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/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rw [if_pos (by rwa [Set.mem_setOf_eq]),
map_eq_map_one_mul_floor hf, this, Int.cast_one, mul_one] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
this : ⌊f x⌋ = 1
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
this : ⌊f x⌋ = 1
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution_of_fract_map_one_pos | [103, 1] | [134, 68] | rwa [Set.mem_setOf_eq] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
this : ⌊f x⌋ = 1
⊢ x ∈ {x | ⌊f x⌋ ≠ 0} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
h1 : ∀ (x : M), 0 ≤ ⌊f x⌋
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
this : ⌊f x⌋ = 1
⊢ x ∈ {x | ⌊f x⌋ ≠ 0}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution | [144, 1] | [166, 38] | simp only [Set.mem_empty_iff_false, and_self] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f = 0
x✝¹ x✝ : M
⊢ x✝¹ * x✝ ∈ ∅ ↔ x✝¹ ∈ ∅ ∧ x✝ ∈ ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f = 0
x✝¹ x✝ : M
⊢ x✝¹ * x✝ ∈ ∅ ↔ x✝¹ ∈ ∅ ∧ x✝ ∈ ∅
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution | [144, 1] | [166, 38] | rw [Set.mem_empty_iff_false, if_false, h] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f = 0
x : M
⊢ f x = if x ∈ ∅ then 1 else 0 | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f = 0
x : M
⊢ 0 x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f = 0
x : M
⊢ f x = if x ∈ ∅ then 1 else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution | [144, 1] | [166, 38] | rfl | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f = 0
x : M
⊢ 0 x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : f = 0
x : M
⊢ 0 x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution | [144, 1] | [166, 38] | rw [h3, ← Int.cast_one, ← h, Int.floor_add_fract] | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
x✝ : Infinitesimal (Int.fract (f 1)) ∧ ∃ φ, ∀ (x : M), f x = f 1 * ↑(φ x)
h1 : Infinitesimal (Int.fract (f 1))
h2 : ∃ φ, ∀ (x : M), f x = f 1 * ↑(φ x)
φ : M →* ℕ
h3 : ∀ (x : M), f x = f 1 * ↑(φ x)
y : M
⊢ f y = (1 + Int.fract (f 1)) * ↑(φ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf : MonoidGood f
h : ⌊f 1⌋ = 1
h0 : 0 < Int.fract (f 1)
x✝ : Infinitesimal (Int.fract (f 1)) ∧ ∃ φ, ∀ (x : M), f x = f 1 * ↑(φ x)
h1 : Infinitesimal (Int.fract (f 1))
h2 : ∃ φ, ∀ (x : M), f x = f 1 * ↑(φ x)
φ : M →* ℕ
h3 : ∀ (x : M), f x = f 1 * ↑(φ x)
y : M
⊢ f y = (1 + Int.fract (f 1)) * ↑(φ y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution | [144, 1] | [166, 38] | rcases hf with ⟨φ, rfl⟩ | ⟨ε, hε, hε0, φ, rfl⟩ | ⟨A, hA, C, hC, rfl⟩ | R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf :
(∃ φ, f = fun x => ↑(φ x)) ∨
(∃ ε, ∃ (_ : 0 < ε) (_ : Infinitesimal ε), ∃ φ, f = fun x => (1 + ε) * ↑(φ x)) ∨
∃ A, ∃ (_ : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A), ∃ C, ∃ (_ : ⌊C⌋ = 1), f = fun x => if x ∈ A then C else 0
⊢ MonoidGood f | case inl.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
φ : M →* ℤ
⊢ MonoidGood fun x => ↑(φ x)
case inr.inl.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
ε : R
hε : 0 < ε
hε0 : Infinitesimal ε
φ : M →* ℕ
⊢ MonoidGood fun x => (1 + ε) * ↑(φ x)
case inr.inr.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
A : Set M
hA : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
C : R
hC : ⌊C⌋ = 1
⊢ MonoidGood fun x => if x ∈ A then C else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
f : M → R
hf :
(∃ φ, f = fun x => ↑(φ x)) ∨
(∃ ε, ∃ (_ : 0 < ε) (_ : Infinitesimal ε), ∃ φ, f = fun x => (1 + ε) * ↑(φ x)) ∨
∃ A, ∃ (_ : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A), ∃ C, ∃ (_ : ⌊C⌋ = 1), f = fun x => if x ∈ A then C else 0
⊢ MonoidGood f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.solution | [144, 1] | [166, 38] | exacts [castMonoidHom_is_MonoidGood φ.toMulHom,
one_add_infinitesimal_mul_is_MonoidGood φ.toMulHom hε.le hε0,
indicator_const_is_good hA hC] | case inl.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
φ : M →* ℤ
⊢ MonoidGood fun x => ↑(φ x)
case inr.inl.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
ε : R
hε : 0 < ε
hε0 : Infinitesimal ε
φ : M →* ℕ
⊢ MonoidGood fun x => (1 + ε) * ↑(φ x)
case inr.inr.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
A : Set M
hA : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
C : R
hC : ⌊C⌋ = 1
⊢ MonoidGood fun x => if x ∈ A then C else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
φ : M →* ℤ
⊢ MonoidGood fun x => ↑(φ x)
case inr.inl.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
ε : R
hε : 0 < ε
hε0 : Infinitesimal ε
φ : M →* ℕ
⊢ MonoidGood fun x => (1 + ε) * ↑(φ x)
case inr.inr.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : MulOneClass M
A : Set M
hA : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
C : R
hC : ⌊C⌋ = 1
⊢ MonoidGood fun x => if x ∈ A then C else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SqSubOneMap.lean | IMOSL.IMO2012A5.sq_sub_one_is_good | [50, 1] | [51, 18] | ring | R : Type u_1
inst✝ : CommRing R
x✝¹ x✝ : R
⊢ (fun x => x ^ 2 - 1) (x✝¹ * x✝ + 1) =
(fun x => x ^ 2 - 1) x✝¹ * (fun x => x ^ 2 - 1) x✝ + (fun x => x ^ 2 - 1) (x✝¹ + x✝) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x✝¹ x✝ : R
⊢ (fun x => x ^ 2 - 1) (x✝¹ * x✝ + 1) =
(fun x => x ^ 2 - 1) x✝¹ * (fun x => x ^ 2 - 1) x✝ + (fun x => x ^ 2 - 1) (x✝¹ + x✝)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_of_singleton | [32, 1] | [33, 62] | rw [NatSeq_ofList, List.length_singleton, Nat.mod_one] | α : Type u_1
inst✝ : Inhabited α
c : α
n : ℕ
⊢ NatSeq_ofList [c] n = c | α : Type u_1
inst✝ : Inhabited α
c : α
n : ℕ
⊢ [c].getI 0 = c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Inhabited α
c : α
n : ℕ
⊢ NatSeq_ofList [c] n = c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_of_singleton | [32, 1] | [33, 62] | rfl | α : Type u_1
inst✝ : Inhabited α
c : α
n : ℕ
⊢ [c].getI 0 = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Inhabited α
c : α
n : ℕ
⊢ [c].getI 0 = c
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | replace hN := List.length_pos.mp <| hN.trans_eq (List_ofNatSeq_length a N).symm | α : Type u_1
inst✝ : Inhabited α
N : ℕ
hN : 0 < N
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
⊢ NatSeq_ofList (List_ofNatSeq a N) n = a n | α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ NatSeq_ofList (List_ofNatSeq a N) n = a n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Inhabited α
N : ℕ
hN : 0 < N
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
⊢ NatSeq_ofList (List_ofNatSeq a N) n = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | rw [NatSeq_nonempty_eq_get hN] | α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ NatSeq_ofList (List_ofNatSeq a N) n = a n | α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ (List_ofNatSeq a N).get ⟨n % (List_ofNatSeq a N).length, ⋯⟩ = a n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ NatSeq_ofList (List_ofNatSeq a N) n = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | unfold List_ofNatSeq | α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ (List_ofNatSeq a N).get ⟨n % (List_ofNatSeq a N).length, ⋯⟩ = a n | α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ (List.map a (List.range N)).get ⟨n % (List.map a (List.range N)).length, ⋯⟩ = a n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ (List_ofNatSeq a N).get ⟨n % (List_ofNatSeq a N).length, ⋯⟩ = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/OfList.lean | IMOSL.Extra.NatSeq_ofList_of_PeriodicNatSeq_eq | [49, 1] | [55, 58] | rw [List.get_map, List.get_range, Fin.val_mk,
List.length_map, List.length_range, ha.map_mod_nat] | α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ (List.map a (List.range N)).get ⟨n % (List.map a (List.range N)).length, ⋯⟩ = a n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Inhabited α
N : ℕ
a : ℕ → α
ha : Function.Periodic a N
n : ℕ
hN : List_ofNatSeq a N ≠ []
⊢ (List.map a (List.range N)).get ⟨n % (List.map a (List.range N)).length, ⋯⟩ = a n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Hom.lean | IMOSL.Extra.CharTwo.pullback_of_inj | [22, 1] | [23, 71] | rw [φ.map_add, φ.map_zero, CharTwo.add_self_eq_zero] | R : Type u_1
R' : Type u_2
inst✝² : AddMonoid R
inst✝¹ : CharTwo R
inst✝ : AddMonoid R'
φ : R' →+ R
h : Function.Injective ⇑φ
x : R'
⊢ φ (x + x) = φ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
R' : Type u_2
inst✝² : AddMonoid R
inst✝¹ : CharTwo R
inst✝ : AddMonoid R'
φ : R' →+ R
h : Function.Injective ⇑φ
x : R'
⊢ φ (x + x) = φ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Hom.lean | IMOSL.Extra.CharTwo.forward_of_surj | [25, 1] | [26, 86] | rw [← h0, ← φ.map_add, add_self_eq_zero, φ.map_zero] | R : Type u_2
R' : Type u_1
inst✝² : AddMonoid R
inst✝¹ : CharTwo R
inst✝ : AddMonoid R'
φ : R →+ R'
h : Function.Surjective ⇑φ
x : R'
c : R
h0 : φ c = x
⊢ x + x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
R' : Type u_1
inst✝² : AddMonoid R
inst✝¹ : CharTwo R
inst✝ : AddMonoid R'
φ : R →+ R'
h : Function.Surjective ⇑φ
x : R'
c : R
h0 : φ c = x
⊢ x + x = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | refine Iff.symm ⟨λ h x y ↦ ?_, λ h ↦ ?_⟩ | f : ℤ → ℤ
⊢ (∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1) ↔ (f = fun x => -1) ∨ f = fun x => x + 1 | case refine_1
f : ℤ → ℤ
h : (f = fun x => -1) ∨ f = fun x => x + 1
x y : ℤ
⊢ f (x - f y) = f (f x) - f y - 1
case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
⊢ (∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1) ↔ (f = fun x => -1) ∨ f = fun x => x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rcases h with rfl | rfl | case refine_1
f : ℤ → ℤ
h : (f = fun x => -1) ∨ f = fun x => x + 1
x y : ℤ
⊢ f (x - f y) = f (f x) - f y - 1 | case refine_1.inl
x y : ℤ
⊢ (fun x => -1) (x - (fun x => -1) y) = (fun x => -1) ((fun x => -1) x) - (fun x => -1) y - 1
case refine_1.inr
x y : ℤ
⊢ (fun x => x + 1) (x - (fun x => x + 1) y) = (fun x => x + 1) ((fun x => x + 1) x) - (fun x => x + 1) y - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℤ → ℤ
h : (f = fun x => -1) ∨ f = fun x => x + 1
x y : ℤ
⊢ f (x - f y) = f (f x) - f y - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rw [sub_sub, neg_add_self, sub_zero] | case refine_1.inl
x y : ℤ
⊢ (fun x => -1) (x - (fun x => -1) y) = (fun x => -1) ((fun x => -1) x) - (fun x => -1) y - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.inl
x y : ℤ
⊢ (fun x => -1) (x - (fun x => -1) y) = (fun x => -1) ((fun x => -1) x) - (fun x => -1) y - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rw [sub_sub, add_sub_add_right_eq_sub, add_sub_right_comm] | case refine_1.inr
x y : ℤ
⊢ (fun x => x + 1) (x - (fun x => x + 1) y) = (fun x => x + 1) ((fun x => x + 1) x) - (fun x => x + 1) y - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.inr
x y : ℤ
⊢ (fun x => x + 1) (x - (fun x => x + 1) y) = (fun x => x + 1) ((fun x => x + 1) x) - (fun x => x + 1) y - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | have h0 := h 0 (f 0) | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = f (f 0) - f (f 0) - 1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rw [sub_self, zero_sub 1] at h0 | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = f (f 0) - f (f 0) - 1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = f (f 0) - f (f 0) - 1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | replace h0 x : f (x + 1) = f (f x) := by
have h1 := h x (0 - f (f 0))
rwa [h0, sub_neg_eq_add (f (f x)), add_sub_cancel_right] at h1 | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
⊢ (f = fun x => -1) ∨ f = fun x => x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | have h1 : ∀ n, f n = (f (-1) + 1) * n + f 0 := by
refine Extra.IntIntLinearSolverAlt λ n ↦ ?_
have h1 := h0 (n - f n - 1)
rw [sub_add_cancel, sub_right_comm, h, ← h0, h (n - 1), ← h0,
sub_add_cancel, sub_self, zero_sub, sub_eq_iff_eq_add] at h1
exact eq_add_of_sub_eq h1 | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
⊢ (f = fun x => -1) ∨ f = fun x => x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | refine (eq_or_ne (f (-1) + 1) 0).imp
(λ h2 ↦ funext λ x ↦ ?_) (λ h2 ↦ funext λ x ↦ ?_) | case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
⊢ (f = fun x => -1) ∨ f = fun x => x + 1 | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 = 0
x : ℤ
⊢ f x = -1
case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
⊢ f x = x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
⊢ (f = fun x => -1) ∨ f = fun x => x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | have h1 := h x (0 - f (f 0)) | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
x : ℤ
⊢ f (x + 1) = f (f x) | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
x : ℤ
h1 : f (x - f (0 - f (f 0))) = f (f x) - f (0 - f (f 0)) - 1
⊢ f (x + 1) = f (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
x : ℤ
⊢ f (x + 1) = f (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rwa [h0, sub_neg_eq_add (f (f x)), add_sub_cancel_right] at h1 | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
x : ℤ
h1 : f (x - f (0 - f (f 0))) = f (f x) - f (0 - f (f 0)) - 1
⊢ f (x + 1) = f (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : f (0 - f (f 0)) = -1
x : ℤ
h1 : f (x - f (0 - f (f 0))) = f (f x) - f (0 - f (f 0)) - 1
⊢ f (x + 1) = f (f x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | refine Extra.IntIntLinearSolverAlt λ n ↦ ?_ | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
⊢ ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0 | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
⊢ f (n + 1) = f (-1) + 1 + f n | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
⊢ ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | have h1 := h0 (n - f n - 1) | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
⊢ f (n + 1) = f (-1) + 1 + f n | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
h1 : f (n - f n - 1 + 1) = f (f (n - f n - 1))
⊢ f (n + 1) = f (-1) + 1 + f n | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
⊢ f (n + 1) = f (-1) + 1 + f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rw [sub_add_cancel, sub_right_comm, h, ← h0, h (n - 1), ← h0,
sub_add_cancel, sub_self, zero_sub, sub_eq_iff_eq_add] at h1 | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
h1 : f (n - f n - 1 + 1) = f (f (n - f n - 1))
⊢ f (n + 1) = f (-1) + 1 + f n | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
h1 : f (n + 1) - f n = f (-1) + 1
⊢ f (n + 1) = f (-1) + 1 + f n | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
h1 : f (n - f n - 1 + 1) = f (f (n - f n - 1))
⊢ f (n + 1) = f (-1) + 1 + f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | exact eq_add_of_sub_eq h1 | f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
h1 : f (n + 1) - f n = f (-1) + 1
⊢ f (n + 1) = f (-1) + 1 + f n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
n : ℤ
h1 : f (n + 1) - f n = f (-1) + 1
⊢ f (n + 1) = f (-1) + 1 + f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rw [h1, h2, Int.zero_mul, zero_add, ← eq_neg_of_add_eq_zero_left h2] | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 = 0
x : ℤ
⊢ f x = -1 | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 = 0
x : ℤ
⊢ f 0 = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 = 0
x : ℤ
⊢ f x = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | specialize h1 (-1) | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 = 0
x : ℤ
⊢ f 0 = f (-1) | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h2 : f (-1) + 1 = 0
x : ℤ
h1 : f (-1) = (f (-1) + 1) * -1 + f 0
⊢ f 0 = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 = 0
x : ℤ
⊢ f 0 = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rw [h2, Int.zero_mul, zero_add] at h1 | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h2 : f (-1) + 1 = 0
x : ℤ
h1 : f (-1) = (f (-1) + 1) * -1 + f 0
⊢ f 0 = f (-1) | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h2 : f (-1) + 1 = 0
x : ℤ
h1 : f (-1) = f 0
⊢ f 0 = f (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h2 : f (-1) + 1 = 0
x : ℤ
h1 : f (-1) = (f (-1) + 1) * -1 + f 0
⊢ f 0 = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | exact h1.symm | case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h2 : f (-1) + 1 = 0
x : ℤ
h1 : f (-1) = f 0
⊢ f 0 = f (-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h2 : f (-1) + 1 = 0
x : ℤ
h1 : f (-1) = f 0
⊢ f 0 = f (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | specialize h0 x | case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
⊢ f x = x + 1 | case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
h0 : f (x + 1) = f (f x)
⊢ f x = x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h0 : ∀ (x : ℤ), f (x + 1) = f (f x)
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
⊢ f x = x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | rw [h1, h1 (f x), add_left_inj, Int.mul_eq_mul_left_iff h2] at h0 | case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
h0 : f (x + 1) = f (f x)
⊢ f x = x + 1 | case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
h0 : x + 1 = f x
⊢ f x = x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
h0 : f (x + 1) = f (f x)
⊢ f x = x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/A2/A2.lean | IMOSL.IMO2015A2.final_solution | [20, 1] | [51, 20] | exact h0.symm | case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
h0 : x + 1 = f x
⊢ f x = x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
f : ℤ → ℤ
h : ∀ (x y : ℤ), f (x - f y) = f (f x) - f y - 1
h1 : ∀ (n : ℤ), f n = (f (-1) + 1) * n + f 0
h2 : f (-1) + 1 ≠ 0
x : ℤ
h0 : x + 1 = f x
⊢ f x = x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | intro n h0 | k c : ℕ
h : good k c
⊢ good (k + 1) (2 * c + 1) | k c : ℕ
h : good k c
n : ℕ
h0 : 0 < n
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(n + (2 * c + 1)) / ↑n = (map (fun m => ↑(m + 1) / ↑m) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
k c : ℕ
h : good k c
⊢ good (k + 1) (2 * c + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rcases n.even_or_odd' with ⟨t, rfl | rfl⟩ | k c : ℕ
h : good k c
n : ℕ
h0 : 0 < n
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(n + (2 * c + 1)) / ↑n = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inl
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod
case intro.inr
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t + 1
⊢ ∃ S,
card S = k + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
k c : ℕ
h : good k c
n : ℕ
h0 : 0 < n
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(n + (2 * c + 1)) / ↑n = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | replace h0 := pos_of_mul_pos_right h0 (Nat.zero_le 2) | case intro.inl
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inl
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < t
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rcases h t h0 with ⟨T, rfl, h1, h2⟩ | case intro.inl
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < t
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ∃ S,
card S = card T + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < t
⊢ ∃ S, card S = k + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | have X := t.add_pos_left h0 c | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ∃ S,
card S = card T + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ∃ S,
card S = card T + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ∃ S,
card S = card T + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | refine ⟨(2 * (t + c)) ::ₘ T, card_cons _ T,
forall_mem_cons.mpr ⟨mul_pos (Nat.succ_pos 1) X, h1⟩, ?_⟩ | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ∃ S,
card S = card T + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) (2 * (t + c) ::ₘ T)).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ∃ S,
card S = card T + 1 ∧ (∀ m ∈ S, 0 < m) ∧ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rw [map_cons, prod_cons, ← h2, ← add_assoc, ← mul_add,
div_mul_div_comm, Nat.cast_mul, Nat.cast_mul, mul_right_comm] | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) (2 * (t + c) ::ₘ T)).prod | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ↑(2 * (t + c) + 1) / (↑2 * ↑t) = ↑(2 * (t + c) + 1) * ↑(t + c) / (↑2 * ↑t * ↑(t + c)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ↑(2 * t + (2 * c + 1)) / ↑(2 * t) = (map (fun m => ↑(m + 1) / ↑m) (2 * (t + c) ::ₘ T)).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | exact (mul_div_mul_right _ _ <| Nat.cast_ne_zero.mpr X.ne.symm).symm | case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ↑(2 * (t + c) + 1) / (↑2 * ↑t) = ↑(2 * (t + c) + 1) * ↑(t + c) / (↑2 * ↑t * ↑(t + c)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inl.intro.intro.intro
c t : ℕ
h0 : 0 < t
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + c) / ↑t = (map (fun m => ↑(m + 1) / ↑m) T).prod
X : 0 < t + c
⊢ ↑(2 * (t + c) + 1) / (↑2 * ↑t) = ↑(2 * (t + c) + 1) * ↑(t + c) / (↑2 * ↑t * ↑(t + c))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | have X := t.succ_pos | case intro.inr
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t + 1
⊢ ∃ S,
card S = k + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inr
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
⊢ ∃ S,
card S = k + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t + 1
⊢ ∃ S,
card S = k + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rcases h (t + 1) X with ⟨T, rfl, h1, h2⟩ | case intro.inr
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
⊢ ∃ S,
card S = k + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ∃ S,
card S = card T + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr
k c : ℕ
h : good k c
t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
⊢ ∃ S,
card S = k + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | refine ⟨(2 * t + 1) ::ₘ T, card_cons _ T,
forall_mem_cons.mpr ⟨(2 * t).succ_pos, h1⟩, ?_⟩ | case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ∃ S,
card S = card T + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod | case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) ((2 * t + 1) ::ₘ T)).prod | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ∃ S,
card S = card T + 1 ∧
(∀ m ∈ S, 0 < m) ∧ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) S).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | rw [map_cons, prod_cons, ← h2, add_add_add_comm, add_right_comm,
add_assoc (2 * t) 1, ← mul_add_one (α := ℕ), ← mul_add, div_mul_div_comm,
Nat.cast_mul, Nat.cast_mul, mul_right_comm] | case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) ((2 * t + 1) ::ₘ T)).prod | case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ↑2 * ↑(t + 1 + c) / ↑(2 * t + 1) = ↑2 * ↑(t + 1 + c) * ↑(t + 1) / (↑(2 * t + 1) * ↑(t + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ↑(2 * t + 1 + (2 * c + 1)) / ↑(2 * t + 1) = (map (fun m => ↑(m + 1) / ↑m) ((2 * t + 1) ::ₘ T)).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.good_two_mul_add_one | [40, 1] | [60, 73] | exact (mul_div_mul_right _ _ <| Nat.cast_ne_zero.mpr X.ne.symm).symm | case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ↑2 * ↑(t + 1 + c) / ↑(2 * t + 1) = ↑2 * ↑(t + 1 + c) * ↑(t + 1) / (↑(2 * t + 1) * ↑(t + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.inr.intro.intro.intro
c t : ℕ
h0 : 0 < 2 * t + 1
X : 0 < t.succ
T : Multiset ℕ
h : good (card T) c
h1 : ∀ m ∈ T, 0 < m
h2 : ↑(t + 1 + c) / ↑(t + 1) = (map (fun m => ↑(m + 1) / ↑m) T).prod
⊢ ↑2 * ↑(t + 1 + c) / ↑(2 * t + 1) = ↑2 * ↑(t + 1 + c) * ↑(t + 1) / (↑(2 * t + 1) * ↑(t + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | rw [pow_zero, Nat.sub_self, add_zero] | n : ℕ
h : 0 < n
⊢ ↑(n + (2 ^ 0 - 1)) / ↑n = (map (fun m => ↑(m + 1) / ↑m) 0).prod | n : ℕ
h : 0 < n
⊢ ↑n / ↑n = (map (fun m => ↑(m + 1) / ↑m) 0).prod | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : 0 < n
⊢ ↑(n + (2 ^ 0 - 1)) / ↑n = (map (fun m => ↑(m + 1) / ↑m) 0).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | exact div_self (Nat.cast_ne_zero.mpr h.ne.symm) | n : ℕ
h : 0 < n
⊢ ↑n / ↑n = (map (fun m => ↑(m + 1) / ↑m) 0).prod | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
h : 0 < n
⊢ ↑n / ↑n = (map (fun m => ↑(m + 1) / ↑m) 0).prod
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | have h := good_two_mul_add_one (final_solution k) | k : ℕ
⊢ good (k + 1) (2 ^ (k + 1) - 1) | k : ℕ
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
⊢ good (k + 1) (2 ^ (k + 1) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
⊢ good (k + 1) (2 ^ (k + 1) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | have h0 := k.one_le_pow 2 (Nat.succ_pos 1) | k : ℕ
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
⊢ good (k + 1) (2 ^ (k + 1) - 1) | k : ℕ
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
h0 : 1 ≤ 2 ^ k
⊢ good (k + 1) (2 ^ (k + 1) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
⊢ good (k + 1) (2 ^ (k + 1) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N2/N2.lean | IMOSL.IMO2013N2.final_solution | [63, 1] | [71, 61] | rwa [two_mul, add_assoc, Nat.sub_add_cancel h0, add_comm _ (2 ^ k),
← Nat.add_sub_assoc h0, ← two_mul, ← pow_succ'] at h | k : ℕ
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
h0 : 1 ≤ 2 ^ k
⊢ good (k + 1) (2 ^ (k + 1) - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
h : good (k + 1) (2 * (2 ^ k - 1) + 1)
h0 : 1 ≤ 2 ^ k
⊢ good (k + 1) (2 ^ (k + 1) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq1 | [38, 1] | [40, 61] | rw [hf.is_good, add_add_cancel_left, hf.map_one, add_zero] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * (x + 1) + 1) = f x * 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✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * (x + 1) + 1) = f x * f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq2 | [42, 1] | [44, 69] | rw [sq, hf.is_good, add_self_eq_zero, hf.map_zero, sub_eq_add_neg] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x + 1) = f x ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x + 1) = f x ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq2_v2 | [46, 1] | [47, 56] | rw [← Eq2 hf, add_one_mul_self, add_add_cancel_right] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x) = f (x + 1) ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x) = f (x + 1) ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3 | [49, 1] | [54, 90] | have h : x * (x + 1) = x * x + x := mul_add_one x x | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1) | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : x * (x + 1) = x * x + x
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 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✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3 | [49, 1] | [54, 90] | rw [← Eq2_v2 hf, ← Eq1 hf, mul_sub_one, ← add_sub_right_comm, h, add_assoc,
← hf.is_good, mul_assoc, hf.is_good, h, add_add_cancel_middle₁, add_sub_cancel_right] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : x * (x + 1) = x * x + x
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * 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✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : x * (x + 1) = x * x + x
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3_v2 | [56, 1] | [59, 78] | have h := Eq3 hf (x + 1) | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f (x + 1) * f ((x + 1) * (x + 1) + (x + 1)) =
(f (x + 1 + 1) ^ 2 - 1) * (f (x + 1 + 1) - 1) + f (x + 1) * f (x + 1 + 1)
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3_v2 | [56, 1] | [59, 78] | rwa [add_add_cancel_right, add_one_mul_self, add_add_add_cancel_right] at h | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f (x + 1) * f ((x + 1) * (x + 1) + (x + 1)) =
(f (x + 1 + 1) ^ 2 - 1) * (f (x + 1 + 1) - 1) + f (x + 1) * f (x + 1 + 1)
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + 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✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f (x + 1) * f ((x + 1) * (x + 1) + (x + 1)) =
(f (x + 1 + 1) ^ 2 - 1) * (f (x + 1 + 1) - 1) + f (x + 1) * f (x + 1 + 1)
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1_ring_id1 | [74, 1] | [77, 63] | ring | R : Type ?u.23673
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ a * ((a ^ 2 - 1) * (a - 1) + b * a) - b * ((b ^ 2 - 1) * (b - 1) + a * b) =
(a ^ 2 + b ^ 2 - 1) * (a + b - 1) * (a - b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.23673
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ a * ((a ^ 2 - 1) * (a - 1) + b * a) - b * ((b ^ 2 - 1) * (b - 1) + a * b) =
(a ^ 2 + b ^ 2 - 1) * (a + b - 1) * (a - b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1_ring_id2 | [79, 1] | [82, 45] | ring | R : Type ?u.29081
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a : S
⊢ a ^ 2 * ((a ^ 2 - 1) ^ 2 + 1) - ((a ^ 2 - 1) * (a - 1) + a * a) ^ 2 = (1 - 2 * a) * (a ^ 2 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.29081
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a : S
⊢ a ^ 2 * ((a ^ 2 - 1) ^ 2 + 1) - ((a ^ 2 - 1) * (a - 1) + a * a) ^ 2 = (1 - 2 * a) * (a ^ 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | have h := Thm1_ring_id1 (f x) (f (x + 1)) | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f x * ((f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x) -
f (x + 1) * ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) =
(f x ^ 2 + f (x + 1) ^ 2 - 1) * (f x + f (x + 1) - 1) * (f x - f (x + 1))
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [← Eq3 hf, ← Eq3_v2 hf, mul_left_comm, sub_self, zero_eq_mul,
mul_eq_zero, sub_eq_zero, sub_eq_zero, sub_eq_zero] at h | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f x * ((f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x) -
f (x + 1) * ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) =
(f x ^ 2 + f (x + 1) ^ 2 - 1) * (f x + f (x + 1) - 1) * (f x - f (x + 1))
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : (f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1) ∨ f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f x * ((f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x) -
f (x + 1) * ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) =
(f x ^ 2 + f (x + 1) ^ 2 - 1) * (f x + f (x + 1) - 1) * (f x - f (x + 1))
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | cases h with | inl h => exact h | inr h => ?_ | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : (f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1) ∨ f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | case inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : (f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1) ∨ f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | right | case inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [← h, ← two_mul] | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x + f (x + 1) = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | have h0 : _ ^ 2 = _ ^ 2 := congrArg (λ x ↦ x ^ 2) (Eq3 hf x) | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ 2 * f x = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : (f x * f (x * x + x)) ^ 2 = ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) ^ 2
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [mul_pow, ← add_eq_of_eq_sub (Eq2 hf (x * x + x)), ← h, add_mul_self,
← mul_add_one (x * x), Eq1 hf, Eq2 hf, Eq2_v2 hf, ← h,
← sub_eq_zero, ← sq, Thm1_ring_id2, mul_eq_zero] at h0 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : (f x * f (x * x + x)) ^ 2 = ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) ^ 2
⊢ 2 * f x = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0 ∨ f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : (f x * f (x * x + x)) ^ 2 = ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) ^ 2
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | cases h0 with | inl h0 => exact (eq_of_sub_eq_zero h0).symm | inr h0 => ?_ | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0 ∨ f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0 ∨ f x ^ 2 - 1 = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | have h1 : f (x * x) = 0 := by rw [Eq2_v2 hf, ← h, h0] | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | replace h := Eq3_v2 hf (x * x) | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : f (x * x + 1) * f (x * x * (x * x) + x * x) = (f (x * x) ^ 2 - 1) * (f (x * x) - 1) + f (x * x + 1) * f (x * x)
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [h1, mul_zero, add_zero, sq, zero_mul, zero_sub,
neg_mul_neg, one_mul, Eq2 hf, h0, zero_mul] at h | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : f (x * x + 1) * f (x * x * (x * x) + x * x) = (f (x * x) ^ 2 - 1) * (f (x * x) - 1) + f (x * x + 1) * f (x * x)
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : 0 = 1
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : f (x * x + 1) * f (x * x * (x * x) + x * x) = (f (x * x) ^ 2 - 1) * (f (x * x) - 1) + f (x * x + 1) * f (x * x)
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [← mul_one (2 * f x), ← h, mul_zero] | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : 0 = 1
⊢ 2 * f x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : 0 = 1
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | exact h | case inl
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 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✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | exact (eq_of_sub_eq_zero h0).symm | case inr.h.inl
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0
⊢ 2 * f x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inl
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [Eq2_v2 hf, ← h, h0] | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ f (x * x) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ f (x * x) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharTwo_map_add_one | [109, 1] | [111, 83] | have h := Thm1 hf x | R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
⊢ f (x + 1) = f x + 1 | R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 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✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharTwo_map_add_one | [109, 1] | [111, 83] | rwa [← CharTwo.add_sq, CharTwo.sq_eq_one_iff, or_self, add_eq_iff_eq_add''] at h | R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f (x + 1) = 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✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.pow_four_Eq1 | [113, 1] | [114, 76] | rw [← add_add_cancel_right (x ^ 2) 1, add_one_sq, sq, sq, Eq2 hf, Eq2 hf] | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2) = (f x ^ 2 - 1) ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2) = (f x ^ 2 - 1) ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.pow_four_Eq2 | [116, 1] | [117, 49] | rw [← pow_four_Eq1 hf, add_one_sq, add_one_sq] | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2 + 1) = (f (x + 1) ^ 2 - 1) ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2 + 1) = (f (x + 1) ^ 2 - 1) ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_main_ring_id | [119, 1] | [122, 55] | ring | R : Type ?u.57818
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ ((a - 1) ^ 2 - 1) * ((b - 1) ^ 2 - 1) - ((a * b - 1) ^ 2 - 1) = 2 * (a * b * (2 + 1 - (a + b))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.57818
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ ((a - 1) ^ 2 - 1) * ((b - 1) ^ 2 - 1) - ((a * b - 1) ^ 2 - 1) = 2 * (a * b * (2 + 1 - (a + b)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | have h0 := pow_four_Eq2 hf (x * (x + 1)) | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f (((x * (x + 1)) ^ 2) ^ 2 + 1) = (f (x * (x + 1) + 1) ^ 2 - 1) ^ 2 - 1
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rw [Eq1 hf, mul_pow, mul_pow, add_one_sq, add_one_sq, Eq1 hf, pow_four_Eq1 hf,
pow_four_Eq2 hf, ← sub_eq_zero, mul_pow, SCharNeTwo_main_ring_id, mul_eq_zero,
or_iff_right h, mul_eq_zero, ← mul_pow, sq_eq_zero_iff] at h0 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f (((x * (x + 1)) ^ 2) ^ 2 + 1) = (f (x * (x + 1) + 1) ^ 2 - 1) ^ 2 - 1
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f (((x * (x + 1)) ^ 2) ^ 2 + 1) = (f (x * (x + 1) + 1) ^ 2 - 1) ^ 2 - 1
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | revert h0 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0 →
(f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | refine Or.imp mul_eq_zero.mp λ h0 ↦ ?_ | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0 →
(f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0 →
(f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rw [sub_eq_zero, eq_comm] at h0 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | refine (Thm1 hf x).elim (λ h1 ↦ Not.elim h ?_) (λ h1 ↦ ⟨h1, ?_⟩) | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x ^ 2 + f (x + 1) ^ 2 = 1
⊢ 2 = 0
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x + f (x + 1) = 1
⊢ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | apply congrArg (λ y ↦ y ^ 2) at h1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x + f (x + 1) = 1
⊢ f x * f (x + 1) = -1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : (f x + f (x + 1)) ^ 2 = 1 ^ 2
⊢ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x + f (x + 1) = 1
⊢ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rw [one_pow, add_sq', h0, add_right_comm, add_left_eq_self,
mul_assoc, ← mul_one_add (2 : S), mul_eq_zero] at h1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : (f x + f (x + 1)) ^ 2 = 1 ^ 2
⊢ f x * f (x + 1) = -1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : 2 = 0 ∨ 1 + f x * f (x + 1) = 0
⊢ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : (f x + f (x + 1)) ^ 2 = 1 ^ 2
⊢ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | exact eq_neg_of_add_eq_zero_right (h1.resolve_left h) | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : 2 = 0 ∨ 1 + f x * f (x + 1) = 0
⊢ f x * f (x + 1) = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : 2 = 0 ∨ 1 + f x * f (x + 1) = 0
⊢ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rwa [h0, add_left_eq_self] at h1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x ^ 2 + f (x + 1) ^ 2 = 1
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x ^ 2 + f (x + 1) ^ 2 = 1
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharTwo.solution | [153, 1] | [161, 60] | rcases CommSubring.oneVarCommLiftDomain_exists hf x with
⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h, hf'⟩ | R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) = f x + 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 1) = f (φ x) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharTwo.solution | [153, 1] | [161, 60] | have R'char := pullback_of_inj φ.toAddMonoidHom hφ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 1) = f (φ x) + 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
⊢ f (φ x + 1) = f (φ x) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 1) = f (φ x) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharTwo.solution | [153, 1] | [161, 60] | have S'char := pullback_of_inj ρ.toAddMonoidHom hρ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
⊢ f (φ x + 1) = f (φ x) + 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : CharTwo S'
⊢ f (φ x + 1) = f (φ x) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
⊢ f (φ x + 1) = f (φ x) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharTwo.solution | [153, 1] | [161, 60] | rw [h, ← φ.map_one, ← φ.map_add, h, ← ρ.map_one, ← ρ.map_add] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : CharTwo S'
⊢ f (φ x + 1) = f (φ x) + 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : CharTwo S'
⊢ ρ (f' (x + 1)) = ρ (f' x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
S'char : CharTwo S'
⊢ f (φ x + 1) = f (φ x) + 1
TACTIC:
|
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