messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem free_map_coe {α β : Type u} {f : α → β} (x : FreeAbelianGroup α) :\n (free.map f) x = f <$> x :=... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_mul_self' [Mul α] [One α] (a : α) {_ : Invertible a} : ⅟ a * a = 1 :="
},
{
"role": "... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_mul_self [Mul α] [One α] (a : α) [Invertible a] : ⅟ a * a = 1 :="
},
{
"role": "assis... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_invOf_self' [Mul α] [One α] (a : α) {_ : Invertible a} : a * ⅟ a = 1 :="
},
{
"role": "... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_invOf_self [Mul α] [One α] (a : α) [Invertible a] : a * ⅟ a = 1 :="
},
{
"role": "assis... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_mul_self_assoc' [Monoid α] (a b : α) {_ : Invertible a} : ⅟ a * (a * b) = b :="
},
{
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_mul_self_assoc [Monoid α] (a b : α) [Invertible a] : ⅟ a * (a * b) = b :="
},
{
"role... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_invOf_self_assoc' [Monoid α] (a b : α) {_ : Invertible a} : a * (⅟ a * b) = b :="
},
{
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_invOf_self_assoc [Monoid α] (a b : α) [Invertible a] : a * (⅟ a * b) = b :="
},
{
"role... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_invOf_mul_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * ⅟ b * b = a :="
},
{
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_invOf_mul_self_cancel [Monoid α] (a b : α) [Invertible b] : a * ⅟ b * b = a :="
},
{
"r... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_mul_invOf_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * b * ⅟ b = a :="
},
{
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_mul_invOf_self_cancel [Monoid α] (a b : α) [Invertible b] : a * b * ⅟ b = a :="
},
{
"r... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_eq_right_inv [Monoid α] {a b : α} [Invertible a] (hac : a * b = 1) : ⅟ a = b :="
},
{
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_eq_left_inv [Monoid α] {a b : α} [Invertible a] (hac : b * a = 1) : ⅟ a = b :="
},
{
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invertible_unique {α : Type u} [Monoid α] (a b : α) [Invertible a] [Invertible b]\n (h : a = b) ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem Invertible.congr [Monoid α] (a b : α) [Invertible a] [Invertible b] (h : a = b) :\n ⅟a = ⅟b :="
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_eq_group_inv [Group α] (a : α) [Invertible a] : ⅟ a = a⁻¹ :="
},
{
"role": "assistant... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_one' [Monoid α] {_ : Invertible (1 : α)} : ⅟ (1 : α) = 1 :="
},
{
"role": "assistant"... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_one [Monoid α] [Invertible (1 : α)] : ⅟ (1 : α) = 1 :="
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_invOf [Monoid α] (a : α) [Invertible a] [Invertible (⅟ a)] : ⅟ (⅟ a) = a :="
},
{
"ro... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_inj [Monoid α] {a b : α} [Invertible a] [Invertible b] : ⅟ a = ⅟ b ↔ a = b :="
},
{
"... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_mul [Monoid α] (a b : α) [Invertible a] [Invertible b] [Invertible (a * b)] :\n ⅟ (a * b) ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_right_inj_of_invertible [Monoid α] (c : α) [Invertible c] :\n a * c = b * c ↔ a = b :="
},... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_left_inj_of_invertible [Monoid α] (c : α) [Invertible c] :\n c * a = c * b ↔ a = b :="
},
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_mul_eq_iff_eq_mul_left [Monoid α] [Invertible (c : α)] :\n ⅟c * a = b ↔ a = c * b :="
},... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_left_eq_iff_eq_invOf_mul [Monoid α] [Invertible (c : α)] :\n c * a = b ↔ a = ⅟c * b :="
},... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_invOf_eq_iff_eq_mul_right [Monoid α] [Invertible (c : α)] :\n a * ⅟c = b ↔ a = b * c :="
}... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem mul_right_eq_iff_eq_mul_invOf [Monoid α] [Invertible (c : α)] :\n a * c = b ↔ a = b * ⅟c :="
}... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_def (n : ℤ) : n.negOnePow = (-1 : ℤˣ) ^ n :="
},
{
"role": "assistant",
"conten... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_add (n₁ n₂ : ℤ) :\n (n₁ + n₂).negOnePow = n₁.negOnePow * n₂.negOnePow :="
},
{
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "@[simp]\nlemma negOnePow_zero : negOnePow 0 = 1 :="
},
{
"role": "assistant",
"content": "@[sim... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "@[simp]\nlemma negOnePow_one : negOnePow 1 = -1 :="
},
{
"role": "assistant",
"content": "@[sim... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_succ (n : ℤ) : (n + 1).negOnePow = - n.negOnePow :="
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_even (n : ℤ) (hn : Even n) : n.negOnePow = 1 :="
},
{
"role": "assistant",
"con... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "@[simp]\nlemma negOnePow_two_mul (n : ℤ) : (2 * n).negOnePow = 1 :="
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_odd (n : ℤ) (hn : Odd n) : n.negOnePow = -1 :="
},
{
"role": "assistant",
"cont... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_eq_one_iff (n : ℤ) : n.negOnePow = 1 ↔ Even n :="
},
{
"role": "assistant",
"co... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_eq_neg_one_iff (n : ℤ) : n.negOnePow = -1 ↔ Odd n :="
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "@[simp]\nlemma negOnePow_neg (n : ℤ) : (-n).negOnePow = n.negOnePow :="
},
{
"role": "assistant",
... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_sub (n₁ n₂ : ℤ) :\n (n₁ - n₂).negOnePow = n₁.negOnePow * n₂.negOnePow :="
},
{
"... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "lemma negOnePow_eq_iff (n₁ n₂ : ℤ) :\n n₁.negOnePow = n₂.negOnePow ↔ Even (n₁ - n₂) :="
},
{
"ro... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "@[simp]\nlemma negOnePow_mul_self (n : ℤ) : n.negOnePow * n.negOnePow = 1 :="
},
{
"role": "assista... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem sq_add_sq_mul_sq_add_sq :\n (x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 + y₂ ^ 2) = (x₁ * y₁ - x₂ * y₂) ^ 2 + (x... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem sq_add_mul_sq_mul_sq_add_mul_sq :\n (x₁ ^ 2 + n * x₂ ^ 2) * (y₁ ^ 2 + n * y₂ ^ 2) =\n (x₁ * y... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem pow_four_add_four_mul_pow_four :\n a ^ 4 + 4 * b ^ 4 = ((a - b) ^ 2 + b ^ 2) * ((a + b) ^ 2 + b ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem pow_four_add_four_mul_pow_four' :\n a ^ 4 + 4 * b ^ 4 = (a ^ 2 - 2 * a * b + 2 * b ^ 2) * (a ^ 2... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem sum_four_sq_mul_sum_four_sq :\n (x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem sum_eight_sq_mul_sum_eight_sq :\n (x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem isUnit_of_invertible [Monoid α] (a : α) [Invertible a] : IsUnit a :="
},
{
"role": "assista... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem invOf_units [Monoid α] (u : αˣ) [Invertible (u : α)] : ⅟ (u : α) = ↑u⁻¹ :="
},
{
"role": "a... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Your goal is to provide a complete, commented proof in Lean4 for the given mathematical statement."
},
{
"role": "user",
"content": "theorem IsUnit.nonempty_invertible [Monoid α] {a : α} (h : IsUnit a) : Nonempty (Invertible a) :="
},
{... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.