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_comp_aux | [366, 1] | [483, 31] | . intro x hx
rw [dpow_zero hI hJ hIJ] exact hx | case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ {x : A}, x ∈ ?m.65386 → dpow hI hJ 0 x = 1
case dpow_add
A : Type u_1
inst✝ : CommRing A
I : ... | case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J → y ∈ I + J → dpow hI hJ n (x + y) = ∑ k ∈ antidiagonal n, dpo... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ {x : A}, x ∈ ?m.65386 → dpow hI h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | . simp only [Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk]
intro n x y hx hy
rw [dpow_add' hI hJ hIJ]
exact hx
exact hy | case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J → y ∈ I + J → dpow hI hJ n (x + y) = ∑ k ∈ antidiagonal n, dpo... | case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ {n : ℕ}, n ≠ 0 → dpow hI hJ n 0 = 0
case hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro k i hi | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n - i)... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
⊢ dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n - i) b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dp... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [cnik] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
⊢ dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n - i) b) =
... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
⊢ dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n - i) b) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
⊢ dpow hI h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | by_cases hi2 : i = n | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
⊢ dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n - i) b) =
... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * h... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
⊢ dpow hI h... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | have hi2' : n - i ≠ 0 := by
intro h; apply hi2
rw [Finset.mem_range, Nat.lt_succ_iff] at hi
rw [← Nat.sub_add_cancel hi, h, zero_add] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
⊢ dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * ... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
⊢ dpow hI hJ (Multiset.count i ↑k... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | by_cases hi1 : i = 0 | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
⊢ dpow hI hJ (Multiset.count i ↑k... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hi2] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * h... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [Nat.sub_self] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [if_neg hn] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [if_pos rfl] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [mchoose_zero', mul_one, Nat.cast_one, MulZeroClass.mul_zero, hJ.dpow_zero hb] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a * h... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [dpow_eq_of_mem_left hI hJ hIJ _ (hI.dpow_mem hn ha)] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ dpow hI hJ (Multiset.count n ↑k) (hI.dpow n a) =
... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ hI.dpow (Multiset.count n ↑k) (hI.dpow n a) = ↑(m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hI.dpow_comp _ hn ha] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : i = n
⊢ hI.dpow (Multiset.count n ↑k) (hI.dpow n a) = ↑(m... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro h | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
⊢ n - i ≠ 0 | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
h : n - i = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply hi2 | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
h : n - i = 0
⊢ False | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
h : n - i = 0
⊢ i = n | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [Finset.mem_range, Nat.lt_succ_iff] at hi | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
h : n - i = 0
⊢ i = n | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ≤ n
hi2 : ¬i = n
h : n - i = 0
⊢ i = n | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [← Nat.sub_add_cancel hi, h, zero_add] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ≤ n
hi2 : ¬i = n
h : n - i = 0
⊢ i = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ≤ n
hi2 : ¬i = n
h : n - i ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hi1] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [mchoose_zero'] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hI.dpow_zero ha] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [Nat.sub_zero] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [one_mul] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [if_pos rfl] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [dpow_eq_of_mem_right hI hJ hIJ _ (hJ.dpow_mem hn hb)] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ dpow hI hJ (Multise... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ hJ.dpow (Multiset.c... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hJ.dpow_comp _ hn hb] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ hJ.dpow (Multiset.c... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ ↑(mchoose (Multiset... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [MulZeroClass.mul_zero] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ ↑(mchoose (Multiset... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ ↑(mchoose (Multiset... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hI.dpow_zero ha] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ ↑(mchoose (Multiset... | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ ↑(mchoose (Multiset... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [Nat.cast_one, one_mul, mul_one] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : i = 0
⊢ ↑(mchoose (Multiset... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [if_neg hi1] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ dpow hI hJ (Multis... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ dpow hI hJ (Multis... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [if_neg hi2] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ dpow hI hJ (Multis... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ dpow hI hJ (Multis... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [mul_comm, dpow_smul hI hJ hIJ, mul_comm] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ dpow hI hJ (Multis... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ dpow hI hJ (Multis... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [dpow_eq_of_mem_left hI hJ hIJ _ (hI.dpow_mem hi1 ha)] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ dpow hI hJ (Multis... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow (Multiset.... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [← hJ.factorial_mul_dpow_eq_pow (Multiset.count i k)] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow (Multiset.... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow (Multiset.... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hI.dpow_comp _ hi1 ha] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow (Multiset.... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hJ.dpow_comp _ hi2' hb] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multise... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [← mul_assoc] | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multise... | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multise... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply congr_arg₂ _ _ rfl | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multise... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [mul_assoc] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [mul_comm (hI.dpow _ a)] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [← mul_assoc] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply congr_arg₂ _ _ rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [Sym.mem_coe, ge_iff_le, Nat.cast_mul] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply congr_arg₂ _ _ rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [mul_comm] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ ↑(mchoose (Multiset.count i... | case neg.hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hJ.dpow (n - i)... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact hJ.dpow_mem hi2' hb | case neg.hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hJ.dpow (n - i)... | case neg.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow i a ∈ I ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply Submodule.mem_sup_left | case neg.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow i a ∈ I ... | case neg.x.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow i a ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact hI.dpow_mem hi1 ha | case neg.x.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1)
hi2 : ¬i = n
hi2' : n - i ≠ 0
hi1 : ¬i = 0
⊢ hI.dpow i a ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.x.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
k : Sym ℕ m
i : ℕ
hi : i ∈ range (n + 1... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [← Finset.sum_fiberwise_of_maps_to hφ _] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | suffices L4 :
∀ (p : ℕ) (_ : p ∈ Finset.range (m * n + 1)),
((Finset.filter (fun x : Sym ℕ m => (fun k : Sym ℕ m => φ k) x = p)
((Finset.range (n + 1)).sym m)).sum
fun x : Sym ℕ m =>
(Finset.range (n + 1)).prod fun i : ℕ =>
dpow hI hJ (Multiset.count i ↑x) (hI.dpow i a * hJ.d... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro p _ | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply Finset.sum_congr rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro k hk | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [Finset.mem_filter] at hk | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [Finset.prod_congr rfl (L1 k), Finset.prod_mul_distrib, Finset.prod_mul_distrib,
hI.prod_dpow_self _ ha, hJ.prod_dpow_self _ hb] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [mul_assoc] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply congr_arg₂ _ rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply congr_arg₂ _ rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [sum_range_sym_mul_compl hk.1] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [← mul_assoc] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [mem_sym_iff, mem_range, hφ_def] at hk | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hk.2] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply congr_arg₂ _ _ rfl | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [mul_comm] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [Finset.sum_congr rfl L4] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [Sym.mem_coe, mem_sym_iff, mem_range, ge_iff_le, Nat.cast_sum, Nat.cast_mul,
Nat.cast_prod] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp_rw [Finset.sum_mul] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro k hk | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [Sym.mem_coe, mem_range, Nat.lt_succ_iff] | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact range_sym_weighted_sum_le hk | A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
dpow hI hJ (Multiset.count i ↑k) (hI.dpow i a * hJ.dpow (n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
L1 :
∀ (k : Sym ℕ m),
∀ i ∈ range (n + 1),
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro x hx | case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ {x : A}, x ∈ ?m.65386 → dpow hI hJ 0 x = 1 | case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
x : A
hx : x ∈ ?m.65386
⊢ dpow hI hJ 0 x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ {x : A}, x ∈ ?m.65386 → 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_zero hI hJ hIJ] | case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
x : A
hx : x ∈ ?m.65386
⊢ dpow hI hJ 0 x = 1 | case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
x : A
hx : x ∈ ?m.65386
⊢ x ∈ I + J | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
x : A
hx : x ∈ ?m.65386
⊢ dpow hI hJ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact hx | case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
x : A
hx : x ∈ ?m.65386
⊢ x ∈ I + J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
x : A
hx : x ∈ ?m.65386
⊢ x ∈ I + J
T... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | simp only [Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J → y ∈ I + J → dpow hI hJ n (x + y) = ∑ k ∈ antidiagonal n, dpo... | case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J → y ∈ I + J → dpow hI hJ n (x + y) = ∑ x_3 ∈ range n.succ, dpo... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro n x y hx hy | case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J → y ∈ I + J → dpow hI hJ n (x + y) = ∑ x_3 ∈ range n.succ, dpo... | case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : y ∈ I + J
⊢ dpow hI hJ n (x + y) = ∑ x_1 ∈ range n.succ, dpow h... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ (n : ℕ) {x y : A},
x ∈ I + J →... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [dpow_add' hI hJ hIJ] | case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : y ∈ I + J
⊢ dpow hI hJ n (x + y) = ∑ x_1 ∈ range n.succ, dpow h... | case dpow_add.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : y ∈ I + J
⊢ x ∈ I + J
case dpow_add.x
A : Type u_1
inst✝ : C... | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : y ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact hx | case dpow_add.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : y ∈ I + J
⊢ x ∈ I + J
case dpow_add.x
A : Type u_1
inst✝ : C... | case dpow_add.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : y ∈ I + J
⊢ y ∈ I + J | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact hy | case dpow_add.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : y ∈ I + J
⊢ y ∈ I + J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_add.x
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
x y : A
hx : x ∈ I + J
hy : ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro n hn | case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ {n : ℕ}, n ≠ 0 → dpow hI hJ n 0 = 0 | case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn✝ : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
hn : n ≠ 0
⊢ dpow hI hJ n 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ {n : ℕ}, n ≠ 0 → dpow hI hJ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [dpow_eq_of_mem_left hI hJ hIJ n I.zero_mem] | case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn✝ : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
hn : n ≠ 0
⊢ dpow hI hJ n 0 = 0 | case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn✝ : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn✝ : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
hn : n ≠ 0
⊢ dpow hI hJ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact dpow_eval_zero hI hn | case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn✝ : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case dpow_eval_zero
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n✝ : ℕ
hn✝ : n✝ ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
n : ℕ
hn : n ≠ 0
⊢ hI.dpow n ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | intro i _ | case hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ i ∈ range (n + 1), hI.dpow i a * hJ.dpow (n - i) b ∈ I + J | case hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
⊢ hI.dpow i a * hJ.dpow (n - i) b ∈ I + J | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
⊢ ∀ i ∈ range (n + 1), hI.dpow i a * hJ.dpow... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | by_cases hi0 : i = 0 | case hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
⊢ hI.dpow i a * hJ.dpow (n - i) b ∈ I + J | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hI.dpow i a * hJ.dpow (n - i) b ∈ I + J
case neg
A : Type u_... | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
⊢ hI.dpow i a *... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | rw [hi0] | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hI.dpow i a * hJ.dpow (n - i) b ∈ I + J | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hI.dpow 0 a * hJ.dpow (n - 0) b ∈ I + J | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply Submodule.mem_sup_right | case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hI.dpow 0 a * hJ.dpow (n - 0) b ∈ I + J | case pos.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hI.dpow 0 a * hJ.dpow (n - 0) b ∈ J | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply Ideal.mul_mem_left | case pos.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hI.dpow 0 a * hJ.dpow (n - 0) b ∈ J | case pos.a.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hJ.dpow (n - 0) b ∈ J | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact hJ.dpow_mem hn hb | case pos.a.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = 0
⊢ hJ.dpow (n - 0) b ∈ J | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : i = ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply Submodule.mem_sup_left | case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i = 0
⊢ hI.dpow i a * hJ.dpow (n - i) b ∈ I + J | case neg.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i = 0
⊢ hI.dpow i a * hJ.dpow (n - i) b ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i = 0
⊢... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | apply Ideal.mul_mem_right | case neg.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i = 0
⊢ hI.dpow i a * hJ.dpow (n - i) b ∈ I | case neg.a.h
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i = 0
⊢ hI.dpow i a ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i = 0... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.dpow_comp_aux | [366, 1] | [483, 31] | exact hI.dpow_mem hi0 ha | case neg.a.h
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i = 0
⊢ hI.dpow i a ∈ I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.h
A : Type u_1
inst✝ : CommRing A
I : Ideal A
hI : DividedPowers I
J : Ideal A
hJ : DividedPowers J
hIJ : ∀ (n : ℕ), ∀ a ∈ I ⊓ J, hI.dpow n a = hJ.dpow n a
m n : ℕ
hn : n ≠ 0
a : A
ha : a ∈ I
b : A
hb : b ∈ J
i : ℕ
a✝ : i ∈ range (n + 1)
hi0 : ¬i =... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | simp only [Ring.inverse] | A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
⊢ Ring.inverse (C a) = C (Ring.inverse a) | A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
⊢ Ring.inverse (C a) = C (Ring.inverse a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | by_cases ha : IsUnit a | A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0) | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0)
case neg
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | simp only [dif_pos ha] | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0) | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C ↑ha.unit⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C (if h : IsUnit a then ↑h.unit⁻¹ else 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | have hCa : IsUnit (C a) := by
rw [← IsUnit.unit_spec ha]
exact RingHom.isUnit_map C ha | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C ↑ha.unit⁻¹ | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C ↑ha.unit⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C ↑ha.unit⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | rw [dif_pos hCa] | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C ↑ha.unit⁻¹ | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ = C ↑ha.unit⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ (if h : IsUnit (C a) then ↑h.unit⁻¹ else 0) = C ↑ha.unit⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | apply IsUnit.mul_right_cancel hCa | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ = C ↑ha.unit⁻¹ | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ * C a = C ↑ha.unit⁻¹ * C a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ = C ↑ha.unit⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | rw [← map_mul] | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ * C a = C ↑ha.unit⁻¹ * C a | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ * C a = C (↑ha.unit⁻¹ * a) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ * C a = C ↑ha.unit⁻¹ * C a
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | simp only [IsUnit.val_inv_mul, map_one] | case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ * C a = C (↑ha.unit⁻¹ * a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
hCa : IsUnit (C a)
⊢ ↑hCa.unit⁻¹ * C a = C (↑ha.unit⁻¹ * a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | rw [← IsUnit.unit_spec ha] | A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ IsUnit (C a) | A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ IsUnit (C ↑ha.unit) | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ IsUnit (C a)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/IdealAdd.lean | DividedPowers.IdealAdd.Polynomial.inv_C_eq_C_inv | [490, 1] | [511, 31] | exact RingHom.isUnit_map C ha | A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ IsUnit (C ↑ha.unit) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.83499
inst✝¹ : CommRing A
I : Ideal A
hI : DividedPowers I
R : Type u_1
inst✝ : CommSemiring R
a : R
ha : IsUnit a
⊢ IsUnit (C ↑ha.unit)
TACTIC:
|
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