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 recC_intro {motive : (a : α) → Acc r a → Sort v}\n (intro : (x : α) → (h : ∀ (y : α), r y x → Ac... |
[
{
"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 rec_eq_recC : @Acc.rec = @Acc.recC :="
},
{
"role": "assistant",
"content": "theorem re... |
[
{
"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 ndrec_eq_ndrecC : @Acc.ndrec = @Acc.ndrecC :="
},
{
"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 ndrecOn_eq_ndrecOnC : @Acc.ndrecOn = @Acc.ndrecOnC :="
},
{
"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 fixF_eq_fixFC : @fixF = @fixFC :="
},
{
"role": "assistant",
"content": "theorem fixF_e... |
[
{
"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 fix_eq_fixC : @fix = @fixC :="
},
{
"role": "assistant",
"content": "theorem fix_eq_fix... |
[
{
"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 $extName:ident : ext_type% $flat $struct:ident :="
},
{
"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 $extIffName:ident : ext_iff_type% $flat $struct:ident :="
},
{
"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 PUnit.ext (x y : PUnit) : x = y :="
},
{
"role": "assistant",
"content": "-- This 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 Unit.ext (x y : Unit) : x = y :="
},
{
"role": "assistant",
"content": "-- Prove that e... |
[
{
"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 Not.intro {a : Prop} (h : a → False) : ¬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 Not.imp {a b : Prop} (H2 : ¬b) (H1 : 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 not_congr (h : a ↔ b) : ¬a ↔ ¬b :="
},
{
"role": "assistant",
"content": "-- If a is eq... |
[
{
"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 not_not_not : ¬¬¬a ↔ ¬a :="
},
{
"role": "assistant",
"content": "theorem not_not_not :... |
[
{
"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 not_not_of_not_imp : ¬(a → b) → ¬¬a :="
},
{
"role": "assistant",
"content": "theorem 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 not_of_not_imp {a : Prop} : ¬(a → b) → ¬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 imp_not_self : (a → ¬a) ↔ ¬a :="
},
{
"role": "assistant",
"content": "theorem imp_not_... |
[
{
"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 iff_def : (a ↔ b) ↔ (a → b) ∧ (b → a) :="
},
{
"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 iff_def' : (a ↔ b) ↔ (b → a) ∧ (a → b) :="
},
{
"role": "assistant",
"content": "theore... |
[
{
"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.to_iff : a = b → (a ↔ b)"
},
{
"role": "assistant",
"content": "theorem Eq.to_iff : ... |
[
{
"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 iff_of_eq : a = b → (a ↔ b) :="
},
{
"role": "assistant",
"content": "-- If a = b, then... |
[
{
"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 neq_of_not_iff : ¬(a ↔ b) → a ≠ b :="
},
{
"role": "assistant",
"content": "theorem neq... |
[
{
"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 iff_iff_eq : (a ↔ b) ↔ a = b :="
},
{
"role": "assistant",
"content": "theorem iff_iff_... |
[
{
"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_iff_iff {p q : Prop} : (p = q) ↔ (p ↔ q) :="
},
{
"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 of_iff_true (h : a ↔ True) : a :="
},
{
"role": "assistant",
"content": "theorem of_iff... |
[
{
"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 not_of_iff_false : (a ↔ False) → ¬a :="
},
{
"role": "assistant",
"content": "theorem 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 iff_of_true (ha : a) (hb : b) : a ↔ b :="
},
{
"role": "assistant",
"content": "-- If 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 iff_of_false (ha : ¬a) (hb : ¬b) : a ↔ b :="
},
{
"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 iff_true_left (ha : a) : (a ↔ b) ↔ b :="
},
{
"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 iff_true_right (ha : a) : (b ↔ a) ↔ b :="
},
{
"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 iff_false_left (ha : ¬a) : (a ↔ b) ↔ ¬b :="
},
{
"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 iff_false_right (ha : ¬a) : (b ↔ a) ↔ ¬b :="
},
{
"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 iff_true_intro (h : a) : a ↔ True :="
},
{
"role": "assistant",
"content": "theorem iff... |
[
{
"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 iff_false_intro (h : ¬a) : a ↔ False :="
},
{
"role": "assistant",
"content": "-- Given... |
[
{
"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 not_iff_false_intro (h : a) : ¬a ↔ False :="
},
{
"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 iff_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a ↔ b) ↔ (c ↔ d) :="
},
{
"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 not_true : (¬True) ↔ False :="
},
{
"role": "assistant",
"content": "-- This theorem st... |
[
{
"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 not_false_iff : (¬False) ↔ True :="
},
{
"role": "assistant",
"content": "-- Prove that... |
[
{
"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_self_iff_false (a : α) : a ≠ a ↔ False :="
},
{
"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 eq_self_iff_true (a : α) : a = a ↔ True :="
},
{
"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 heq_self_iff_true (a : α) : HEq a a ↔ True :="
},
{
"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 iff_not_self : ¬(a ↔ ¬a)"
},
{
"role": "assistant",
"content": "theorem iff_not_self : ... |
[
{
"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 not_iff_self : ¬(¬a ↔ a)"
},
{
"role": "assistant",
"content": "theorem not_iff_self : ... |
[
{
"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 true_iff_false : (True ↔ False) ↔ False :="
},
{
"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 false_iff_true : (False ↔ True) ↔ False :="
},
{
"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 false_of_true_iff_false : (True ↔ False) → False :="
},
{
"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 false_of_true_eq_false : (True = False) → False :="
},
{
"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 true_eq_false_of_false : False → (True = False) :="
},
{
"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 eq_comm {a b : α} : a = b ↔ b = a :="
},
{
"role": "assistant",
"content": "theorem eq_... |
[
{
"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 imp_intro {α β : Prop} (h : α) : β → α :="
},
{
"role": "assistant",
"content": "theore... |
[
{
"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 imp_imp_imp {a b c d : Prop} (h₀ : c → a) (h₁ : b → d) : (a → b) → (c → 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 imp_iff_right {a : Prop} (ha : a) : (a → b) ↔ 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 imp_true_iff (α : Sort u) : (α → True) ↔ True :="
},
{
"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 false_imp_iff (a : Prop) : (False → a) ↔ True :="
},
{
"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 true_imp_iff (α : Prop) : (True → α) ↔ α :="
},
{
"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 imp_self : (a → a) ↔ True :="
},
{
"role": "assistant",
"content": "-- Prove that if p ... |
[
{
"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 imp_false : (a → False) ↔ ¬a :="
},
{
"role": "assistant",
"content": "-- Proof by 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 imp.swap : (a → b → c) ↔ (b → a → c) :="
},
{
"role": "assistant",
"content": "-- This ... |
[
{
"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 imp_not_comm : (a → ¬b) ↔ (b → ¬a) :="
},
{
"role": "assistant",
"content": "theorem im... |
[
{
"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 imp_congr_left (h : a ↔ b) : (a → c) ↔ (b → c) :="
},
{
"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 imp_congr_right (h : a → (b ↔ c)) : (a → b) ↔ (a → c) :="
},
{
"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 imp_congr_ctx (h₁ : a ↔ c) (h₂ : c → (b ↔ d)) : (a → b) ↔ (c → d) :="
},
{
"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 imp_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a → b) ↔ (c → d) :="
},
{
"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 imp_iff_not (hb : ¬b) : a → b ↔ ¬a :="
},
{
"role": "assistant",
"content": "theorem im... |
[
{
"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 And.symm : a ∧ b → b ∧ a"
},
{
"role": "assistant",
"content": "theorem And.symm : 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 And.imp (f : a → c) (g : b → d) (h : a ∧ b) : c ∧ d :="
},
{
"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 And.imp_left (h : a → b) : a ∧ c → b ∧ c :="
},
{
"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 And.imp_right (h : a → b) : c ∧ a → c ∧ 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 and_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : a ∧ b ↔ c ∧ d :="
},
{
"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 and_comm : a ∧ b ↔ b ∧ a :="
},
{
"role": "assistant",
"content": "-- Prove that 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 and_congr_right (h : a → (b ↔ c)) : a ∧ b ↔ a ∧ c :="
},
{
"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 and_congr_left (h : c → (a ↔ b)) : a ∧ c ↔ b ∧ c :="
},
{
"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 and_congr_left' (h : a ↔ b) : a ∧ c ↔ b ∧ c :="
},
{
"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 and_congr_right' (h : b ↔ c) : a ∧ b ↔ a ∧ c :="
},
{
"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 and_congr_right_eq (h : a → b = c) : (a ∧ b) = (a ∧ c) :="
},
{
"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 and_congr_left_eq (h : c → a = b) : (a ∧ c) = (b ∧ c) :="
},
{
"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 and_assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :="
},
{
"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 and_left_comm : a ∧ (b ∧ c) ↔ b ∧ (a ∧ c) :="
},
{
"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 and_right_comm : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b :="
},
{
"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 and_rotate : a ∧ b ∧ c ↔ b ∧ c ∧ a :="
},
{
"role": "assistant",
"content": "theorem an... |
[
{
"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 and_and_and_comm : (a ∧ b) ∧ c ∧ d ↔ (a ∧ c) ∧ b ∧ d :="
},
{
"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 and_and_left : a ∧ b ∧ c ↔ (a ∧ b) ∧ a ∧ c :="
},
{
"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 and_and_right : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b ∧ c :="
},
{
"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 and_iff_left_of_imp (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 and_iff_right_of_imp (h : b → a) : (a ∧ b) ↔ 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 and_iff_left (hb : b) : a ∧ b ↔ a :="
},
{
"role": "assistant",
"content": "theorem and... |
[
{
"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 and_iff_right (ha : a) : a ∧ b ↔ b :="
},
{
"role": "assistant",
"content": "theorem an... |
[
{
"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 and_iff_left_iff_imp : ((a ∧ b) ↔ a) ↔ (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 and_iff_right_iff_imp : ((a ∧ b) ↔ b) ↔ (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 iff_self_and : (p ↔ p ∧ q) ↔ (p → q) :="
},
{
"role": "assistant",
"content": "-- This ... |
[
{
"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 iff_and_self : (p ↔ q ∧ p) ↔ (p → q) :="
},
{
"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 and_congr_right_iff : (a ∧ b ↔ a ∧ c) ↔ (a → (b ↔ c)) :="
},
{
"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 and_congr_left_iff : (a ∧ c ↔ b ∧ c) ↔ c → (a ↔ 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 and_self_left : a ∧ a ∧ b ↔ a ∧ b :="
},
{
"role": "assistant",
"content": "theorem and... |
[
{
"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 and_self_right : (a ∧ b) ∧ b ↔ a ∧ b :="
},
{
"role": "assistant",
"content": "-- Prove... |
[
{
"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 not_and_of_not_left (b : Prop) : ¬a → ¬(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 not_and_of_not_right (a : Prop) {b : Prop} : ¬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 and_not_self : ¬(a ∧ ¬a)"
},
{
"role": "assistant",
"content": "-- Proposition 1.6. If ... |
[
{
"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 not_and_self : ¬(¬a ∧ a)"
},
{
"role": "assistant",
"content": "theorem not_and_self : ... |
[
{
"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 and_not_self_iff (a : Prop) : a ∧ ¬a ↔ False :="
},
{
"role": "assistant",
"content": "... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.