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 : Ξ±) β Acc r a β Sort v}\n (intro : (x : Ξ±) β (h : β (y : Ξ±), r y x β 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 β False) : Β¬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 : Β¬b) (H1 : a β b) : Β¬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 β b) : Β¬a β Β¬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 β Β¬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 : Β¬(a β b) β ¬¬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} : Β¬(a β b) β Β¬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 β Β¬a) β Β¬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 β b) β (a β b) β§ (b β 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 β b) β (b β a) β§ (a β 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 β (a β 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 β (a β 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 : Β¬(a β b) β a β 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 β b) β 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) β (p β 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 β 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 β False) β Β¬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 β 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 : Β¬a) (hb : Β¬b) : a β 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 β b) β 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 β a) β 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 : Β¬a) : (a β b) β Β¬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 : Β¬a) : (b β a) β Β¬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 β 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 : Β¬a) : a β 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) : Β¬a β 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 (hβ : a β c) (hβ : b β d) : (a β b) β (c β 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 : (Β¬True) β 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 : (Β¬False) β 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 : Ξ±) : a β a β 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 : Ξ±) : a = a β 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 : Ξ±) : HEq a a β 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 : Β¬(a β Β¬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 : Β¬(Β¬a β 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 β False) β 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 β True) β 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 β False) β 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) β 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 β (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 : Ξ±} : a = b β 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 {Ξ± Ξ² : Prop} (h : Ξ±) : Ξ² β Ξ± :="
},
{
"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} (hβ : c β a) (hβ : b β d) : (a β b) β (c β 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 β b) β 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 (Ξ± : Sort u) : (Ξ± β True) β 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 β a) β 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 (Ξ± : Prop) : (True β Ξ±) β Ξ± :="
},
{
"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 β a) β 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 β False) β Β¬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 β b β c) β (b β a β 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 β Β¬b) β (b β Β¬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 β b) : (a β c) β (b β 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 β (b β c)) : (a β b) β (a β 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 (hβ : a β c) (hβ : c β (b β d)) : (a β b) β (c β 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 (hβ : a β c) (hβ : b β d) : (a β b) β (c β 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 : Β¬b) : a β b β Β¬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 β§ b β b β§ a"
},
{
"role": "assistant",
"content": "theorem And.symm : a β§ ... |
[
{
"role": "system",
"content": "You 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 β c) (g : b β d) (h : a β§ b) : c β§ 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 β b) : a β§ c β b β§ 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 β b) : c β§ a β c β§ 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 (hβ : a β c) (hβ : b β d) : a β§ b β c β§ 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 β§ b β b β§ a :="
},
{
"role": "assistant",
"content": "-- Prove that a β§ 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 β (b β c)) : a β§ b β a β§ 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 β (a β b)) : a β§ c β b β§ 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 β b) : a β§ c β b β§ 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 β c) : a β§ b β a β§ 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 β b = c) : (a β§ b) = (a β§ 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 β a = b) : (a β§ c) = (b β§ 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 β§ b) β§ c β a β§ (b β§ 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 β§ (b β§ c) β b β§ (a β§ 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 β§ b) β§ c β (a β§ c) β§ 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 β§ b β§ c β b β§ c β§ 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 β§ b) β§ c β§ d β (a β§ c) β§ b β§ 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 β§ b β§ c β (a β§ b) β§ a β§ 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 β§ b) β§ c β (a β§ c) β§ b β§ 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 β b) : (a β§ b) β 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 β a) : (a β§ b) β 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 β§ b β 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 β§ b β 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 β§ b) β a) β (a β 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 β§ b) β b) β (b β 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 β p β§ q) β (p β 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 β q β§ p) β (p β 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 β§ b β a β§ c) β (a β (b β 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 β§ c β b β§ c) β c β (a β 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 β§ a β§ b β a β§ 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 β§ b) β§ b β a β§ 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) : Β¬a β Β¬(a β§ 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} : Β¬b β Β¬(a β§ 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 : Β¬(a β§ Β¬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 : Β¬(Β¬a β§ 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 β§ Β¬a β False :="
},
{
"role": "assistant",
"content": "... |
End of preview. Expand
in Data Studio
No dataset card yet
- Downloads last month
- 8