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 Decidable.not_imp_not [Decidable a] : (¬a → ¬b) ↔ (b → 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 sub_sub_self {n m : Nat} (h : m ≤ n) : n - (n - m) = m :="
},
{
"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 sub_add_comm {n m k : Nat} (h : k ≤ n) : n + m - k = n - k + m :="
},
{
"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_one_sub_lt (h : i < n) : n - 1 - i < n :="
},
{
"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 sub_lt_self {a b : Nat} (h₀ : 0 < a) (h₁ : a ≤ b) : 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 add_sub_cancel' {n m : Nat} (h : m ≤ n) : m + (n - m) = 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 add_le_of_le_sub_left {n k m : Nat} (H : m ≤ k) (h : n ≤ k - m) : m + n ≤ k :="
},
{
"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 le_sub_iff_add_le {x y k : Nat} (h : k ≤ y) : x ≤ y - k ↔ x + k ≤ y :="
},
{
"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 sub_le_iff_le_add {a b c : Nat} : a - b ≤ c ↔ a ≤ c + 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_le_iff_le_add' {a b c : Nat} : a - b ≤ c ↔ a ≤ b + 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 sub_le_sub_iff_left {n m k : Nat} (hn : n ≤ k) : k - m ≤ k - n ↔ n ≤ m :="
},
{
"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 sub_add_lt_sub {n m k : Nat} (h₁ : m + k ≤ n) (h₂ : 0 < k) :\n n - (m + k) < 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 le_min {a b c : Nat} : a ≤ min b c ↔ a ≤ b ∧ a ≤ c :="
},
{
"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 lt_min {a b c : Nat} : a < min b c ↔ a < b ∧ a < c :="
},
{
"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 min_eq_left {a b : Nat} (h : a ≤ b) : min a b = a :="
},
{
"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": "theorem min_eq_right {a b : Nat} (h : b ≤ a) : min a b = b :="
},
{
"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 zero_min (a : Nat) : min 0 a = 0 :="
},
{
"role": "assistant",
"content": "theorem zero... |
[
{
"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_zero (a : Nat) : min a 0 = 0 :="
},
{
"role": "assistant",
"content": "-- If 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 max_le {a b c : Nat} : max a b ≤ c ↔ a ≤ c ∧ b ≤ c :="
},
{
"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 max_eq_right {a b : Nat} (h : a ≤ b) : max a b = b :="
},
{
"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 max_eq_left {a b : Nat} (h : b ≤ a) : max a b = a :="
},
{
"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": "theorem min_succ_succ (x y : Nat) : min (succ x) (succ y) = succ (min x y) :="
},
{
"role": "assist... |
[
{
"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_eq_sub_min (n m : Nat) : n - m = n - min n m :="
},
{
"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 sub_add_min_cancel (n m : Nat) : n - m + min n m = n :="
},
{
"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 sub_add_eq_max {a b : Nat} : a - b + b = max a b :="
},
{
"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 mul_right_comm (n m k : Nat) : n * m * k = n * k * m :="
},
{
"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 mul_mul_mul_comm (a b c d : Nat) : (a * b) * (c * d) = (a * c) * (b * d) :="
},
{
"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_two (n : Nat) : n * 2 = n + n :="
},
{
"role": "assistant",
"content": "theorem mul... |
[
{
"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 two_mul (n : Nat) : 2 * n = n + n :="
},
{
"role": "assistant",
"content": "theorem two... |
[
{
"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_eq_zero {n m : Nat} : n * m = 0 ↔ n = 0 ∨ m = 0 :="
},
{
"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": "theorem mul_ne_zero_iff : n * m ≠ 0 ↔ n ≠ 0 ∧ m ≠ 0 :="
},
{
"role": "assistant",
"content": "t... |
[
{
"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_ne_zero (n0 : n ≠ 0) (m0 : m ≠ 0) : n * m ≠ 0 :="
},
{
"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": "theorem mul_le_mul_of_nonneg_left {a b c : Nat} (h₁ : a ≤ b) : c * a ≤ c * 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 mul_le_mul_of_nonneg_right {a b c : Nat} (h₁ : a ≤ b) : a * c ≤ b * c :="
},
{
"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_lt_mul (hac : a < c) (hbd : b ≤ d) (pos_b : 0 < b) : a * b < c * d :="
},
{
"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_lt_mul' (h1 : a ≤ c) (h2 : b < d) (h3 : 0 < c) : a * b < c * d :="
},
{
"role": "assist... |
[
{
"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 succ_mul_succ_eq (a b : Nat) : succ a * succ b = a * b + a + b + 1 :="
},
{
"role": "assist... |
[
{
"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_self_sub_mul_self_eq (a b : Nat) : a * a - b * b = (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 mod_add_div (m k : Nat) : m % k + k * (m / k) = m :="
},
{
"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": "theorem div_one (n : Nat) : n / 1 = n :="
},
{
"role": "assistant",
"content": "theorem div_one... |
[
{
"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_zero (n : Nat) : n / 0 = 0 :="
},
{
"role": "assistant",
"content": "theorem div_ze... |
[
{
"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_div (b : Nat) : 0 / b = 0 :="
},
{
"role": "assistant",
"content": "theorem zero_d... |
[
{
"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 le_div_iff_mul_le (k0 : 0 < k) : x ≤ y / k ↔ x * k ≤ y :="
},
{
"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 div_le_of_le_mul {m n : Nat} : ∀ {k}, m ≤ k * n → m / k ≤ 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 div_eq_sub_div (h₁ : 0 < b) (h₂ : b ≤ a) : a / b = (a - b) / b + 1 :="
},
{
"role": "assist... |
[
{
"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_eq_of_lt (h₀ : a < b) : a / b = 0 :="
},
{
"role": "assistant",
"content": "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 div_lt_iff_lt_mul (Hk : 0 < k) : x / k < y ↔ x < y * k :="
},
{
"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 sub_mul_div (x n p : Nat) (h₁ : n*p ≤ x) : (x - n*p) / n = x / n - p :="
},
{
"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 div_mul_le_self : ∀ (m n : Nat), m / n * n ≤ m"
},
{
"role": "assistant",
"content": "t... |
[
{
"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_div_right (x : Nat) {z : Nat} (H : 0 < z) : (x + z) / z = succ (x / z) :="
},
{
"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 add_div_left (x : Nat) {z : Nat} (H : 0 < z) : (z + x) / z = succ (x / z) :="
},
{
"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_div_right (n : Nat) {m : Nat} (H : 0 < m) : m * n / m = 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 mul_div_left (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m :="
},
{
"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 div_self (H : 0 < n) : n / n = 1 :="
},
{
"role": "assistant",
"content": "theorem div_... |
[
{
"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_mul_div_left (x z : Nat) {y : Nat} (H : 0 < y) : (x + y * z) / y = x / y + z :="
},
{
"... |
[
{
"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_mul_div_right (x y : Nat) {z : Nat} (H : 0 < z) : (x + y * z) / z = x / z + 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 mul_div_cancel (m : Nat) {n : Nat} (H : 0 < n) : m * n / n = m :="
},
{
"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 mul_div_cancel_left (m : Nat) {n : Nat} (H : 0 < n) : n * m / n = m :="
},
{
"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 div_eq_of_eq_mul_left (H1 : 0 < n) (H2 : m = k * n) : m / n = k :="
},
{
"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 div_eq_of_eq_mul_right (H1 : 0 < n) (H2 : m = n * k) : m / n = k :="
},
{
"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 div_eq_of_lt_le (lo : k * n ≤ m) (hi : m < succ k * n) : m / n = k :="
},
{
"role": "assist... |
[
{
"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_sub_div (x n p : Nat) (h₁ : x < n*p) : (n * p - succ x) / n = p - succ (x / 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 div_div_eq_div_mul (m n k : Nat) : m / n / k = m / (n * k) :="
},
{
"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 mul_div_mul_left {m : Nat} (n k : Nat) (H : 0 < m) :\n m * n / (m * k) = n / k :="
},
{
... |
[
{
"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_mul_right {m : Nat} (n k : Nat) (H : 0 < m) :\n n * m / (k * m) = n / k :="
},
{
... |
[
{
"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_le (m n : Nat) : n * (m / n) ≤ m :="
},
{
"role": "assistant",
"content": "theo... |
[
{
"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 mod_two_eq_zero_or_one (n : Nat) : n % 2 = 0 ∨ n % 2 = 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 le_of_mod_lt {a b : Nat} (h : a % b < a) : 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 add_mod_right (x z : Nat) : (x + z) % z = x % z :="
},
{
"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 add_mod_left (x z : Nat) : (x + z) % x = z % x :="
},
{
"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 add_mul_mod_self_left (x y z : Nat) : (x + y * z) % y = x % y :="
},
{
"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_mul_mod_self_right (x y z : Nat) : (x + y * z) % z = x % z :="
},
{
"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 mul_mod_right (m n : Nat) : (m * n) % m = 0 :="
},
{
"role": "assistant",
"content": "t... |
[
{
"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_mod_left (m n : Nat) : (m * n) % n = 0 :="
},
{
"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 mul_mod_mul_left (z x y : Nat) : (z * x) % (z * y) = z * (x % y) :="
},
{
"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 mul_mod_mul_right (z x y : Nat) : (x * z) % (y * z) = (x % y) * z :="
},
{
"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 sub_mul_mod {x k n : Nat} (h₁ : n*k ≤ x) : (x - n*k) % n = x % 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 mod_mod (a n : Nat) : (a % n) % n = a % n :="
},
{
"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 mul_mod (a b n : Nat) : a * b % n = (a % n) * (b % n) % 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 mod_add_mod (m n k : Nat) : (m % n + k) % n = (m + k) % 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 add_mod_mod (m n k : Nat) : (m + n % k) % k = (m + n) % k :="
},
{
"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_mod (a b n : Nat) : (a + b) % n = ((a % n) + (b % n)) % 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 pow_succ' {m n : Nat} : m ^ n.succ = m * m ^ n :="
},
{
"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 pow_eq {m n : Nat} : m.pow n = m ^ n :="
},
{
"role": "assistant",
"content": "-- Show ... |
[
{
"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 shiftLeft_eq (a b : Nat) : a <<< b = a * 2 ^ b :="
},
{
"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 one_shiftLeft (n : Nat) : 1 <<< n = 2 ^ n :="
},
{
"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 {n : Nat} (H : 0 < n) : 0 ^ n = 0 :="
},
{
"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 one_pow (n : Nat) : 1 ^ n = 1 :="
},
{
"role": "assistant",
"content": "theorem one_pow... |
[
{
"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_one (a : Nat) : a ^ 1 = a :="
},
{
"role": "assistant",
"content": "-- Prove that 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 pow_two (a : Nat) : a ^ 2 = a * a :="
},
{
"role": "assistant",
"content": "theorem pow... |
[
{
"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_add (a m n : Nat) : a ^ (m + n) = a ^ m * a ^ n :="
},
{
"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": "theorem pow_add' (a m n : Nat) : a ^ (m + n) = a ^ n * a ^ m :="
},
{
"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 pow_mul (a m n : Nat) : a ^ (m * n) = (a ^ m) ^ 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": "theorem pow_mul' (a m n : Nat) : a ^ (m * n) = (a ^ n) ^ m :="
},
{
"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 pow_right_comm (a m n : Nat) : (a ^ m) ^ n = (a ^ n) ^ m :="
},
{
"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 mul_pow (a b n : Nat) : (a * b) ^ n = a ^ n * b ^ n :="
},
{
"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": "theorem le_log2 (h : n ≠ 0) : k ≤ n.log2 ↔ 2 ^ k ≤ n :="
},
{
"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 log2_lt (h : n ≠ 0) : n.log2 < k ↔ n < 2 ^ k :="
},
{
"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 log2_self_le (h : n ≠ 0) : 2 ^ n.log2 ≤ n :="
},
{
"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 lt_log2_self (h : n ≠ 0) : n < 2 ^ (n.log2 + 1) :="
},
{
"role": "assistant",
"content"... |
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