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/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rw [posPart_eq_zero.mpr h, zero_mul, eq_comm, posPart_eq_zero, min_le_iff, min_le_iff] | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | right | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | left | case inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | exact max_le h (mul_nonpos_of_nonpos_of_nonneg h (sq_nonneg b)) | case inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ max a (a * b ^ 2) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rw [posPart_eq_zero.mpr h0, mul_zero, eq_comm, posPart_eq_zero, min_le_iff, min_le_iff] | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | right | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inr.inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a * b ≤ 0 ∨ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | right | case inr.inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0 | case inr.inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max b (a ^ 2 * b) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max a (a * b ^ 2) ≤ 0 ∨ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | exact max_le h0 (mul_nonpos_of_nonneg_of_nonpos (sq_nonneg a) h0) | case inr.inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max b (a ^ 2 * b) ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.h.h
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ max b (a ^ 2 * b) ≤ 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | have h1 := mul_le_mul_of_nonneg_left (le_max_one_sq b) h | case inr.inr.refine_1
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max a (a * b ^ 2) | case inr.inr.refine_1
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
h1 : a * b ≤ a * max 1 (b ^ 2)
⊢ a * b ≤ max a (a * b ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.refine_1
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max a (a * b ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rwa [mul_max_of_nonneg _ _ h, mul_one] at h1 | case inr.inr.refine_1
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
h1 : a * b ≤ a * max 1 (b ^ 2)
⊢ a * b ≤ max a (a * b ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.refine_1
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
h1 : a * b ≤ a * max 1 (b ^ 2)
⊢ a * b ≤ max a (a * b ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | have h1 := mul_le_mul_of_nonneg_right (le_max_one_sq a) h0 | case inr.inr.refine_2
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max b (a ^ 2 * b) | case inr.inr.refine_2
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
h1 : a * b ≤ max 1 (a ^ 2) * b
⊢ a * b ≤ max b (a ^ 2 * b) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.refine_2
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max b (a ^ 2 * b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rwa [max_mul_of_nonneg _ _ h0, one_mul] at h1 | case inr.inr.refine_2
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
h1 : a * b ≤ max 1 (a ^ 2) * b
⊢ a * b ≤ max b (a ^ 2 * b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.refine_2
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
h1 : a * b ≤ max 1 (a ^ 2) * b
⊢ a * b ≤ max b (a ^ 2 * b)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | rw [inf_pos_part, Pi_inf_mul_pos_part] | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) ... | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) ... | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaC... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | exact ofInf | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaC... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | rw [sup_pos_part, Pi_sup_mul_pos_part] | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) ... | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) ... | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaC... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | exact ofSup | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg✝ : MetaClosure (fun x => x ∈ S) g
a✝¹ : (i : I) → R i
hg : a✝¹ ∈ S
a✝ b✝ : (i : I) → R i
x✝¹ : MetaC... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | rw [inf_pos_part, Pi_pos_part_mul_inf] | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) ... | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) ... | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : Me... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | exact ofInf | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : Me... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | rw [sup_pos_part, Pi_pos_part_mul_sup] | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) ... | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) ... | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : Me... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_closure_pos_part_mul_closure_pos_part_mem | [144, 1] | [153, 69] | exact ofSup | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : MetaClosure (fun x => x ∈ S) b✝
⊢ MetaClosure (fun x => x ∈ S) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
a✝ b✝ : (i : I) → R i
x✝¹ : MetaClosure (fun x => x ∈ S) a✝
x✝ : Me... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_mul_mem | [155, 1] | [164, 70] | rw [← posPart_sub_negPart f, sub_mul,
← posPart_sub_negPart g, mul_sub, mul_sub] | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
⊢ MetaClosure (fun x => x ∈ S) (f * g) | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
⊢ MetaClosure (fun x => x ∈ S) (f⁺ * g⁺ - f⁺ * g⁻ - (f⁻ * g⁺ - f⁻ * g⁻)) | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
⊢ MetaClosure (fun x => x ∈ S) (f * g)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_mul_mem | [155, 1] | [164, 70] | let T := S.toAddSubgroup | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
⊢ MetaClosure (fun x => x ∈ S) (f⁺ * g⁺ - f⁺ * g⁻ - (f⁻ * g⁺ - f⁻ * g⁻)) | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
⊢ MetaClosure (fun x => x ∈ S) (f⁺ * g⁺ - f⁺ * g⁻ - (f⁻ * g⁺ - f⁻ * g⁻)) | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
⊢ MetaClosure (fun x => x ∈ S) (f⁺ * g⁺ - f⁺ * g⁻ - (f⁻ * g⁺ - f⁻ *... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_mul_mem | [155, 1] | [164, 70] | have hf₀ : MetaClosure (λ x ↦ x ∈ S) (-f) := neg_mem T hf | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
⊢ MetaClosure (fun x => x ∈ S) (f⁺ * g⁺ - f⁺ * g⁻ - (f⁻ * g⁺ - f⁻ * g⁻)) | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
hf₀ : MetaClosure (fun x => x ∈ S) (-f)
⊢ MetaClosure (fun x => x ∈ S) (f⁺ * ... | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
⊢ MetaClosure (f... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_mul_mem | [155, 1] | [164, 70] | have hg₀ : MetaClosure (λ x ↦ x ∈ S) (-g) := neg_mem T hg | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
hf₀ : MetaClosure (fun x => x ∈ S) (-f)
⊢ MetaClosure (fun x => x ∈ S) (f⁺ * ... | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
hf₀ : MetaClosure (fun x => x ∈ S) (-f)
hg₀ : MetaClosure (fun x => x ∈ S) (-... | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
hf₀ : MetaClosur... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.MetaClosure.Pi_mul_mem | [155, 1] | [164, 70] | apply sub_mem T <;> apply sub_mem T <;>
apply Pi_closure_pos_part_mul_closure_pos_part_mem <;> assumption | I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
hf₀ : MetaClosure (fun x => x ∈ S) (-f)
hg₀ : MetaClosure (fun x => x ∈ S) (-... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type u_2
R : I → Type u_1
inst✝ : (i : I) → LinearOrderedRing (R i)
S : Subring ((i : I) → R i)
f g : (i : I) → R i
hf : MetaClosure (fun x => x ∈ S) f
hg : MetaClosure (fun x => x ∈ S) g
T : AddSubgroup ((i : I) → R i) := S.toAddSubgroup
hf₀ : MetaClosur... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_add_cancel_left | [29, 1] | [30, 47] | rw [← add_assoc, add_self_eq_zero, zero_add] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + (x + y) = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + (x + y) = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_add_cancel_right | [32, 1] | [33, 45] | rw [add_assoc, add_self_eq_zero, add_zero] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y + y = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y + y = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_add_add_cancel_middle | [35, 1] | [36, 41] | rw [← add_assoc, add_add_cancel_right] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y + (y + z) = x + z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y + (y + z) = x + z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_comm | [38, 1] | [40, 59] | rw [← add_add_cancel_right (x + y) (y + x),
add_add_add_cancel_middle, add_self_eq_zero, zero_add] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y = y + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y = y + x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_add_cancel_middle₁ | [42, 1] | [43, 38] | rw [add_comm, add_add_cancel_right] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + (y + x) = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + (y + x) = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_add_cancel_middle₂ | [45, 1] | [46, 37] | rw [add_comm, add_add_cancel_left] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y + x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y + x = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_right_inj | [48, 1] | [49, 86] | rw [← add_add_cancel_left x y, h, add_add_cancel_left] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
h : x + y = x + z
⊢ y = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
h : x + y = x + z
⊢ y = z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_left_inj | [51, 1] | [52, 43] | rw [add_comm, add_comm y, add_right_inj] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + z = y + z ↔ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + z = y + z ↔ x = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_eq_iff_eq_add | [54, 1] | [55, 53] | rw [← add_left_inj (z := y), add_add_cancel_right] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ x = z + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ x = z + y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_eq_iff_eq_add' | [57, 1] | [58, 35] | rw [add_eq_iff_eq_add, add_comm] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ x = y + z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ x = y + z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_eq_iff_eq_add'' | [60, 1] | [61, 36] | rw [add_comm, add_eq_iff_eq_add'] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ y = x + z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ y = x + z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_eq_iff_eq_add''' | [63, 1] | [64, 37] | rw [add_eq_iff_eq_add'', add_comm] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ y = z + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y = z ↔ y = z + x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_eq_zero_iff_eq | [66, 1] | [67, 35] | rw [add_eq_iff_eq_add, zero_add] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y = 0 ↔ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y : M
⊢ x + y = 0 ↔ x = y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_add_add_cancel_left | [69, 1] | [70, 43] | rw [← add_assoc, add_add_cancel_middle₂] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y + (x + z) = y + z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + y + (x + z) = y + z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.add_add_add_cancel_right | [72, 1] | [73, 41] | rw [add_assoc, add_add_cancel_middle₁] | M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + z + (y + z) = x + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝¹ : AddMonoid M
inst✝ : CharTwo M
x y z : M
⊢ x + z + (y + z) = x + y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Defs.lean | IMOSL.Extra.CharTwo.sub_eq_add | [93, 1] | [94, 35] | rw [sub_eq_add_neg, neg_eq_self] | G : Type u_1
inst✝¹ : AddGroup G
inst✝ : CharTwo G
x y : G
⊢ x - y = x + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝¹ : AddGroup G
inst✝ : CharTwo G
x y : G
⊢ x - y = x + y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5Hom.lean | IMOSL.IMO2012A5.good_comp_hom | [22, 1] | [25, 53] | simp only [Function.comp_apply] | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
R₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring R₀
φ : R₀ →+* R
x y : R₀
⊢ (f ∘ ⇑φ) (x * y + 1) = (f ∘ ⇑φ) x * (f ∘ ⇑φ) y + (f ∘ ⇑φ) (x + y) | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
R₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring R₀
φ : R₀ →+* R
x y : R₀
⊢ f (φ (x * y + 1)) = f (φ x) * f (φ y) + f (φ (x + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
R₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring R₀
φ : R₀ →+* R
x y : R₀
⊢ (f ∘ ⇑φ) (x * y + 1) = (f ∘ ⇑φ) x * (f ∘ ⇑φ) y + (f ∘ ⇑φ) (x + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5Hom.lean | IMOSL.IMO2012A5.good_comp_hom | [22, 1] | [25, 53] | rw [φ.map_add, φ.map_mul, φ.map_one, h, φ.map_add] | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
R₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring R₀
φ : R₀ →+* R
x y : R₀
⊢ f (φ (x * y + 1)) = f (φ x) * f (φ y) + f (φ (x + y)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
R₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring R₀
φ : R₀ →+* R
x y : R₀
⊢ f (φ (x * y + 1)) = f (φ x) * f (φ y) + f (φ (x + y))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5Hom.lean | IMOSL.IMO2012A5.good_hom_comp | [27, 1] | [30, 31] | simp only [Function.comp_apply] | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
x y : R
⊢ (⇑φ ∘ f) (x * y + 1) = (⇑φ ∘ f) x * (⇑φ ∘ f) y + (⇑φ ∘ f) (x + y) | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
x y : R
⊢ φ (f (x * y + 1)) = φ (f x) * φ (f y) + φ (f (x + y)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
x y : R
⊢ (⇑φ ∘ f) (x * y + 1) = (⇑φ ∘ f) x * (⇑φ ∘ f) y + (⇑φ ∘ f) (x + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5Hom.lean | IMOSL.IMO2012A5.good_hom_comp | [27, 1] | [30, 31] | rw [h, φ.map_add, φ.map_mul] | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
x y : R
⊢ φ (f (x * y + 1)) = φ (f x) * φ (f y) + φ (f (x + y)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
h : good f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
x y : R
⊢ φ (f (x * y + 1)) = φ (f x) * φ (f y) + φ (f (x + y))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5Hom.lean | IMOSL.IMO2012A5.NontrivialGood.hom_comp | [37, 1] | [40, 78] | rw [← φ.map_one, ← φ.map_zero, ← hf.map_zero_add_one, φ.map_add] | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
hf : NontrivialGood f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
⊢ (⇑φ ∘ f) 0 + 1 = 0 | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
hf : NontrivialGood f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
⊢ (⇑φ ∘ f) 0 + φ 1 = φ (f 0) + φ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
hf : NontrivialGood f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
⊢ (⇑φ ∘ f) 0 + 1 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5General/A5Hom.lean | IMOSL.IMO2012A5.NontrivialGood.hom_comp | [37, 1] | [40, 78] | rfl | R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
hf : NontrivialGood f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
⊢ (⇑φ ∘ f) 0 + φ 1 = φ (f 0) + φ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
S : Type u_3
inst✝² : NonAssocSemiring R
inst✝¹ : NonAssocSemiring S
f : R → S
S₀ : Type u_1
hf : NontrivialGood f
inst✝ : NonAssocSemiring S₀
φ : S →+* S₀
⊢ (⇑φ ∘ f) 0 + φ 1 = φ (f 0) + φ 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_comm | [96, 11] | [98, 71] | rw [mul_def, mul_def, mul_comm a, mul_comm b, add_comm (a * b')] | a b a' b' : ℤ
⊢ { re := a, im := b } * { re := a', im := b' } = { re := a', im := b' } * { re := a, im := b } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' : ℤ
⊢ { re := a, im := b } * { re := a', im := b' } = { re := a', im := b' } * { re := a, im := b }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_assoc | [100, 11] | [103, 39] | rw [mul_def, mul_def, mul_def, mul_def, ℤφ.mk.injEq] | a b a' b' a'' b'' : ℤ
⊢ { re := a, im := b } * { re := a', im := b' } * { re := a'', im := b'' } =
{ re := a, im := b } * ({ re := a', im := b' } * { re := a'', im := b'' }) | a b a' b' a'' b'' : ℤ
⊢ { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * a'' +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' ... | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : ℤ
⊢ { re := a, im := b } * { re := a', im := b' } * { re := a'', im := b'' } =
{ re := a, im := b } * ({ re := a', im := b' } * { re := a'', im := b'' })
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_assoc | [100, 11] | [103, 39] | dsimp only | a b a' b' a'' b'' : ℤ
⊢ { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * a'' +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re +
b * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' ... | a b a' b' a'' b'' : ℤ
⊢ (a * a' + b * b') * a'' + (a * b' + a' * b + b * b') * b'' =
a * (a' * a'' + b' * b'') + b * (a' * b'' + a'' * b' + b' * b'') ∧
(a * a' + b * b') * b'' + a'' * (a * b' + a' * b + b * b') + (a * b' + a' * b + b * b') * b'' =
a * (a' * b'' + a'' * b' + b' * b'') + (a' * a'' + b' * ... | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : ℤ
⊢ { re := a * a' + b * b', im := a * b' + a' * b + b * b' }.re * a'' +
{ re := a * a' + b * b', im := a * b' + a' * b + b * b' }.im * b'' =
a * { re := a' * a'' + b' * b'', im := a' * b'' + a'' * b' + b' * b'' }.re +
... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_assoc | [100, 11] | [103, 39] | constructor <;> ring | a b a' b' a'' b'' : ℤ
⊢ (a * a' + b * b') * a'' + (a * b' + a' * b + b * b') * b'' =
a * (a' * a'' + b' * b'') + b * (a' * b'' + a'' * b' + b' * b'') ∧
(a * a' + b * b') * b'' + a'' * (a * b' + a' * b + b * b') + (a * b' + a' * b + b * b') * b'' =
a * (a' * b'' + a'' * b' + b' * b'') + (a' * a'' + b' * ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : ℤ
⊢ (a * a' + b * b') * a'' + (a * b' + a' * b + b * b') * b'' =
a * (a' * a'' + b' * b'') + b * (a' * b'' + a'' * b' + b' * b'') ∧
(a * a' + b * b') * b'' + a'' * (a * b' + a' * b + b * b') + (a * b' + a' * b + b * b') * b'' =
... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.zero_mul | [105, 11] | [107, 46] | rw [mul_def, zero_def, zero_mul,
zero_mul, mul_zero, add_zero, add_zero] | a b : ℤ
⊢ 0 * { re := a, im := b } = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℤ
⊢ 0 * { re := a, im := b } = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.one_mul | [112, 11] | [114, 56] | rw [mul_def, one_def, one_mul, one_mul,
zero_mul, mul_zero, add_zero, add_zero, add_zero] | a b : ℤ
⊢ 1 * { re := a, im := b } = { re := a, im := b } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℤ
⊢ 1 * { re := a, im := b } = { re := a, im := b }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_add | [119, 11] | [124, 81] | rw [mul_def, mul_def, mul_def, add_def, add_def, ℤφ.mk.injEq] | a b a' b' a'' b'' : ℤ
⊢ { re := a, im := b } * ({ re := a', im := b' } + { re := a'', im := b'' }) =
{ re := a, im := b } * { re := a', im := b' } + { re := a, im := b } * { re := a'', im := b'' } | a b a' b' a'' b'' : ℤ
⊢ a * { re := a' + a'', im := b' + b'' }.re + b * { re := a' + a'', im := b' + b'' }.im =
a * a' + b * b' + (a * a'' + b * b'') ∧
a * { re := a' + a'', im := b' + b'' }.im + { re := a' + a'', im := b' + b'' }.re * b +
b * { re := a' + a'', im := b' + b'' }.im =
a * b' + a' ... | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : ℤ
⊢ { re := a, im := b } * ({ re := a', im := b' } + { re := a'', im := b'' }) =
{ re := a, im := b } * { re := a', im := b' } + { re := a, im := b } * { re := a'', im := b'' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_add | [119, 11] | [124, 81] | dsimp only | a b a' b' a'' b'' : ℤ
⊢ a * { re := a' + a'', im := b' + b'' }.re + b * { re := a' + a'', im := b' + b'' }.im =
a * a' + b * b' + (a * a'' + b * b'') ∧
a * { re := a' + a'', im := b' + b'' }.im + { re := a' + a'', im := b' + b'' }.re * b +
b * { re := a' + a'', im := b' + b'' }.im =
a * b' + a' ... | a b a' b' a'' b'' : ℤ
⊢ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') ∧
a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : ℤ
⊢ a * { re := a' + a'', im := b' + b'' }.re + b * { re := a' + a'', im := b' + b'' }.im =
a * a' + b * b' + (a * a'' + b * b'') ∧
a * { re := a' + a'', im := b' + b'' }.im + { re := a' + a'', im := b' + b'' }.re * b +
b... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_add | [119, 11] | [124, 81] | constructor | a b a' b' a'' b'' : ℤ
⊢ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') ∧
a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | case left
a b a' b' a'' b'' : ℤ
⊢ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'')
case right
a b a' b' a'' b'' : ℤ
⊢ a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | Please generate a tactic in lean4 to solve the state.
STATE:
a b a' b' a'' b'' : ℤ
⊢ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') ∧
a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_add | [119, 11] | [124, 81] | rw [add_add_add_comm, ← mul_add, ← mul_add] | case left
a b a' b' a'' b'' : ℤ
⊢ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
a b a' b' a'' b'' : ℤ
⊢ a * (a' + a'') + b * (b' + b'') = a * a' + b * b' + (a * a'' + b * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.mul_add | [119, 11] | [124, 81] | rw [add_add_add_comm, ← mul_add, add_add_add_comm, ← mul_add, ← add_mul] | case right
a b a' b' a'' b'' : ℤ
⊢ a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
a b a' b' a'' b'' : ℤ
⊢ a * (b' + b'') + (a' + a'') * b + b * (b' + b'') = a * b' + a' * b + b * b' + (a * b'' + a'' * b + b * b'')
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.add_mul | [126, 11] | [127, 55] | rw [ℤφ.mul_comm, ℤφ.mul_add, z.mul_comm, z.mul_comm] | x y z : ℤφ
⊢ (x + y) * z = x * z + y * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : ℤφ
⊢ (x + y) * z = x * z + y * z
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_zero | [149, 1] | [150, 77] | change ((0 : ℤ) : R) + 0 • r = 0 | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ cast r 0 = 0 | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑0 + 0 • r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ cast r 0 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_zero | [149, 1] | [150, 77] | rw [Int.cast_zero, zero_add, zero_zsmul] | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑0 + 0 • r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑0 + 0 • r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_one | [152, 1] | [153, 76] | change ((1 : ℤ) : R) + 0 • r = 1 | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ cast r 1 = 1 | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑1 + 0 • r = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ cast r 1 = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_one | [152, 1] | [153, 76] | rw [Int.cast_one, zero_zsmul, add_zero] | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑1 + 0 • r = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑1 + 0 • r = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_φ | [155, 1] | [156, 76] | change ((0 : ℤ) : R) + 1 • r = r | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ cast r φ = r | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑0 + 1 • r = r | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ cast r φ = r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_φ | [155, 1] | [156, 76] | rw [Int.cast_zero, zero_add, one_zsmul] | R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑0 + 1 • r = r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddGroupWithOne R
r : R
⊢ ↑0 + 1 • r = r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_add | [158, 1] | [162, 58] | change ((a + a' : ℤ) : R) + (b + b') • r = _ | R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ cast r ({ re := a, im := b } + { re := a', im := b' }) = cast r { re := a, im := b } + cast r { re := a', im := b' } | R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ ↑(a + a') + (b + b') • r = cast r { re := a, im := b } + cast r { re := a', im := b' } | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ cast r ({ re := a, im := b } + { re := a', im := b' }) = cast r { re := a, im := b } + cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_add | [158, 1] | [162, 58] | rw [Int.cast_add, add_zsmul, add_add_add_comm] | R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ ↑(a + a') + (b + b') • r = cast r { re := a, im := b } + cast r { re := a', im := b' } | R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ ↑a + b • r + (↑a' + b' • r) = cast r { re := a, im := b } + cast r { re := a', im := b' } | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ ↑(a + a') + (b + b') • r = cast r { re := a, im := b } + cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_add | [158, 1] | [162, 58] | rfl | R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ ↑a + b • r + (↑a' + b' • r) = cast r { re := a, im := b } + cast r { re := a', im := b' } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : AddCommGroupWithOne R
r : R
a b a' b' : ℤ
⊢ ↑a + b • r + (↑a' + b' • r) = cast r { re := a, im := b } + cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_mul | [166, 1] | [174, 76] | change ↑(a * a' + b * b') + (a * b' + a' * b + b * b') • r
= (↑a + b • r) * (↑a' + b' • r) | R : Type u_1
inst✝ : Ring R
r : R
h : r * r = r + 1
a b a' b' : ℤ
⊢ cast r ({ re := a, im := b } * { re := a', im := b' }) = cast r { re := a, im := b } * cast r { re := a', im := b' } | R : Type u_1
inst✝ : Ring R
r : R
h : r * r = r + 1
a b a' b' : ℤ
⊢ ↑(a * a' + b * b') + (a * b' + a' * b + b * b') • r = (↑a + b • r) * (↑a' + b' • r) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
r : R
h : r * r = r + 1
a b a' b' : ℤ
⊢ cast r ({ re := a, im := b } * { re := a', im := b' }) = cast r { re := a, im := b } * cast r { re := a', im := b' }
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/Extra/ExplicitRings/Zphi.lean | IMOSL.IMO2012A5.ℤφ.cast_mul | [166, 1] | [174, 76] | rw [add_zsmul, Int.cast_add, add_add_add_comm, ← zsmul_one (b * b'),
← zsmul_add, add_comm 1 r, ← h, add_zsmul, ← add_assoc, mul_zsmul,
zsmul_eq_mul, Int.cast_mul, ← mul_add, add_mul, add_assoc, add_right_inj,
mul_add, ← zsmul_eq_mul', mul_zsmul, add_right_inj, mul_zsmul, zsmul_eq_mul,
zsmul_eq_mul', zsmul_eq_m... | R : Type u_1
inst✝ : Ring R
r : R
h : r * r = r + 1
a b a' b' : ℤ
⊢ ↑(a * a' + b * b') + (a * b' + a' * b + b * b') • r = (↑a + b • r) * (↑a' + b' • r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
r : R
h : r * r = r + 1
a b a' b' : ℤ
⊢ ↑(a * a' + b * b') + (a * b' + a' * b + b * b') • r = (↑a + b • r) * (↑a' + b' • r)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/F2X/toList.lean | IMOSL.IMO2012A5.𝔽₂X.toList_inj | [33, 1] | [35, 23] | rw [← 𝔽₂X.toFinset_toList, h, 𝔽₂X.toFinset_toList] | P Q : 𝔽₂X
h : P.toList = Q.toList
⊢ P.toFinset = Q.toFinset | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P Q : 𝔽₂X
h : P.toList = Q.toList
⊢ P.toFinset = Q.toFinset
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.admissible_mem_sq_mul | [32, 1] | [36, 62] | have h1 := h z z h0 h0 (k - 2) | R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
⊢ k * z ^ 2 ∈ A | R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
h1 : z ^ 2 + (k - 2) * z * z + z ^ 2 ∈ A
⊢ k * z ^ 2 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
⊢ k * z ^ 2 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.admissible_mem_sq_mul | [32, 1] | [36, 62] | rwa [mul_assoc, ← sq, ← one_add_mul (α := R), ← add_one_mul (α := R),
add_right_comm, one_add_one_eq_two, add_sub_cancel] at h1 | R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
h1 : z ^ 2 + (k - 2) * z * z + z ^ 2 ∈ A
⊢ k * z ^ 2 ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
z : R
inst✝ : CommRing R
A : Set R
h : admissible A
h0 : z ∈ A
k : R
h1 : z ^ 2 + (k - 2) * z * z + z ^ 2 ∈ A
⊢ k * z ^ 2 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | specialize h (Ideal.span {x, y}) (admissible_ideal _)
(Ideal.subset_span (Set.mem_insert x _))
(Ideal.subset_span (Set.mem_insert_of_mem x rfl)) 1 | R : Type u_1
inst✝ : CommRing R
x y : R
h : ∀ (A : Set R), admissible A → x ∈ A → y ∈ A → ∀ (z : R), z ∈ A
⊢ IsCoprime x y | R : Type u_1
inst✝ : CommRing R
x y : R
h : 1 ∈ ↑(Ideal.span {x, y})
⊢ IsCoprime x y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : ∀ (A : Set R), admissible A → x ∈ A → y ∈ A → ∀ (z : R), z ∈ A
⊢ IsCoprime x y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | rwa [SetLike.mem_coe, Ideal.mem_span_pair] at h | R : Type u_1
inst✝ : CommRing R
x y : R
h : 1 ∈ ↑(Ideal.span {x, y})
⊢ IsCoprime x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : 1 ∈ ↑(Ideal.span {x, y})
⊢ IsCoprime x y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | suffices 1 ∈ A from λ z ↦ mul_one z ▸
one_pow (M := R) 2 ▸ admissible_mem_sq_mul h0 this z | R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ ∀ (z : R), z ∈ A | R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ 1 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ ∀ (z : R), z ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | obtain ⟨a, b, h⟩ : IsCoprime (x ^ 2) (y ^ 2) := IsCoprime.pow h | R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ 1 ∈ A | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ 1 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : CommRing R
x y : R
h : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
⊢ 1 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | rw [← one_pow 2, ← h] | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ 1 ∈ A | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ (a * x ^ 2 + b * y ^ 2) ^ 2 ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ 1 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/N1/N1.lean | IMOSL.IMO2012N1.final_solution | [43, 1] | [58, 39] | refine admissible_add_sq h0 (admissible_mem_sq_mul h0 h1 a)
(admissible_mem_sq_mul h0 h2 b) | case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ (a * x ^ 2 + b * y ^ 2) ^ 2 ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
R : Type u_1
inst✝ : CommRing R
x y : R
h✝ : IsCoprime x y
A : Set R
h0 : admissible A
h1 : x ∈ A
h2 : y ∈ A
a b : R
h : a * x ^ 2 + b * y ^ 2 = 1
⊢ (a * x ^ 2 + b * y ^ 2) ^ 2 ∈ A
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_is_good | [22, 1] | [24, 33] | rw [sub_one_mul, mul_sub_one, sub_sub, ← add_sub_assoc x, sub_add_cancel] | R : Type u_1
inst✝ : NonAssocRing R
x y : R
⊢ (fun x => x - 1) (x * y + 1) = (fun x => x - 1) x * (fun x => x - 1) y + (fun x => x - 1) (x + y) | R : Type u_1
inst✝ : NonAssocRing R
x y : R
⊢ (fun x => x - 1) (x * y + 1) = x * y | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
x y : R
⊢ (fun x => x - 1) (x * y + 1) = (fun x => x - 1) x * (fun x => x - 1) y + (fun x => x - 1) (x + y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_is_good | [22, 1] | [24, 33] | exact add_sub_cancel_right _ _ | R : Type u_1
inst✝ : NonAssocRing R
x y : R
⊢ (fun x => x - 1) (x * y + 1) = x * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : NonAssocRing R
x y : R
⊢ (fun x => x - 1) (x * y + 1) = x * y
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | suffices ∀ x y, f (x + y) = f x + f y + 1 by
have h0 (x y) : f (x + y + 1) = f (x + 1) + f (y + 1) := by
rw [h, h, h, this, add_assoc, add_add_add_comm]
have h1 (x y) : f (x * y + 1) = f (x + 1) * f (y + 1) := by
rw [hf.is_good, h, h, this, add_assoc, ← add_assoc,
← mul_add_one (f x), ← add_one_mul (f... | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
⊢ ∃ φ, ∀ (x : R), f x = φ (x - 1) | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
⊢ ∃ φ, ∀ (x : R), f x = φ (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h0 (x y) : f (x * y) + 1 = f x * f y + f (x + y) :=
(h _).symm.trans (hf.is_good x y) | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1 | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h1 (x y) : f (x * y) = f x * f y + f (x + y) - 1 :=
eq_sub_of_add_eq (h0 x y) | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1 | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h2 (x y) : f ((x + 1) * y) = f (x * y) + (f y + 1) := by
rw [h1, h, add_right_comm, add_one_mul (f x), h, add_add_add_comm,
← h0, add_right_comm, add_sub_cancel_right] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
⊢ ∀ (x y : R), f (x + y) = f x + f y + 1 | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
⊢ ∀ (x y :... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
⊢ ∀ (x ... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | intro x y | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
⊢ ∀ (x y :... | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ ... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h3 : f (x * ((x + 1) * y)) = f ((x + 1) * (x * y)) := by
rw [add_one_mul x, add_one_mul x, mul_add] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
⊢ ... | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
h3... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ ... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rwa [h2, h1, h2, mul_add, ← add_assoc, add_right_comm _ _ 1, h0, ← h,
sub_eq_iff_eq_add, add_assoc, add_assoc, add_assoc, add_right_inj,
← add_left_inj 1, add_assoc, ← h, add_right_comm, ← mul_one_add (x + 1),
h2, add_left_comm, mul_one_add x, add_assoc, add_right_inj, add_comm 1,
h, mul_add_one (f x), h0, add_... | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
h3... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ ... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h0 (x y) : f (x + y + 1) = f (x + 1) + f (y + 1) := by
rw [h, h, h, this, add_assoc, add_add_add_comm] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
⊢ ∃ φ, ∀ (x : R), f x = φ (x - 1) | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
⊢ ∃ φ, ∀ (x : R), f x = φ (x - 1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
⊢ ∃ φ, ∀ (x : R), f x = φ (x - 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | have h1 (x y) : f (x * y + 1) = f (x + 1) * f (y + 1) := by
rw [hf.is_good, h, h, this, add_assoc, ← add_assoc,
← mul_add_one (f x), ← add_one_mul (f x)] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
⊢ ∃ φ, ∀ (x : R), f x = φ (x - 1) | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : ∀ (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
⊢ ∃ φ, ∀ (x : R), f x ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
⊢ ∃ φ, ∀ (x : R),... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | exact ⟨⟨⟨⟨λ x ↦ f (x + 1), (h 1).trans <| by rw [hf.map_one, zero_add]⟩, h1⟩,
((h 0).trans hf.map_zero_add_one), h0⟩,
λ x ↦ congrArg f (sub_add_cancel x 1).symm⟩ | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : ∀ (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
⊢ ∃ φ, ∀ (x : R), f x ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : ∀ (x y : R),... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [h, h, h, this, add_assoc, add_add_add_comm] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
x y : R
⊢ f (x + y + 1) = f (x + 1) + f (y + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
x y : R
⊢ f (x + y + 1) = f (x + 1) + f (y + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [hf.is_good, h, h, this, add_assoc, ← add_assoc,
← mul_add_one (f x), ← add_one_mul (f x)] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
x y : R
⊢ f (x * y + 1) = f (x + 1) * f (y + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
x y : R
⊢ f (x * ... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [hf.map_one, zero_add] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : ∀ (x y : R), f (x * y + 1) = f (x + 1) * f (y + 1)
⊢ f 1 + 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
this : ∀ (x y : R), f (x + y) = f x + f y + 1
h0 : ∀ (x y : R), f (x + y + 1) = f (x + 1) + f (y + 1)
h1 : ∀ (x y : R),... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [h1, h, add_right_comm, add_one_mul (f x), h, add_add_add_comm,
← h0, add_right_comm, add_sub_cancel_right] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
x y : R
⊢ f ((x + 1) * y) = f (x * y) + (f y + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
x y : R... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Answers/SubOneMap.lean | IMOSL.IMO2012A5.sub_one_solver | [29, 1] | [59, 59] | rw [add_one_mul x, add_one_mul x, mul_add] | R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ (x y : R), f ((x + 1) * y) = f (x * y) + (f y + 1)
x y : R
⊢ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_2
inst✝¹ : NonAssocRing R
S : Type u_1
inst✝ : NonAssocRing S
f : R → S
hf : NontrivialGood f
h : ∀ (x : R), f (x + 1) = f x + 1
h0 : ∀ (x y : R), f (x * y) + 1 = f x * f y + f (x + y)
h1 : ∀ (x y : R), f (x * y) = f x * f y + f (x + y) - 1
h2 : ∀ ... |
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.symmDiff_singleton_eq_erase | [28, 1] | [32, 21] | have h0 : {i} \ S = ∅ :=
Finset.sdiff_eq_empty_iff_subset.mpr λ c h0 ↦ Finset.mem_singleton.mp h0 ▸ h | ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
⊢ symmDiff {i} S = S.erase i | ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
h0 : {i} \ S = ∅
⊢ symmDiff {i} S = S.erase i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
⊢ symmDiff {i} S = S.erase i
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.symmDiff_singleton_eq_erase | [28, 1] | [32, 21] | rw [symmDiff, Finset.sdiff_singleton_eq_erase, h0] | ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
h0 : {i} \ S = ∅
⊢ symmDiff {i} S = S.erase i | ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
h0 : {i} \ S = ∅
⊢ ∅ ⊔ S.erase i = S.erase i | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
h0 : {i} \ S = ∅
⊢ symmDiff {i} S = S.erase i
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.symmDiff_singleton_eq_erase | [28, 1] | [32, 21] | exact bot_sup_eq _ | ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
h0 : {i} \ S = ∅
⊢ ∅ ⊔ S.erase i = S.erase i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
inst✝ : DecidableEq ι
i : ι
S : Finset ι
h : i ∈ S
h0 : {i} \ S = ∅
⊢ ∅ ⊔ S.erase i = S.erase i
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/CharTwo/Finset.lean | IMOSL.Extra.CharTwo.symmDiff_singleton_sum_eq | [48, 1] | [52, 61] | by_cases h : i ∈ S | M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
⊢ (symmDiff {i} S).sum f = f i + S.sum f | case pos
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
h : i ∈ S
⊢ (symmDiff {i} S).sum f = f i + S.sum f
case neg
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
h : i ∉ ... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
ι : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : CharTwo M
inst✝ : DecidableEq ι
f : ι → M
i : ι
S : Finset ι
⊢ (symmDiff {i} S).sum f = f i + S.sum f
TACTIC:
|
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