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/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [reduced_eq_zero_iff hf h, h2, eq_neg_iff_add_eq_zero, one_add_one_eq_two, h3] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3✝ : 2 = 0 ∨ f (d * x + 1) = 0
h3 : 2 = 0
⊢ c = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
h3✝ : 2 = 0 ∨ f (d * x + 1) = 0
h3 : 2 = 0
⊢ c = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [iff_left hf.toNontrivialGood] at h | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1) | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_mul_left_eq_zero_imp | [129, 1] | [138, 84] | rw [hf.is_good, h, add_left_comm, h, h2, neg_one_mul,
neg_one_mul, mul_neg, ← neg_add, hf.is_good] | R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : NonAssocRing R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : ∀ (x : R), f (c + x) = -f c * f x
h0 : c ≠ 0
d : R
h1 : d * c = 0
x : R
h2 : f c = 1
⊢ f (d * (c + x) + 1) = -f (d * x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5QuasiPeriodic.lean | IMOSL.IMO2012A5.QuasiPeriodic.reduced_QuasiPeriod_equiv_cases | [148, 1] | [152, 68] | rwa [sub_one_mul, sub_eq_zero] | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : ∀ (d : R), d * c = 0 → QuasiPeriodic f d := fun d => reduced_mul_left_eq_zero_imp hf h h0
h2 : d * c = c
⊢ (d - 1) * c = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : NonAssocRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : ReducedGood f
c : R
h : QuasiPeriodic f c
h0 : c ≠ 0
d : R
h1 : ∀ (d : R), d * c = 0 → QuasiPeriodic f d := fun d => reduced_mul_left_eq_zero_imp hf h h0
h2 : d * c = c
⊢ (d - 1) * c = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [add_inf] | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) (a✝¹ + a✝ ⊓ b✝) | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) ((a✝¹ + a✝) ⊓ (a✝¹ + b✝)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) (a✝¹ + a✝ ⊓ b✝)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofInf | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) ((a✝¹ + a✝) ⊓ (a✝¹ + b✝)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) ((a✝¹ + a✝) ⊓ (a✝¹ + b✝))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [add_sup] | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) (a✝¹ + a✝ ⊔ b✝) | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) ((a✝¹ + a✝) ⊔ (a✝¹ + b✝)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) (a✝¹ + a✝ ⊔ b✝)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofSup | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) ((a✝¹ + a✝) ⊔ (a✝¹ + b✝)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha✝ : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝¹ : G
ha : a✝¹ ∈ S
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝¹ + a✝) →
MetaClosure (fun x => x ∈ S) (a✝¹ + b✝) → MetaClosure (fun x => x ∈ S) ((a✝¹ + a✝) ⊔ (a✝¹ + b✝))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [inf_add] | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) (a✝ ⊓ b✝ + b) | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) ((a✝ + b) ⊓ (b✝ + b)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) (a✝ ⊓ b✝ + b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofInf | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) ((a✝ + b) ⊓ (b✝ + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) ((a✝ + b) ⊓ (b✝ + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | rw [sup_add] | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) (a✝ ⊔ b✝ + b) | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) ((a✝ + b) ⊔ (b✝ + b)) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) (a✝ ⊔ b✝ + b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.add_mem | [30, 1] | [38, 43] | exact ofSup | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) ((a✝ + b) ⊔ (b✝ + b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a b : G
ha : MetaClosure (fun x => x ∈ S) a
hb : MetaClosure (fun x => x ∈ S) b
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (a✝ + b) →
MetaClosure (fun x => x ∈ S) (b✝ + b) → MetaClosure (fun x => x ∈ S) ((a✝ + b) ⊔ (b✝ + b))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | rw [neg_inf] | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-(a✝ ⊓ b✝)) | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-a✝ ⊔ -b✝) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-(a✝ ⊓ b✝))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | exact ofSup | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-a✝ ⊔ -b✝) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-a✝ ⊔ -b✝)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | rw [neg_sup] | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-(a✝ ⊔ b✝)) | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-a✝ ⊓ -b✝) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-(a✝ ⊔ b✝))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7Group.lean | IMOSL.IMO2012A7.MetaClosure.neg_mem | [42, 1] | [45, 43] | exact ofInf | G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-a✝ ⊓ -b✝) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝³ : Lattice G
inst✝² : AddGroup G
inst✝¹ : CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
S : AddSubgroup G
a : G
ha : MetaClosure (fun x => x ∈ S) a
a✝ b✝ : G
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) (-a✝) → MetaClosure (fun x => x ∈ S) (-b✝) → MetaClosure (fun x => x ∈ S) (-a✝ ⊓ -b✝)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | have h0 : ∃ n, ∃ x, f x = n := ⟨f 0, 0, rfl⟩ | f g : ℕ → ℕ
h : good f g
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | refine ⟨Nat.find h0, λ k ↦
⟨Nat.find_min' h0, Nat.le_induction (Nat.find_spec h0) ?_ k⟩⟩ | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k : ℕ
⊢ ∀ (n : ℕ), Nat.find h0 ≤ n → (∃ x, f x = n) → ∃ x, f x = n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
⊢ ∃ a, ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | rintro n h1 ⟨x, rfl⟩ | f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k : ℕ
⊢ ∀ (n : ℕ), Nat.find h0 ≤ n → (∃ x, f x = n) → ∃ x, f x = n + 1 | case intro
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k x : ℕ
h1 : Nat.find h0 ≤ f x
⊢ ∃ x_1, f x_1 = f x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k : ℕ
⊢ ∀ (n : ℕ), Nat.find h0 ≤ n → (∃ x, f x = n) → ∃ x, f x = n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem1 | [30, 1] | [34, 41] | exact ⟨g x, h x⟩ | case intro
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k x : ℕ
h1 : Nat.find h0 ≤ f x
⊢ ∃ x_1, f x_1 = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℕ → ℕ
h : good f g
h0 : ∃ n x, f x = n
k x : ℕ
h1 : Nat.find h0 ≤ f x
⊢ ∃ x_1, f x_1 = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | rcases h1 with ⟨x, rfl⟩ | f g : ℕ → ℕ
h : good f g
h0 : good g f
k m : ℕ
h1 : ∃ x, f x = k
h2 : ∃ y, f y = m
h3 : f k = f m
⊢ k = m | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
m : ℕ
h2 : ∃ y, f y = m
x : ℕ
h3 : f (f x) = f m
⊢ f x = m | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
k m : ℕ
h1 : ∃ x, f x = k
h2 : ∃ y, f y = m
h3 : f k = f m
⊢ k = m
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | rcases h2 with ⟨y, rfl⟩ | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
m : ℕ
h2 : ∃ y, f y = m
x : ℕ
h3 : f (f x) = f m
⊢ f x = m | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : f (f x) = f (f y)
⊢ f x = f y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
m : ℕ
h2 : ∃ y, f y = m
x : ℕ
h3 : f (f x) = f m
⊢ f x = m
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | replace h3 := lem2 h0 h3 | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : f (f x) = f (f y)
⊢ f x = f y | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g (f x) = g (f y)
⊢ f x = f y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : f (f x) = f (f y)
⊢ f x = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | rw [h0, h0, Nat.succ_inj'] at h3 | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g (f x) = g (f y)
⊢ f x = f y | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g x = g y
⊢ f x = f y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g (f x) = g (f y)
⊢ f x = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem4 | [45, 1] | [49, 18] | exact lem2 h h3 | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g x = g y
⊢ f x = f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
x y : ℕ
h3 : g x = g y
⊢ f x = f y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | wlog h3 : a ≤ b | f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
⊢ a = b | case inr
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
this :
∀ {f g : ℕ → ℕ},
good f g →
good g f → ∀ {a b : ℕ}, (∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k) → (∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k) → a ≤ b → a = b
h3 : ¬a ≤ b
⊢ a = b
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | refine h3.antisymm ((h2 _).mp ?_) | f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ a = b | f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | obtain ⟨k, h4⟩ := Nat.exists_eq_add_of_le (lem3 h0 h1) | f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ ∃ x, g x = a | case intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | obtain ⟨d, h5⟩ := (h1 (a + k)).mpr (a.le_add_right k) | case intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
⊢ ∃ x, g x = a | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
d : ℕ
h5 : f d = a + k
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | rw [Nat.succ_add, ← h5, ← h, eq_comm] at h4 | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
d : ℕ
h5 : f d = a + k
⊢ ∃ x, g x = a | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k d : ℕ
h4 : f (g d) = f a
h5 : f d = a + k
⊢ ∃ x, g x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k : ℕ
h4 : f a = a.succ + k
d : ℕ
h5 : f d = a + k
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | exact ⟨d, lem4 h h0 ((h1 _).mpr <| h3.trans <| (h2 _).mp ⟨d, rfl⟩)
((h1 a).mpr a.le_refl) h4⟩ | case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k d : ℕ
h4 : f (g d) = f a
h5 : f d = a + k
⊢ ∃ x, g x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
h3 : a ≤ b
k d : ℕ
h4 : f (g d) = f a
h5 : f d = a + k
⊢ ∃ x, g x = a
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem5 | [51, 1] | [60, 31] | exact (this h0 h h2 h1 (le_of_not_le h3)).symm | case inr
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
this :
∀ {f g : ℕ → ℕ},
good f g →
good g f → ∀ {a b : ℕ}, (∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k) → (∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k) → a ≤ b → a = b
h3 : ¬a ≤ b
⊢ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f g : ℕ → ℕ
h : good f g
h0 : good g f
a b : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k
this :
∀ {f g : ℕ → ℕ},
good f g →
good g f → ∀ {a b : ℕ}, (∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k) → (∀ (k : ℕ), (∃ x, g x = k) ↔ b ≤ k) → a ≤ b → a = b
h3 : ¬a ≤ b
⊢ a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | refine (Nat.succ_le_of_lt (lem3 h0 h1)).eq_or_gt.resolve_right λ h3 ↦ ?_ | f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f a = a.succ | f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : a.succ < f a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f a = a.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | obtain ⟨t, h3⟩ := Nat.exists_eq_add_of_lt h3 | f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : a.succ < f a
⊢ False | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : a.succ < f a
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | obtain ⟨x, h4⟩ := (h1 (a + t)).mpr (a.le_add_right t) | case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
⊢ False | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | obtain ⟨y, h5⟩ := (h1 (g x)).mpr <| (h2 _).mp ⟨x, rfl⟩ | case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
⊢ False | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
y : ℕ
h5 : f y = g x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | rw [Nat.succ_add, ← h4, ← h, ← Nat.succ_eq_add_one, ← h, ← h5] at h3 | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
y : ℕ
h5 : f y = g x
⊢ False | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t : ℕ
h3 : f a = a.succ + t + 1
x : ℕ
h4 : f x = a + t
y : ℕ
h5 : f y = g x
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | refine (lem4 h h0 ((h1 a).mpr a.le_refl)
((h1 _).mpr <| (h2 _).mp ⟨f y, rfl⟩) h3).not_lt ?_ | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ False | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ a < g (f y) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | rw [h0, Nat.lt_succ_iff, ← h2] | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ a < g (f y) | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ ∃ x, g x = g y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ a < g (f y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.lem7 | [67, 1] | [77, 17] | exact ⟨y, rfl⟩ | case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ ∃ x, g x = g y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3✝ : a.succ < f a
t x : ℕ
h4 : f x = a + t
y : ℕ
h3 : f a = f (g (f y))
h5 : f y = g x
⊢ ∃ x, g x = g y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | obtain ⟨a, h1, h2⟩ := lem6 h h0 | f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
⊢ f = g | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f = g | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
⊢ f = g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | suffices h3 : ∀ n, a ≤ n → f n = n.succ ∧ g n = n.succ by
ext x; rw [← Nat.succ_inj', ← h, (h3 _ <| (h2 _).mp ⟨x, rfl⟩).1] | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f = g | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ f = g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | refine Nat.le_induction ⟨lem7 h h0 h1 h2, lem7 h0 h h2 h1⟩
(λ n _ h3 ↦ ⟨?_, ?_⟩) | case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ | case intro.intro.refine_1
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ f (n + 1) = (n + 1).succ
case intro.intro.refine_2
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ g (n + 1) = (n + 1).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
⊢ ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | ext x | f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
⊢ f = g | case h
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
x : ℕ
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
⊢ f = g
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | rw [← Nat.succ_inj', ← h, (h3 _ <| (h2 _).mp ⟨x, rfl⟩).1] | case h
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
x : ℕ
⊢ f x = g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
h3 : ∀ (n : ℕ), a ≤ n → f n = n.succ ∧ g n = n.succ
x : ℕ
⊢ f x = g x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | rw [← Nat.succ_eq_add_one, ← h3.2, h, h3.1, h3.2] | case intro.intro.refine_1
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ f (n + 1) = (n + 1).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_1
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ f (n + 1) = (n + 1).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2010/A6/A6.lean | IMOSL.IMO2010A6.final_solution | [80, 1] | [87, 55] | rw [← Nat.succ_eq_add_one, ← h3.1, h0, h3.1, h3.2] | case intro.intro.refine_2
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ g (n + 1) = (n + 1).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
f✝ g✝ : ℕ → ℕ
h✝ : good f✝ g✝
h0✝ : good g✝ f✝
f g : ℕ → ℕ
h : good f g
h0 : good g f
a : ℕ
h1 : ∀ (k : ℕ), (∃ x, f x = k) ↔ a ≤ k
h2 : ∀ (k : ℕ), (∃ x, g x = k) ↔ a ≤ k
n : ℕ
x✝ : a ≤ n
h3 : f n = n.succ ∧ g n = n.succ
⊢ g (n + 1) = (n + 1).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | refine ⟨λ h ↦ funext λ n ↦ le_antisymm ?_ ?_,
λ h m n ↦ h.symm ▸ dvd_refl (m * m + n)⟩ | f : ℕ+ → ℕ+
⊢ (∀ (m n : ℕ+), m * m + f n ∣ m * f m + n) ↔ f = id | case refine_1
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ f n ≤ id n
case refine_2
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ id n ≤ f n | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ+ → ℕ+
⊢ (∀ (m n : ℕ+), m * m + f n ∣ m * f m + n) ↔ f = id
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | specialize h (f n) n | case refine_1
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * f n + f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rw [← mul_add_one (α := ℕ+)] at h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * f n + f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * (f n + 1) ∣ f n * f (f n) + n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * f n + f n ∣ f n * f (f n) + n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | replace h := (dvd_mul_right _ _).trans h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * (f n + 1) ∣ f n * f (f n) + n
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * (f n + 1) ∣ f n * f (f n) + n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rw [PNat.dvd_iff, PNat.add_coe, PNat.mul_coe,
Nat.dvd_add_right ⟨_, rfl⟩, ← PNat.dvd_iff] at h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ f n * f (f n) + n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | exact PNat.le_of_dvd h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ n
⊢ f n ≤ id n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rcases eq_or_ne n 1 with rfl | h0 | case refine_2
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ id n ≤ f n | case refine_2.inl
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
⊢ id 1 ≤ f 1
case refine_2.inr
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
h0 : n ≠ 1
⊢ id n ≤ f n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ id n ≤ f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | exact (f 1).one_le | case refine_2.inl
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
⊢ id 1 ≤ f 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
⊢ id 1 ≤ f 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rcases PNat.exists_eq_succ_of_ne_one h0 with ⟨k, rfl⟩ | case refine_2.inr
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
h0 : n ≠ 1
⊢ id n ≤ f n | case refine_2.inr.intro
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
k : ℕ+
h0 : k + 1 ≠ 1
⊢ id (k + 1) ≤ f (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
h0 : n ≠ 1
⊢ id n ≤ f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | replace h := PNat.le_of_dvd (h (k + 1) (k + 1)) | case refine_2.inr.intro
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
k : ℕ+
h0 : k + 1 ≠ 1
⊢ id (k + 1) ≤ f (k + 1) | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : (k + 1) * (k + 1) + f (k + 1) ≤ (k + 1) * f (k + 1) + (k + 1)
⊢ id (k + 1) ≤ f (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
k : ℕ+
h0 : k + 1 ≠ 1
⊢ id (k + 1) ≤ f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rw [add_one_mul (α := ℕ+), add_right_comm,
add_assoc, add_one_mul (α := ℕ+), add_assoc] at h | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : (k + 1) * (k + 1) + f (k + 1) ≤ (k + 1) * f (k + 1) + (k + 1)
⊢ id (k + 1) ≤ f (k + 1) | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : k * (k + 1) + (f (k + 1) + (k + 1)) ≤ k * f (k + 1) + (f (k + 1) + (k + 1))
⊢ id (k + 1) ≤ f (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : (k + 1) * (k + 1) + f (k + 1) ≤ (k + 1) * f (k + 1) + (k + 1)
⊢ id (k + 1) ≤ f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | exact le_of_mul_le_mul_left' <| le_of_add_le_add_right (α := ℕ+) h | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : k * (k + 1) + (f (k + 1) + (k + 1)) ≤ k * f (k + 1) + (f (k + 1) + (k + 1))
⊢ id (k + 1) ≤ f (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : k * (k + 1) + (f (k + 1) + (k + 1)) ≤ k * f (k + 1) + (f (k + 1) + (k + 1))
⊢ id (k + 1) ≤ f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq1 | [38, 1] | [40, 61] | rw [hf.is_good, add_add_cancel_left, hf.map_one, add_zero] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * (x + 1) + 1) = f x * f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * (x + 1) + 1) = f x * f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq2 | [42, 1] | [44, 69] | rw [sq, hf.is_good, add_self_eq_zero, hf.map_zero, sub_eq_add_neg] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x + 1) = f x ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x + 1) = f x ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq2_v2 | [46, 1] | [47, 56] | rw [← Eq2 hf, add_one_mul_self, add_add_cancel_right] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x) = f (x + 1) ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x * x) = f (x + 1) ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3 | [49, 1] | [54, 90] | have h : x * (x + 1) = x * x + x := mul_add_one x x | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1) | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : x * (x + 1) = x * x + x
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3 | [49, 1] | [54, 90] | rw [← Eq2_v2 hf, ← Eq1 hf, mul_sub_one, ← add_sub_right_comm, h, add_assoc,
← hf.is_good, mul_assoc, hf.is_good, h, add_add_cancel_middle₁, add_sub_cancel_right] | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : x * (x + 1) = x * x + x
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : x * (x + 1) = x * x + x
⊢ f x * f (x * x + x) = (f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3_v2 | [56, 1] | [59, 78] | have h := Eq3 hf (x + 1) | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f (x + 1) * f ((x + 1) * (x + 1) + (x + 1)) =
(f (x + 1 + 1) ^ 2 - 1) * (f (x + 1 + 1) - 1) + f (x + 1) * f (x + 1 + 1)
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.Eq3_v2 | [56, 1] | [59, 78] | rwa [add_add_cancel_right, add_one_mul_self, add_add_add_cancel_right] at h | R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f (x + 1) * f ((x + 1) * (x + 1) + (x + 1)) =
(f (x + 1 + 1) ^ 2 - 1) * (f (x + 1 + 1) - 1) + f (x + 1) * f (x + 1 + 1)
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : Semiring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f (x + 1) * f ((x + 1) * (x + 1) + (x + 1)) =
(f (x + 1 + 1) ^ 2 - 1) * (f (x + 1 + 1) - 1) + f (x + 1) * f (x + 1 + 1)
⊢ f (x + 1) * f (x * x + x) = (f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1_ring_id1 | [74, 1] | [77, 63] | ring | R : Type ?u.23673
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ a * ((a ^ 2 - 1) * (a - 1) + b * a) - b * ((b ^ 2 - 1) * (b - 1) + a * b) =
(a ^ 2 + b ^ 2 - 1) * (a + b - 1) * (a - b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.23673
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ a * ((a ^ 2 - 1) * (a - 1) + b * a) - b * ((b ^ 2 - 1) * (b - 1) + a * b) =
(a ^ 2 + b ^ 2 - 1) * (a + b - 1) * (a - b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1_ring_id2 | [79, 1] | [82, 45] | ring | R : Type ?u.29081
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a : S
⊢ a ^ 2 * ((a ^ 2 - 1) ^ 2 + 1) - ((a ^ 2 - 1) * (a - 1) + a * a) ^ 2 = (1 - 2 * a) * (a ^ 2 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.29081
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a : S
⊢ a ^ 2 * ((a ^ 2 - 1) ^ 2 + 1) - ((a ^ 2 - 1) * (a - 1) + a * a) ^ 2 = (1 - 2 * a) * (a ^ 2 - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | have h := Thm1_ring_id1 (f x) (f (x + 1)) | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f x * ((f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x) -
f (x + 1) * ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) =
(f x ^ 2 + f (x + 1) ^ 2 - 1) * (f x + f (x + 1) - 1) * (f x - f (x + 1))
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [← Eq3 hf, ← Eq3_v2 hf, mul_left_comm, sub_self, zero_eq_mul,
mul_eq_zero, sub_eq_zero, sub_eq_zero, sub_eq_zero] at h | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f x * ((f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x) -
f (x + 1) * ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) =
(f x ^ 2 + f (x + 1) ^ 2 - 1) * (f x + f (x + 1) - 1) * (f x - f (x + 1))
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : (f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1) ∨ f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h :
f x * ((f x ^ 2 - 1) * (f x - 1) + f (x + 1) * f x) -
f (x + 1) * ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) =
(f x ^ 2 + f (x + 1) ^ 2 - 1) * (f x + f (x + 1) - 1) * (f x - f (x + 1))
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | cases h with | inl h => exact h | inr h => ?_ | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : (f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1) ∨ f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | case inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : (f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1) ∨ f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | right | case inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x + f (x + 1) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [← h, ← two_mul] | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x + f (x + 1) = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | have h0 : _ ^ 2 = _ ^ 2 := congrArg (λ x ↦ x ^ 2) (Eq3 hf x) | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ 2 * f x = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : (f x * f (x * x + x)) ^ 2 = ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) ^ 2
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [mul_pow, ← add_eq_of_eq_sub (Eq2 hf (x * x + x)), ← h, add_mul_self,
← mul_add_one (x * x), Eq1 hf, Eq2 hf, Eq2_v2 hf, ← h,
← sub_eq_zero, ← sq, Thm1_ring_id2, mul_eq_zero] at h0 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : (f x * f (x * x + x)) ^ 2 = ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) ^ 2
⊢ 2 * f x = 1 | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0 ∨ f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : (f x * f (x * x + x)) ^ 2 = ((f (x + 1) ^ 2 - 1) * (f (x + 1) - 1) + f x * f (x + 1)) ^ 2
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | cases h0 with | inl h0 => exact (eq_of_sub_eq_zero h0).symm | inr h0 => ?_ | case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0 ∨ f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0 ∨ f x ^ 2 - 1 = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | have h1 : f (x * x) = 0 := by rw [Eq2_v2 hf, ← h, h0] | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | replace h := Eq3_v2 hf (x * x) | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : f (x * x + 1) * f (x * x * (x * x) + x * x) = (f (x * x) ^ 2 - 1) * (f (x * x) - 1) + f (x * x + 1) * f (x * x)
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [h1, mul_zero, add_zero, sq, zero_mul, zero_sub,
neg_mul_neg, one_mul, Eq2 hf, h0, zero_mul] at h | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : f (x * x + 1) * f (x * x * (x * x) + x * x) = (f (x * x) ^ 2 - 1) * (f (x * x) - 1) + f (x * x + 1) * f (x * x)
⊢ 2 * f x = 1 | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : 0 = 1
⊢ 2 * f x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : f (x * x + 1) * f (x * x * (x * x) + x * x) = (f (x * x) ^ 2 - 1) * (f (x * x) - 1) + f (x * x + 1) * f (x * x)
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [← mul_one (2 * f x), ← h, mul_zero] | case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : 0 = 1
⊢ 2 * f x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inr
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h0 : f x ^ 2 - 1 = 0
h1 : f (x * x) = 0
h : 0 = 1
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | exact h | case inl
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | exact (eq_of_sub_eq_zero h0).symm | case inr.h.inl
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0
⊢ 2 * f x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h.inl
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : 1 - 2 * f x = 0
⊢ 2 * f x = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.Thm1 | [85, 1] | [106, 7] | rw [Eq2_v2 hf, ← h, h0] | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ f (x * x) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
h : f x = f (x + 1)
h0 : f x ^ 2 - 1 = 0
⊢ f (x * x) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharTwo_map_add_one | [109, 1] | [111, 83] | have h := Thm1 hf x | R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
⊢ f (x + 1) = f x + 1 | R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f (x + 1) = f x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharTwo_map_add_one | [109, 1] | [111, 83] | rwa [← CharTwo.add_sq, CharTwo.sq_eq_one_iff, or_self, add_eq_iff_eq_add''] at h | R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f (x + 1) = f x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : CharTwo R
inst✝² : CommRing S
inst✝¹ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
inst✝ : CharTwo S
x : R
h : f x ^ 2 + f (x + 1) ^ 2 = 1 ∨ f x + f (x + 1) = 1
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.pow_four_Eq1 | [113, 1] | [114, 76] | rw [← add_add_cancel_right (x ^ 2) 1, add_one_sq, sq, sq, Eq2 hf, Eq2 hf] | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2) = (f x ^ 2 - 1) ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2) = (f x ^ 2 - 1) ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.pow_four_Eq2 | [116, 1] | [117, 49] | rw [← pow_four_Eq1 hf, add_one_sq, add_one_sq] | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2 + 1) = (f (x + 1) ^ 2 - 1) ^ 2 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
x : R
⊢ f ((x ^ 2) ^ 2 + 1) = (f (x + 1) ^ 2 - 1) ^ 2 - 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_main_ring_id | [119, 1] | [122, 55] | ring | R : Type ?u.57818
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ ((a - 1) ^ 2 - 1) * ((b - 1) ^ 2 - 1) - ((a * b - 1) ^ 2 - 1) = 2 * (a * b * (2 + 1 - (a + b))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type ?u.57818
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
a b : S
⊢ ((a - 1) ^ 2 - 1) * ((b - 1) ^ 2 - 1) - ((a * b - 1) ^ 2 - 1) = 2 * (a * b * (2 + 1 - (a + b)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | have h0 := pow_four_Eq2 hf (x * (x + 1)) | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f (((x * (x + 1)) ^ 2) ^ 2 + 1) = (f (x * (x + 1) + 1) ^ 2 - 1) ^ 2 - 1
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rw [Eq1 hf, mul_pow, mul_pow, add_one_sq, add_one_sq, Eq1 hf, pow_four_Eq1 hf,
pow_four_Eq2 hf, ← sub_eq_zero, mul_pow, SCharNeTwo_main_ring_id, mul_eq_zero,
or_iff_right h, mul_eq_zero, ← mul_pow, sq_eq_zero_iff] at h0 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f (((x * (x + 1)) ^ 2) ^ 2 + 1) = (f (x * (x + 1) + 1) ^ 2 - 1) ^ 2 - 1
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f (((x * (x + 1)) ^ 2) ^ 2 + 1) = (f (x * (x + 1) + 1) ^ 2 - 1) ^ 2 - 1
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | revert h0 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0 →
(f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ (f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | refine Or.imp mul_eq_zero.mp λ h0 ↦ ?_ | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0 →
(f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
⊢ f x * f (x + 1) = 0 ∨ 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0 →
(f x = 0 ∨ f (x + 1) = 0) ∨ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rw [sub_eq_zero, eq_comm] at h0 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : 2 + 1 - (f x ^ 2 + f (x + 1) ^ 2) = 0
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | refine (Thm1 hf x).elim (λ h1 ↦ Not.elim h ?_) (λ h1 ↦ ⟨h1, ?_⟩) | R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x ^ 2 + f (x + 1) ^ 2 = 1
⊢ 2 = 0
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x + f (x + 1) = 1
⊢ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
⊢ f x + f (x + 1) = 1 ∧ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | apply congrArg (λ y ↦ y ^ 2) at h1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x + f (x + 1) = 1
⊢ f x * f (x + 1) = -1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : (f x + f (x + 1)) ^ 2 = 1 ^ 2
⊢ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x + f (x + 1) = 1
⊢ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rw [one_pow, add_sq', h0, add_right_comm, add_left_eq_self,
mul_assoc, ← mul_one_add (2 : S), mul_eq_zero] at h1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : (f x + f (x + 1)) ^ 2 = 1 ^ 2
⊢ f x * f (x + 1) = -1 | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : 2 = 0 ∨ 1 + f x * f (x + 1) = 0
⊢ f x * f (x + 1) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : (f x + f (x + 1)) ^ 2 = 1 ^ 2
⊢ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | exact eq_neg_of_add_eq_zero_right (h1.resolve_left h) | case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : 2 = 0 ∨ 1 + f x * f (x + 1) = 0
⊢ f x * f (x + 1) = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : 2 = 0 ∨ 1 + f x * f (x + 1) = 0
⊢ f x * f (x + 1) = -1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.CommCase.SCharNeTwo_cases | [125, 1] | [140, 56] | rwa [h0, add_left_eq_self] at h1 | case refine_1
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x ^ 2 + f (x + 1) ^ 2 = 1
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
R : Type u_2
S : Type u_1
inst✝³ : CommSemiring R
inst✝² : CharTwo R
inst✝¹ : CommRing S
inst✝ : NoZeroDivisors S
f : R → S
hf : NontrivialGood f
h : 2 ≠ 0
x : R
h0 : f x ^ 2 + f (x + 1) ^ 2 = 2 + 1
h1 : f x ^ 2 + f (x + 1) ^ 2 = 1
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharTwo.solution | [153, 1] | [161, 60] | rcases CommSubring.oneVarCommLiftDomain_exists hf x with
⟨R', R'comm, φ, hφ, ⟨x, rfl⟩, S', S'comm, S'nzd, ρ, hρ, f', h, hf'⟩ | R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) = f x + 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 1) = f (φ x) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
x : R
⊢ f (x + 1) = f x + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharTwo.solution | [153, 1] | [161, 60] | have R'char := pullback_of_inj φ.toAddMonoidHom hφ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 1) = f (φ x) + 1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
R'char : CharTwo R'
⊢ f (φ x + 1) = f (φ x) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
S : Type u_1
inst✝⁴ : Ring R
inst✝³ : CharTwo R
inst✝² : Ring S
inst✝¹ : NoZeroDivisors S
inst✝ : CharTwo S
f : R → S
hf : NontrivialGood f
R' : Type u_2
R'comm : CommRing R'
φ : R' →+* R
hφ : Function.Injective ⇑φ
x : R'
S' : Type u_1
S'comm : CommRing S'
S'nzd : NoZeroDivisors S'
ρ : S' →+* S
hρ : Function.Injective ⇑ρ
f' : R' → S'
h : ∀ (a : R'), f (φ a) = ρ (f' a)
hf' : NontrivialGood f'
⊢ f (φ x + 1) = f (φ x) + 1
TACTIC:
|
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