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/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | apply congr_argβ _ _ rfl | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | apply congr_arg | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only [Finset.mem_antidiagonal] at h | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only [hs_def, Prod.mk.injEq] | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [rewriting_4_fold_sums h.symm (fun x => u.choose x.fst * v.choose x.snd) rfl _] | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rintro β¨β¨i, jβ©, β¨k, lβ©β© | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
β’ β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1 + x.... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow ((i, j), k, l).1.1 a * hJ.dpow ((i, j), k, l).1.2 b *
(hI.dpow ((i, j), k, l).2.1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
β’ β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | dsimp | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow ((i, j), k, l).1.1 a * hJ.dpow ((i, j), k, l).1.2 b *
(hI.dpow ((i, j), k, l).2.1 ... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * hJ.dpow j b * (hI.dpow k a * hJ.dpow l b) =
β((i + k).choose i) * β((j + l).ch... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow ((i, j), k, l).1.1 a * hJ.d... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [mul_assoc] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * hJ.dpow j b * (hI.dpow k a * hJ.dpow l b) =
β((i + k).choose i) * β((j + l).ch... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hJ.dpow j b * (hI.dpow k a * hJ.dpow l b)) =
β((i + k).choose i) * β((j + l).... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * hJ.dpow j b * (hI.dpo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [β mul_assoc (hJ.dpow j b) _ _] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hJ.dpow j b * (hI.dpow k a * hJ.dpow l b)) =
β((i + k).choose i) * β((j + l).... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hJ.dpow j b * hI.dpow k a * hJ.dpow l b) =
β((i + k).choose i) * β((j + l).ch... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hJ.dpow j b * (hI.dp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [mul_comm (hJ.dpow j b)] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hJ.dpow j b * hI.dpow k a * hJ.dpow l b) =
β((i + k).choose i) * β((j + l).ch... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * hJ.dpow j b * hJ.dpow l b) =
β((i + k).choose i) * β((j + l).ch... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hJ.dpow j b * hI.dpo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [mul_assoc] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * hJ.dpow j b * hJ.dpow l b) =
β((i + k).choose i) * β((j + l).ch... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * (hJ.dpow j b * hJ.dpow l b)) =
β((i + k).choose i) * β((j + l).... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * hJ.dpo... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [hJ.dpow_mul j l hb] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * (hJ.dpow j b * hJ.dpow l b)) =
β((i + k).choose i) * β((j + l).... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * (β((j + l).choose j) * hJ.dpow (j + l) b)) =
β((i + k).choose i... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * (hJ.dp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [β mul_assoc] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * (β((j + l).choose j) * hJ.dpow (j + l) b)) =
β((i + k).choose i... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * hI.dpow k a * (β((j + l).choose j) * hJ.dpow (j + l) b) =
β((i + k).choose i) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * (hI.dpow k a * (β((j ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [hI.dpow_mul i k ha] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * hI.dpow k a * (β((j + l).choose j) * hJ.dpow (j + l) b) =
β((i + k).choose i) ... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ β((i + k).choose i) * hI.dpow (i + k) a * (β((j + l).choose j) * hJ.dpow (j + l) b) =
β((i +... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ hI.dpow i a * hI.dpow k a * (β((j +... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | ring | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ β((i + k).choose i) * hI.dpow (i + k) a * (β((j + l).choose j) * hJ.dpow (j + l) b) =
β((i +... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
i j k l : β
β’ β((i + k).choose i) * hI.dpow (i + ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rintro β¨β¨i, jβ©, β¨k, lβ©β© h | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only [Finset.mem_antidiagonal, Finset.mem_product] at h β’ | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [add_assoc, β add_assoc k j l, add_comm k _, add_assoc, h.2, β add_assoc, h.1] | case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | apply Finset.sum_congr rfl | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rintro β¨u, vβ© _ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only [Prod.mk.injEq, mem_product, mem_antidiagonal, and_imp, Prod.forall, Nat.cast_sum, Nat.cast_mul] | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only [Finset.sum_mul] | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | apply Finset.sum_congr rfl | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rintro β¨β¨i, jβ©, β¨k, lβ©β© hx | case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only [hs_def, mem_product, mem_antidiagonal, and_imp, Prod.forall, mem_filter,
Prod.mk.injEq] at hx | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [hx.2.1] | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [hx.2.2] | case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [β Nat.add_choose_eq] | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | rw [h] | case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | intro x h | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
β((x.1.1... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mul | [163, 1] | [238, 93] | simp only [Nat.choose_eq_zero_of_lt h, MulZeroClass.zero_mul, MulZeroClass.mul_zero] | case inr
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * hJ.dpow x.1.2 b * (hI.dpow x.2.1 a * hJ.dpow x.2.2 b) =
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
a : A
ha : a β I
b : A
hb : b β J
hf :
β (x : (β Γ β) Γ β Γ β),
hI.dpow x.1.1 a * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | rw [Ideal.add_eq_sup, Submodule.mem_sup] at hx | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x : A
hn : n β 0
hx : x β I + J
β’ dpow hI hJ n x β I + J | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x : A
hn : n β 0
hx : β y β I, β z β J, y + z = x
β’ dpow hI hJ n x β I + J | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x : A
hn : n β 0
hx : x β I + J
β’ dpow hI hJ n x β I + J
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | obtain β¨a, ha, b, hb, rflβ© := hx | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x : A
hn : n β 0
hx : β y β I, β z β J, y + z = x
β’ dpow hI hJ n x β I + J | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (a + b) β I + J | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x : A
hn : n β 0
hx : β y β I, β z β J, y + z = x
β’ dpow hI hJ n x β I + J
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | rw [dpow_eq hI hJ hIJ _ ha hb] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (a + b) β I + J | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k a * hJ.dpow (n - k) b β I + J | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (a + b) β ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | apply Submodule.sum_mem (I β J) | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k a * hJ.dpow (n - k) b β I + J | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ β c β range (n + 1), hI.dpow c a * hJ.dpow (n - c) b β I β J | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | intro k _ | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ β c β range (n + 1), hI.dpow c a * hJ.dpow (n - c) b β I β J | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
β’ hI.dpow k a * hJ.dpow (n - k) b β I β J | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ β c β range (n + 1), hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | by_cases hk0 : k = 0 | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
β’ hI.dpow k a * hJ.dpow (n - k) b β I β J | case pos
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI.dpow k a * hJ.dpow (n - k) b β I β J
case neg
A : Type u_1
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n +... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | rw [hk0] | case pos
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI.dpow k a * hJ.dpow (n - k) b β I β J | case pos
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI.dpow 0 a * hJ.dpow (n - 0) b β I β J | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | apply Submodule.mem_sup_right | case pos
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI.dpow 0 a * hJ.dpow (n - 0) b β I β J | case pos.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI.dpow 0 a * hJ.dpow (n - 0) b β J | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | apply Ideal.mul_mem_left | case pos.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hI.dpow 0 a * hJ.dpow (n - 0) b β J | case pos.a.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hJ.dpow (n - 0) b β J | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | exact hJ.dpow_mem hn hb | case pos.a.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
β’ hJ.dpow (n - 0) b β J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : k = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | apply Submodule.mem_sup_left | case neg
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0
β’ hI.dpow k a * hJ.dpow (n - k) b β I β J | case neg.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0
β’ hI.dpow k a * hJ.dpow (n - k) b β I | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0
β’ h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | apply Ideal.mul_mem_right | case neg.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0
β’ hI.dpow k a * hJ.dpow (n - k) b β I | case neg.a.h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0
β’ hI.dpow k a β I | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0
β’... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_mem | [241, 1] | [258, 29] | exact hI.dpow_mem hk0 ha | case neg.a.h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0
β’ hI.dpow k a β I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.h
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
k : β
aβ : k β range (n + 1)
hk0 : Β¬k = 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | intro n c x | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {c x : A}, x β I + J β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c x : A
β’ x β I + J β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {c x : A}, x β I + J β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [Ideal.add_eq_sup, Submodule.mem_sup] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c x : A
β’ x β I + J β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c x : A
β’ (β y β I, β z β J, y + z = x) β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c x : A
β’ x β I + J β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rintro β¨a, ha, b, hb, rflβ© | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c x : A
β’ (β y β I, β z β J, y + z = x) β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * (a + b)) = c ^ n * dpow hI hJ n (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c x : A
β’ (β y β I, β z β J, y + z = x) β dpow hI hJ n (c * x) = c ^ n * dpow hI hJ n x
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [dpow_eq hI hJ hIJ n ha hb] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * (a + b)) = c ^ n * dpow hI hJ n (a + b) | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * (a + b)) = c ^ n * β k β range (n + 1), hI.dpow k a * hJ.dpow (n - k) b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * (a + b)) = c ^... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [mul_add] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * (a + b)) = c ^ n * β k β range (n + 1), hI.dpow k a * hJ.dpow (n - k) b | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * a + c * b) = c ^ n * β k β range (n + 1), hI.dpow k a * hJ.dpow (n - k) b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * (a + b)) = c ^... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [dpow_eq hI hJ hIJ n (Ideal.mul_mem_left I c ha) (Ideal.mul_mem_left J c hb)] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * a + c * b) = c ^ n * β k β range (n + 1), hI.dpow k a * hJ.dpow (n - k) b | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) =
c ^ n * β k β range (n... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ n (c * a + c * b) = c... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [Finset.mul_sum] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) =
c ^ n * β k β range (n... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) =
β i β range (n + 1), c... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | apply Finset.sum_congr rfl | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) =
β i β range (n + 1), c... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β x β range (n + 1), hI.dpow x (c * a) * hJ.dpow (n - x) (c * b) = c ^ n * (hI.dpow x a * hJ.... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β k β range (n + 1), hI.dpow k (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | intro k hk | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β x β range (n + 1), hI.dpow x (c * a) * hJ.dpow (n - x) (c * b) = c ^ n * (hI.dpow x a * hJ.... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β range (n + 1)
β’ hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) = c ^ n * (hI.dpow k... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
β’ β x β range (n + 1), hI.dpow x (... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | simp only [Finset.mem_range, Nat.lt_succ_iff] at hk | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β range (n + 1)
β’ hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) = c ^ n * (hI.dpow k... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) = c ^ n * (hI.dpow k a * hJ.dpow... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β range (n + 1)
β’ hI.... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [hI.dpow_smul] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ hI.dpow k (c * a) * hJ.dpow (n - k) (c * b) = c ^ n * (hI.dpow k a * hJ.dpow... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow k a * hJ.dpow (n - k) (c * b) = c ^ n * (hI.dpow k a * hJ.dp... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ hI.dpow k (c * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [hJ.dpow_smul] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow k a * hJ.dpow (n - k) (c * b) = c ^ n * (hI.dpow k a * hJ.dp... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow k a * (c ^ (n - k) * hJ.dpow (n - k) b) = c ^ n * (hI.dpow k... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | simp only [β mul_assoc] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow k a * (c ^ (n - k) * hJ.dpow (n - k) b) = c ^ n * (hI.dpow k... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow k a * c ^ (n - k) * hJ.dpow (n - k) b = c ^ n * hI.dpow k a ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [mul_comm, β mul_assoc] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow k a * c ^ (n - k) = c ^ n * hI.dpow k a
case intro.intro.intro.intro.x
A : Type u_1
inst... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ (n - k) * c ^ k * hI.dpow k a = c ^ n * hI.dpow k a
case intro.intro.intro.intro.x
A : Type u_1
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ k * hI.dpow k a * c ^ (n - k) = c ^ n * ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | rw [β pow_add, Nat.sub_add_cancel hk] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ (n - k) * c ^ k = c ^ n
case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : Ideal A
h... | case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ b β J
case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ c ^ (n - k) * c ^ k = c ^ n
case intro.intr... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | exact hb | case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ b β J
case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : ... | case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ a β I | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ b β J
case i... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_smul | [261, 1] | [282, 11] | exact ha | case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ a β I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.x
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
c a : A
ha : a β I
b : A
hb : b β J
k : β
hk : k β€ n
β’ a β I
TACTIC:... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | intro n x y | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {x y : A},
x β I + J β y β I + J β dpow hI hJ n (x + y) = β k β range (n + 1), dpow hI hJ k x * dpow hI hJ (n - k) y | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x y : A
β’ x β I + J β y β I + J β dpow hI hJ n (x + y) = β k β range (n + 1), dpow hI hJ k x * dpow hI hJ (n - k) y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {x y : A},
x β I + J β y β I + J β dpow hI hJ n (x + y) = β k β range (n + 1), dpow hI hJ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Ideal.add_eq_sup, Submodule.mem_sup] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x y : A
β’ x β I + J β y β I + J β dpow hI hJ n (x + y) = β k β range (n + 1), dpow hI hJ k x * dpow hI hJ (n - k) y | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x y : A
β’ (β y β I, β z β J, y + z = x) β
y β I β J β dpow hI hJ n (x + y) = β k β range (n + 1), dpow hI hJ k x * dpow hI hJ (n - k) y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x y : A
β’ x β I + J β y β I + J β dpow hI hJ n (x + y) = β k β range (n + 1), dpow hI hJ k x * dpow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rintro β¨a, ha, b, hb, rflβ© | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x y : A
β’ (β y β I, β z β J, y + z = x) β
y β I β J β dpow hI hJ n (x + y) = β k β range (n + 1), dpow hI hJ k x * dpow hI hJ (n - k) y | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
y a : A
ha : a β I
b : A
hb : b β J
β’ y β I β J β dpow hI hJ n (a + b + y) = β k β range (n + 1), dpow hI hJ k (a + b) * dpow hI hJ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
x y : A
β’ (β y β I, β z β J, y + z = x) β
y β I β J β dpow hI hJ n (x + y) = β k β range (n + 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Submodule.mem_sup] | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
y a : A
ha : a β I
b : A
hb : b β J
β’ y β I β J β dpow hI hJ n (a + b + y) = β k β range (n + 1), dpow hI hJ k (a + b) * dpow hI hJ... | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
y a : A
ha : a β I
b : A
hb : b β J
β’ (β y_1 β I, β z β J, y_1 + z = y) β
dpow hI hJ n (a + b + y) = β k β range (n + 1), dpow ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
y a : A
ha : a β I
b : A
hb : b β J
β’ y β I β J β dpow hI hJ n (a + b ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rintro β¨a', ha', b', hb', rflβ© | case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
y a : A
ha : a β I
b : A
hb : b β J
β’ (β y_1 β I, β z β J, y_1 + z = y) β
dpow hI hJ n (a + b + y) = β k β range (n + 1), dpow ... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
β’ dpow hI hJ n (a + b + (a' + b')... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
y a : A
ha : a β I
b : A
hb : b β J
β’ (β y_1 β I, β z β J, y_1 + z = y... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [add_add_add_comm a b a' b'] | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
β’ dpow hI hJ n (a + b + (a' + b')... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
β’ dpow hI hJ n (a + a' + (b + b')... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [dpow_eq hI hJ hIJ n (Submodule.add_mem I ha ha') (Submodule.add_mem J hb hb')] | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
β’ dpow hI hJ n (a + a' + (b + b')... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
β’ β k β range (n + 1), hI.dpow k ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | let f : β Γ β Γ β Γ β β A := fun β¨i, j, k, lβ© =>
hI.dpow i a * hI.dpow j a' * hJ.dpow k b * hJ.dpow l b' | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
β’ β k β range (n + 1), hI.dpow k ... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | have hf1 :
β k β Finset.range (n + 1),
hI.dpow k (a + a') * hJ.dpow (n - k) (b + b') =
(Finset.range (k + 1)).sum fun i =>
(Finset.range (n - k + 1)).sum fun l =>
hI.dpow i a * hI.dpow (k - i) a' * hJ.dpow l b * hJ.dpow (n - k - l) b' := by
intro k _
rw [hI.dpow_add' k ha ha']; rw [hJ.... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Finset.sum_congr rfl hf1] | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | have hf2 :
β k β Finset.range (n + 1),
dpow hI hJ k (a + b) * dpow hI hJ (n - k) (a' + b') =
(Finset.range (k + 1)).sum fun i =>
(Finset.range (n - k + 1)).sum fun l =>
hI.dpow i a * hI.dpow l a' * hJ.dpow (k - i) b * hJ.dpow (n - k - l) b' :=
by
intro k _
rw [dpow_eq hI hJ hIJ k ha ... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Finset.sum_congr rfl hf2] | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | convert Finset.sum_4_rw f n | case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | intro k _ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [hI.dpow_add' k ha ha'] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [hJ.dpow_add' (n - k) hb hb'] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Finset.sum_mul] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | apply Finset.sum_congr rfl | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | intro i _ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Finset.mul_sum] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | apply Finset.sum_congr rfl | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | intro l _ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | ring | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | intro k _ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [dpow_eq hI hJ hIJ k ha hb] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [dpow_eq hI hJ hIJ (n - k) ha' hb'] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Finset.sum_mul] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | apply Finset.sum_congr rfl | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | intro i _ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | rw [Finset.mul_sum] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | apply Finset.sum_congr rfl | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | intro j _ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add' | [285, 1] | [333, 30] | ring | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
fun x =>
match x with
| (i, j, k, l) => hI.dpow i a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
n : β
a : A
ha : a β I
b : A
hb : b β J
a' : A
ha' : a' β I
b' : A
hb' : b' β J
f : β Γ β Γ β Γ β β A :=
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add | [336, 1] | [342, 28] | simp only [Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {x y : A},
x β I + J β
y β I + J β
dpow hI hJ n (x + y) =
β x_3 β antidiagonal n,
match x_3 with
| (k,... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {x y : A},
x β I + J β y β I + J β dpow hI hJ n (x + y) = β x_3 β range n.succ, dpow hI hJ x_3 x * dpow hI hJ (n - x_3) y | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {x y : A},
x β I + J β
y β I + J β
dpow hI hJ n (x + y) =
β x_3 β... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_add | [336, 1] | [342, 28] | exact dpow_add' hI hJ hIJ | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {x y : A},
x β I + J β y β I + J β dpow hI hJ n (x + y) = β x_3 β range n.succ, dpow hI hJ x_3 x * dpow hI hJ (n - x_3) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
β’ β (n : β) {x y : A},
x β I + J β y β I + J β dpow hI hJ n (x + y) = β x_3 β range n.succ, dpow hI h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [dpow_eq hI hJ hIJ n ha hb, dpow_sum_aux (dpow hI hJ)] | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ m (dpow hI hJ n (a + b)) =
β p β range (m * n + 1),
β(β x β filter (fun l => β i β range (... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ β k β (range (n + 1)).sym m, β i β range (n + 1), dpow hI hJ (Multiset.count i βk) (hI.dpow i a * hJ.dpow (n - ... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
β’ dpow hI hJ m (dpow hI hJ n (a + b)) =
β p β ra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | set Ο : Sym β m β β := fun k => (Finset.range (n + 1)).sum fun i => Multiset.count i βk * i
with hΟ_def | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
L1 :
β (k : Sym β m),
β i β range (n + 1),
dpow hI hJ (Multiset.count i βk) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
L1 :
β (k : Sym β m),
β i β range (n + 1),
dpow hI hJ (Multiset.count i βk) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
L1 :
β (k : Sym β m),
β i β range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | suffices hΟ : β k : Sym β m, k β (Finset.range (n + 1)).sym m β Ο k β Finset.range (m * n + 1) by
rw [β Finset.sum_fiberwise_of_maps_to hΟ _]
suffices L4 :
β (p : β) (_ : p β Finset.range (m * n + 1)),
((Finset.filter (fun x : Sym β m => (fun k : Sym β m => Ο k) x = p)
((Finset.range (n + 1)... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
L1 :
β (k : Sym β m),
β i β range (n + 1),
dpow hI hJ (Multiset.count i βk) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
L1 :
β (k : Sym β m),
β i β range (n + 1),
dpow hI hJ (Multiset.count i βk) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
instβ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : β (n : β), β a β I β J, hI.dpow n a = hJ.dpow n a
m n : β
hn : n β 0
a : A
ha : a β I
b : A
hb : b β J
L1 :
β (k : Sym β m),
β i β range (n + 1),
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.