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/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.mul_assoc | [210, 1] | [213, 68] | rw [𝔽₂X.mul_add, 𝔽₂X.mul_add, 𝔽₂X.mul_add, h, h0] | P Q P✝ Q✝ : 𝔽₂X
h : P * Q * P✝ = P * (Q * P✝)
h0 : P * Q * Q✝ = P * (Q * Q✝)
⊢ P * Q * (P✝ + Q✝) = P * (Q * (P✝ + Q✝)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P Q P✝ Q✝ : 𝔽₂X
h : P * Q * P✝ = P * (Q * P✝)
h0 : P * Q * Q✝ = P * (Q * Q✝)
⊢ P * Q * (P✝ + Q✝) = P * (Q * (P✝ + Q✝))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.square_eq_mul_self | [235, 1] | [238, 73] | rw [square_add, square_Xpow, 𝔽₂X.add_mul, 𝔽₂X.mul_add, ← Xpow_add, Nat.two_mul,
𝔽₂X.mul_add, ← h, ← P.mul_comm, CharTwo.add_add_add_cancel_middle] | n : ℕ
P : 𝔽₂X
h : P.square = P * P
⊢ (Xpow n + P).square = (Xpow n + P) * (Xpow n + P) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
P : 𝔽₂X
h : P.square = P * P
⊢ (Xpow n + P).square = (Xpow n + P) * (Xpow n + P)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.square_XpowMul | [240, 1] | [243, 82] | unfold square XpowMul | n : ℕ
P : 𝔽₂X
⊢ (XpowMul n P).square = XpowMul (2 * n) P.square | n : ℕ
P : 𝔽₂X
⊢ { toFinset := Finset.image (fun n => 2 * n) { toFinset := Finset.image (fun k => k + n) P.toFinset }.toFinset } =
{ toFinset := Finset.image (fun k => k + 2 * n) { toFinset := Finset.image (fun n => 2 * n) P.toFinset }.toFinset } | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
P : 𝔽₂X
⊢ (XpowMul n P).square = XpowMul (2 * n) P.square
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.square_XpowMul | [240, 1] | [243, 82] | rw [Finset.image_image, Finset.image_image] | n : ℕ
P : 𝔽₂X
⊢ { toFinset := Finset.image (fun n => 2 * n) { toFinset := Finset.image (fun k => k + n) P.toFinset }.toFinset } =
{ toFinset := Finset.image (fun k => k + 2 * n) { toFinset := Finset.image (fun n => 2 * n) P.toFinset }.toFinset } | n : ℕ
P : 𝔽₂X
⊢ { toFinset := Finset.image ((fun n => 2 * n) ∘ fun k => k + n) P.toFinset } =
{ toFinset := Finset.image ((fun k => k + 2 * n) ∘ fun n => 2 * n) P.toFinset } | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
P : 𝔽₂X
⊢ { toFinset := Finset.image (fun n => 2 * n) { toFinset := Finset.image (fun k => k + n) P.toFinset }.toFinset } =
{ toFinset := Finset.image (fun k => k + 2 * n) { toFinset := Finset.image (fun n => 2 * n) P.toFinset }.toFinset }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.square_XpowMul | [240, 1] | [243, 82] | exact 𝔽₂X.ext _ _ (congrArg P.toFinset.image <| funext λ n ↦ Nat.mul_add 2 _ _) | n : ℕ
P : 𝔽₂X
⊢ { toFinset := Finset.image ((fun n => 2 * n) ∘ fun k => k + n) P.toFinset } =
{ toFinset := Finset.image ((fun k => k + 2 * n) ∘ fun n => 2 * n) P.toFinset } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
P : 𝔽₂X
⊢ { toFinset := Finset.image ((fun n => 2 * n) ∘ fun k => k + n) P.toFinset } =
{ toFinset := Finset.image ((fun k => k + 2 * n) ∘ fun n => 2 * n) P.toFinset }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.square_mul | [245, 1] | [249, 77] | rw [← XpowMul_eq_mul_Xpow, square_XpowMul,
square_Xpow, XpowMul_eq_mul_Xpow] | P : 𝔽₂X
n : ℕ
⊢ (P * Xpow n).square = P.square * (Xpow n).square | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P : 𝔽₂X
n : ℕ
⊢ (P * Xpow n).square = P.square * (Xpow n).square
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.square_mul | [245, 1] | [249, 77] | rw [P.mul_add, square_add, h, h0, square_add, 𝔽₂X.mul_add] | P P✝ Q✝ : 𝔽₂X
h : (P * P✝).square = P.square * P✝.square
h0 : (P * Q✝).square = P.square * Q✝.square
⊢ (P * (P✝ + Q✝)).square = P.square * (P✝ + Q✝).square | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P P✝ Q✝ : 𝔽₂X
h : (P * P✝).square = P.square * P✝.square
h0 : (P * Q✝).square = P.square * Q✝.square
⊢ (P * (P✝ + Q✝)).square = P.square * (P✝ + Q✝).square
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_of_ne_zero | [266, 1] | [269, 28] | rw [𝔽₂X.natPow, if_neg h] | n : ℕ
P : 𝔽₂X
h : n ≠ 0
⊢ P.natPow n = if n % 2 = 0 then P.square.natPow (n / 2) else P.square.natPow (n / 2) * P | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
P : 𝔽₂X
h : n ≠ 0
⊢ P.natPow n = if n % 2 = 0 then P.square.natPow (n / 2) else P.square.natPow (n / 2) * P
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul | [271, 1] | [277, 62] | rcases Decidable.eq_or_ne n 0 with rfl | h | P : 𝔽₂X
n : ℕ
⊢ P.natPow (2 * n) = P.square.natPow n | case inl
P : 𝔽₂X
⊢ P.natPow (2 * 0) = P.square.natPow 0
case inr
P : 𝔽₂X
n : ℕ
h : n ≠ 0
⊢ P.natPow (2 * n) = P.square.natPow n | Please generate a tactic in lean4 to solve the state.
STATE:
P : 𝔽₂X
n : ℕ
⊢ P.natPow (2 * n) = P.square.natPow n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul | [271, 1] | [277, 62] | rw [Nat.mul_zero, 𝔽₂X.natPow_zero, 𝔽₂X.natPow_zero] | case inl
P : 𝔽₂X
⊢ P.natPow (2 * 0) = P.square.natPow 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
P : 𝔽₂X
⊢ P.natPow (2 * 0) = P.square.natPow 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul | [271, 1] | [277, 62] | have h0 : 0 < 2 := Nat.two_pos | case inr
P : 𝔽₂X
n : ℕ
h : n ≠ 0
⊢ P.natPow (2 * n) = P.square.natPow n | case inr
P : 𝔽₂X
n : ℕ
h : n ≠ 0
h0 : 0 < 2
⊢ P.natPow (2 * n) = P.square.natPow n | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
P : 𝔽₂X
n : ℕ
h : n ≠ 0
⊢ P.natPow (2 * n) = P.square.natPow n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul | [271, 1] | [277, 62] | rw [P.natPow_of_ne_zero (Nat.mul_ne_zero h0.ne.symm h),
if_pos (Nat.mul_mod_right _ _), Nat.mul_div_right _ h0] | case inr
P : 𝔽₂X
n : ℕ
h : n ≠ 0
h0 : 0 < 2
⊢ P.natPow (2 * n) = P.square.natPow n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
P : 𝔽₂X
n : ℕ
h : n ≠ 0
h0 : 0 < 2
⊢ P.natPow (2 * n) = P.square.natPow n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_two_mul_add_one | [279, 1] | [283, 51] | rw [P.natPow_of_ne_zero (2 * n).add_one_ne_zero, Nat.mul_add_mod,
Nat.mul_add_div Nat.two_pos, if_neg Nat.one_ne_zero,
Nat.div_eq_of_lt Nat.one_lt_two, Nat.add_zero] | P : 𝔽₂X
n : ℕ
⊢ P.natPow (2 * n + 1) = P.square.natPow n * P | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P : 𝔽₂X
n : ℕ
⊢ P.natPow (2 * n + 1) = P.square.natPow n * P
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | rw [← n.div_add_mod 2] | P : 𝔽₂X
n : ℕ
⊢ P.natPow n.succ = P.natPow n * P | P : 𝔽₂X
n : ℕ
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P | Please generate a tactic in lean4 to solve the state.
STATE:
P : 𝔽₂X
n : ℕ
⊢ P.natPow n.succ = P.natPow n * P
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | rcases n.mod_two_eq_zero_or_one with h0 | h0 | P : 𝔽₂X
n : ℕ
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P | case inl
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 0
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
case inr
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 1
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P | Please generate a tactic in lean4 to solve the state.
STATE:
P : 𝔽₂X
n : ℕ
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | rw [h0, Nat.add_zero, 𝔽₂X.natPow_two_mul_add_one, 𝔽₂X.natPow_two_mul] | case inl
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 0
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 0
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | rw [h0, Nat.succ_eq_add_one, ← Nat.mul_succ 2, 𝔽₂X.natPow_two_mul,
𝔽₂X.natPow_two_mul_add_one, 𝔽₂X.mul_assoc, ← P.square_eq_mul_self] | case inr
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 1
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P | case inr
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 1
⊢ P.square.natPow (n / 2).succ = P.square.natPow (n / 2) * P.square | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 1
⊢ P.natPow (2 * (n / 2) + n % 2).succ = P.natPow (2 * (n / 2) + n % 2) * P
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | exact (square P).natPow_succ (n / 2) | case inr
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 1
⊢ P.square.natPow (n / 2).succ = P.square.natPow (n / 2) * P.square | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
P : 𝔽₂X
n : ℕ
h0 : n % 2 = 1
⊢ P.square.natPow (n / 2).succ = P.square.natPow (n / 2) * P.square
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | apply Nat.bitwise_rec_lemma | P : 𝔽₂X
n : ℕ
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : n % 2 = 1
⊢ (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 ⟨P.square, n / 2⟩ ⟨P, n⟩ | case hNe
P : 𝔽₂X
n : ℕ
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : n % 2 = 1
⊢ n ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
P : 𝔽₂X
n : ℕ
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : n % 2 = 1
⊢ (invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 ⟨P.square, n / 2⟩ ⟨P, n⟩
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | rintro rfl | case hNe
P : 𝔽₂X
n : ℕ
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : n % 2 = 1
⊢ n ≠ 0 | case hNe
P : 𝔽₂X
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, 0⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : 0 % 2 = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case hNe
P : 𝔽₂X
n : ℕ
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, n⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : n % 2 = 1
⊢ n ≠ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/Defs.lean | IMOSL.IMO2012A5.𝔽₂X.natPow_succ | [291, 1] | [299, 63] | exact absurd h0.symm Nat.one_ne_zero | case hNe
P : 𝔽₂X
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, 0⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : 0 % 2 = 1
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hNe
P : 𝔽₂X
a✝ :
∀ (y : (_ : 𝔽₂X) ×' ℕ),
(invImage (fun x => PSigma.casesOn x fun P n => n) instWellFoundedRelationOfSizeOf).1 y ⟨P, 0⟩ →
y.1.natPow y.2.succ = y.1.natPow y.2 * y.1
h0 : 0 % 2 = 1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.add_mul | [231, 11] | [232, 58] | rw [ℤ₄.mul_comm, ℤ₄.mul_add, z.mul_comm, z.mul_comm] | x y z : ℤ₄
⊢ (x + y) * z = x * z + y * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : ℤ₄
⊢ (x + y) * z = x * z + y * z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_add | [259, 1] | [274, 47] | rw [← h, ← Nat.cast_two, ← Nat.cast_add] | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0 | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_add | [259, 1] | [274, 47] | rfl | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_add | [259, 1] | [274, 47] | rwa [neg_eq_iff_add_eq_zero, ← add_assoc, h0] | R : Type u_1
inst✝ : NonAssocRing R
h✝ : 4 = 0
x y : ℤ₄
h : 2 + 2 = 0
h0 : 1 + 1 = 2
⊢ -1 = 1 + 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h✝ : 4 = 0
x y : ℤ₄
h : 2 + 2 = 0
h0 : 1 + 1 = 2
⊢ -1 = 1 + 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_mul | [276, 1] | [287, 64] | rw [← h, ← Nat.cast_two, ← Nat.cast_add] | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0 | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_mul | [276, 1] | [287, 64] | rfl | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.castRingHom_injective | [296, 1] | [303, 43] | rw [← one_mul (2 : R), h1, zero_mul] | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
h0 : 2 ≠ 0
h1 : 1 = 0
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
h0 : 2 ≠ 0
h1 : 1 = 0
⊢ 2 = 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, ← sub_eq_add_neg] | R : Type u_1
inst✝ : NonAssocRing R
x y : R
⊢ (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
inst✝ : NonAssocRing R
x y : R
⊢ (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
inst✝ : NonAssocRing R
f : R → R
h : good f
x y : R
⊢ (-f) ((-f) x * (-f) y) + (-f) (x + y) = (-f) (x * y) | R : Type u_1
inst✝ : NonAssocRing R
f : R → R
h : good f
x y : R
⊢ -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
inst✝ : NonAssocRing R
f : R → R
h : good f
x y : R
⊢ (-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, ← neg_add, h] | R : Type u_1
inst✝ : NonAssocRing R
f : R → R
h : good f
x y : R
⊢ -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
inst✝ : NonAssocRing R
f : R → R
h : good f
x y : R
⊢ -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 [← add_left_eq_self, h, add_one_mul x, mul_add_one x, h0, add_comm 1 x] | R : Type u_1
inst✝ : NonAssocRing R
f : R → R
h : good f
x y : R
h0 : x * y = 1
⊢ 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
inst✝ : NonAssocRing R
f : R → R
h : good f
x y : R
h0 : x * y = 1
⊢ 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
inst✝ : Ring R
f : R → R
h : good f
⊢ f (f 0 ^ 2) = 0 | R : Type u_1
inst✝ : Ring R
f : R → R
h : f (f 0 * f 0) + f (0 + 0) = f (0 * 0)
⊢ f (f 0 ^ 2) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
⊢ 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, ← sq] at h | R : Type u_1
inst✝ : Ring R
f : R → R
h : f (f 0 * f 0) + f (0 + 0) = f (0 * 0)
⊢ f (f 0 ^ 2) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
f : R → R
h : f (f 0 * f 0) + f (0 + 0) = f (0 * 0)
⊢ 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 [← h0, ← mul_zero x, ← h, add_zero, h0, mul_one] | R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
x : R
⊢ f (f x) + f x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
x : R
⊢ 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', ← h2 x, add_left_inj] | R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ f x = 1 - x | R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ x = f (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ 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
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ x = f (f x) | case a
R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ f x = f (f (f x)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ 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)), ← h2 x, add_sub_cancel_left] | case a
R : Type u_1
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ 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
inst✝ : Ring R
f : R → R
h : good f
h0 : f 0 = 1
h1 : Function.Injective f
h2 : ∀ (x : R), f (f x) + f x = 1
x : R
⊢ 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
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
⊢ f = 0 | D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f (f (c - 1 + 1) * f ((c - 1)⁻¹ + 1)) = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
⊢ 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
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f (f (c - 1 + 1) * f ((c - 1)⁻¹ + 1)) = 0
⊢ f = 0 | D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f 0 = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f (f (c - 1 + 1) * f ((c - 1)⁻¹ + 1)) = 0
⊢ 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
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f 0 = 0
⊢ f = 0 | case h
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f 0 = 0
x : D
⊢ f x = 0 x | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f 0 = 0
⊢ 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, ← h3, ← mul_zero x, ← h,
h3, mul_zero, h3, zero_add, add_zero] | case h
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f 0 = 0
x : D
⊢ f x = 0 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f ≠ 0
c : D
h1 : f c = 0
h2 : ¬c = 1
h3 : f 0 = 0
x : D
⊢ 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
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
c : D
h1 : f = 0
⊢ 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
c : D
h1 : f = 0
⊢ 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
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
⊢ f (x + 1) + 1 = f x | D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
h1 : f (f x * f 1) + f (x + 1) = f (x * 1)
⊢ f (x + 1) + 1 = f x | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
⊢ 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
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
h1 : f (f x * f 1) + f (x + 1) = f (x * 1)
⊢ f (x + 1) + 1 = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
h1 : f (f x * f 1) + f (x + 1) = f (x * 1)
⊢ 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 [← good_shift h h0, sub_add_cancel] | D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
⊢ f (x - 1) = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
⊢ 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
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
⊢ f (x + 1) = 0 ↔ x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h : good f
h0 : f 0 = 1
x : D
⊢ f (x + 1) = 0 ↔ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
⊢ f = 0 ∨ (f = fun x => 1 - x) ∨ f = fun x => x - 1 | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
⊢ ¬f = 0 → (f = fun x => 1 - x) ∨ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
⊢ f = 0 ∨ (f = fun x => 1 - x) ∨ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
⊢ ¬f = 0 → (f = fun x => 1 - x) ∨ f = fun x => x - 1 | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1 : ¬f = 0
⊢ (f = fun x => 1 - x) ∨ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
⊢ ¬f = 0 → (f = fun x => 1 - x) ∨ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1 : ¬f = 0
⊢ (f = fun x => 1 - x) ∨ f = fun x => x - 1 | case f
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : f 0 = 1
⊢ f = fun x => 1 - x
case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : f 0 = -1
⊢ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1 : ¬f = 0
⊢ (f = fun x => 1 - x) ∨ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : f 0 = 1
⊢ f = fun x => 1 - x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : f 0 = 1
⊢ 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 [← neg_eq_iff_eq_neg] at h1 | case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : f 0 = -1
⊢ f = fun x => x - 1 | case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
⊢ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : f 0 = -1
⊢ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
⊢ f = fun x => x - 1 | case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
⊢ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
⊢ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
⊢ f = fun x => x - 1 | case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
⊢ f = fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
⊢ 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 [← neg_inj, h3] | case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
⊢ f = fun x => x - 1 | case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
⊢ (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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
⊢ 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 λ x ↦ (neg_sub x 1).symm | case g
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
⊢ (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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : good f
h1✝ : ¬f = 0
h1 : -f 0 = 1
h2 : good (-f)
h3 : -f = fun x => 1 - x
⊢ (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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : f = 0 ∨ (f = fun x => 1 - x) ∨ f = fun x => x - 1
⊢ good f | case inl
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good 0
case inr.inl
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good fun x => 1 - x
case inr.inr
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good fun x => x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
f : D → D
h0 : f = 0 ∨ (f = fun x => 1 - x) ∨ f = fun x => x - 1
⊢ 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
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good 0
case inr.inl
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good fun x => 1 - x
case inr.inr
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good fun x => x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good 0
case inr.inl
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ good fun x => 1 - x
case inr.inr
D : Type u_1
inst✝ : DivisionRing D
f : D → D
h✝ : good f
h : ∀ (f : D → D), good f → f 0 = 1 → Function.Injective f
⊢ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
⊢ Function.Injective f | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
⊢ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
⊢ 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 : ∀ y, f y = f (-y) → y = 0 := λ y h4 ↦ by
rwa [← h3, self_eq_add_left, ← h2, good_map_eq_zero_iff h0 h1,
sub_eq_iff_eq_add, one_add_one_eq_two, mul_right_eq_self₀,
or_iff_left h, ← 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
⊢ Function.Injective f | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
⊢ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
⊢ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
⊢ Function.Injective f | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
⊢ 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
⊢ a = b | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
⊢ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
⊢ 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 : ∀ y z, f y = f z → f (-y) = f (-z) :=
λ y z h5 ↦ by rw [← h3, h5, h3] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
⊢ f (a - b) = f (-(a - b)) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
⊢ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
⊢ 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 [← h0, ← h0 b, h4, add_comm a] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
⊢ f (a - b) = f (-(a - b)) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
⊢ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
⊢ 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 h8 := h0 a (-b) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
⊢ f (a - b) = f (-(a - b)) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
h8 : f (f a * f (-b)) + f (a + -b) = f (a * -b)
⊢ f (a - b) = f (-(a - b)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
⊢ 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] | rwa [mul_neg, h5 _ _ h6, ← mul_neg, ← h0 b, h4, h5 a b h4,
add_right_inj, ← sub_eq_add_neg, ← sub_eq_add_neg, ← neg_sub a] at h8 | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
h8 : f (f a * f (-b)) + f (a + -b) = f (a * -b)
⊢ f (a - b) = f (-(a - b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
h6 : f (a * b) = f (b * a)
h8 : f (f a * f (-b)) + f (a + -b) = f (a * -b)
⊢ 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] | rw [add_assoc, ← neg_one_mul, ← h0 (-1)] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + 1 + f y = f (-y) | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + (1 + f y) = f (f (-1) * f y) + f (-1 + y) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + 1 + f y = f (-y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | refine congr_arg₂ _ ?_ ?_ | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + (1 + f y) = f (f (-1) * f y) + f (-1 + y) | case refine_1
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) = f (f (-1) * f y)
case refine_2
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ 1 + f y = f (-1 + y) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) + (1 + f y) = f (f (-1) * f y) + f (-1 + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [← zero_sub, h2, h1, one_add_one_eq_two] | case refine_1
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) = f (f (-1) * f y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ f (2 * f y) = f (f (-1) * f y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [add_comm, ← h2, neg_add_eq_sub] | case refine_2
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ 1 + f y = f (-1 + y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
y : D
⊢ 1 + f y = f (-1 + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rwa [← h3, self_eq_add_left, ← h2, good_map_eq_zero_iff h0 h1,
sub_eq_iff_eq_add, one_add_one_eq_two, mul_right_eq_self₀,
or_iff_left h, ← 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
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
y : D
h4 : f y = f (-y)
⊢ y = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : D), f (x - 1) = f x + 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
y : D
h4 : f y = f (-y)
⊢ y = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [← h3, h5, h3] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
y z : D
h5 : f y = f z
⊢ f (-y) = f (-z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
y z : D
h5 : f y = f z
⊢ f (-y) = f (-z)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case1_injective | [138, 1] | [161, 74] | rw [← h0, ← h0 b, h4, add_comm a] | D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
⊢ f (a * b) = f (b * a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type u_1
inst✝ : DivisionRing D
f✝ : D → D
h✝ : good f✝
h : 2 ≠ 0
f : D → D
h0 : good f
h1 : f 0 = 1
h3 : ∀ (y : D), f (2 * f y) + 1 + f y = f (-y)
h2 : ∀ (y : D), f y = f (-y) → y = 0
a b : D
h4 : f a = f b
h5 : ∀ (y z : D), f y = f z → f (-y) = f (-z)
⊢ f (a * b) = f (b * a)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h2 := good_shift h0 h1 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
⊢ Function.Injective f | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
⊢ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
⊢ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h3 : ∀ c d : F, d ≠ 0 → f (c + 1) = f (d + 1) →
f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1) := λ c d h3 h4 ↦ by
rw [good_shift2 h0 h1, ← h0, h4, add_assoc, ← add_assoc (c + 1), h2,
good_special_equality h0 (mul_inv_cancel h3), zero_add, add_right_comm] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
⊢ Function.Injective f | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
⊢ Function.Injective f | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
⊢ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | intros a b h4 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
⊢ Function.Injective f | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f a = f b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
⊢ Function.Injective f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [← h2 a, ← h2 b, add_left_inj] at h4 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f a = f b
⊢ a = b | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f a = f b
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h5 := good_map_add_one_eq_zero_iff h0 h1 | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
⊢ a = b | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rcases eq_or_ne a 0 with rfl | ha | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
⊢ a = b | case inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
b : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
h4 : f (0 + 1) = f (b + 1)
⊢ 0 = b
case inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rcases eq_or_ne b 0 with rfl | hb | case inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
⊢ a = b | case inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
h4 : f (a + 1) = f (0 + 1)
⊢ a = 0
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹))
+ a * a⁻¹ * (b * b⁻¹) := by ring | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | have h7 : ∀ c d : F, (c + 1) * (d + 1) = c * d + c + d + 1 := λ c d ↦ by
rw [add_one_mul (α := F), mul_add_one (α := F), ← add_assoc] | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [mul_inv_cancel ha, mul_inv_cancel hb, mul_one, mul_one, mul_one,
add_comm b b⁻¹, add_add_add_comm, add_comm a⁻¹ b, ← add_assoc, ← h7] at h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) = (a + b⁻¹ + 1) * (b + a⁻¹ + 1)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | replace h6 := congr_arg f h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) = (a + b⁻¹ + 1) * (b + a⁻¹ + 1)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1)) = f ((a + b⁻¹ + 1) * (b + a⁻¹ + 1))
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) = (a + b⁻¹ + 1) * (b + a⁻¹ + 1)
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [← h0, h3 a b hb h4, h3 b a ha h4.symm, h7, add_sub_cancel_right,
h7, add_sub_cancel_right, ← h0 (a + b⁻¹ + 1), add_right_inj] at h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1)) = f ((a + b⁻¹ + 1) * (b + a⁻¹ + 1))
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1)) = f ((a + b⁻¹ + 1) * (b + a⁻¹ + 1))
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | replace h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) =
a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1) := by ring | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [mul_inv_cancel ha, mul_inv_cancel hb, one_add_one_eq_two, h, zero_add] at h7 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [← h2, ← h7, add_add_add_comm, add_add_add_comm a, ← add_add_add_comm,
← h0, add_right_comm, add_left_eq_self, ← h2, add_assoc,
one_add_one_eq_two, h, add_zero, h5, mul_eq_zero, h5, h5] at h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | replace h3 : ∀ c : F, -c = c := λ c ↦ by
rw [neg_eq_iff_add_eq_zero, ← two_mul, h, zero_mul] | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rcases h6 with h6 | h6 | case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
⊢ a = b | case inr.inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : a + b = 0
⊢ a = b
case inr.inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : b⁻¹ + a⁻¹ = 0
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [good_shift2 h0 h1, ← h0, h4, add_assoc, ← add_assoc (c + 1), h2,
good_special_equality h0 (mul_inv_cancel h3), zero_add, add_right_comm] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
c d : F
h3 : d ≠ 0
h4 : f (c + 1) = f (d + 1)
⊢ f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
c d : F
h3 : d ≠ 0
h4 : f (c + 1) = f (d + 1)
⊢ f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [zero_add, good_map_one h0, eq_comm, h5, eq_comm] at h4 | case inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
b : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
h4 : f (0 + 1) = f (b + 1)
⊢ 0 = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
b : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
h4 : f (0 + 1) = f (b + 1)
⊢ 0 = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [zero_add, good_map_one h0, h5] at h4 | case inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
h4 : f (a + 1) = f (0 + 1)
⊢ a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a : F
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
h4 : f (a + 1) = f (0 + 1)
⊢ a = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | ring | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
⊢ ((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [add_one_mul (α := F), mul_add_one (α := F), ← add_assoc] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
c d : F
⊢ (c + 1) * (d + 1) = c * d + c + d + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 :
((a + 1) * (b⁻¹ + 1) - 1) * ((b + 1) * (a⁻¹ + 1) - 1) =
(a + b⁻¹) * (b + a⁻¹) + ((a + a⁻¹) * (b * b⁻¹) + (b + b⁻¹) * (a * a⁻¹)) + a * a⁻¹ * (b * b⁻¹)
c d : F
⊢ (c + 1) * (d + 1) = c * d + c + d + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | ring | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : ∀ (c d : F), (c + 1) * (d + 1) = c * d + c + d + 1
h6 : f (a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹)) = f (a + b⁻¹ + 1 + (b + a⁻¹ + 1))
⊢ (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + (a * a⁻¹ + b * b⁻¹ + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rw [neg_eq_iff_add_eq_zero, ← two_mul, h, zero_mul] | F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
c : F
⊢ -c = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
h3 : ∀ (c d : F), d ≠ 0 → f (c + 1) = f (d + 1) → f ((c + 1) * (d⁻¹ + 1) - 1) = f (c + d⁻¹ + 1)
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h6 : a + b = 0 ∨ b⁻¹ + a⁻¹ = 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
c : F
⊢ -c = c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [add_eq_zero_iff_eq_neg, h3] at h6 | case inr.inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : a + b = 0
⊢ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : a + b = 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A6/A6.lean | IMOSL.IMO2017A6.case2_injective | [170, 1] | [206, 61] | rwa [add_eq_zero_iff_eq_neg, h3, inv_inj, eq_comm] at h6 | case inr.inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : b⁻¹ + a⁻¹ = 0
⊢ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
F : Type u_1
inst✝ : Field F
h : 2 = 0
f : F → F
h0 : good f
h1 : f 0 = 1
h2 : ∀ (x : F), f (x + 1) + 1 = f x
a b : F
h4 : f (a + 1) = f (b + 1)
h5 : ∀ (x : F), f (x + 1) = 0 ↔ x = 0
ha : a ≠ 0
hb : b ≠ 0
h7 : (a + b + 1) * (b⁻¹ + a⁻¹ + 1) = a * b⁻¹ + a + b⁻¹ + (b * a⁻¹ + b + a⁻¹) + 1
h3 : ∀ (c : F), -c = c
h6 : b⁻¹ + a⁻¹ = 0
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/Infinitesimal/FloorRing.lean | IMOSL.Extra.Infinitesimal.FloorRing_mul_left | [26, 1] | [31, 22] | apply (hε (k * ⌈|r|⌉.natAbs)).trans_le' | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| < 1 | R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| ≤ (k * ⌈|r|⌉.natAbs) • |ε| | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝¹ : LinearOrderedRing R
inst✝ : FloorRing R
ε : R
hε : Infinitesimal ε
r : R
k : ℕ
⊢ k • |r * ε| < 1
TACTIC:
|
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