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/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rintro ⟨⟨k, a⟩, c⟩ h | α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
⊢ ∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)),
... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
⊢ (fun m x... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
⊢ ∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | simp only [Finset.mem_sigma, Finset.mem_range, Nat.lt_succ_iff] at h | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
⊢ (fun m x... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => r... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | simp_rw [Finset.mem_sigma, Finset.mem_range, Nat.lt_succ_iff] | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ ... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | constructor | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ ... | case mk.mk.left
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | constructor | case mk.mk.left
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : ... | case mk.mk.left.left
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.left
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | apply le_trans (add_le_add h.1.2 h.2) _ | case mk.mk.left.left
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)... | α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ n ∧ a ≤ k) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.left.left
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rw [add_comm] | α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ n ∧ a ≤ k) ... | α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ n ∧ a ≤ k) ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rw [Nat.sub_add_cancel h.1.1] | α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h : (k ≤ n ∧ a ≤ k) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + ... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | exact le_self_add | case mk.mk.left.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.left.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigm... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rw [add_comm a c] | case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h :... | case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h :... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rw [← Nat.sub_sub n c a] | case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h :... | case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h :... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | apply Nat.sub_le_sub_right | case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h :... | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.right
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rw [Nat.le_sub_iff_add_le] | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rw [Nat.le_sub_iff_add_le h.1.1, add_comm] at h | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | exact h.2 | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | exact le_trans h.2 (Nat.sub_le n k) | case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1)
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.right.h
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
c k a : ℕ
h✝ : ⟨⟨k, a⟩, c⟩ ∈ ((range (n + 1)).sigma f... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rintro ⟨⟨k, a⟩, c⟩ h | α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigm... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | simp only [Finset.mem_sigma, Finset.mem_range, Nat.lt_succ_iff] at h | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rw [Sigma.mk.inj_iff] | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | simp only [Sigma.mk.inj_iff, heq_eq_eq, and_true, add_tsub_cancel_left] | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | refine add_tsub_cancel_of_le h.1.2 | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | rintro ⟨⟨k, a⟩, c⟩ h | α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n - x.fst + 1... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigm... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | simp only [mem_sigma, mem_range, Nat.lt_succ_iff] at h | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | Finset.sum_4_rw | [139, 1] | [185, 63] | simp only [Nat.add_sub_self_left a c, Nat.sub_sub,
add_comm (a + c), ← add_assoc, Nat.sub_add_cancel h.1.2] | case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n + 1)).sigma fun l => range (l + 1)).sigma fun x => range (n ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk
α : Type u_1
inst✝ : AddCommMonoid α
f : ℕ × ℕ × ℕ × ℕ → α
n : ℕ
φ : (_ : (_ : ℕ) × ℕ) × ℕ → (_ : (_ : ℕ) × ℕ) × ℕ :=
fun x =>
match x with
| ⟨⟨k, a⟩, c⟩ => ⟨⟨a + c, a⟩, k - a⟩
h1 :
∀ (a : (_ : (_ : ℕ) × ℕ) × ℕ)
(ha : a ∈ ((range (n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | simp only [Finset.mem_sym_iff] at hk | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = m | m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = m | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | simp_rw [← k.prop] | m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = m | m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = Multiset.card ↑k | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | rw [← Multiset.toFinset_sum_count_eq] | m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = Multiset.card ↑k | m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = ∑ a ∈ (↑k).toFinset, Multiset.count a ↑k | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = Multiset.card ↑k
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | apply symm | m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = ∑ a ∈ (↑k).toFinset, Multiset.count a ↑k | case a
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ a ∈ (↑k).toFinset, Multiset.count a ↑k = ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k = ∑ a ∈ (↑k).toFinset, Multiset.count a ↑k
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | apply Finset.sum_subset_zero_on_sdiff | case a
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ a ∈ (↑k).toFinset, Multiset.count a ↑k = ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k | case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ (↑k).toFinset ⊆ Finset.range (n + 1)
case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∀ x ∈ Finset.range (n + 1) \ (↑k).toFinset, Multiset.count x ↑k = 0
case a.hfg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∑ a ∈ (↑k).toFinset, Multiset.count a ↑k = ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | intro i hi | case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ (↑k).toFinset ⊆ Finset.range (n + 1) | case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
i : ℕ
hi : i ∈ (↑k).toFinset
⊢ i ∈ Finset.range (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ (↑k).toFinset ⊆ Finset.range (n + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | simp only [Sym.val_eq_coe, Multiset.mem_toFinset, Sym.mem_coe] at hi | case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
i : ℕ
hi : i ∈ (↑k).toFinset
⊢ i ∈ Finset.range (n + 1) | case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
i : ℕ
hi : i ∈ k
⊢ i ∈ Finset.range (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
i : ℕ
hi : i ∈ (↑k).toFinset
⊢ i ∈ Finset.range (n + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | exact hk i hi | case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
i : ℕ
hi : i ∈ k
⊢ i ∈ Finset.range (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
i : ℕ
hi : i ∈ k
⊢ i ∈ Finset.range (n + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | intro x hx | case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∀ x ∈ Finset.range (n + 1) \ (↑k).toFinset, Multiset.count x ↑k = 0 | case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x ∈ Finset.range (n + 1) \ (↑k).toFinset
⊢ Multiset.count x ↑k = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∀ x ∈ Finset.range (n + 1) \ (↑k).toFinset, Multiset.count x ↑k = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | simp only [Sym.val_eq_coe, Finset.mem_sdiff, Finset.mem_range, Multiset.mem_toFinset, Sym.mem_coe] at hx | case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x ∈ Finset.range (n + 1) \ (↑k).toFinset
⊢ Multiset.count x ↑k = 0 | case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x < n + 1 ∧ x ∉ k
⊢ Multiset.count x ↑k = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x ∈ Finset.range (n + 1) \ (↑k).toFinset
⊢ Multiset.count x ↑k = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | simp only [Multiset.count_eq_zero, Sym.mem_coe] | case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x < n + 1 ∧ x ∉ k
⊢ Multiset.count x ↑k = 0 | case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x < n + 1 ∧ x ∉ k
⊢ x ∉ k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x < n + 1 ∧ x ∉ k
⊢ Multiset.count x ↑k = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | exact hx.2 | case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x < n + 1 ∧ x ∉ k
⊢ x ∉ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
hx : x < n + 1 ∧ x ∉ k
⊢ x ∉ k
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | intro x _ | case a.hfg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∀ x ∈ (↑k).toFinset, Multiset.count x ↑k = Multiset.count x ↑k | case a.hfg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
a✝ : x ∈ (↑k).toFinset
⊢ Multiset.count x ↑k = Multiset.count x ↑k | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hfg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
⊢ ∀ x ∈ (↑k).toFinset, Multiset.count x ↑k = Multiset.count x ↑k
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_prop | [190, 1] | [205, 19] | rfl | case a.hfg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
a✝ : x ∈ (↑k).toFinset
⊢ Multiset.count x ↑k = Multiset.count x ↑k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hfg
m n : ℕ
k : Sym ℕ m
hk : ∀ a ∈ k, a ∈ Finset.range (n + 1)
x : ℕ
a✝ : x ∈ (↑k).toFinset
⊢ Multiset.count x ↑k = Multiset.count x ↑k
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_weighted_sum_le | [208, 1] | [216, 52] | suffices h : ∀ i ∈ Finset.range (n + 1), Multiset.count i k * i ≤ Multiset.count i k * n by
apply le_trans (Finset.sum_le_sum h)
rw [← Finset.sum_mul, range_sym_prop hk] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ m * n | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ m * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_weighted_sum_le | [208, 1] | [216, 52] | intro i hi | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_weighted_sum_le | [208, 1] | [216, 52] | apply Nat.mul_le_mul_left | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n | case h
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ i ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_weighted_sum_le | [208, 1] | [216, 52] | exact Nat.lt_succ_iff.mp (Finset.mem_range.mp hi) | case h
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ i ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ i ≤ n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_weighted_sum_le | [208, 1] | [216, 52] | apply le_trans (Finset.sum_le_sum h) | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
h : ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ m * n | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
h : ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * n ≤ m * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
h : ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ m * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | range_sym_weighted_sum_le | [208, 1] | [216, 52] | rw [← Finset.sum_mul, range_sym_prop hk] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
h : ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * n ≤ m * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
h : ∀ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i ≤ Multiset.count i ↑k * n
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * n ≤ m * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | suffices h : (((Finset.range (n + 1)).sum fun i => Multiset.count i k * (n - i)) +
(Finset.range (n + 1)).sum fun i => Multiset.count i k * i) = m * n by
rw [← h, Nat.add_sub_cancel] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) =
m * n - ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) + ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i =
m * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) =
m * n - ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | rw [← Finset.sum_add_distrib] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) + ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i =
m * n | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ x ∈ Finset.range (n + 1), (Multiset.count x ↑k * (n - x) + Multiset.count x ↑k * x) = m * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) + ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i =
m * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | simp_rw [← mul_add] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ x ∈ Finset.range (n + 1), (Multiset.count x ↑k * (n - x) + Multiset.count x ↑k * x) = m * n | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = m * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ x ∈ Finset.range (n + 1), (Multiset.count x ↑k * (n - x) + Multiset.count x ↑k * x) = m * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | have :
∀ x ∈ Finset.range (n + 1),
Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n := by
intro x hx
rw [Nat.sub_add_cancel (Nat.lt_succ_iff.mp (Finset.mem_range.mp hx))] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = m * n | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
this : ∀ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n
⊢ ∑ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = m * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∑ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = m * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | rw [Finset.sum_congr rfl this, ← Finset.sum_mul, range_sym_prop hk] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
this : ∀ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n
⊢ ∑ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = m * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
this : ∀ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n
⊢ ∑ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = m * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | rw [← h, Nat.add_sub_cancel] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
h :
∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) + ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i =
m * n
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) =
m * n - ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
h :
∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) + ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * i =
m * n
⊢ ∑ i ∈ Finset.range (n + 1), Multiset.count i ↑k * (n - i) =
m * n... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | intro x hx | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∀ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
x : ℕ
hx : x ∈ Finset.range (n + 1)
⊢ Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
⊢ ∀ x ∈ Finset.range (n + 1), Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/BasicLemmas.lean | sum_range_sym_mul_compl | [219, 1] | [232, 70] | rw [Nat.sub_add_cancel (Nat.lt_succ_iff.mp (Finset.mem_range.mp hx))] | m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
x : ℕ
hx : x ∈ Finset.range (n + 1)
⊢ Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
k : Sym ℕ m
hk : k ∈ (Finset.range (n + 1)).sym m
x : ℕ
hx : x ∈ Finset.range (n + 1)
⊢ Multiset.count x ↑k * (n - x + x) = Multiset.count x ↑k * n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.algHom_apply_apply | [111, 1] | [113, 16] | simp [algHom] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝⁶ : DecidableEq α
inst✝⁵ : Semiring N
inst✝⁴ : Semiring P
inst✝³ : Monoid α
inst✝² : CommSemiring R
inst✝¹ : Algebra R N
inst✝ : Algebra R P
e : N →ₐ[R] P
x : MonoidAlgebra N α
a : α
⊢ ((algHom e) x) a = e (x a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝⁶ : DecidableEq α
inst✝⁵ : Semiring N
inst✝⁴ : Semiring P
inst✝³ : Monoid α
inst✝² : CommSemiring R
inst✝¹ : Algebra R N
inst✝ : Algebra R P
e : N →ₐ[R] P
x : MonoidAlgebra N α
a : α
⊢ ((al... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.algHom_apply_single | [115, 1] | [117, 16] | simp [algHom] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝⁶ : DecidableEq α
inst✝⁵ : Semiring N
inst✝⁴ : Semiring P
inst✝³ : Monoid α
inst✝² : CommSemiring R
inst✝¹ : Algebra R N
inst✝ : Algebra R P
e : N →ₐ[R] P
a : α
n : N
⊢ (algHom e) (single a n) = single a (e n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝⁶ : DecidableEq α
inst✝⁵ : Semiring N
inst✝⁴ : Semiring P
inst✝³ : Monoid α
inst✝² : CommSemiring R
inst✝¹ : Algebra R N
inst✝ : Algebra R P
e : N →ₐ[R] P
a : α
n : N
⊢ (algHom e) (single a... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_tmul_apply | [162, 1] | [170, 68] | simp only [rTensorAlgHom] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_tmul_apply | [162, 1] | [170, 68] | simp only [Algebra.TensorProduct.lift_tmul] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_tmul_apply | [162, 1] | [170, 68] | rw [AlgHom.comp_apply, singleOneAlgHom_apply, mul_single_one_apply] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_tmul_apply | [162, 1] | [170, 68] | simp only [Algebra.TensorProduct.includeRight_apply] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_tmul_apply | [162, 1] | [170, 68] | simp only [algHom_apply_apply, Algebra.TensorProduct.includeLeft_apply] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_tmul_apply | [162, 1] | [170, 68] | simp only [Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap | [172, 1] | [182, 6] | ext x n | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | case a.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : A... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap | [172, 1] | [182, 6] | dsimp only [AlgebraTensorModule.curry_apply, TensorProduct.curry_apply,
LinearMap.coe_restrictScalars, AlgHom.toLinearMap_apply] | case a.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : A... | case a.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : A... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap | [172, 1] | [182, 6] | apply Finsupp.ext | case a.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : A... | case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap | [172, 1] | [182, 6] | intro a | case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ :... | case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap | [172, 1] | [182, 6] | rw [rTensorAlgHom_apply_tmul_apply, ← finsuppLeft_apply_tmul_apply] | case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ :... | case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap | [172, 1] | [182, 6] | rfl | case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.h.h
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap' | [184, 1] | [189, 6] | rw [rTensorAlgHom_toLinearMap] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_toLinearMap' | [184, 1] | [189, 6] | rfl | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_eq | [191, 1] | [194, 6] | rw [← AlgHom.toLinearMap_apply, rTensorAlgHom_toLinearMap] | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/RingTheory/TensorProduct/MonoidAlgebra.lean | MonoidAlgebra.rTensorAlgHom_apply_eq | [191, 1] | [194, 6] | rfl | α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² : IsScalarTower R S M
inst✝¹ : Semiring N
inst✝ : Algebra R N
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
inst✝¹⁰ : Monoid α
inst✝⁹ : DecidableEq α
inst✝⁸ : CommSemiring R
S : Type u_6
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
inst✝⁵ : Semiring M
inst✝⁴ : Algebra R M
inst✝³ : Algebra S M
inst✝² :... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | obtain ⟨a, ha⟩ := h | α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
h : Summable f
⊢ (Function.support f).Finite | case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : HasSum f a
⊢ (Function.support f).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
h : Summable f
⊢ (Function.support f).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | simp only [HasSum] at ha | case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : HasSum f a
⊢ (Function.support f).Finite | case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : Filter.Tendsto (fun s => ∑ b ∈ s, f b) Filter.atTop (nhds a)
⊢ (Function.support f).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : HasSum f a
⊢ (Function.support f).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | classical
simp_rw [tendsto_atTop_nhds] at ha
obtain ⟨s, hs⟩ := ha {a} rfl (isOpen_discrete _)
apply Set.Finite.subset s.finite_toSet
intro b
rw [Function.mem_support, not_imp_comm]
intro hb
let hs' := hs (insert b s) (s.subset_insert b)
specialize hs s (subset_of_eq rfl)
simp only [Set.mem_singleton_iff] at hs hs'
simp... | case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : Filter.Tendsto (fun s => ∑ b ∈ s, f b) Filter.atTop (nhds a)
⊢ (Function.support f).Finite | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : Filter.Tendsto (fun s => ∑ b ∈ s, f b) Filter.atTop (nhds a)
⊢ (Function.support f).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | simp_rw [tendsto_atTop_nhds] at ha | case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : Filter.Tendsto (fun s => ∑ b ∈ s, f b) Filter.atTop (nhds a)
⊢ (Function.support f).Finite | case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
⊢ (Function.support f).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : Filter.Tendsto (fun s => ∑ b ∈ s, f b) Filter.atTop (nhds a)
⊢ (Function.support f).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | obtain ⟨s, hs⟩ := ha {a} rfl (isOpen_discrete _) | case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
⊢ (Function.support f).Finite | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
⊢ (Function.support f).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
⊢ (Function.support f).Finite
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | apply Set.Finite.subset s.finite_toSet | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
⊢ (Function.support f).Finite | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
⊢ Function.support f ⊆ ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | intro b | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
⊢ Function.support f ⊆ ↑s | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
⊢ b ∈ Function.support f →... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | rw [Function.mem_support, not_imp_comm] | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
⊢ b ∈ Function.support f →... | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
⊢ b ∉ ↑s → f b = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | intro hb | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
⊢ b ∉ ↑s → f b = 0 | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
⊢ f b = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | let hs' := hs (insert b s) (s.subset_insert b) | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
⊢ f b = 0 | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
hs' : ∑ b ∈ in... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | specialize hs s (subset_of_eq rfl) | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
hs' : ∑ b ∈ in... | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs✝ : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
hs' : ∑ b ∈ i... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs : ∀ (n : Finset β)... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | simp only [Set.mem_singleton_iff] at hs hs' | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs✝ : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
hs' : ∑ b ∈ i... | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs✝ : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
hs : ∑ b ∈ s,... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs✝ : ∀ (n : Finset β... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | Set.Finite.support_of_summable | [26, 1] | [41, 63] | simpa [Finset.sum_insert hb, hs, add_left_eq_self] using hs' | case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs✝ : ∀ (n : Finset β), s ≤ n → ∑ b ∈ n, f b ∈ {a}
b : β
hb : b ∉ ↑s
hs : ∑ b ∈ s,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : TopologicalSpace α
inst✝ : DiscreteTopology α
β : Type u_2
f : β → α
a : α
ha : ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : Finset β), N ≤ n → ∑ b ∈ n, f b ∈ U
s : Finset β
hs✝ : ∀ (n : Finset β... |
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | rw [add_pow] | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ (a + b) ^ (m + n).pred = 0 | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ ∑ m_1 ∈ Finset.range ((m + n).pred + 1), a ^ m_1 * b ^ ((m + n).pred - m_1) * ↑((m + n).pred.choose m_1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ (a + b) ^ (m + n).pred = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | apply Finset.sum_eq_zero | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ ∑ m_1 ∈ Finset.range ((m + n).pred + 1), a ^ m_1 * b ^ ((m + n).pred - m_1) * ↑((m + n).pred.choose m_1) = 0 | case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ ∀ x ∈ Finset.range ((m + n).pred + 1), a ^ x * b ^ ((m + n).pred - x) * ↑((m + n).pred.choose x) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ ∑ m_1 ∈ Finset.range ((m + n).pred + 1), a ^ m_1 * b ^ ((m + n).pred - m_1) * ↑((m + n).pred.choose m_1) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | intro k hk | case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ ∀ x ∈ Finset.range ((m + n).pred + 1), a ^ x * b ^ ((m + n).pred - x) * ↑((m + n).pred.choose x) = 0 | case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k ∈ Finset.range ((m + n).pred + 1)
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
⊢ ∀ x ∈ Finset.range ((m + n).pred + 1), a ^ x * b ^ ((m + n).pred - x) * ↑((m + n).pred.choose x) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | simp only [Finset.mem_range] at hk | case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k ∈ Finset.range ((m + n).pred + 1)
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k ∈ Finset.range ((m + n).pred + 1)
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | by_cases h : k < m | case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0
case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | have : n ≤ (m + n).pred - k := by
rw [Nat.le_sub_iff_add_le (Nat.le_of_lt_succ hk), add_comm]
rw [Nat.le_pred_iff_lt (lt_of_le_of_lt (zero_le k) (Nat.lt_add_right n h))]
exact Nat.add_lt_add_right h n | case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
this : n ≤ (m + n).pred - k
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | rw [← Nat.add_sub_of_le this, pow_add, hb] | case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
this : n ≤ (m + n).pred - k
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
this : n ≤ (m + n).pred - k
⊢ a ^ k * (0 * b ^ ((m + n).pred - k - n)) * ↑((m + n).pred.choose k) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
this : n ≤ (m + n).pred - k
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | simp only [zero_mul, mul_zero] | case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
this : n ≤ (m + n).pred - k
⊢ a ^ k * (0 * b ^ ((m + n).pred - k - n)) * ↑((m + n).pred.choose k) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
this : n ≤ (m + n).pred - k
⊢ a ^ k * (0 * b ^ ((m + n).pred - k - n)) * ↑((m + n).pred.choose k) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | rw [Nat.le_sub_iff_add_le (Nat.le_of_lt_succ hk), add_comm] | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ n ≤ (m + n).pred - k | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ k + n ≤ (m + n).pred | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ n ≤ (m + n).pred - k
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | rw [Nat.le_pred_iff_lt (lt_of_le_of_lt (zero_le k) (Nat.lt_add_right n h))] | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ k + n ≤ (m + n).pred | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ k + n < m + n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ k + n ≤ (m + n).pred
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | exact Nat.add_lt_add_right h n | α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ k + n < m + n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : k < m
⊢ k + n < m + n
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | simp only [not_lt] at h | case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : ¬k < m
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : m ≤ k
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : ¬k < m
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | rw [← Nat.add_sub_of_le h, pow_add, ha] | case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : m ≤ k
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0 | case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : m ≤ k
⊢ 0 * a ^ (k - m) * b ^ ((m + n).pred - (m + (k - m))) * ↑((m + n).pred.choose (m + (k - m))) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : m ≤ k
⊢ a ^ k * b ^ ((m + n).pred - k) * ↑((m + n).pred.choose k) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | add_pow_add_pred_eq_zero_of_pow_eq_zero | [43, 1] | [59, 25] | simp only [zero_mul] | case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : m ≤ k
⊢ 0 * a ^ (k - m) * b ^ ((m + n).pred - (m + (k - m))) * ↑((m + n).pred.choose (m + (k - m))) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : CommSemiring α
a b : α
m n : ℕ
ha : a ^ m = 0
hb : b ^ n = 0
k : ℕ
hk : k < (m + n).pred + 1
h : m ≤ k
⊢ 0 * a ^ (k - m) * b ^ ((m + n).pred - (m + (k - m))) * ↑((m + n).pred.choose (m + (k - m))) = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | IsNilpotent.add | [61, 1] | [66, 59] | obtain ⟨m, ha⟩ := ha | α : Type u_1
inst✝ : CommSemiring α
a b : α
ha : IsNilpotent a
hb : IsNilpotent b
⊢ IsNilpotent (a + b) | case intro
α : Type u_1
inst✝ : CommSemiring α
a b : α
hb : IsNilpotent b
m : ℕ
ha : a ^ m = 0
⊢ IsNilpotent (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : CommSemiring α
a b : α
ha : IsNilpotent a
hb : IsNilpotent b
⊢ IsNilpotent (a + b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | IsNilpotent.add | [61, 1] | [66, 59] | obtain ⟨n, hb⟩ := hb | case intro
α : Type u_1
inst✝ : CommSemiring α
a b : α
hb : IsNilpotent b
m : ℕ
ha : a ^ m = 0
⊢ IsNilpotent (a + b) | case intro.intro
α : Type u_1
inst✝ : CommSemiring α
a b : α
m : ℕ
ha : a ^ m = 0
n : ℕ
hb : b ^ n = 0
⊢ IsNilpotent (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝ : CommSemiring α
a b : α
hb : IsNilpotent b
m : ℕ
ha : a ^ m = 0
⊢ IsNilpotent (a + b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/DividedPowers4.git | 18a13603ed0158d2880b6b0b0369d78417040a1d | DividedPowers/ForMathlib/MvPowerSeries/Substitutions.lean | IsNilpotent.add | [61, 1] | [66, 59] | exact ⟨_, add_pow_add_pred_eq_zero_of_pow_eq_zero ha hb⟩ | case intro.intro
α : Type u_1
inst✝ : CommSemiring α
a b : α
m : ℕ
ha : a ^ m = 0
n : ℕ
hb : b ^ n = 0
⊢ IsNilpotent (a + b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝ : CommSemiring α
a b : α
m : ℕ
ha : a ^ m = 0
n : ℕ
hb : b ^ n = 0
⊢ IsNilpotent (a + b)
TACTIC:
|
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