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/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase23.solution | [754, 1] | [760, 40] | refine ⟨R', _, φ, ι, λ x ↦ ?_⟩ | case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
⊢ ∃ R' x φ ι, ∀ (x_1 : R), f x_1 = ι (RestrictedSq (φ x_1) - 1) | case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
x : R
⊢ f x = ι (RestrictedSq (φ x) - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
⊢ ∃ R' x φ ι, ∀ (x_1 : R), f x_1 = ι (RestrictedSq (φ x_1) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase23.solution | [754, 1] | [760, 40] | rw [ι.map_sub, ← h0, ι.map_one] | case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
x : R
⊢ f x = ι (RestrictedSq (φ x) - 1) | case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
x : R
⊢ f x = (f + 1) x - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
x : R
⊢ f x = ι (RestrictedSq (φ x) - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.RGoodSubcase23.solution | [754, 1] | [760, 40] | exact (add_sub_cancel_right _ _).symm | case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
x : R
⊢ f x = (f + 1) x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : RGoodCase2 f
h : f 2 = 3
R' : Type u
w✝ : CommRing R'
φ : R →+* R'
ι : ↥(SqSubring R') →+* S
h0 : ∀ (x : R), (f + 1) x = ι (RestrictedSq (φ x))
x : R
⊢ f x = (f + 1) x - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | refine hf.period_imp_zero λ x ↦ ?_ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
⊢ 2 = 0 | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
x : R
⊢ f (x + 2) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rcases CommSubring.oneVarCommLiftDomain_exists hf.toNontrivialGood x with
⟨R', R'comm, φ, -, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h1, hf'⟩ | S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
x : R
⊢ f (x + 2) = f x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 2) = f (φ x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
x : R
⊢ f (x + 2) = f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rw [← map_ofNat φ 2, ← φ.map_add, h1, h1, CommCase.two_periodic_of_map_two hf'] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 2) = f (φ x) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 = -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
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 2) = f (φ x)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | refine map_even_of_map_one hf'.is_good (hρ ?_) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' (-1)) = ρ 0 | 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.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ∀ (x : R'), f' (-x) = f' x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rw [← h1, φ.map_neg, φ.map_one, h, ρ.map_zero] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' (-1)) = ρ 0 | no goals | 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.h
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' (-1)) = ρ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | apply hρ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 = -1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0.a
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' 2) = ρ (-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.h0
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f' 2 = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case2.lean | IMOSL.IMO2012A5.Case2.CharTwo'_of_map_two | [773, 1] | [781, 61] | rw [← h1, map_ofNat, h0, ρ.map_neg, ρ.map_one] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h0.a
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' 2) = ρ (-1) | no goals | 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.h0.a
S : Type u_1
R : Type u
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
h : f (-1) = 0
h0 : f 2 = -1
R' : Type u
R'comm : CommRing R'
φ : R' →+* R
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h1 : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ ρ (f' 2) = ρ (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.add_mul | [231, 11] | [232, 58] | rw [ℤ₄.mul_comm, ℤ₄.mul_add, z.mul_comm, z.mul_comm] | x y z : ℤ₄
⊢ (x + y) * z = x * z + y * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : ℤ₄
⊢ (x + y) * z = x * z + y * z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_add | [259, 1] | [274, 47] | rw [← h, ← Nat.cast_two, ← Nat.cast_add] | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0 | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_add | [259, 1] | [274, 47] | rfl | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_add | [259, 1] | [274, 47] | rwa [neg_eq_iff_add_eq_zero, ← add_assoc, h0] | R : Type u_1
inst✝ : NonAssocRing R
h✝ : 4 = 0
x y : ℤ₄
h : 2 + 2 = 0
h0 : 1 + 1 = 2
⊢ -1 = 1 + 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h✝ : 4 = 0
x y : ℤ₄
h : 2 + 2 = 0
h0 : 1 + 1 = 2
⊢ -1 = 1 + 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_mul | [276, 1] | [287, 64] | rw [← h, ← Nat.cast_two, ← Nat.cast_add] | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0 | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ 2 + 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.cast_mul | [276, 1] | [287, 64] | rfl | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
x y : ℤ₄
⊢ ↑(2 + 2) = 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Z4.lean | IMOSL.IMO2012A5.ℤ₄.castRingHom_injective | [296, 1] | [303, 43] | rw [← one_mul (2 : R), h1, zero_mul] | R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
h0 : 2 ≠ 0
h1 : 1 = 0
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
h : 4 = 0
h0 : 2 ≠ 0
h1 : 1 = 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.castMonoidHom_is_MonoidGood | [33, 1] | [34, 65] | rw [Int.floor_intCast, ← Int.cast_mul, ← φ.map_mul] | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℤ
m n : M
⊢ (fun x => ↑(φ x)) (m * n) = (fun x => ↑(φ x)) m * ↑⌊(fun x => ↑(φ x)) n⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℤ
m n : M
⊢ (fun x => ↑(φ x)) (m * n) = (fun x => ↑(φ x)) m * ↑⌊(fun x => ↑(φ x)) n⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | change (1 + ε) * _ = (1 + ε) * _ * ⌊(1 + ε) * _⌋ | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (fun x => (1 + ε) * ↑(φ x)) (m * n) = (fun x => (1 + ε) * ↑(φ x)) m * ↑⌊(fun x => (1 + ε) * ↑(φ x)) n⌋ | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ (m * n)) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (fun x => (1 + ε) * ↑(φ x)) (m * n) = (fun x => (1 + ε) * ↑(φ x)) m * ↑⌊(fun x => (1 + ε) * ↑(φ x)) n⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | rw [φ.map_mul, Nat.cast_mul, ← mul_assoc] | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ (m * n)) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ m) * ↑(φ n) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ (m * n)) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | apply congrArg | R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ m) * ↑(φ n) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋ | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ ↑(φ n) = ↑⌊(1 + ε) * ↑(φ n)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ (1 + ε) * ↑(φ m) * ↑(φ n) = (1 + ε) * ↑(φ m) * ↑⌊(1 + ε) * ↑(φ n)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | rw [one_add_mul ε, Int.floor_nat_add, Int.cast_add, Int.cast_natCast,
← nsmul_eq_mul', self_eq_add_right, Int.cast_eq_zero, Int.floor_eq_zero_iff] | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ ↑(φ n) = ↑⌊(1 + ε) * ↑(φ n)⌋ | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ φ n • ε ∈ Set.Ico 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ ↑(φ n) = ↑⌊(1 + ε) * ↑(φ n)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.one_add_infinitesimal_mul_is_MonoidGood | [36, 1] | [43, 57] | exact ⟨nsmul_nonneg h _, abs_eq_self.mpr h ▸ h0 (φ n)⟩ | case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ φ n • ε ∈ Set.Ico 0 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_2
M : Type u_1
inst✝² : LinearOrderedRing R
inst✝¹ : FloorRing R
inst✝ : Mul M
φ : M →ₙ* ℕ
ε : R
h : 0 ≤ ε
h0 : Infinitesimal ε
m n : M
⊢ φ n • ε ∈ Set.Ico 0 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | simp only [h] | R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (fun n => if n ∈ A then C else 0) (m * n) =
(fun n => if n ∈ A then C else 0) m * ↑⌊(fun n => if n ∈ A then C else 0) n⌋ | R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (fun n => if n ∈ A then C else 0) (m * n) =
(fun n => if n ∈ A then C else 0) m * ↑⌊(fun n => if n ∈ A then C else 0) n⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | by_cases h1 : n ∈ A | R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | rw [if_pos h1, h0, Int.cast_one, mul_one] | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = if m ∈ A then C else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | exact if_congr (and_iff_left h1) rfl rfl | case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = if m ∈ A then C else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∈ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = if m ∈ A then C else 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | rw [if_neg h1, Int.floor_zero, Int.cast_zero, mul_zero] | case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋ | case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = (if m ∈ A then C else 0) * ↑⌊if n ∈ A then C else 0⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.indicator_const_is_good | [45, 1] | [52, 32] | exact if_neg λ h2 ↦ h1 h2.2 | case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_2
M : Type u_1
inst✝³ : LinearOrderedRing R
inst✝² : FloorRing R
inst✝¹ : Mul M
C : R
A : Set M
inst✝ : DecidablePred fun x => x ∈ A
h : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A
h0 : ⌊C⌋ = 1
m n : M
h1 : n ∉ A
⊢ (if m ∈ A ∧ n ∈ A then C else 0) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.map_eq_map_one_mul_floor | [66, 1] | [67, 21] | rw [← hf, one_mul] | 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
x : M
⊢ f x = f 1 * ↑⌊f 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
x : M
⊢ f x = f 1 * ↑⌊f x⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | have h := map_eq_map_one_mul_floor hf 1 | 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
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | 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 = f 1 * ↑⌊f 1⌋
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | 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
⊢ f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | rw [← sub_eq_zero, ← mul_one_sub, mul_eq_zero] at 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 1 = f 1 * ↑⌊f 1⌋
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | 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 = 0 ∨ 1 - ↑⌊f 1⌋ = 0
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | 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 = f 1 * ↑⌊f 1⌋
⊢ f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | revert 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 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0
⊢ f = 0 ∨ ⌊f 1⌋ = 1 | 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
⊢ f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0 → f = 0 ∨ ⌊f 1⌋ = 1 | 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 = 0 ∨ 1 - ↑⌊f 1⌋ = 0
⊢ f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | refine Or.imp (λ h ↦ funext λ n ↦ ?_)
(λ h ↦ Int.cast_eq_one.mp (eq_of_sub_eq_zero h).symm) | 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
⊢ f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0 → f = 0 ∨ ⌊f 1⌋ = 1 | 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 = 0
n : M
⊢ f n = 0 n | 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
⊢ f 1 = 0 ∨ 1 - ↑⌊f 1⌋ = 0 → f = 0 ∨ ⌊f 1⌋ = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | rw [map_eq_map_one_mul_floor hf, h, zero_mul] | 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 = 0
n : M
⊢ f n = 0 n | 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 = 0
n : M
⊢ 0 = 0 n | 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 = 0
n : M
⊢ f n = 0 n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.eq_zero_or_floor_map_one_eq_one | [69, 1] | [76, 53] | 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 1 = 0
n : M
⊢ 0 = 0 n | 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 = 0
n : M
⊢ 0 = 0 n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.fract_eq_eps_mul_floor | [80, 1] | [81, 89] | rw [Int.fract, Int.fract, h, Int.cast_one, sub_one_mul, ← map_eq_map_one_mul_floor hf] | 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
x : M
⊢ Int.fract (f x) = Int.fract (f 1) * ↑⌊f 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 1⌋ = 1
x : M
⊢ Int.fract (f x) = Int.fract (f 1) * ↑⌊f x⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | have h0 : f 1 ≠ 0 := λ h0 ↦ Int.zero_ne_one <| by rw [← h, h0, Int.floor_zero] | 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
x y : M
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | 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
x y : M
h0 : f 1 ≠ 0
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | 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
x y : M
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | have h1 := hf x y | 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
x y : M
h0 : f 1 ≠ 0
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | 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
x y : M
h0 : f 1 ≠ 0
h1 : f (x * y) = f x * ↑⌊f y⌋
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋ | 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
x y : M
h0 : f 1 ≠ 0
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | rwa [map_eq_map_one_mul_floor hf, map_eq_map_one_mul_floor hf x, mul_assoc, ← sub_eq_zero,
← mul_sub, mul_eq_zero, or_iff_right h0, ← Int.cast_mul, sub_eq_zero, Int.cast_inj] at h1 | 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
x y : M
h0 : f 1 ≠ 0
h1 : f (x * y) = f x * ↑⌊f y⌋
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f 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
x y : M
h0 : f 1 ≠ 0
h1 : f (x * y) = f x * ↑⌊f y⌋
⊢ ⌊f (x * y)⌋ = ⌊f x⌋ * ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_map_mul | [83, 1] | [87, 94] | rw [← h, h0, Int.floor_zero] | 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
x y : M
h0 : f 1 = 0
⊢ 0 = 1 | 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
x y : M
h0 : f 1 = 0
⊢ 0 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | rcases floor_unbounded_of_one_lt h0 N with ⟨y, h1⟩ | 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
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋ | case intro
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
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋ | 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
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | use x * y | case intro
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
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋ | case 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 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ↑(N + 1) < ⌊f (x * y)⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ∃ y, ↑(N + 1) < ⌊f y⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | rw [floor_map_mul hf h, Nat.cast_succ, ← one_mul ((N : ℤ) + 1)] | case 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 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ↑(N + 1) < ⌊f (x * y)⌋ | case 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 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ 1 * (↑N + 1) < ⌊f x⌋ * ⌊f y⌋ | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ ↑(N + 1) < ⌊f (x * y)⌋
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Monoid.lean | IMOSL.IMO2010A1.MonoidGood.floor_unbounded_of_one_lt | [94, 1] | [99, 89] | exact mul_lt_mul_of_nonneg_of_pos h0 h1 Int.one_nonneg (N.cast_nonneg.trans_lt h1) | case 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 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ 1 * (↑N + 1) < ⌊f x⌋ * ⌊f y⌋ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 1⌋ = 1
x : M
h0 : 1 < ⌊f x⌋
N : ℕ
y : M
h1 : ↑N < ⌊f y⌋
⊢ 1 * (↑N + 1) < ⌊f x⌋ * ⌊f y⌋
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] | rw [map_eq_map_one_mul_floor hf,
← Int.natAbs_of_nonneg (h1 x), Int.cast_natCast] | 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
⊢ f x = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x) | 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
⊢ f 1 * ↑⌊f x⌋.natAbs = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x) | 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
⊢ f x = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x)
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] | 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 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
⊢ f 1 * ↑⌊f x⌋.natAbs = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } 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 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
⊢ f 1 * ↑⌊f x⌋.natAbs = f 1 * ↑({ toFun := fun x => ⌊f x⌋.natAbs, map_one' := ⋯, map_mul' := ⋯ } x)
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] | refine ⟨{x : M | ⌊f x⌋ ≠ 0}, λ x y ↦ ?_, λ x ↦ ?_⟩ | 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
⊢ ∃ A, ∃ (_ : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A), ∀ (x : M), f x = if x ∈ A then f 1 else 0 | case refine_1
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 y : M
⊢ x * y ∈ {x | ⌊f x⌋ ≠ 0} ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0}
case refine_2
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
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 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 : 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
⊢ ∃ A, ∃ (_ : ∀ (m n : M), m * n ∈ A ↔ m ∈ A ∧ n ∈ A), ∀ (x : M), f x = if x ∈ A 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] | by_cases h3 : ⌊f x⌋ = 0 | case refine_2
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
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case pos
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
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0
case neg
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
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
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
⊢ 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] | suffices ⌊f x⌋ = 1 by rw [if_pos (by rwa [Set.mem_setOf_eq]),
map_eq_map_one_mul_floor hf, this, Int.cast_one, mul_one] | case neg
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
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case neg
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
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
⊢ 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] | specialize h1 x | case neg
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
⊢ ⌊f x⌋ = 1 | case neg
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 0 ≤ ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
⊢ ⌊f x⌋ = 1
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] | rw [le_iff_eq_or_lt, eq_comm, or_iff_right h3,
Int.lt_iff_add_one_le, zero_add, le_iff_eq_or_lt] at h1 | case neg
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 0 ≤ ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋ ∨ 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 0 ≤ ⌊f x⌋
⊢ ⌊f x⌋ = 1
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] | rcases h1 with h1 | h1 | case neg
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋ ∨ 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg.inl
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋
⊢ ⌊f x⌋ = 1
case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋ ∨ 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1
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] | exact h1.symm | case neg.inl
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋
⊢ ⌊f x⌋ = 1
case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 = ⌊f x⌋
⊢ ⌊f x⌋ = 1
case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1
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] | refine h2.elim λ N ↦ ?_ | case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1 | case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
⊢ N • Int.fract (f 1) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
⊢ ⌊f x⌋ = 1
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] | rcases floor_unbounded_of_one_lt hf h h1 N with ⟨y, h4⟩ | case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
⊢ N • Int.fract (f 1) < 1 | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ N • Int.fract (f 1) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
⊢ N • Int.fract (f 1) < 1
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] | rw [nsmul_eq_mul', ← Int.cast_natCast] | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ N • Int.fract (f 1) < 1 | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑↑N < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ N • Int.fract (f 1) < 1
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] | apply (mul_lt_mul_of_pos_left (Int.cast_lt.mpr h4) h0).trans | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑↑N < 1 | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑⌊f y⌋ < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑↑N < 1
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] | rw [← fract_eq_eps_mul_floor hf h] | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑⌊f y⌋ < 1 | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f y) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f 1) * ↑⌊f y⌋ < 1
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] | exact Int.fract_lt_one _ | case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f y) < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.intro
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)
h2 : ¬∀ (k : ℕ), k • Int.fract (f 1) < 1
x : M
h3 : ¬⌊f x⌋ = 0
h1 : 1 < ⌊f x⌋
N : ℕ
y : M
h4 : ↑N < ⌊f y⌋
⊢ Int.fract (f y) < 1
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] | rw [Set.mem_setOf_eq, floor_map_mul hf h, mul_ne_zero_iff] | case refine_1
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 y : M
⊢ x * y ∈ {x | ⌊f x⌋ ≠ 0} ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0} | case refine_1
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 y : M
⊢ ⌊f x⌋ ≠ 0 ∧ ⌊f y⌋ ≠ 0 ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
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 y : M
⊢ x * y ∈ {x | ⌊f x⌋ ≠ 0} ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 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] | rfl | case refine_1
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 y : M
⊢ ⌊f x⌋ ≠ 0 ∧ ⌊f y⌋ ≠ 0 ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 0} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
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 y : M
⊢ ⌊f x⌋ ≠ 0 ∧ ⌊f y⌋ ≠ 0 ↔ x ∈ {x | ⌊f x⌋ ≠ 0} ∧ y ∈ {x | ⌊f x⌋ ≠ 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] | rw [map_eq_map_one_mul_floor hf, h3, Int.cast_zero, mul_zero] | case pos
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
⊢ f x = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case pos
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
⊢ 0 = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
⊢ 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] | refine (if_neg ?_).symm | case pos
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
⊢ 0 = if x ∈ {x | ⌊f x⌋ ≠ 0} then f 1 else 0 | case pos
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
⊢ x ∉ {x | ⌊f x⌋ ≠ 0} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
⊢ 0 = 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, not_not] | case pos
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
⊢ x ∉ {x | ⌊f x⌋ ≠ 0} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
⊢ x ∉ {x | ⌊f x⌋ ≠ 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] | 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/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.iterate_add_mul_eq | [31, 1] | [35, 59] | rw [k.mul_succ, ← Nat.add_assoc, f.iterate_add,
iterate_add_mul_eq h t, ← f.iterate_add, h] | n k : ℕ
S : Type u_1
f : S → S
h : f^[n + k] = f^[n]
t : ℕ
⊢ f^[n + k * (t + 1)] = f^[n] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
S : Type u_1
f : S → S
h : f^[n + k] = f^[n]
t : ℕ
⊢ f^[n + k * (t + 1)] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | apply (Set.range_comp_subset_range f f).antisymm | S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f^[2] = Set.range f | S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f^[2] = Set.range f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | rcases Nat.exists_eq_add_of_le h with ⟨k, rfl⟩ | S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
m : ℕ
f : S → S
h : 2 ≤ m
h0 : f^[m] = f
⊢ Set.range f ⊆ Set.range (f ∘ f)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | nth_rw 1 [← h0, f.iterate_add] | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range f ⊆ Set.range (f ∘ f) | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range (f^[2] ∘ f^[k]) ⊆ Set.range (f ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range f ⊆ Set.range (f ∘ f)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.range_iter_two_eq_of_exists_iter_eq_self | [37, 1] | [42, 72] | exact Set.range_comp_subset_range _ _ | case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range (f^[2] ∘ f^[k]) ⊆ Set.range (f ∘ f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type u_1
f : S → S
k : ℕ
h : 2 ≤ 2 + k
h0 : f^[2 + k] = f
⊢ Set.range (f^[2] ∘ f^[k]) ⊆ Set.range (f ∘ f)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_lt_iterate_eq | [46, 1] | [49, 73] | obtain ⟨a, b, h, h0⟩ : ∃ a b : ℕ, a ≠ b ∧ f^[a] = f^[b] :=
Finite.exists_ne_map_eq_of_infinite _ | S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ a b, a < b ∧ f^[a] = f^[b] | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a ≠ b
h0 : f^[a] = f^[b]
⊢ ∃ a b, a < b ∧ f^[a] = f^[b] | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ a b, a < b ∧ f^[a] = f^[b]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_lt_iterate_eq | [46, 1] | [49, 73] | exact h.lt_or_lt.elim (λ h ↦ ⟨a, b, h, h0⟩) (λ h ↦ ⟨b, a, h, h0.symm⟩) | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a ≠ b
h0 : f^[a] = f^[b]
⊢ ∃ a b, a < b ∧ f^[a] = f^[b] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a ≠ b
h0 : f^[a] = f^[b]
⊢ ∃ a b, a < b ∧ f^[a] = f^[b]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | obtain ⟨a, b, h, h0⟩ := exists_lt_iterate_eq f | S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a < b
h0 : f^[a] = f^[b]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rcases Nat.exists_eq_add_of_le h.le with ⟨c, rfl⟩ | case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a < b
h0 : f^[a] = f^[b]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : a < a + c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a b : ℕ
h : a < b
h0 : f^[a] = f^[b]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rw [Nat.lt_add_right_iff_pos] at h | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : a < a + c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : a < a + c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | refine ⟨c * a.succ, Nat.mul_pos h a.succ_pos, ?_⟩ | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[2 * (c * a.succ)] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ ∃ n, 0 < n ∧ f^[2 * n] = f^[n]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rw [Nat.two_mul] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[2 * (c * a.succ)] = f^[c * a.succ] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c * a.succ] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[2 * (c * a.succ)] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | refine iterate_add_mul_eq ?_ _ | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c * a.succ] = f^[c * a.succ] | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c * a.succ] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | obtain ⟨k, h1⟩ := Nat.exists_eq_add_of_le
(Nat.le_mul_of_pos_left a.succ h) | case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c] = f^[c * a.succ] | case intro.intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
k : ℕ
h1 : c * a.succ = a.succ + k
⊢ f^[c * a.succ + c] = f^[c * a.succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
⊢ f^[c * a.succ + c] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.exists_iterate_idempotent | [51, 1] | [61, 59] | rw [h1, Nat.add_right_comm, f.iterate_add, Nat.succ_add,
f.iterate_succ, ← h0, ← f.iterate_succ, f.iterate_add] | case intro.intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
k : ℕ
h1 : c * a.succ = a.succ + k
⊢ f^[c * a.succ + c] = f^[c * a.succ] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
a c : ℕ
h : 0 < c
h0 : f^[a] = f^[a + c]
k : ℕ
h1 : c * a.succ = a.succ + k
⊢ f^[c * a.succ + c] = f^[c * a.succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | rcases exists_iterate_idempotent f with ⟨n, h0, h1⟩ | S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
⊢ Set.range f^[2] = Set.range f | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ Set.range f^[2] = Set.range f | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
⊢ Set.range f^[2] = Set.range f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | refine range_iter_two_eq_of_exists_iter_eq_self
(Nat.lt_add_of_pos_left h0) (h _ ?_) | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ Set.range f^[2] = Set.range f | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1] ∘ f ∘ f^[n + 1] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ Set.range f^[2] = Set.range f
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | rw [← f.iterate_succ', ← f.iterate_add] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1] ∘ f ∘ f^[n + 1] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1 + (n + 1).succ] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1] ∘ f ∘ f^[n + 1]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | change (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1 + (n + 1).succ] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ f ∘ f^[n + 1] ∘ f = f^[n + 1 + (n + 1).succ]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2017/A3/A3.lean | IMOSL.IMO2017A3.final_solution | [66, 1] | [73, 64] | rw [Nat.succ_add, ← Nat.two_mul, f.iterate_succ' (2 * n), h1] | case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type u_1
inst✝¹ : Fintype S
inst✝ : DecidableEq S
f : S → S
h : ∀ (g : S → S), f ∘ g ∘ f = g ∘ f ∘ g → g = f
n : ℕ
h0 : 0 < n
h1 : f^[2 * n] = f^[n]
⊢ (f ∘ f^[n]) ∘ f^[2] = f^[n + 1 + n] ∘ f^[2]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0 := by
have h0 := h 0 0
rw [zsmul_zero, ← sub_eq_zero, ← mul_one_sub, mul_eq_zero] at h0
exact h0.symm.imp_left λ h0 ↦ Int.cast_eq_one.mp (eq_of_sub_eq_zero h0).symm | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | revert h0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 → (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1 ∨ f 0 = 0
⊢ (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | refine Or.imp (λ h0 ↦ ?_) (λ h0 ↦ ?_) | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 → (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0 | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
⊢ ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C
case refine_2
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 → (∃ C, ⌊C⌋ = 1 ∧ f = fun x => C) ∨ f = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | have h0 := h 0 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f (⌊0⌋ • 0) = f 0 * ↑⌊f 0⌋
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | rw [zsmul_zero, ← sub_eq_zero, ← mul_one_sub, mul_eq_zero] at h0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f (⌊0⌋ • 0) = f 0 * ↑⌊f 0⌋
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0 ∨ 1 - ↑⌊f 0⌋ = 0
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f (⌊0⌋ • 0) = f 0 * ↑⌊f 0⌋
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | exact h0.symm.imp_left λ h0 ↦ Int.cast_eq_one.mp (eq_of_sub_eq_zero h0).symm | R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0 ∨ 1 - ↑⌊f 0⌋ = 0
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : f 0 = 0 ∨ 1 - ↑⌊f 0⌋ = 0
⊢ ⌊f 0⌋ = 1 ∨ f 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A1/A1Dense.lean | IMOSL.IMO2010A1.good_iff_of_DenselyOrdered | [30, 1] | [65, 39] | refine ⟨f 0, h0, funext λ x ↦ ?_⟩ | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
⊢ ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C | case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
x : R
⊢ f x = f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
inst✝⁴ : LinearOrderedRing R
inst✝³ : FloorRing R
S : Type u_1
inst✝² : DenselyOrdered R
inst✝¹ : LinearOrderedRing S
inst✝ : FloorRing S
f : R → S
h : good f
h0 : ⌊f 0⌋ = 1
⊢ ∃ C, ⌊C⌋ = 1 ∧ f = fun x => C
TACTIC:
|
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