messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf it is not the case that A implies B, then B implies A. Show that this is true.... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $m \\le n$, prove that $n-(n-m) = m$.\n\nIf $m\\le n$, then by the definition o... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet $n$, $m$, and $k$ be natural numbers such that $k ≤ n$. Show that $n + m - k =... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $i < n$, then show that $n - 1 - i < n$.\n\nIf $i < n$, then $n - i > 1$, so $n... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $a$ and $b$ are natural numbers and $a$ is less than $b$, what is $b-a$? Prove ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $m$ and $n$ are natural numbers with $m ≤ n$, prove that $m + (n - m) = n$.\n\n... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet $n,k,m$ be natural numbers. If $m\\leq k$ and $n\\leq k-m$, show that $m+n\\le... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet \\( x \\), \\( y \\), and \\( k\\) be natural numbers such that \\( k \\leq y ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor all natural numbers $a,b,c$, prove that $a-b \\le c$ if and only if $a \\le c+... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that the following statement is true:\n\n$$ a - b ≤ c \\text{ if and only if... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:**\nIf $n ≤ k$, then prove that $k - m ≤ k - n$ if and only if $n ≤ m$.... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf \\(m+k\\le n\\) and \\(k>0\\), then \\(n-(m+k)<n-m\\).\n\nIf \\(k\\) is 0, then... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:** Prove that for any natural numbers $a$, $b$, and $c$, $a \\leq \\min... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for natural numbers $a$, $b$, and $c$, $a$ is less than the minimum of ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf a number, $a$, is less than or equal to another number, $b$, then the minimum o... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:**\nIf $b \\leq a$, prove that $\\min(a, b) = b$.\n\nLean4 statement to... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nThe minimum of 0 and any natural number is 0. Prove this.\n\nThe minimum of 0 and ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $a * b = a^b + b^a$, for all positive integer values of $a$ and $b$, then what ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:** Let \\(a,b,c\\) be natural numbers. Show that \\(max(a,b)\\leq c\\) ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Theorem:** For any two natural numbers $a$ and $b$, if $a \\leq b$, then $\\max(... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf \\( b \\le a \\), show that \\( \\max (a,b) = a \\).\n\nIf \\( b \\le a \\), th... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nWhat is the minimum of the successors of two natural numbers $x$ and $y$? Show th... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor all natural numbers $n$ and $m$, show that $n - m = n - \\min(n, m)$.\n\nThe n... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nWhat is $n - m + \\min(n, m)$? Show that it is equal to $n$.\n\nSince $\\min(n, m)... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor any natural numbers $a$ and $b$, show that $a - b + b = \\max(a, b)$.\n\nIf $a... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that for any natural numbers $n$, $m$, and $k$, $n * m * k = n * k * m$.\n\nN... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that for any natural numbers $a$, $b$, $c$, and $d$, $(a \\times b) \\times (... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for any natural number $n$, $n \\times 2 = n + n$.\n\nWe prove this by ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that $2*n = n+n$ for any natural number $n$.\n\nThe proof is by induction on ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve: For all natural numbers n and m, n * m = 0 if and only if n = 0 or m = 0.\n... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor any integers $n$ and $m$, show that $n\\cdot m \\neq 0$ if and only if $n\\neq... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet $n$ and $m$ be two nonzero natural numbers. Prove that $n \\times m$ is not ze... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $a\\le b$ and $c$ is a non-negative integer, then $ac\\le bc$.\n\nWe consider t... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $a \\leq b$ and $c$ is a nonnegative integer, then $a \\times c \\leq b \\times... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $a<c$ and $b\\le d$ and $b>0$, then $ab<cd$.\n\nLean4 statement to prove:\n```l... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $a\\le c$ and $b<d$, and $c>0$, then $ab<cd$.\n\nSince $a\\le c$, $ab\\le cb$. ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for all natural numbers $a$ and $b$, $(a + 1)(b + 1) = ab + a + b + 1$.... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question**: Prove that for all natural numbers a and b, $a^2 - b^2 = (a+b)(a-b)$... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that $m \\mod k + k \\cdot \\lfloor \\frac{m}{k} \\rfloor = m$ for natural nu... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for any natural number $n$, $n$ divided by $1$ is equal to $n$.\n\nIf w... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve the following: For all integers $n$, $n/0=0$.\n\nLean4 statement to prove:\n... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that $0/b=0$ for any natural number $b$.\n\nWe can use the fact that $0/b=0$ ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for any positive integer $k$ and any integers $x$ and $y$, if $k > 0$, ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:**\n\nIf $m \\leq kn$, prove that $\\frac{m}{k} \\leq n$.\n\nLean4 stat... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet $a$ and $b$ be positive integers such that $a \\geq b$. Show that $a / b = (a ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $a < b$, then $a / b = 0$. Show that this is true.\n\nSuppose $a < b$. By the d... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Theorem:** Let $k$ be a positive integer. Then $x / k < y$ if and only if $x < y... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $x$, $n$, and $p$ are natural numbers such that $n \\times p \\leq x$, then sho... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor all natural numbers $m$ and $n$, show that $m / n * n \\leq m$.\n\nIf $n = 0$,... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor any natural number $x$ and any natural number $z$ greater than $0$, prove that... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nState a theorem about the division of a sum by one of its addends. Prove it.\nShow... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet $m, n \\in \\mathbb{N}$. Prove that $m \\cdot n / m = n$.\n\nWe can prove this... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for all natural numbers \\(m\\) and \\(n\\), if \\(n>0\\), then\n$$\\fr... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve the following statement: If $n$ is a positive integer, then $n$ divided by $... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for all natural numbers $x$, $y$, and $z$ for which $y$ is nonzero, the... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nGiven natural numbers $x$, $y$, and $z$, with $z \\gt 0$, prove that $(x + y z) / ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that for any natural number $n>0$, we have $m\\times n/n=m$.\n\nWe have $m\\t... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nThe natural language version of the question and proof is:\n\n**Question:**\nProve... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet $m$, $n$, and $k$ be natural numbers such that $n>0$ and $m=kn$. Prove that $\... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf \\( m = nk\\) and \\(n>0\\), then prove \\(m/n=k\\).\n\nSince \\( m = nk \\), \... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $kn \\leq m < (k+1)n$, then $m/n = k$. Prove this.\n\nSuppose $m/n = q$ for som... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nLet $x$, $n$, and $p$ be natural numbers such that $x < n \\times p$. Prove that $... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $m$, $n$, and $k$ are positive integers, then $\\frac{m}{n/k} = \\frac{m}{n \\t... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor all natural numbers m, n, and k, where m is positive, show that $$\\frac{m\\cd... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:**\n\nLet $m$, $n$, and $k$ be natural numbers with $0 < m$. Prove that... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nSuppose $m$ and $n$ are natural numbers and $n$ is positive. Prove that $n$ times ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for any natural number n, n mod 2 is either 0 or 1.\n\nSuppose n is eve... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:**\n\nLet $a$ and $b$ be natural numbers such that $a\\bmod b < a$. Pro... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor any natural numbers $x$ and $z$, show that $(x+z) \\text{ mod } z = x \\text{ ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor any natural numbers $x$ and $z$, prove that $(x + z) \\text{ mod } x = z \\tex... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that for any natural numbers $x$, $y$, and $z$, $(x + y * z) \\mod y = x \\mo... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor all natural numbers $x$, $y$ and $z$, we have $(x + y * z) \\% z = x \\% z$.\n... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor any natural numbers $m$ and $n$, show that $(mn) \\mod m = 0$.\n\nLet $m$ and ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf $m$ and $n$ are natural numbers, what is $(mn) \\pmod{n}$? Show that it is 0.\n... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nFor all natural numbers $z$, $x$, and $y$, if $y \\neq 0$ and $z \\neq 0$,\n$$(z \... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:** Prove that for any natural numbers $x$, $y$, and $z$, we have $(x * ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nIf \\( n \\times k \\le x \\), then $$(x - n \\times k) \\text{ mod } n = x \\tex... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nWhat is the value of $a \\text{ mod } n \\text{ mod } n$? Show that it is $a \\tex... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nGiven three natural numbers $a$, $b$, and $n$, prove that\n$a * b \\text{ mod } n ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nThe remainder of $m$ divided by $n$ plus $k$ is equal to the remainder of $m + k$ ... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:** Prove that $(m + n) \\% k = (m + n \\% k) \\% k$ for all natural num... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that for any natural numbers $a$, $b$, and $n$, $(a + b) \\% n = ((a \\% n) +... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that m ^ (n + 1) = m * m ^ n.\n\nThis is just the definition of exponentiati... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:** Show that the infix operator `pow` in Lean4 is equivalent to the `^`... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that for any natural numbers $a$ and $b$, $a \\lll b = a \\times 2^b$.\n\nIf... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nWhat is the value of $1$ shifted left by $n$ for any natural number $n$? Show tha... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that $0^n=0$ for any natural number $n>0$.\n\nIf $n=0$, then the statement i... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that $1^n = 1$ for every natural number $n$.\n\nLean4 statement to prove:\n`... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that $a^1 = a$.\n\nThe proof is by induction on $a$. The base case is $a = 0... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that for any natural number $a$, $a^2 = a \\times a$.\n\nBy induction on $a$.... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that $a^{m + n} = a^m \\times a^n$.\n\nWe can prove it by induction on n.\n-... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nWhat is the value of $a^(m+n)$? Show that it is $a^n * a^m$.\n\nWe use the followi... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:** Prove the following property of exponentiation: $a^{(m \\times n)} =... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:** Prove that $a^{mn} = (a^n)^m$.\n\nLean4 statement to prove:\n```lean... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that for all natural numbers $a$, $m$, and $n$, $(a^m)^n = (a^n)^m$.\n\nWe ha... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**Question:**\nFor all natural numbers $a$, $b$, and $n$, prove that $(a \\times b... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\n**[Question]** Show that for any natural number $n \\neq 0$ and any integer $k$, w... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nShow that $n < 2^k$ if and only if $\\log_2(n) < k$.\n\nThe proof is by a chain of... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that $2^{log_2(n)} \\leq n$ for any positive integer $n$.\nWe know that $log... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that any positive integer $n$ is less than $2^{n+1}$.\n\nTo prove this, we u... |
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