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 IsLowerSet.compl (hs : IsLowerSet s) : IsUpperSet sᶜ :="
},
{
"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": "theorem nonempty_invertible_iff_isUnit [Monoid α] (a : α) : Nonempty (Invertible a) ↔ IsUnit 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 invOf_neg [Monoid α] [HasDistribNeg α] (a : α) [Invertible a] [Invertible (-a)] :\n ⅟ (-a) = -⅟ ... |
[
{
"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 one_sub_invOf_two [Ring α] [Invertible (2 : α)] : 1 - (⅟ 2 : α) = ⅟ 2 :="
},
{
"role": "ass... |
[
{
"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_two_add_invOf_two [NonAssocSemiring α] [Invertible (2 : α)] :\n (⅟ 2 : α) + (⅟ 2 : α) = 1 ... |
[
{
"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 Commute.invOf_right [Monoid α] {a b : α} [Invertible b] (h : Commute a b) :\n Commute a (⅟ 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 Commute.invOf_left [Monoid α] {a b : α} [Invertible b] (h : Commute b a) :\n Commute (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 commute_invOf {M : Type*} [One M] [Mul M] (m : M) [Invertible m] : Commute m (⅟ m) :="
},
{
... |
[
{
"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 pos_of_invertible_cast [Semiring α] [Nontrivial α] (n : ℕ) [Invertible (n : α)] : 0 < n :="
},
... |
[
{
"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 Ring.inverse_invertible (x : α) [Invertible x] : Ring.inverse x = ⅟ x :="
},
{
"role": "ass... |
[
{
"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 div_mul_cancel_of_invertible (a b : α) [Invertible b] : a / b * b = a :="
},
{
"role": "ass... |
[
{
"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_div_cancel_of_invertible (a b : α) [Invertible b] : a * b / b = a :="
},
{
"role": "ass... |
[
{
"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 div_self_of_invertible (a : α) [Invertible a] : 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_div (a b : α) [Invertible a] [Invertible b] [Invertible (a / b)] :\n ⅟ (a / 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 map_invOf {R : Type*} {S : Type*} {F : Type*} [MulOneClass R] [Monoid S]\n [MonoidHomClass F R S... |
[
{
"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 image.fac : factorThruImage f ≫ image.ι f = f :="
},
{
"role": "assistant",
"content": ... |
[
{
"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 image.lift_fac (F' : MonoFactorisation f) : image.lift F' ≫ F'.m = image.ι f :="
},
{
"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 commutatorElement_def {G : Type*} [Group G] (g₁ g₂ : G) :\n ⁅g₁, g₂⁆ = g₁ * g₂ * g₁⁻¹ * g₂⁻¹ :="... |
[
{
"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 zero_pow : ∀ {n : ℕ}, 0 < n → (0 : M) ^ n = 0"
},
{
"role": "assistant",
"content": "--... |
[
{
"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 zero_pow' : ∀ n : ℕ, n ≠ 0 → (0 : M) ^ n = 0"
},
{
"role": "assistant",
"content": "the... |
[
{
"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 zero_pow_eq (n : ℕ) : (0 : M) ^ n = if n = 0 then 1 else 0 :="
},
{
"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 pow_eq_zero_of_le {x : M} {n m : ℕ} (hn : n ≤ m) (hx : x ^ n = 0) : x ^ m = 0 :="
},
{
"rol... |
[
{
"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_eq_zero [NoZeroDivisors M] {x : M} {n : ℕ} (H : x ^ n = 0) : x = 0 :="
},
{
"role": "as... |
[
{
"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_eq_zero_iff [NoZeroDivisors M] {a : M} {n : ℕ} (hn : 0 < n) : a ^ n = 0 ↔ a = 0 :="
},
{
... |
[
{
"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_eq_zero_iff' [NoZeroDivisors M] [Nontrivial M] {a : M} {n : ℕ} :\n a ^ n = 0 ↔ a = 0 ∧ n ≠ 0... |
[
{
"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_ne_zero_iff [NoZeroDivisors M] {a : M} {n : ℕ} (hn : 0 < n) : a ^ n ≠ 0 ↔ a ≠ 0 :="
},
{
... |
[
{
"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 ne_zero_pow {a : M} {n : ℕ} (hn : n ≠ 0) : a ^ n ≠ 0 → a ≠ 0 :="
},
{
"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 pow_ne_zero [NoZeroDivisors M] {a : M} (n : ℕ) (h : a ≠ 0) : a ^ n ≠ 0 :="
},
{
"role": "as... |
[
{
"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_eq_zero_iff [NoZeroDivisors M] {a : M} : a ^ 2 = 0 ↔ a = 0 :="
},
{
"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 zero_pow_eq_zero [Nontrivial M] {n : ℕ} : (0 : M) ^ n = 0 ↔ 0 < n :="
},
{
"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 Ring.inverse_pow (r : M) : ∀ n : ℕ, Ring.inverse r ^ n = Ring.inverse (r ^ n)"
},
{
"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 coe_powMonoidWithZeroHom : (powMonoidWithZeroHom hn : M → M) = fun x ↦ (x^n : M) :="
},
{
"... |
[
{
"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 powMonoidWithZeroHom_apply (a : M) : powMonoidWithZeroHom hn a = a ^ n :="
},
{
"role": "as... |
[
{
"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_dvd_pow_iff [CancelCommMonoidWithZero R] {x : R} {n m : ℕ} (h0 : x ≠ 0)\n (h1 : ¬IsUnit 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 RingHom.map_pow (f : R →+* S) (a) : ∀ n : ℕ, f (a ^ n) = f a ^ n :="
},
{
"role": "assistan... |
[
{
"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 min_pow_dvd_add {n m : ℕ} {a b c : R} (ha : c ^ n ∣ a) (hb : c ^ m ∣ b) :\n c ^ min n m ∣ 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 add_sq (a b : R) : (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2 :="
},
{
"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 add_sq' (a b : R) : (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b :="
},
{
"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 neg_one_pow_eq_or : ∀ n : ℕ, (-1 : R) ^ n = 1 ∨ (-1 : R) ^ n = -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 neg_pow (a : R) (n : ℕ) : (-a) ^ n = (-1) ^ n * a ^ 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": "theorem neg_pow' (a : R) (n : ℕ) : (-a) ^ n = a ^ n * (-1) ^ n :="
},
{
"role": "assistant",
"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 neg_pow_bit0 (a : R) (n : ℕ) : (-a) ^ bit0 n = a ^ bit0 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": "theorem neg_pow_bit1 (a : R) (n : ℕ) : (-a) ^ bit1 n = -a ^ bit1 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": "theorem neg_sq (a : R) : (-a) ^ 2 = a ^ 2 :="
},
{
"role": "assistant",
"content": "theorem neg... |
[
{
"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 neg_one_sq : (-1 : R) ^ 2 = 1 :="
},
{
"role": "assistant",
"content": "-- This theorem... |
[
{
"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 Commute.sq_sub_sq (h : Commute a b) : a ^ 2 - b ^ 2 = (a + b) * (a - b) :="
},
{
"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 neg_one_pow_mul_eq_zero_iff {n : ℕ} {r : R} : (-1) ^ n * r = 0 ↔ r = 0 :="
},
{
"role": "as... |
[
{
"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_neg_one_pow_eq_zero_iff {n : ℕ} {r : R} : r * (-1) ^ n = 0 ↔ r = 0 :="
},
{
"role": "as... |
[
{
"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 Commute.sq_eq_sq_iff_eq_or_eq_neg (h : Commute a b) :\n a ^ 2 = b ^ 2 ↔ a = 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 sq_eq_one_iff : a ^ 2 = 1 ↔ a = 1 ∨ a = -1 :="
},
{
"role": "assistant",
"content": "th... |
[
{
"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_ne_one_iff : a ^ 2 ≠ 1 ↔ a ≠ 1 ∧ a ≠ -1 :="
},
{
"role": "assistant",
"content": "th... |
[
{
"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_sub_sq (a b : R) : a ^ 2 - b ^ 2 = (a + b) * (a - b) :="
},
{
"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 sub_sq (a b : R) : (a - b) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2 :="
},
{
"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 sub_sq' (a b : R) : (a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * a * b :="
},
{
"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 sq_eq_sq_iff_eq_or_eq_neg : a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b :="
},
{
"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 eq_or_eq_neg_of_sq_eq_sq (a b : R) : a ^ 2 = b ^ 2 → a = b ∨ a = -b :="
},
{
"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 sq_eq_sq_iff_eq_or_eq_neg {a b : Rˣ} : a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b :="
},
{
"role": "ass... |
[
{
"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 eq_or_eq_neg_of_sq_eq_sq (a b : Rˣ) (h : a ^ 2 = b ^ 2) : a = b ∨ a = -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 IsConj.refl (a : α) : IsConj a a :="
},
{
"role": "assistant",
"content": "theorem IsCo... |
[
{
"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 IsConj.symm {a b : α} : IsConj a b → IsConj b a"
},
{
"role": "assistant",
"content": "... |
[
{
"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 isConj_comm {g h : α} : IsConj g h ↔ IsConj h g :="
},
{
"role": "assistant",
"content"... |
[
{
"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 IsConj.trans {a b c : α} : IsConj a b → IsConj b c → IsConj a c"
},
{
"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 isConj_iff_eq {α : Type*} [CommMonoid α] {a b : α} : IsConj a b ↔ a = b :="
},
{
"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 MonoidHom.map_isConj (f : α →* β) {a b : α} : IsConj a b → IsConj (f a) (f 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 isConj_one_right {a : α} : IsConj 1 a ↔ a = 1 :="
},
{
"role": "assistant",
"content": ... |
[
{
"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 isConj_one_left {a : α} : IsConj a 1 ↔ a = 1 :="
},
{
"role": "assistant",
"content": "... |
[
{
"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 isConj_iff {a b : α} : IsConj a b ↔ ∃ c : α, c * a * c⁻¹ = b :="
},
{
"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 conj_inv {a b : α} : (b * a * b⁻¹)⁻¹ = b * a⁻¹ * b⁻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": "theorem conj_mul {a b c : α} : b * a * b⁻¹ * (b * c * b⁻¹) = b * (a * c) * b⁻¹ :="
},
{
"role": "as... |
[
{
"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 conj_pow {i : ℕ} {a b : α} : (a * b * a⁻¹) ^ i = a * b ^ i * a⁻¹ :="
},
{
"role": "assistan... |
[
{
"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 conj_zpow {i : ℤ} {a b : α} : (a * b * a⁻¹) ^ i = a * b ^ i * a⁻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 conj_injective {x : α} : Function.Injective fun g : α => x * g * x⁻¹ :="
},
{
"role": "assi... |
[
{
"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 isConj_iff₀ [GroupWithZero α] {a b : α} : IsConj a b ↔ ∃ c : α, c ≠ 0 ∧ c * a * c⁻¹ = 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 mk_eq_mk_iff_isConj {a b : α} : ConjClasses.mk a = ConjClasses.mk b ↔ IsConj 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 quotient_mk_eq_mk (a : α) : ⟦a⟧ = ConjClasses.mk a :="
},
{
"role": "assistant",
"conte... |
[
{
"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 quot_mk_eq_mk (a : α) : Quot.mk Setoid.r a = ConjClasses.mk a :="
},
{
"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 forall_isConj {p : ConjClasses α → Prop} : (∀ a, p a) ↔ ∀ a, p (ConjClasses.mk 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 mk_surjective : Function.Surjective (@ConjClasses.mk α _) :="
},
{
"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 one_eq_mk_one : (1 : ConjClasses α) = ConjClasses.mk 1 :="
},
{
"role": "assistant",
"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 exists_rep (a : ConjClasses α) : ∃ a0 : α, ConjClasses.mk a0 = a :="
},
{
"role": "assistan... |
[
{
"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 map_surjective {f : α →* β} (hf : Function.Surjective f) :\n Function.Surjective (ConjClasses.ma... |
[
{
"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 mk_injective : Function.Injective (@ConjClasses.mk α _) :="
},
{
"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 mk_bijective : Function.Bijective (@ConjClasses.mk α _) :="
},
{
"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 mem_conjugatesOf_self {a : α} : a ∈ conjugatesOf a :="
},
{
"role": "assistant",
"conte... |
[
{
"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 IsConj.conjugatesOf_eq {a b : α} (ab : IsConj a b) : conjugatesOf a = conjugatesOf 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 isConj_iff_conjugatesOf_eq {a b : α} : IsConj a b ↔ conjugatesOf a = conjugatesOf 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 mem_carrier_mk {a : α} : a ∈ carrier (ConjClasses.mk a) :="
},
{
"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 mem_carrier_iff_mk_eq {a : α} {b : ConjClasses α} :\n a ∈ carrier b ↔ ConjClasses.mk 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 carrier_eq_preimage_mk {a : ConjClasses α} : a.carrier = ConjClasses.mk ⁻¹' {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 toDual_one [One α] : toDual (1 : α) = 1 :="
},
{
"role": "assistant",
"content": "theor... |
[
{
"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 ofDual_one [One α] : (ofDual 1 : α) = 1 :="
},
{
"role": "assistant",
"content": "-- Le... |
[
{
"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 toDual_mul [Mul α] (a b : α) : toDual (a * b) = toDual a * toDual b :="
},
{
"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 ofDual_mul [Mul α] (a b : αᵒᵈ) : ofDual (a * b) = ofDual a * ofDual b :="
},
{
"role": "ass... |
[
{
"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 toDual_inv [Inv α] (a : α) : toDual a⁻¹ = (toDual a)⁻¹ :="
},
{
"role": "assistant",
"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 ofDual_inv [Inv α] (a : αᵒᵈ) : ofDual a⁻¹ = (ofDual 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 toDual_div [Div α] (a b : α) : toDual (a / b) = toDual a / toDual b :="
},
{
"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 ofDual_div [Div α] (a b : αᵒᵈ) : ofDual (a / b) = ofDual a / ofDual b :="
},
{
"role": "ass... |
[
{
"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 toDual_pow [Pow α β] (a : α) (b : β) : toDual (a ^ b) = toDual a ^ b :="
},
{
"role": "assi... |
[
{
"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 ofDual_pow [Pow α β] (a : αᵒᵈ) (b : β) : ofDual (a ^ b) = ofDual a ^ b :="
},
{
"role": "as... |
[
{
"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_toDual [Pow α β] (a : α) (b : β) : a ^ toDual b = a ^ b :="
},
{
"role": "assistant",
... |
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