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:\n**Theorem:** If we have a recursive definition of a predicate `motive` for a type ... |
[
{
"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:\nIn the following Lean4 code,\n\n```lean4\ntheorem rec_eq_recC : @Acc.rec = @Acc.re... |
[
{
"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 the notion of accessibility predicate `Acc.ndrec` is equivalent to the 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 the definition of `Acc.ndrecOn` is equivalent to `Acc.ndrecOnC`.\n\nThe... |
[
{
"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 `fixF` is equal to `fixFC`.\n\nSince the type of `fixF` and `fixFC` are... |
[
{
"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 the fixed point combinator is the same as the fixed point combinator def... |
[
{
"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 the following Lean4 expression is valid:\n\n```lean4\ntheorem $extName: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:\n**Theorem:**\nThe following two statements are equivalent:\n1. For all `x`, if `P`... |
[
{
"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 two points in the unit interval are equal.\n\nClearly, any two poin... |
[
{
"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 every pair of unit elements in a type is equal.\n\nThe proof is just 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:\nWhat is the definition of logical negation in Lean4? Show that it is $$\\lnot 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:\n**Theorem:** If not B, and if A implies B, then not A.\n\nLean4 statement to 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 a is equivalent to b, then the negation of a is equivalent to the negation of 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:\nFor any proposition $a$, prove that $¬¬¬a$ is equivalent to $¬a$.\n\nIn other word... |
[
{
"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 if the implication $a \\Rightarrow b$ is false, then $a$ is true.\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:\n**Question:** If $a$ does not imply $b$, then what can you conclude?\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:\nProve: $(a \\to \\lnot a) \\leftrightarrow \\lnot a$.\n\nLean4 statement to 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:\nThe definition of the equivalence of two propositions is that the first implies 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:\nProve that the following propositions are logically equivalent:\n1) a if and only ... |
[
{
"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$ if and only if $b$ is true. Show that this is true.\n\nThis 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:\nIf $a=b$, prove that $a\\leftrightarrow b$.\n\nFrom the hypothesis, $a=b$. 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:\nShow that if two statements, $a$ and $b$, are not equivalent, then $a$ is not equa... |
[
{
"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 $a \\Leftrightarrow b$ if and only if $a = b$.\n\nWe have $(a \\Leftrigh... |
[
{
"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 $p=q$ if and only if $p \\leftrightarrow q$.\n\nThe proof is straightfo... |
[
{
"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 an equality between a proposition and the constant `True`, the proposition 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 the proposition \"if P then Q\" is false, what can you conclude about P? Show 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$ and $b$ are both true, then $a$ is equivalent to $b$.\n\nThe proof is strai... |
[
{
"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 if $a$ and $b$ are both false, then $a$ is equivalent to $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:\n**Theorem:** If a is true, then (a if and only if b) is equivalent to b.\n\nLean4 ... |
[
{
"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**Problem:** Prove the equivalence $(b \\Leftrightarrow a) \\Leftrightarrow b$ for... |
[
{
"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\\leftrightarrow b$ is equivalent to $\\lnot b$ when $\\lnot a$ is gi... |
[
{
"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 propositions $a$ and $b$, $a ↔ b$ if and only if $¬b$.\n\nIf $a$ is false,... |
[
{
"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$ is true, then $a$ is equivalent to True.\n\nSuppose that $a$ is true. 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:\nGiven that ¬a is true, prove that a ↔ False.\n\nIf ¬a is true, then a must be fals... |
[
{
"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$ is true, then prove that not $a$ is equivalent to false.\n\nLet $a$ be a pr... |
[
{
"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$ if and only if $c$ and $b$ if and only if $d$, then $a$ if and only if $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:\nWhat is the truth table of $¬True$ and $False$?\n\n$\\begin{array}{|c|c|} \\hline ... |
[
{
"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 $\\lnot \\lnot False$ is equivalent to $True$.\n\nLean4 statement to pr... |
[
{
"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:\nA number is not equal to itself if and only if False.\n\nLean4 statement to 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:\nProve that $a=a$ if and only if True.\n\nSuppose $a=a$. Then $True$ is true, so $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 for any element `a` of type `α`, `a` is equal to itself 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:\nProve the following statement:\n\n$$\n\\lnot (P \\Leftrightarrow \\lnot P)\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:\nProve that the statement \"not a is equivalent to a\" is false.\n\nLean4 statement... |
[
{
"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 $T \\iff F \\iff F$.\n\n$T \\iff F$ is equivalent to both $T \\implies ... |
[
{
"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: $False \\leftrightarrow True \\leftrightarrow False$.\n\nThis is a simple 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:\nProve that if $True$ is equivalent to $False$, then $False$ is true.\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:\nAssume True equals False. Then the assumption leads to a contradiction, so the ass... |
[
{
"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 `True = False` is provable if `False` is provable.\n\nIf `False` is prov... |
[
{
"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 = b$ if and only if $b = a$.\n\nThis follows immediately from the sy... |
[
{
"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:** If \\(α\\) and \\(β\\) are propositions, and \\(α\\) is true, 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:\nLet \\(P\\), \\(Q\\), \\(R\\), and \\(S\\) be propositions. Suppose that \\(P \\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 \\(a\\to b\\) if and only if \\(b\\). Show that this is equivalent 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:\nProve that for any type α, the type α → True is equivalent to True.\n\nLean4 state... |
[
{
"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 equivalence $(False \\rightarrow a) \\iff True$.\n\nAssume $(False \\rig... |
[
{
"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 logical equivalence of \"if True, then α\"? Show that it is just α.\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 if $p$ is a proposition, then $p\\to p$ is always true.\n\nIf $p$ is tr... |
[
{
"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 by contradiction that if a → False then ¬a.\n\nAssume that a → False and 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 the equivalence of the following two statements:\n\n1. For all $a$ and $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:\nProve the equivalence of the two logical statements:\n1. If A then not B.\n2. If 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**Theorem**. If \\(a\\)\\( \\Leftrightarrow b \\), then \\( (a \\Rightarrow c) \\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:\n**Claim:** If $a \\implies (b \\iff c)$, then $a \\implies b$ is equivalent to $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:\n**(Question)**\nProve that $(a → b) ↔ (c → d)$ if and only if $a ↔ c$ and $c → (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**: If $a \\leftrightarrow c$ and $b \\leftrightarrow d$, show that $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 the following statement is true:\n\n$$a\\Rightarrow b \\Leftrightarrow ... |
[
{
"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 proof is a one-liner. It simply states that the conjunction of two proposition... |
[
{
"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 $f$ is a function from $a$ to $c$ and $g$ is a function from $b$... |
[
{
"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:\nIf \\(a \\Rightarrow b\\) and \\(a \\wedge c\\) then \\(b \\wedge c\\).\n\nSuppose... |
[
{
"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 propositions $a$, $b$, and $c$, if $a$ implies $b$, then $c$ 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 $a \\wedge b \\leftrightarrow c \\wedge d$ if $a \\leftrightarrow c$ an... |
[
{
"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 $a \\wedge b$ is equivalent to $b \\wedge a$.\n\nSince the conjunction ... |
[
{
"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:** If \\(a\\) implies \\(b\\) is equivalent to \\(a\\) implies \\(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**Proposition:** For all propositions \\(a\\), \\(b\\), and \\(c\\), if \\(a\\) is... |
[
{
"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 propositions $a$, $b$, and $c$, the equivalence $a \\leftrightarrow b$ can... |
[
{
"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 $a$ and $b$, prove that $a \\land b$ is equivalent to $a \\land c$ if and 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:\n**Question:** Prove that if $a \\land b = c$, then $a \\land b = a \\land c$.\n\nL... |
[
{
"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:** If \\(c\\), \\(a\\), and \\(b\\) are propositions, and \\(c\\rightar... |
[
{
"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**Associativity of Logical And**.\n\nShow that the following two logical expressio... |
[
{
"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 $a \\wedge (b \\wedge c)$ and $b \\wedge (a \\wedge c)$ are equivalent.\... |
[
{
"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 three propositions $a$, $b$, and $c$, $(a \\wedge b) \\wedge 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:\nIs the following statement true or false?\n\n$$(p \\wedge q) \\wedge r \\Leftright... |
[
{
"role": "system",
"content": "You are a Lean4 expert. Given a mathematical theorem statement in Lean4, provide a complete, compilable Lean4 proof."
},
{
"role": "user",
"content": "Natural language description:\nProve that the following equivalence holds:\n\n$$(a \\land b) \\land c \\land 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:\nProve the associative property of the conjunction operator.\n\nThe conjunction ope... |
[
{
"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 \\wedge b) \\wedge c \\Leftrightarrow (a \\wedge c) \\wedge b \\we... |
[
{
"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 logical equivalence of `(a ∧ b) ↔ a`? Prove it.\n\nIt is given that `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:\nWhat is the logical equivalence of `a ∧ b`? Show that it is `b`.\n\nWe can prove 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:\nProve the following statement. For all a and b, if a and b are true, then a is tr... |
[
{
"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 proposition $a$, $a \\wedge b$ is equivalent to $b$.\n\nAssume $... |
[
{
"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:** $(a \\land b) \\leftrightarrow a$ if and only if $a \\rightarrow b$\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**Theorem:** (a ∧ b) ↔ b if and only if b → a.\n\nLean4 statement to prove:\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:\nProve that the following two statements are equivalent:\n1. $p$ if and only if $p$... |
[
{
"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 $(p \\leftrightarrow q \\wedge p) \\leftrightarrow (p \\to q)$.\n\nBy 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:\nLet $a, b, c$ be propositions. Prove that the following are equivalent:\n\n$(a \\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 \\Leftrightarrow b$ and $a \\wedge c \\Leftrightarrow b \\wedge c$, prove 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:\nProve the following equivalence:\n$$a \\land a \\land b \\quad \\Leftrightarrow \\... |
[
{
"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 $(a ∧ b) ∧ b$ is logically equivalent to $a ∧ b$.\n\nLet $P$ and $Q$ be... |
[
{
"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 the first proposition is false, then the conjunction of the first and second pr... |
[
{
"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 $a$ is false, show that $a\\wedge b$ is false for any proposition $b$.\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:\nProposition 1.6. If $a$ is a proposition, then $\\lnot (a \\land \\lnot a)$.\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:\nShow that it is false that \\( ¬a ∧ a \\).\nIf this were true, by the principle 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 ¬(p ∧ ¬p).\n\nLean4 statement to prove:\n```lean4\ntheorem and_not_self... |
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