» h1 β¦ by
obtain β¨p, h2, h3β© := K.exists_infinite_primes
exact h2.not_lt (h0 p h3 β¨0, 0, h1β©) | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
β’ β C, f = fun x => C | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
β’ β C, f = fun x => C | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
β’ β C, f = fun x => C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | refine const_of_infinite_map_eq_map_zero h
Ξ» k β¦ β¨(f 0).natAbs * (4 * K.factorial) * k, ?_, ?_β© | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
β’ β C, f = fun x => C | case refine_1
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
β’ k β€ (f 0).natAbs * (4 * K.factorial) * k
case refine_2
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
β’ β C, f = fun x => C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | obtain β¨p, h2, h3β© := K.exists_infinite_primes | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 = 0
β’ False | case intro.intro
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 = 0
p : β
h2 : K β€ p
h3 : p.Prime
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 = 0
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | exact h2.not_lt (h0 p h3 β¨0, 0, h1β©) | case intro.intro
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 = 0
p : β
h2 : K β€ p
h3 : p.Prime
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 = 0
p : β
h2 : K β€ p
h3 : p.Prime
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | exact Nat.le_mul_of_pos_left _ <| Nat.mul_pos (Int.natAbs_pos.mpr h1)
(Nat.mul_pos (Nat.succ_pos 3) K.factorial_pos) | case refine_1
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
β’ k β€ (f 0).natAbs * (4 * K.factorial) * k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
β’ k β€ (f 0).natAbs * (4 * K.factorial) * k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | specialize h ((f 0).natAbs * (4 * K.factorial) * k) 0 | case refine_2
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial) * k) - β0 β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | rw [Nat.cast_zero, Int.sub_zero, Nat.cast_mul] at h | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial) * k) - β0 β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial)) * βk β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial) * k) - β0 β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | apply (dvd_mul_right _ _).trans at h | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial)) * βk β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial)) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial)) * βk β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | rw [Nat.cast_mul, β abs_dvd, Int.natCast_natAbs,
abs_mul, abs_abs, β abs_mul, abs_dvd] at h | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial)) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : f 0 * β(4 * K.factorial) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : β((f 0).natAbs * (4 * K.factorial)) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | obtain β¨m, h2β© := dvd_sub_self_right.mp (dvd_of_mul_right_dvd h) | case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : f 0 * β(4 * K.factorial) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : f 0 * β(4 * K.factorial) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : f 0 * β(4 * K.factorial) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | rw [h2, β mul_sub_one, mul_dvd_mul_iff_left h1, Int.natCast_dvd] at h | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : f 0 * β(4 * K.factorial) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
h : f 0 * β(4 * K.factorial) β£ f ((f 0).natAbs * (4 * K.factorial) * k) - f 0
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | rw [h2, mul_right_eq_selfβ, or_iff_left h1] | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ f ((f 0).natAbs * (4 * K.factorial) * k) = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | clear h1 | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ m = 1 | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
h1 : f 0 β 0
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | by_cases h1 : m.natAbs = 1 | case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ m = 1 | case pos
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : m.natAbs = 1
β’ m = 1
case neg
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : Β¬m.natAbs = 1
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | rw [Int.natAbs_eq_iff, Nat.cast_one] at h1 | case pos
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : m.natAbs = 1
β’ m = 1 | case pos
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : m = 1 β¨ m = -1
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : m.natAbs = 1
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | exact h1.resolve_right Ξ» h1 β¦ absurd ((dvd_mul_right _ _).trans (h1 βΈ h))
(Nat.not_dvd_of_pos_of_lt (Nat.succ_pos 1) (Nat.le_succ 3)) | case pos
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : m = 1 β¨ m = -1
β’ m = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : m = 1 β¨ m = -1
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | apply Nat.exists_prime_and_dvd at h1 | case neg
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : Β¬m.natAbs = 1
β’ m = 1 | case neg
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : β p, p.Prime β§ p β£ m.natAbs
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : Β¬m.natAbs = 1
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | rcases h1 with β¨p, h1, h3β© | case neg
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : β p, p.Prime β§ p β£ m.natAbs
β’ m = 1 | case neg.intro.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
h1 : β p, p.Prime β§ p β£ m.natAbs
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | specialize h0 p h1 β¨(f 0).natAbs * (4 * Nat.factorial K) * k,
h2 βΈ dvd_mul_of_dvd_right (Int.natCast_dvd.mpr h3) _β© | case neg.intro.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
β’ m = 1 | case neg.intro.intro
f : β β β€
K k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
h0 : p < K
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
f : β β β€
K : β
h0 : β (p : β), p.Prime β (β c, βp β£ f c) β p < K
k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | replace h := (Nat.dvd_factorial h1.pos h0.le).trans (dvd_of_mul_left_dvd h) | case neg.intro.intro
f : β β β€
K k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
h0 : p < K
β’ m = 1 | case neg.intro.intro
f : β β β€
K k : β
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
h0 : p < K
h : p β£ (m - 1).natAbs
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
f : β β β€
K k : β
m : β€
h : 4 * K.factorial β£ (m - 1).natAbs
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
h0 : p < K
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | rw [β Int.natCast_dvd] at h h3 | case neg.intro.intro
f : β β β€
K k : β
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
h0 : p < K
h : p β£ (m - 1).natAbs
β’ m = 1 | case neg.intro.intro
f : β β β€
K k : β
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : βp β£ m
h0 : p < K
h : βp β£ m - 1
β’ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
f : β β β€
K k : β
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : p β£ m.natAbs
h0 : p < K
h : p β£ (m - 1).natAbs
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.const_of_finite_prime_divisor | [42, 1] | [74, 61] | exact absurd (Int.natCast_dvd.mp <|
(Int.dvd_iff_dvd_of_dvd_sub h).mp h3) h1.not_dvd_one | case neg.intro.intro
f : β β β€
K k : β
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : βp β£ m
h0 : p < K
h : βp β£ m - 1
β’ m = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
f : β β β€
K k : β
m : β€
h2 : f ((f 0).natAbs * (4 * K.factorial) * k) = f 0 * m
p : β
h1 : p.Prime
h3 : βp β£ m
h0 : p < K
h : βp β£ m - 1
β’ m = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.final_solution | [79, 1] | [85, 68] | refine by_contra Ξ» h1 β¦ absurd h0 (not_forall.mpr β¨f 0, ?_β©) | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
β’ β p, K β€ p β§ p.Prime β§ β c, βp β£ f c | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ Β¬β b, f b β f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
β’ β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.final_solution | [79, 1] | [85, 68] | rw [β not_forall, not_not] | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ Β¬β b, f b β f 0 | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ β (x : β), f x = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ Β¬β b, f b β f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.final_solution | [79, 1] | [85, 68] | suffices β C, f = Ξ» _ β¦ C from this.elim Ξ» C this β¦ this βΈ Ξ» x β¦ rfl | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ β (x : β), f x = f 0 | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ β C, f = fun x => C | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ β (x : β), f x = f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N3/N3.lean | IMOSL.IMO2009N3.final_solution | [79, 1] | [85, 68] | exact const_of_finite_prime_divisor h (K := K)
Ξ» p h2 h3 β¦ lt_of_not_le Ξ» h4 β¦ not_exists.mp h1 p β¨h4, h2, h3β© | f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ β C, f = fun x => C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β€
h : β (a b : β), βa - βb β£ f a - f b
h0 : β (C : β€), β b, f b β C
K : β
h1 : Β¬β p, K β€ p β§ p.Prime β§ β c, βp β£ f c
β’ β C, f = fun x => C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean | IMOSL.IMO2012A5.zero_is_good | [21, 1] | [22, 37] | rw [add_zero, zero_mul] | R : Type u_1
S : Type u_2
instβΒΉ : NonAssocSemiring R
instβ : NonAssocSemiring S
xβΒΉ xβ : R
β’ (fun x => 0) (xβΒΉ * xβ + 1) = (fun x => 0) xβΒΉ * (fun x => 0) xβ + (fun x => 0) (xβΒΉ + xβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
instβΒΉ : NonAssocSemiring R
instβ : NonAssocSemiring S
xβΒΉ xβ : R
β’ (fun x => 0) (xβΒΉ * xβ + 1) = (fun x => 0) xβΒΉ * (fun x => 0) xβ + (fun x => 0) (xβΒΉ + xβ)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean | IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero | [24, 1] | [34, 89] | rw [zero_add, β h, one_mul] | R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
β’ 0 + f (1 + 1) = f 1 * f 1 + f (1 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
β’ 0 + f (1 + 1) = f 1 * f 1 + f (1 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean | IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero | [24, 1] | [34, 89] | have h1 := h x 0 | R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
x : R
β’ 0 = f x * (f 0 + 1) | R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
x : R
h1 : f (x * 0 + 1) = f x * f 0 + f (x + 0)
β’ 0 = f x * (f 0 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
x : R
β’ 0 = f x * (f 0 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean | IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero | [24, 1] | [34, 89] | rwa [mul_zero, zero_add, h0, add_zero, β mul_add_one (f x)] at h1 | R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
x : R
h1 : f (x * 0 + 1) = f x * f 0 + f (x + 0)
β’ 0 = f x * (f 0 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
x : R
h1 : f (x * 0 + 1) = f x * f 0 + f (x + 0)
β’ 0 = f x * (f 0 + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean | IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero | [24, 1] | [34, 89] | specialize h1 x | R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
h1 : β (x : R), 0 = f x * (f 0 + 1)
h2 : f 0 = 0
x : R
β’ f x = 0 | R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
h2 : f 0 = 0
x : R
h1 : 0 = f x * (f 0 + 1)
β’ f x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
h1 : β (x : R), 0 = f x * (f 0 + 1)
h2 : f 0 = 0
x : R
β’ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/ZeroMap.lean | IMOSL.IMO2012A5.good_Nontrivial_or_eq_zero | [24, 1] | [34, 89] | rwa [h2, zero_add, mul_one, eq_comm] at h1 | R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
h2 : f 0 = 0
x : R
h1 : 0 = f x * (f 0 + 1)
β’ f x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
instβΒ³ : NonAssocSemiring R
instβΒ² : NonAssocSemiring S
instβΒΉ : IsCancelAdd S
instβ : NoZeroDivisors S
f : R β S
h : good f
h0 : f 1 = 0
h2 : f 0 = 0
x : R
h1 : 0 = f x * (f 0 + 1)
β’ f x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.one_sub_is_good | [36, 1] | [38, 68] | rw [sub_add_sub_comm, mul_one_sub, one_sub_mul, sub_sub, sub_add, add_sub_sub_cancel,
add_sub_add_left_eq_sub, sub_sub_cancel_left, β sub_eq_add_neg] | R : Type u_1
instβ : NonAssocRing R
x y : R
β’ (fun x => 1 - x) ((fun x => 1 - x) x * (fun x => 1 - x) y) + (fun x => 1 - x) (x + y) = (fun x => 1 - x) (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 => 1 - x) ((fun x => 1 - x) x * (fun x => 1 - x) y) + (fun x => 1 - x) (x + y) = (fun x => 1 - x) (x * y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_neg | [42, 1] | [43, 59] | simp only [Pi.neg_apply] | R : Type u_1
instβ : NonAssocRing R
f : R β R
h : good f
x y : R
β’ (-f) ((-f) x * (-f) y) + (-f) (x + y) = (-f) (x * y) | R : Type u_1
instβ : NonAssocRing R
f : R β R
h : good f
x y : R
β’ -f (-f x * -f y) + -f (x + y) = -f (x * y) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : NonAssocRing R
f : R β R
h : good f
x y : R
β’ (-f) ((-f) x * (-f) y) + (-f) (x + y) = (-f) (x * y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_neg | [42, 1] | [43, 59] | rw [neg_mul_neg, β neg_add, h] | R : Type u_1
instβ : NonAssocRing R
f : R β R
h : good f
x y : R
β’ -f (-f x * -f y) + -f (x + y) = -f (x * y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : NonAssocRing R
f : R β R
h : good f
x y : R
β’ -f (-f x * -f y) + -f (x + y) = -f (x * y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_special_equality | [49, 1] | [51, 77] | rw [β add_left_eq_self, h, add_one_mul x, mul_add_one x, h0, add_comm 1 x] | R : Type u_1
instβ : NonAssocRing R
f : R β R
h : good f
x y : R
h0 : x * y = 1
β’ f (f (x + 1) * f (y + 1)) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : NonAssocRing R
f : R β R
h : good f
x y : R
h0 : x * y = 1
β’ f (f (x + 1) * f (y + 1)) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_map_zero_sq | [61, 1] | [62, 74] | specialize h 0 0 | R : Type u_1
instβ : Ring R
f : R β R
h : good f
β’ f (f 0 ^ 2) = 0 | R : Type u_1
instβ : Ring R
f : R β R
h : f (f 0 * f 0) + f (0 + 0) = f (0 * 0)
β’ f (f 0 ^ 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : Ring R
f : R β R
h : good f
β’ f (f 0 ^ 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_map_zero_sq | [61, 1] | [62, 74] | rwa [add_zero, mul_zero, add_left_eq_self, β sq] at h | R : Type u_1
instβ : Ring R
f : R β R
h : f (f 0 * f 0) + f (0 + 0) = f (0 * 0)
β’ f (f 0 ^ 2) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : Ring R
f : R β R
h : f (f 0 * f 0) + f (0 + 0) = f (0 * 0)
β’ f (f 0 ^ 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_eq_of_inj | [64, 1] | [70, 66] | rw [β h0, β mul_zero x, β h, add_zero, h0, mul_one] | R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
x : R
β’ f (f x) + f x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
x : R
β’ f (f x) + f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_eq_of_inj | [64, 1] | [70, 66] | rw [eq_sub_iff_add_eq', β h2 x, add_left_inj] | R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ f x = 1 - x | R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ x = f (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ f x = 1 - x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_eq_of_inj | [64, 1] | [70, 66] | apply h1 | R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ x = f (f x) | case a
R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ f x = f (f (f x)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ x = f (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_eq_of_inj | [64, 1] | [70, 66] | rw [eq_sub_of_add_eq (h2 (f x)), β h2 x, add_sub_cancel_left] | case a
R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ f x = f (f (f x)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
R : Type u_1
instβ : Ring R
f : R β R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : β (x : R), f (f x) + f x = 1
x : R
β’ f x = f (f (f x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_eq_zero | [80, 1] | [88, 44] | have h3 := good_special_equality h (mul_inv_cancel <| sub_ne_zero_of_ne h2) | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
β’ f = 0 | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f (f (c - 1 + 1) * f ((c - 1)β»ΒΉ + 1)) = 0
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_eq_zero | [80, 1] | [88, 44] | rw [sub_add_cancel, h1, zero_mul] at h3 | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f (f (c - 1 + 1) * f ((c - 1)β»ΒΉ + 1)) = 0
β’ f = 0 | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f 0 = 0
β’ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f (f (c - 1 + 1) * f ((c - 1)β»ΒΉ + 1)) = 0
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_eq_zero | [80, 1] | [88, 44] | ext x | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f 0 = 0
β’ f = 0 | case h
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f 0 = 0
x : D
β’ f x = 0 x | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f 0 = 0
β’ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_eq_zero | [80, 1] | [88, 44] | rw [Pi.zero_apply, β h3, β mul_zero x, β h,
h3, mul_zero, h3, zero_add, add_zero] | case h
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f 0 = 0
x : D
β’ f x = 0 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f β 0
c : D
h1 : f c = 0
h2 : Β¬c = 1
h3 : f 0 = 0
x : D
β’ f x = 0 x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_eq_zero_iff | [101, 1] | [103, 42] | rwa [h1] at h0 | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
c : D
h1 : f = 0
β’ 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
c : D
h1 : f = 0
β’ 0 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_shift | [106, 1] | [108, 62] | have h1 := h x 1 | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
β’ f (x + 1) + 1 = f x | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
h1 : f (f x * f 1) + f (x + 1) = f (x * 1)
β’ f (x + 1) + 1 = f x | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
β’ f (x + 1) + 1 = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_shift | [106, 1] | [108, 62] | rwa [good_map_one h, mul_zero, h0, add_comm, mul_one] at h1 | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
h1 : f (f x * f 1) + f (x + 1) = f (x * 1)
β’ f (x + 1) + 1 = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
h1 : f (f x * f 1) + f (x + 1) = f (x * 1)
β’ f (x + 1) + 1 = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_shift2 | [110, 1] | [111, 41] | rw [β good_shift h h0, sub_add_cancel] | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
β’ f (x - 1) = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
β’ f (x - 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.good_map_add_one_eq_zero_iff | [113, 1] | [115, 51] | rw [good_map_eq_zero_iff h h0, add_left_eq_self] | D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
β’ f (x + 1) = 0 β x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
f : D β D
h : good f
h0 : f 0 = 1
x : D
β’ f (x + 1) = 0 β x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | rw [or_iff_not_imp_left] | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
β’ f = 0 β¨ (f = fun x => 1 - x) β¨ f = fun x => x - 1 | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
β’ Β¬f = 0 β (f = fun x => 1 - x) β¨ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
β’ f = 0 β¨ (f = fun x => 1 - x) β¨ f = fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | intros h1 | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
β’ Β¬f = 0 β (f = fun x => 1 - x) β¨ f = fun x => x - 1 | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1 : Β¬f = 0
β’ (f = fun x => 1 - x) β¨ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
β’ Β¬f = 0 β (f = fun x => 1 - x) β¨ f = fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | apply (good_map_zero h0 h1).imp <;> intro h1 | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1 : Β¬f = 0
β’ (f = fun x => 1 - x) β¨ f = fun x => x - 1 | case f
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : f 0 = 1
β’ f = fun x => 1 - x
case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : f 0 = -1
β’ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1 : Β¬f = 0
β’ (f = fun x => 1 - x) β¨ f = fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | exact good_eq_of_inj h0 h1 (h f h0 h1) | case f
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : f 0 = 1
β’ f = fun x => 1 - x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : f 0 = 1
β’ f = fun x => 1 - x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | rw [β neg_eq_iff_eq_neg] at h1 | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : f 0 = -1
β’ f = fun x => x - 1 | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
β’ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : f 0 = -1
β’ f = fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | have h2 := good_neg h0 | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
β’ f = fun x => x - 1 | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
β’ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
β’ f = fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | have h3 := good_eq_of_inj h2 h1 (h (-f) h2 h1) | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
β’ f = fun x => x - 1 | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
β’ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
β’ f = fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | rw [β neg_inj, h3] | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
β’ f = fun x => x - 1 | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
β’ (fun x => 1 - x) = -fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
β’ f = fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | exact funext Ξ» x β¦ (neg_sub x 1).symm | case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
β’ (fun x => 1 - x) = -fun x => x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : good f
h1β : Β¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
β’ (fun x => 1 - x) = -fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | rcases h0 with rfl | rfl | rfl | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : f = 0 β¨ (f = fun x => 1 - x) β¨ f = fun x => x - 1
β’ good f | case inl
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good 0
case inr.inl
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good fun x => 1 - x
case inr.inr
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
f : D β D
h0 : f = 0 β¨ (f = fun x => 1 - x) β¨ f = fun x => x - 1
β’ good f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.solution_of_map_zero_eq_one_imp_injective | [120, 1] | [135, 61] | exacts [zero_is_good, one_sub_is_good, sub_one_is_good] | case inl
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good 0
case inr.inl
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good fun x => 1 - x
case inr.inr
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good fun x => x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good 0
case inr.inl
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good fun x => 1 - x
case inr.inr
D : Type u_1
instβ : DivisionRing D
f : D β D
hβ : good f
h : β (f : D β D), good f β f 0 = 1 β Function.Injective f
β’ good fun x => x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | have h2 := good_shift2 h0 h1 | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
β’ Function.Injective f | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h2 : β (x : D), f (x - 1) = f x + 1
β’ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
β’ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | replace h2 : β y, f y = f (-y) β y = 0 := Ξ» y h4 β¦ by
rwa [β h3, self_eq_add_left, β h2, good_map_eq_zero_iff h0 h1,
sub_eq_iff_eq_add, one_add_one_eq_two, mul_right_eq_selfβ,
or_iff_left h, β add_sub_cancel_right y 1, h2, add_left_eq_self,
good_map_add_one_eq_zero_iff h0 h1] at h4 | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h2 : β (x : D), f (x - 1) = f x + 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
β’ Function.Injective f | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
β’ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h2 : β (x : D), f (x - 1) = f x + 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
β’ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | intros a b h4 | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
β’ Function.Injective f | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
β’ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
β’ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | refine eq_of_sub_eq_zero (h2 _ ?_) | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
β’ a = b | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
β’ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
β’ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | have h5 : β y z, f y = f z β f (-y) = f (-z) :=
Ξ» y z h5 β¦ by rw [β h3, h5, h3] | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
β’ f (a - b) = f (-(a - b)) | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
h5 : β (y z : D), f y = f z β f (-y) = f (-z)
β’ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
β’ f (a - b) = f (-(a - b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | have h6 : f (a * b) = f (b * a) := by rw [β h0, β h0 b, h4, add_comm a] | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
h5 : β (y z : D), f y = f z β f (-y) = f (-z)
β’ f (a - b) = f (-(a - b)) | D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
h5 : β (y z : D), f y = f z β f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
β’ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
instβ : DivisionRing D
fβ : D β D
hβ : good fβ
h : 2 β 0
f : D β D
h0 : good f
h1 : f 0 = 1
h3 : β (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : β (y : D), f y = f (-y) β y = 0
a b : D
h4 : f a = f b
h5 : β (y z : D), f y = f z β f (-y) = f (-z)
β’ f (a - b) = f (-(a - b))
TACTIC:
|
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