url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2006/A1/A1.lean | IMOSL.IMO2006A1.final_solution | [174, 1] | [189, 87] | exact h0.resolve_right (add_pos (Nat.cast_pos.mpr (pos_of_gt hC)) one_pos).ne.symm | case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : f^[k + N + 2] r = f^[k + N] r ∨ ↑C + 1 = 0
⊢ f^[k + N + 2] r = f^[k + N] r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Archimedean R
r : R
C : ℕ
hC : 1 < C
N w✝ : ℕ
h : ∀ (k : ℕ), (↑C + 1) * f^[k + N] r = -↑C ^ 2
k : ℕ
h0 : f^[k + N + 2] r = f^[k + N] r ∨ ↑C + 1 = 0
⊢ f^[k + N + 2] r = f^[k + N] r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_general_lower_bound | [43, 1] | [50, 42] | rw [map_cons, prod_cons, targetSum, pow_add,
length_cons, Nat.factorial_succ, mul_mul_mul_comm] | a : ℕ
l : List ℕ
⊢ (map Nat.succ (a :: l)).prod ≤ 2 ^ targetSum (a :: l) * (a :: l).length.factorial | a : ℕ
l : List ℕ
⊢ a.succ * (map Nat.succ l).prod ≤ 2 ^ (a / l.length.succ) * (l.length + 1) * (2 ^ targetSum l * l.length.factorial) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
l : List ℕ
⊢ (map Nat.succ (a :: l)).prod ≤ 2 ^ targetSum (a :: l) * (a :: l).length.factorial
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_general_lower_bound | [43, 1] | [50, 42] | exact Nat.mul_le_mul (succ_le_mul_two_pow_div a l.length.succ_pos)
(targetSum_general_lower_bound l) | a : ℕ
l : List ℕ
⊢ a.succ * (map Nat.succ l).prod ≤ 2 ^ (a / l.length.succ) * (l.length + 1) * (2 ^ targetSum l * l.length.factorial) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ
l : List ℕ
⊢ a.succ * (map Nat.succ l).prod ≤ 2 ^ (a / l.length.succ) * (l.length + 1) * (2 ^ targetSum l * l.length.factorial)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_perm_iota_n_lower_bound | [52, 1] | [57, 72] | have h0 := targetSum_general_lower_bound l | l : List ℕ
n : ℕ
h : l ~ iota n
⊢ n.size ≤ targetSum l | l : List ℕ
n : ℕ
h : l ~ iota n
h0 : (map Nat.succ l).prod ≤ 2 ^ targetSum l * l.length.factorial
⊢ n.size ≤ targetSum l | Please generate a tactic in lean4 to solve the state.
STATE:
l : List ℕ
n : ℕ
h : l ~ iota n
⊢ n.size ≤ targetSum l
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_perm_iota_n_lower_bound | [52, 1] | [57, 72] | rw [(h.map Nat.succ).prod_eq, h.length_eq,
prod_map_succ_iota, length_iota] at h0 | l : List ℕ
n : ℕ
h : l ~ iota n
h0 : (map Nat.succ l).prod ≤ 2 ^ targetSum l * l.length.factorial
⊢ n.size ≤ targetSum l | l : List ℕ
n : ℕ
h : l ~ iota n
h0 : (n + 1).factorial ≤ 2 ^ targetSum l * n.factorial
⊢ n.size ≤ targetSum l | Please generate a tactic in lean4 to solve the state.
STATE:
l : List ℕ
n : ℕ
h : l ~ iota n
h0 : (map Nat.succ l).prod ≤ 2 ^ targetSum l * l.length.factorial
⊢ n.size ≤ targetSum l
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_perm_iota_n_lower_bound | [52, 1] | [57, 72] | exact Nat.size_le.mpr (Nat.le_of_mul_le_mul_right h0 n.factorial_pos) | l : List ℕ
n : ℕ
h : l ~ iota n
h0 : (n + 1).factorial ≤ 2 ^ targetSum l * n.factorial
⊢ n.size ≤ targetSum l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
l : List ℕ
n : ℕ
h : l ~ iota n
h0 : (n + 1).factorial ≤ 2 ^ targetSum l * n.factorial
⊢ n.size ≤ targetSum l
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.iota_map_add_append_iota_eq_iota | [81, 1] | [85, 76] | rw [iota_succ, map_cons, cons_append] | n k : ℕ
⊢ map n.add (iota (k + 1)) ++ iota n = iota (n + (k + 1)) | n k : ℕ
⊢ n.add (k + 1) :: (map n.add (iota k) ++ iota n) = iota (n + (k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
⊢ map n.add (iota (k + 1)) ++ iota n = iota (n + (k + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.iota_map_add_append_iota_eq_iota | [81, 1] | [85, 76] | exact congr_arg₂ _ rfl (iota_map_add_append_iota_eq_iota n k) | n k : ℕ
⊢ n.add (k + 1) :: (map n.add (iota k) ++ iota n) = iota (n + (k + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
⊢ n.add (k + 1) :: (map n.add (iota k) ++ iota n) = iota (n + (k + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_perm_iota | [87, 1] | [99, 59] | rw [lowerBoundMk_bit0_succ, Nat.bit_false, Nat.bit0_val] | b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ lowerBoundMk (Nat.bit false (k + 1)) ~ iota (Nat.bit false (k + 1)) | b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ map (k + 1).add (iota k) ++ (2 * k + 2) :: lowerBoundMk (k + 1) ~ iota (2 * (k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ lowerBoundMk (Nat.bit false (k + 1)) ~ iota (Nat.bit false (k + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_perm_iota | [87, 1] | [99, 59] | refine perm_middle.trans (((h.append_left _).trans ?_).cons _) | b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ map (k + 1).add (iota k) ++ (2 * k + 2) :: lowerBoundMk (k + 1) ~ iota (2 * (k + 1)) | b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ map (k + 1).add (iota k) ++ iota (k + 1) ~ iota (Nat.mul 2 k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ map (k + 1).add (iota k) ++ (2 * k + 2) :: lowerBoundMk (k + 1) ~ iota (2 * (k + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_perm_iota | [87, 1] | [99, 59] | rw [iota_map_add_append_iota_eq_iota, add_right_comm, ← two_mul] | b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ map (k + 1).add (iota k) ++ iota (k + 1) ~ iota (Nat.mul 2 k + 1) | b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ iota (2 * k + 1) ~ iota (Nat.mul 2 k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ map (k + 1).add (iota k) ++ iota (k + 1) ~ iota (Nat.mul 2 k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_perm_iota | [87, 1] | [99, 59] | exact Perm.refl _ | b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ iota (2 * k + 1) ~ iota (Nat.mul 2 k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
n k : ℕ
h : lowerBoundMk (k + 1) ~ iota (k + 1)
⊢ iota (2 * k + 1) ~ iota (Nat.mul 2 k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_perm_iota | [87, 1] | [99, 59] | rw [lowerBoundMk_bit1, Nat.bit_true, Nat.bit1_val] | b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ lowerBoundMk (Nat.bit true n) ~ iota (Nat.bit true n) | b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ map n.add (iota n) ++ (2 * n + 1) :: lowerBoundMk n ~ iota (2 * n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ lowerBoundMk (Nat.bit true n) ~ iota (Nat.bit true n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_perm_iota | [87, 1] | [99, 59] | refine perm_middle.trans (((h.append_left _).trans ?_).cons _) | b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ map n.add (iota n) ++ (2 * n + 1) :: lowerBoundMk n ~ iota (2 * n + 1) | b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ map n.add (iota n) ++ iota n ~ iota (2 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ map n.add (iota n) ++ (2 * n + 1) :: lowerBoundMk n ~ iota (2 * n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_perm_iota | [87, 1] | [99, 59] | rw [iota_map_add_append_iota_eq_iota, ← two_mul] | b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ map n.add (iota n) ++ iota n ~ iota (2 * n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
n✝ n : ℕ
h : lowerBoundMk n ~ iota n
⊢ map n.add (iota n) ++ iota n ~ iota (2 * n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_map_add_iota_length_succ | [104, 1] | [112, 50] | rw [iota_succ, map_cons, cons_append, targetSum, length_cons,
targetSum_map_add_iota_length_succ h k, Nat.add_eq_right] | n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ targetSum (map n.add (iota (k + 1)) ++ l) = targetSum l | n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ n.add (k + 1) / (map n.add (iota k) ++ l).length.succ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ targetSum (map n.add (iota (k + 1)) ++ l) = targetSum l
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_map_add_iota_length_succ | [104, 1] | [112, 50] | refine Nat.div_eq_of_lt ?_ | n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ n.add (k + 1) / (map n.add (iota k) ++ l).length.succ = 0 | n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ n.add (k + 1) < (map n.add (iota k) ++ l).length.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ n.add (k + 1) / (map n.add (iota k) ++ l).length.succ = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.targetSum_map_add_iota_length_succ | [104, 1] | [112, 50] | rw [length_append, length_map, length_iota, h] | n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ n.add (k + 1) < (map n.add (iota k) ++ l).length.succ | n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ n.add (k + 1) < (k + n.succ).succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
l : List ℕ
h : l.length = n.succ
k : ℕ
⊢ n.add (k + 1) < (map n.add (iota k) ++ l).length.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_targetSum | [114, 1] | [137, 78] | have h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ :=
congr_arg Nat.succ (lowerBoundMk_length (k + 1)) | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
⊢ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ
⊢ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size | Please generate a tactic in lean4 to solve the state.
STATE:
X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
⊢ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_targetSum | [114, 1] | [137, 78] | rw [lowerBoundMk_bit0_succ, Nat.bit_false, Nat.size_bit0 k.succ_ne_zero,
(k + 1).size.succ_eq_add_one, targetSum_map_add_iota_length_succ h0,
targetSum, h0, h, add_comm, Nat.add_right_inj] | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ
⊢ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ
⊢ (2 * k + 2) / (k + 1).succ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ
⊢ targetSum (lowerBoundMk (Nat.bit false (k + 1))) = (Nat.bit false (k + 1)).size
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_targetSum | [114, 1] | [137, 78] | exact Nat.div_eq_of_lt_le
((one_mul _).trans_le <| Nat.add_le_add_right (X0 _) 2)
(Nat.mul_lt_mul_of_pos_left (k + 1).lt_succ_self X) | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ
⊢ (2 * k + 2) / (k + 1).succ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n k : ℕ
h : targetSum (lowerBoundMk (k + 1)) = (k + 1).size
h0 : ((2 * k + 2) :: lowerBoundMk (k + 1)).length = (k + 1).succ
⊢ (2 * k + 2) / (k + 1).succ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_targetSum | [114, 1] | [137, 78] | have h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ :=
congr_arg Nat.succ (lowerBoundMk_length n) | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
⊢ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ
⊢ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size | Please generate a tactic in lean4 to solve the state.
STATE:
X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
⊢ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_targetSum | [114, 1] | [137, 78] | rw [lowerBoundMk_bit1, Nat.bit_true, Nat.size_bit1,
targetSum_map_add_iota_length_succ h0, targetSum, h0, h,
n.size.succ_eq_add_one, add_comm, Nat.add_right_inj] | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ
⊢ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ
⊢ (2 * n + 1) / n.succ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ
⊢ targetSum (lowerBoundMk (Nat.bit true n)) = (Nat.bit true n).size
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2021/A3/A3.lean | IMOSL.IMO2021A3.lowerBoundMk_targetSum | [114, 1] | [137, 78] | exact Nat.div_eq_of_lt_le
((one_mul _).trans_le <| Nat.succ_le_succ (X0 _)) (Nat.le_refl _) | X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ
⊢ (2 * n + 1) / n.succ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : 0 < Nat.succ 1
X0 : ∀ (m : ℕ), m ≤ 2 * m
b : Bool
n✝ n : ℕ
h : targetSum (lowerBoundMk n) = n.size
h0 : ((2 * n + 1) :: lowerBoundMk n).length = n.succ
⊢ (2 * n + 1) / n.succ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.d_zero | [29, 1] | [30, 72] | rw [d, sum_range_one, Int.Nat.cast_ofNat_Int, Int.zero_mul, sub_zero] | z : ℕ → ℤ
⊢ d z 0 = z 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
⊢ d z 0 = z 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.d_succ | [32, 1] | [35, 52] | rw [d, sum_range_succ, Int.natCast_add, Int.natCast_one,
add_one_mul (α := ℤ), add_sub_add_right_eq_sub] | z : ℕ → ℤ
n : ℕ
⊢ d z (n + 1) = (range (n + 1)).sum z - ↑n * z (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
n : ℕ
⊢ d z (n + 1) = (range (n + 1)).sum z - ↑n * z (n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.d_one | [37, 1] | [38, 64] | rw [d_succ, sum_range_one, Nat.cast_zero, zero_mul, sub_zero] | z : ℕ → ℤ
⊢ d z 1 = z 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
⊢ d z 1 = z 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.main_lemma | [42, 1] | [45, 79] | rw [d_succ, d, sub_sub, ← mul_add_one (α := ℤ)] | z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ d z (n + 1) ≤ d z n - ↑n | z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ (range (n + 1)).sum z - ↑n * z (n + 1) ≤ (range (n + 1)).sum z - ↑n * (z n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ d z (n + 1) ≤ d z n - ↑n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.main_lemma | [42, 1] | [45, 79] | exact Int.sub_le_sub_left
(Int.mul_le_mul_of_nonneg_left (h n.lt_succ_self) (Int.ofNat_zero_le n)) _ | z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ (range (n + 1)).sum z - ↑n * z (n + 1) ≤ (range (n + 1)).sum z - ↑n * (z n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ (range (n + 1)).sum z - ↑n * z (n + 1) ≤ (range (n + 1)).sum z - ↑n * (z n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.binom_bound | [47, 1] | [53, 76] | rw [Nat.choose, Nat.choose_one_right, Nat.cast_add,
← sub_sub, sub_right_comm] | z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ d z (n + 1) ≤ z 0 - ↑((n + 1).choose 2) | z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ d z (n + 1) ≤ z 0 - ↑(n.choose (1 + 1)) - ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ d z (n + 1) ≤ z 0 - ↑((n + 1).choose 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.binom_bound | [47, 1] | [53, 76] | exact Int.le_sub_right_of_add_le <|
(Int.add_le_of_le_sub_right (main_lemma h n)).trans (binom_bound n) | z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ d z (n + 1) ≤ z 0 - ↑(n.choose (1 + 1)) - ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
h : StrictMono z
n : ℕ
⊢ d z (n + 1) ≤ z 0 - ↑(n.choose (1 + 1)) - ↑n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.final_solution_part2 | [103, 1] | [107, 37] | rw [eq_greatestDPos_iff h h0, d_succ] | z : ℕ → ℤ
h : StrictMono z
h0 : 0 < z 0
N : ℕ
⊢ N = greatestDPos h ↔ ↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≤ ↑N * z (N + 1) | z : ℕ → ℤ
h : StrictMono z
h0 : 0 < z 0
N : ℕ
⊢ 0 < d z N ∧ (range (N + 1)).sum z - ↑N * z (N + 1) ≤ 0 ↔
↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≤ ↑N * z (N + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
h : StrictMono z
h0 : 0 < z 0
N : ℕ
⊢ N = greatestDPos h ↔ ↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≤ ↑N * z (N + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2014/A1/A1.lean | IMOSL.IMO2014A1.final_solution_part2 | [103, 1] | [107, 37] | exact and_congr sub_pos sub_nonpos | z : ℕ → ℤ
h : StrictMono z
h0 : 0 < z 0
N : ℕ
⊢ 0 < d z N ∧ (range (N + 1)).sum z - ↑N * z (N + 1) ≤ 0 ↔
↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≤ ↑N * z (N + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℕ → ℤ
h : StrictMono z
h0 : 0 < z 0
N : ℕ
⊢ 0 < d z N ∧ (range (N + 1)).sum z - ↑N * z (N + 1) ≤ 0 ↔
↑N * z N < (range (N + 1)).sum z ∧ (range (N + 1)).sum z ≤ ↑N * z (N + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq1 | [30, 1] | [32, 87] | have h0 := hf.is_good (x + 1) (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = f (x + 1) * f (-1) + f x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f (x + 1) * f (-1) + f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = f (x + 1) * f (-1) + f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq1 | [30, 1] | [32, 87] | rwa [add_neg_cancel_right, mul_neg_one (x + 1), neg_add, neg_add_cancel_right] at h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f (x + 1) * f (-1) + f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f ((x + 1) * -1 + 1) = f (x + 1) * f (-1) + f (x + 1 + -1)
⊢ f (-x) = f (x + 1) * f (-1) + f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | replace hf := Eq1 hf | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = -f (x + 2) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
⊢ f (-x) = -f (x + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | have h0 := hf (-(x + 1)) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
⊢ f (-x) = -f (x + 2) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (- -(x + 1)) = f (-(x + 1) + 1) * f (-1) + f (-(x + 1))
⊢ f (-x) = -f (x + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | rw [neg_neg, hf (x + 1), neg_add, neg_add_cancel_right, ← add_assoc, self_eq_add_left,
← add_mul, mul_eq_zero, or_iff_left h, add_assoc, one_add_one_eq_two] at h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (- -(x + 1)) = f (-(x + 1) + 1) * f (-1) + f (-(x + 1))
⊢ f (-x) = -f (x + 2) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (-x) + f (x + 2) = 0
⊢ f (-x) = -f (x + 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (- -(x + 1)) = f (-(x + 1) + 1) * f (-1) + f (-(x + 1))
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq2 | [35, 1] | [40, 38] | exact eq_neg_of_add_eq_zero_left h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (-x) + f (x + 2) = 0
⊢ f (-x) = -f (x + 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
h : f (-1) ≠ 0
x : R
hf : ∀ (x : R), f (-x) = f (x + 1) * f (-1) + f x
h0 : f (-x) + f (x + 2) = 0
⊢ f (-x) = -f (x + 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_two | [42, 1] | [43, 66] | rw [← zero_add 2, ← neg_inj, ← Eq2 hf h, neg_zero, hf.map_zero] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f 2 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq3 | [46, 1] | [47, 54] | rw [neg_add, ← hf.is_good, neg_mul_neg, hf.is_good] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x y : R
⊢ f (-x) * f (-y) + f (-(x + y)) = f x * f y + f (x + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x y : R
⊢ f (-x) * f (-y) + f (-(x + y)) = f x * f y + f (x + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq4 | [50, 1] | [51, 79] | rw [hf.is_good, map_two hf h, one_mul, add_comm 2, Eq2 hf h, sub_neg_eq_add] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (2 * x + 1) = f x - f (-x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (2 * x + 1) = f x - f (-x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | have h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) := by
have h0 : x * -x = -x * x := by rw [mul_neg, neg_mul]
rw [sub_mul, mul_add, mul_add, map_commute_of_commute hf.is_good h0,
add_comm, add_sub_add_left_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | have h1 := Eq3 hf x x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f (-x) * f (-x) + f (-(x + x)) = f x * f x + f (x + x)
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [add_comm, ← sub_eq_sub_iff_add_eq_add, h0, Eq1 hf, add_sub_cancel_right,
← two_mul, Eq4 hf h, ← sub_eq_zero, ← mul_sub, mul_eq_zero] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f (-x) * f (-x) + f (-(x + x)) = f x * f x + f (x + x)
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f (-x) * f (-x) + f (-(x + x)) = f x * f x + f (x + x)
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | refine h1.imp (λ h1 ↦ ?_) (λ h1 ↦ (eq_of_sub_eq_zero h1).symm) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f x - f (-x) = 0
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1 : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
⊢ f (x + 1) = 0 ∨ f x + f (-x) = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [Eq1 hf, sub_add_cancel_right, neg_eq_zero, mul_eq_zero] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f x - f (-x) = 0
⊢ f (x + 1) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f (x + 1) = 0 ∨ f (-1) = 0
⊢ f (x + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f x - f (-x) = 0
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | exact h1.resolve_right h | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f (x + 1) = 0 ∨ f (-1) = 0
⊢ f (x + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
h1✝ : f x - f (-x) = 0 ∨ f (-1) - (f x + f (-x)) = 0
h1 : f (x + 1) = 0 ∨ f (-1) = 0
⊢ f (x + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | have h0 : x * -x = -x * x := by rw [mul_neg, neg_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : x * -x = -x * x
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [sub_mul, mul_add, mul_add, map_commute_of_commute hf.is_good h0,
add_comm, add_sub_add_left_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : x * -x = -x * x
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : x * -x = -x * x
⊢ f x * f x - f (-x) * f (-x) = (f x - f (-x)) * (f x + f (-x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq5 | [54, 1] | [66, 27] | rw [mul_neg, neg_mul] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ x * -x = -x * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
⊢ x * -x = -x * x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | have h1 := Eq1 hf x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [h0, zero_mul, zero_add] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | have h2 := Eq3 hf x (-(x + 1)) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : f (-x) * f (- -(x + 1)) + f (-(x + -(x + 1))) = f x * f (-(x + 1)) + f (x + -(x + 1))
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [neg_neg, h0, mul_zero, zero_add, ← sub_eq_add_neg,
sub_add_cancel_left, neg_neg, hf.map_one] at h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : f (-x) * f (- -(x + 1)) + f (-(x + -(x + 1))) = f x * f (-(x + 1)) + f (x + -(x + 1))
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : f (-x) * f (- -(x + 1)) + f (-(x + -(x + 1))) = f x * f (-(x + 1)) + f (x + -(x + 1))
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | have h3 := Eq3 hf x (x + 1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-x) * f (-(x + 1)) + f (-(x + (x + 1))) = f x * f (x + 1) + f (x + (x + 1))
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [h0, mul_zero, zero_add, ← add_assoc, ← two_mul,
Eq4 hf h, h1, sub_self, h2, add_right_inj] at h3 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-x) * f (-(x + 1)) + f (-(x + (x + 1))) = f x * f (x + 1) + f (x + (x + 1))
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-(2 * x + 1)) = f (-1)
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-x) * f (-(x + 1)) + f (-(x + (x + 1))) = f x * f (x + 1) + f (x + (x + 1))
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | replace h2 := Eq4 hf h (-x - 1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-(2 * x + 1)) = f (-1)
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f (2 * (-x - 1) + 1) = f (-x - 1) - f (-(-x - 1))
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h2 : 0 = f x * f (-(x + 1)) + f (-1)
h3 : f (-(2 * x + 1)) = f (-1)
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | rw [two_mul, add_assoc, sub_add_cancel, ← add_sub_right_comm, ← two_mul, mul_neg,
← neg_add', h3, eq_comm, Eq1 hf, sub_add_cancel_right, sub_add_cancel,
neg_eq_iff_add_eq_zero, h1, ← add_one_mul (f x), mul_eq_zero] at h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f (2 * (-x - 1) + 1) = f (-x - 1) - f (-(-x - 1))
⊢ f x = -1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f x + 1 = 0 ∨ f (-1) = 0
⊢ f x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f (2 * (-x - 1) + 1) = f (-x - 1) - f (-(-x - 1))
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Eq6 | [69, 1] | [86, 56] | exact eq_neg_of_add_eq_zero_left (h2.resolve_right h) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f x + 1 = 0 ∨ f (-1) = 0
⊢ f x = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
x : R
h0 : f (x + 1) = 0
h1 : f (-x) = f x
h3 : f (-(2 * x + 1)) = f (-1)
h2 : f x + 1 = 0 ∨ f (-1) = 0
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | rw [← sub_eq_zero (b := 1), eq_neg_iff_add_eq_zero, ← mul_eq_zero,
mul_sub_one, add_mul, two_mul, add_sub_assoc, add_sub_add_left_eq_sub] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) = -2 ∨ f (-1) = 1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) = -2 ∨ f (-1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | have h0 := Eq5 hf h (-1 + -1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1 + 1) = 0 ∨ f (-1 + -1) + f (-(-1 + -1)) = f (-1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | rw [neg_add_cancel_right, or_iff_right h, ← eq_sub_iff_add_eq] at h0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1 + 1) = 0 ∨ f (-1 + -1) + f (-(-1 + -1)) = f (-1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1 + 1) = 0 ∨ f (-1 + -1) + f (-(-1 + -1)) = f (-1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | have h1 := hf.is_good (-1) (-1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
h1 : f (-1 * -1 + 1) = f (-1) * f (-1) + f (-1 + -1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.map_neg_one_cases | [88, 1] | [95, 78] | rwa [h0, neg_mul_neg, one_mul, ← neg_add, neg_neg, one_add_one_eq_two, map_two hf h,
eq_comm, ← sub_eq_zero, add_sub_assoc, sub_sub, one_add_one_eq_two] at h1 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
h1 : f (-1 * -1 + 1) = f (-1) * f (-1) + f (-1 + -1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1 + -1) = f (-1) - f (-(-1 + -1))
h1 : f (-1 * -1 + 1) = f (-1) * f (-1) + f (-1 + -1)
⊢ f (-1) * f (-1) + (f (-1) - 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | rcases Eq5 hf h x with h2 | h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
⊢ f (x + 1) = f x + 1 | case inl
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f (x + 1) = 0
⊢ f (x + 1) = f x + 1
case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f x + f (-x) = f (-1)
⊢ f (x + 1) = f x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | rw [Eq6 hf h h2, h2, neg_add_self] | case inl
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f (x + 1) = 0
⊢ f (x + 1) = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f (x + 1) = 0
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | rw [Eq1 hf, h0, add_left_comm, ← mul_two, mul_neg, ← neg_mul, ← add_mul,
eq_neg_iff_add_eq_zero, ← add_one_mul _ (2 : S), mul_eq_zero, add_assoc] at h2 | case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f x + f (-x) = f (-1)
⊢ f (x + 1) = f x + 1 | case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : -f (x + 1) + (f x + 1) = 0 ∨ 2 = 0
⊢ f (x + 1) = f x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : f x + f (-x) = f (-1)
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.Subcase11_solution | [101, 1] | [110, 88] | exact neg_add_eq_zero.mp (h2.resolve_right (neg_ne_zero.mp (h0.symm.trans_ne h))) | case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : -f (x + 1) + (f x + 1) = 0 ∨ 2 = 0
⊢ f (x + 1) = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : f (-1) ≠ 0
h0 : f (-1) = -2
x : R
h2 : -f (x + 1) + (f x + 1) = 0 ∨ 2 = 0
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.map_neg_one_ne_zero | [126, 1] | [127, 77] | rw [← mul_one (3 : S), ← hf.map_neg_one, h, mul_zero] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : f (-1) = 0
⊢ 3 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : f (-1) = 0
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | have h0 (x) : f (-x) = f (x + 1) + f x := by
rw [Eq1 hf.toNontrivialGood, hf.map_neg_one, mul_one] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | have h1 : f (-x) = f x := by rw [h0, h, zero_add] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | refine hf.period_imp_zero λ y ↦ ?_ | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
⊢ x = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
⊢ f (y + x) = f y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
⊢ x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | have h2 := Eq3 hf.toNontrivialGood x (y - 1) | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
⊢ f (y + x) = f y | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
h2 : f (-x) * f (-(y - 1)) + f (-(x + (y - 1))) = f x * f (y - 1) + f (x + (y - 1))
⊢ f (y + x) = f y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
⊢ f (y + x) = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | rwa [h1, Case1.Eq6 hf.toNontrivialGood (map_neg_one_ne_zero hf) h, neg_one_mul,
neg_one_mul, h0, h0 (x + _), add_assoc, sub_add_cancel, ← add_assoc, add_left_inj,
neg_add_rev, add_assoc, add_right_eq_self, neg_add_eq_zero, eq_comm, add_comm] at h2 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
h2 : f (-x) * f (-(y - 1)) + f (-(x + (y - 1))) = f x * f (y - 1) + f (x + (y - 1))
⊢ f (y + x) = f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
h1 : f (-x) = f x
y : R
h2 : f (-x) * f (-(y - 1)) + f (-(x + (y - 1))) = f x * f (y - 1) + f (x + (y - 1))
⊢ f (y + x) = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | rw [Eq1 hf.toNontrivialGood, hf.map_neg_one, mul_one] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : f (x✝ + 1) = 0
x : R
⊢ f (-x) = f (x + 1) + f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : f (x✝ + 1) = 0
x : R
⊢ f (-x) = f (x + 1) + f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.eq_zero_of_map_add_one | [129, 1] | [138, 89] | rw [h0, h, zero_add] | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ f (-x) = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (x + 1) = 0
h0 : ∀ (x : R), f (-x) = f (x + 1) + f x
⊢ f (-x) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.triple_sum_eq_zero | [144, 1] | [147, 68] | have h := Eq1 hf.toNontrivialGood x | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f x + f (x + 1) + f (x + 2) = 0 | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x + f (x + 1) + f (x + 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f x + f (x + 1) + f (x + 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.triple_sum_eq_zero | [144, 1] | [147, 68] | rwa [Case1.Eq2 hf.toNontrivialGood (map_neg_one_ne_zero hf), hf.map_neg_one,
mul_one, neg_eq_iff_add_eq_zero, add_comm, add_comm (f _)] at h | R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x + f (x + 1) + f (x + 2) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : f (-x) = f (x + 1) * f (-1) + f x
⊢ f x + f (x + 1) + f (x + 2) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | refine hf.period_imp_zero λ x ↦ ?_ | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
⊢ 3 = 0 | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f (x + 3) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
⊢ 3 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | have h := triple_sum_eq_zero hf | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f (x + 3) = f x | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
⊢ f (x + 3) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ f (x + 3) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | have h0 := h (x + 1) | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
⊢ f (x + 3) = f x | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
h0 : f (x + 1) + f (x + 1 + 1) + f (x + 1 + 2) = 0
⊢ f (x + 3) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
⊢ f (x + 3) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.Rchar | [149, 1] | [154, 69] | rwa [add_assoc x, one_add_one_eq_two, ← add_rotate, ← h x, add_left_inj,
add_left_inj, add_assoc, add_comm 1, two_add_one_eq_three] at h0 | R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
h0 : f (x + 1) + f (x + 1 + 1) + f (x + 1 + 2) = 0
⊢ f (x + 3) = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.32555
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), f x + f (x + 1) + f (x + 2) = 0
h0 : f (x + 1) + f (x + 1 + 1) + f (x + 1 + 2) = 0
⊢ f (x + 3) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | have h := eq_zero_or_map_neg_add_self hf | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ x = 0 ∨ x = 1 ∨ x = -1 | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
⊢ x = 0 ∨ x = 1 ∨ x = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
⊢ x = 0 ∨ x = 1 ∨ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | refine (h x).imp_right λ h0 ↦ (h (x - 1)).imp eq_of_sub_eq_zero
λ h1 ↦ eq_neg_of_add_eq_zero_left <| (h (x + 1)).resolve_right λ h2 ↦ ?_ | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
⊢ x = 0 ∨ x = 1 ∨ x = -1 | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
⊢ x = 0 ∨ x = 1 ∨ x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | replace h (x) : f x + f (x + 1) + f (x - 1) = 0 := by
have h3 : (2 : R) = -1 := by
rw [eq_neg_iff_add_eq_zero, two_add_one_eq_three, Rchar hf]
rw [sub_eq_add_neg, ← h3, triple_sum_eq_zero hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
⊢ False | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : ∀ (x : R), f x + f (x + 1) + f (x - 1) = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | rw [add_zero, ← neg_add', add_right_comm (f _), add_add_add_comm,
add_add_add_comm (f _), h0, h2, neg_add_eq_sub, ← neg_sub x, h1,
one_add_one_eq_two, two_add_one_eq_three] at h | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : f x + f (x + 1) + f (x - 1) + (f (-x) + f (-x + 1) + f (-x - 1)) = 0 + 0
⊢ False | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : 3 = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : f x + f (x + 1) + f (x - 1) + (f (-x) + f (-x + 1) + f (-x - 1)) = 0 + 0
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | exact hf.Schar h | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : 3 = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x : R
h0 : f x + f (-x) = 1
h1 : f (x - 1) + f (-(x - 1)) = 1
h2 : f (x + 1) + f (-(x + 1)) = 1
h : 3 = 0
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | have h3 : (2 : R) = -1 := by
rw [eq_neg_iff_add_eq_zero, two_add_one_eq_three, Rchar hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ f x + f (x + 1) + f (x - 1) = 0 | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
h3 : 2 = -1
⊢ f x + f (x + 1) + f (x - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ f x + f (x + 1) + f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | rw [sub_eq_add_neg, ← h3, triple_sum_eq_zero hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
h3 : 2 = -1
⊢ f x + f (x + 1) + f (x - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
h3 : 2 = -1
⊢ f x + f (x + 1) + f (x - 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.value_bash | [156, 1] | [171, 19] | rw [eq_neg_iff_add_eq_zero, two_add_one_eq_three, Rchar hf] | R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ 2 = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type ?u.34262
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
x✝ : R
h : ∀ (x : R), x = 0 ∨ f x + f (-x) = 1
h0 : f x✝ + f (-x✝) = 1
h1 : f (x✝ - 1) + f (-(x✝ - 1)) = 1
h2 : f (x✝ + 1) + f (-(x✝ + 1)) = 1
x : R
⊢ 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.solution | [173, 1] | [186, 78] | rw [h, neg_zero, ← h, hf.map_one] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : 1 = 0
⊢ f (-1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : 1 = 0
⊢ f (-1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.solution | [173, 1] | [186, 78] | change f 0 = ((-1 : ℤ) : S) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 𝔽₃.𝔽₃0.cast = ↑(𝔽₃Map1 𝔽₃.𝔽₃0) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 𝔽₃.𝔽₃0.cast = ↑(𝔽₃Map1 𝔽₃.𝔽₃0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.GoodSubcase12.solution | [173, 1] | [186, 78] | rw [hf.map_zero, Int.cast_neg, Int.cast_one] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : GoodSubcase12 f
h : Function.Bijective 𝔽₃.cast
⊢ f 0 = ↑(-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case1.lean | IMOSL.IMO2012A5.Case1.solution | [194, 1] | [203, 51] | rwa [← two_add_one_eq_three, Ne,
← neg_eq_iff_add_eq_zero, ← h1, eq_comm] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) ≠ 0
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : ¬f (-1) = -2
h1 : f (-1) = 1
⊢ 3 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) ≠ 0
hf' : NontrivialGood f := hf.toNontrivialGood
h0 : ¬f (-1) = -2
h1 : f (-1) = 1
⊢ 3 ≠ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/Basic.lean | IMOSL.Extra.Infinitesimal.add | [30, 1] | [35, 33] | apply (nsmul_le_nsmul_right (abs_add ε₁ ε₂) k).trans_lt | R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ k • |ε₁ + ε₂| < 1 | R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ k • (|ε₁| + |ε₂|) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ k • |ε₁ + ε₂| < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/Basic.lean | IMOSL.Extra.Infinitesimal.add | [30, 1] | [35, 33] | apply lt_of_nsmul_lt_nsmul_right 2 | R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ k • (|ε₁| + |ε₂|) < 1 | R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ 2 • k • (|ε₁| + |ε₂|) < 2 • 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ k • (|ε₁| + |ε₂|) < 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/Basic.lean | IMOSL.Extra.Infinitesimal.add | [30, 1] | [35, 33] | rw [← mul_nsmul', nsmul_add, two_nsmul] | R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ 2 • k • (|ε₁| + |ε₂|) < 2 • 1 | R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ (2 * k) • |ε₁| + (2 * k) • |ε₂| < 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ 2 • k • (|ε₁| + |ε₂|) < 2 • 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/Basic.lean | IMOSL.Extra.Infinitesimal.add | [30, 1] | [35, 33] | exact add_lt_add (h₁ _) (h₂ _) | R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ (2 * k) • |ε₁| + (2 * k) • |ε₂| < 1 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
ε₁ ε₂ : R
h₁ : Infinitesimal ε₁
h₂ : Infinitesimal ε₂
k : ℕ
⊢ (2 * k) • |ε₁| + (2 * k) • |ε₂| < 1 + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/Basic.lean | IMOSL.Extra.Infinitesimal.mul_of_abs_le_one_left | [53, 1] | [56, 87] | rw [abs_mul, nsmul_eq_mul', mul_assoc, ← nsmul_eq_mul'] | R : Type u_1
inst✝ : LinearOrderedRing R
ε r : R
hε : Infinitesimal ε
hr : |r| ≤ 1
k : ℕ
⊢ k • |r * ε| < 1 | R : Type u_1
inst✝ : LinearOrderedRing R
ε r : R
hε : Infinitesimal ε
hr : |r| ≤ 1
k : ℕ
⊢ |r| * k • |ε| < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
ε r : R
hε : Infinitesimal ε
hr : |r| ≤ 1
k : ℕ
⊢ k • |r * ε| < 1
TACTIC:
|
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