messages listlengths 3 3 |
|---|
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"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 Acc r y) β\n ((y : Ξ±) β (hr :... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem rec_eq_recC : @Acc.rec = @Acc.recC :=",
"role": "user"
},
{
"content": "theorem rec_eq_recC : @Acc.rec = @Acc.recC := ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem ndrec_eq_ndrecC : @Acc.ndrec = @Acc.ndrecC :=",
"role": "user"
},
{
"content": "theorem ndrec_eq_ndrecC : @Acc.ndrec =... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem ndrecOn_eq_ndrecOnC : @Acc.ndrecOn = @Acc.ndrecOnC :=",
"role": "user"
},
{
"content": "theorem ndrecOn_eq_ndrecOnC : ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem fixF_eq_fixFC : @fixF = @fixFC :=",
"role": "user"
},
{
"content": "theorem fixF_eq_fixFC : @fixF = @fixFC := by\n fu... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem fix_eq_fixC : @fix = @fixC :=",
"role": "user"
},
{
"content": "theorem fix_eq_fixC : @fix = @fixC := by\n funext Ξ± C... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem $extName:ident : ext_type% $flat $struct:ident :=",
"role": "user"
},
{
"content": "theorem $extName:ident : ext_type%... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem $extIffName:ident : ext_iff_type% $flat $struct:ident :=",
"role": "user"
},
{
"content": "theorem $extIffName:ident :... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem PUnit.ext (x y : PUnit) : x = y :=",
"role": "user"
},
{
"content": "theorem PUnit.ext (x y : PUnit) : x = y := rfl",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem Unit.ext (x y : Unit) : x = y :=",
"role": "user"
},
{
"content": "theorem Unit.ext (x y : Unit) : x = y := rfl",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem Not.intro {a : Prop} (h : a β False) : Β¬a :=",
"role": "user"
},
{
"content": "theorem Not.intro {a : Prop} (h : a β F... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem Not.imp {a b : Prop} (H2 : Β¬b) (H1 : a β b) : Β¬a :=",
"role": "user"
},
{
"content": "theorem Not.imp {a b : Prop} (H2... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_congr (h : a β b) : Β¬a β Β¬b :=",
"role": "user"
},
{
"content": "theorem not_congr (h : a β b) : Β¬a β Β¬b := β¨mt h.... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_not_not : ¬¬¬a β Β¬a :=",
"role": "user"
},
{
"content": "theorem not_not_not : ¬¬¬a β Β¬a := β¨mt not_not_intro, not... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_not_of_not_imp : Β¬(a β b) β ¬¬a :=",
"role": "user"
},
{
"content": "theorem not_not_of_not_imp : Β¬(a β b) β ¬¬a :... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_of_not_imp {a : Prop} : Β¬(a β b) β Β¬b :=",
"role": "user"
},
{
"content": "theorem not_of_not_imp {a : Prop} : Β¬(a... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_not_self : (a β Β¬a) β Β¬a :=",
"role": "user"
},
{
"content": "theorem imp_not_self : (a β Β¬a) β Β¬a := β¨fun h ha =>... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_def : (a β b) β (a β b) β§ (b β a) :=",
"role": "user"
},
{
"content": "theorem iff_def : (a β b) β (a β b) β§ (b β ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_def' : (a β b) β (b β a) β§ (a β b) :=",
"role": "user"
},
{
"content": "theorem iff_def' : (a β b) β (b β a) β§ (a ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem Eq.to_iff : a = b β (a β b)",
"role": "user"
},
{
"content": "theorem Eq.to_iff : a = b β (a β b) | rfl => Iff.rfl",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_of_eq : a = b β (a β b) :=",
"role": "user"
},
{
"content": "theorem iff_of_eq : a = b β (a β b) := Eq.to_iff",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem neq_of_not_iff : Β¬(a β b) β a β b :=",
"role": "user"
},
{
"content": "theorem neq_of_not_iff : Β¬(a β b) β a β b := mt... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_iff_eq : (a β b) β a = b :=",
"role": "user"
},
{
"content": "theorem iff_iff_eq : (a β b) β a = b := β¨propext, if... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem eq_iff_iff {p q : Prop} : (p = q) β (p β q) :=",
"role": "user"
},
{
"content": "theorem eq_iff_iff {p q : Prop} : (p ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem of_iff_true (h : a β True) : a :=",
"role": "user"
},
{
"content": "theorem of_iff_true (h : a β True) : a := h.2 β¨β©",... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_of_iff_false : (a β False) β Β¬a :=",
"role": "user"
},
{
"content": "theorem not_of_iff_false : (a β False) β Β¬a :... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_of_true (ha : a) (hb : b) : a β b :=",
"role": "user"
},
{
"content": "theorem iff_of_true (ha : a) (hb : b) : a β... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_of_false (ha : Β¬a) (hb : Β¬b) : a β b :=",
"role": "user"
},
{
"content": "theorem iff_of_false (ha : Β¬a) (hb : Β¬b)... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_true_left (ha : a) : (a β b) β b :=",
"role": "user"
},
{
"content": "theorem iff_true_left (ha : a) : (a β b) β b... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_true_right (ha : a) : (b β a) β b :=",
"role": "user"
},
{
"content": "theorem iff_true_right (ha : a) : (b β a) β... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_false_left (ha : Β¬a) : (a β b) β Β¬b :=",
"role": "user"
},
{
"content": "theorem iff_false_left (ha : Β¬a) : (a β b... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_false_right (ha : Β¬a) : (b β a) β Β¬b :=",
"role": "user"
},
{
"content": "theorem iff_false_right (ha : Β¬a) : (b β... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_true_intro (h : a) : a β True :=",
"role": "user"
},
{
"content": "theorem iff_true_intro (h : a) : a β True := if... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_false_intro (h : Β¬a) : a β False :=",
"role": "user"
},
{
"content": "theorem iff_false_intro (h : Β¬a) : a β False... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_iff_false_intro (h : a) : Β¬a β False :=",
"role": "user"
},
{
"content": "theorem not_iff_false_intro (h : a) : Β¬a... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_congr (hβ : a β c) (hβ : b β d) : (a β b) β (c β d) :=",
"role": "user"
},
{
"content": "theorem iff_congr (hβ : a... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_true : (Β¬True) β False :=",
"role": "user"
},
{
"content": "theorem not_true : (Β¬True) β False := iff_false_intro ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_false_iff : (Β¬False) β True :=",
"role": "user"
},
{
"content": "theorem not_false_iff : (Β¬False) β True := iff_tr... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem ne_self_iff_false (a : Ξ±) : a β a β False :=",
"role": "user"
},
{
"content": "theorem ne_self_iff_false (a : Ξ±) : a β ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem eq_self_iff_true (a : Ξ±) : a = a β True :=",
"role": "user"
},
{
"content": "theorem eq_self_iff_true (a : Ξ±) : a = a ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem heq_self_iff_true (a : Ξ±) : HEq a a β True :=",
"role": "user"
},
{
"content": "theorem heq_self_iff_true (a : Ξ±) : HE... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_not_self : Β¬(a β Β¬a)",
"role": "user"
},
{
"content": "theorem iff_not_self : Β¬(a β Β¬a) | H => let f h := H.1 h h;... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_iff_self : Β¬(Β¬a β a)",
"role": "user"
},
{
"content": "theorem not_iff_self : Β¬(Β¬a β a) | H => iff_not_self H.symm... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem true_iff_false : (True β False) β False :=",
"role": "user"
},
{
"content": "theorem true_iff_false : (True β False) β... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem false_iff_true : (False β True) β False :=",
"role": "user"
},
{
"content": "theorem false_iff_true : (False β True) β... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem false_of_true_iff_false : (True β False) β False :=",
"role": "user"
},
{
"content": "theorem false_of_true_iff_false ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem false_of_true_eq_false : (True = False) β False :=",
"role": "user"
},
{
"content": "theorem false_of_true_eq_false : ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem true_eq_false_of_false : False β (True = False) :=",
"role": "user"
},
{
"content": "theorem true_eq_false_of_false : ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem eq_comm {a b : Ξ±} : a = b β b = a :=",
"role": "user"
},
{
"content": "theorem eq_comm {a b : Ξ±} : a = b β b = a := β¨E... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_intro {Ξ± Ξ² : Prop} (h : Ξ±) : Ξ² β Ξ± :=",
"role": "user"
},
{
"content": "theorem imp_intro {Ξ± Ξ² : Prop} (h : Ξ±) : Ξ²... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"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": "user"
},
{
"content": "theore... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_iff_right {a : Prop} (ha : a) : (a β b) β b :=",
"role": "user"
},
{
"content": "theorem imp_iff_right {a : Prop} ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_true_iff (Ξ± : Sort u) : (Ξ± β True) β True :=",
"role": "user"
},
{
"content": "theorem imp_true_iff (Ξ± : Sort u) :... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem false_imp_iff (a : Prop) : (False β a) β True :=",
"role": "user"
},
{
"content": "theorem false_imp_iff (a : Prop) : ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem true_imp_iff (Ξ± : Prop) : (True β Ξ±) β Ξ± :=",
"role": "user"
},
{
"content": "theorem true_imp_iff (Ξ± : Prop) : (True ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_self : (a β a) β True :=",
"role": "user"
},
{
"content": "theorem imp_self : (a β a) β True := iff_true_intro id"... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_false : (a β False) β Β¬a :=",
"role": "user"
},
{
"content": "theorem imp_false : (a β False) β Β¬a := Iff.rfl",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp.swap : (a β b β c) β (b β a β c) :=",
"role": "user"
},
{
"content": "theorem imp.swap : (a β b β c) β (b β a β c)... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_not_comm : (a β Β¬b) β (b β Β¬a) :=",
"role": "user"
},
{
"content": "theorem imp_not_comm : (a β Β¬b) β (b β Β¬a) := ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_congr_left (h : a β b) : (a β c) β (b β c) :=",
"role": "user"
},
{
"content": "theorem imp_congr_left (h : a β b)... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_congr_right (h : a β (b β c)) : (a β b) β (a β c) :=",
"role": "user"
},
{
"content": "theorem imp_congr_right (h ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_congr_ctx (hβ : a β c) (hβ : c β (b β d)) : (a β b) β (c β d) :=",
"role": "user"
},
{
"content": "theorem imp_con... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_congr (hβ : a β c) (hβ : b β d) : (a β b) β (c β d) :=",
"role": "user"
},
{
"content": "theorem imp_congr (hβ : a... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem imp_iff_not (hb : Β¬b) : a β b β Β¬a :=",
"role": "user"
},
{
"content": "theorem imp_iff_not (hb : Β¬b) : a β b β Β¬a := ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem And.symm : a β§ b β b β§ a",
"role": "user"
},
{
"content": "theorem And.symm : a β§ b β b β§ a | β¨ha, hbβ© => β¨hb, haβ©",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem And.imp (f : a β c) (g : b β d) (h : a β§ b) : c β§ d :=",
"role": "user"
},
{
"content": "theorem And.imp (f : a β c) (... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem And.imp_left (h : a β b) : a β§ c β b β§ c :=",
"role": "user"
},
{
"content": "theorem And.imp_left (h : a β b) : a β§ c... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem And.imp_right (h : a β b) : c β§ a β c β§ b :=",
"role": "user"
},
{
"content": "theorem And.imp_right (h : a β b) : c β§... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr (hβ : a β c) (hβ : b β d) : a β§ b β c β§ d :=",
"role": "user"
},
{
"content": "theorem and_congr (hβ : a β c... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_comm : a β§ b β b β§ a :=",
"role": "user"
},
{
"content": "theorem and_comm : a β§ b β b β§ a := And.comm",
"role... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_right (h : a β (b β c)) : a β§ b β a β§ c :=",
"role": "user"
},
{
"content": "theorem and_congr_right (h : a ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_left (h : c β (a β b)) : a β§ c β b β§ c :=",
"role": "user"
},
{
"content": "theorem and_congr_left (h : c β ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_left' (h : a β b) : a β§ c β b β§ c :=",
"role": "user"
},
{
"content": "theorem and_congr_left' (h : a β b) :... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_right' (h : b β c) : a β§ b β a β§ c :=",
"role": "user"
},
{
"content": "theorem and_congr_right' (h : b β c)... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_right_eq (h : a β b = c) : (a β§ b) = (a β§ c) :=",
"role": "user"
},
{
"content": "theorem and_congr_right_eq... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_left_eq (h : c β a = b) : (a β§ c) = (b β§ c) :=",
"role": "user"
},
{
"content": "theorem and_congr_left_eq (... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_assoc : (a β§ b) β§ c β a β§ (b β§ c) :=",
"role": "user"
},
{
"content": "theorem and_assoc : (a β§ b) β§ c β a β§ (b β§ ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_left_comm : a β§ (b β§ c) β b β§ (a β§ c) :=",
"role": "user"
},
{
"content": "theorem and_left_comm : a β§ (b β§ c) β b... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_right_comm : (a β§ b) β§ c β (a β§ c) β§ b :=",
"role": "user"
},
{
"content": "theorem and_right_comm : (a β§ b) β§ c β... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_rotate : a β§ b β§ c β b β§ c β§ a :=",
"role": "user"
},
{
"content": "theorem and_rotate : a β§ b β§ c β b β§ c β§ a := ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_and_and_comm : (a β§ b) β§ c β§ d β (a β§ c) β§ b β§ d :=",
"role": "user"
},
{
"content": "theorem and_and_and_comm : (... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_and_left : a β§ b β§ c β (a β§ b) β§ a β§ c :=",
"role": "user"
},
{
"content": "theorem and_and_left : a β§ b β§ c β (a ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_and_right : (a β§ b) β§ c β (a β§ c) β§ b β§ c :=",
"role": "user"
},
{
"content": "theorem and_and_right : (a β§ b) β§ c... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_iff_left_of_imp (h : a β b) : (a β§ b) β a :=",
"role": "user"
},
{
"content": "theorem and_iff_left_of_imp (h : a ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_iff_right_of_imp (h : b β a) : (a β§ b) β b :=",
"role": "user"
},
{
"content": "theorem and_iff_right_of_imp (h : ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_iff_left (hb : b) : a β§ b β a :=",
"role": "user"
},
{
"content": "theorem and_iff_left (hb : b) : a β§ b β a := an... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_iff_right (ha : a) : a β§ b β b :=",
"role": "user"
},
{
"content": "theorem and_iff_right (ha : a) : a β§ b β b := ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_iff_left_iff_imp : ((a β§ b) β a) β (a β b) :=",
"role": "user"
},
{
"content": "theorem and_iff_left_iff_imp : ((a... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_iff_right_iff_imp : ((a β§ b) β b) β (b β a) :=",
"role": "user"
},
{
"content": "theorem and_iff_right_iff_imp : (... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_self_and : (p β p β§ q) β (p β q) :=",
"role": "user"
},
{
"content": "theorem iff_self_and : (p β p β§ q) β (p β q)... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem iff_and_self : (p β q β§ p) β (p β q) :=",
"role": "user"
},
{
"content": "theorem iff_and_self : (p β q β§ p) β (p β q)... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_right_iff : (a β§ b β a β§ c) β (a β (b β c)) :=",
"role": "user"
},
{
"content": "theorem and_congr_right_iff... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_congr_left_iff : (a β§ c β b β§ c) β c β (a β b) :=",
"role": "user"
},
{
"content": "theorem and_congr_left_iff : (... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_self_left : a β§ a β§ b β a β§ b :=",
"role": "user"
},
{
"content": "theorem and_self_left : a β§ a β§ b β a β§ b :=\n ... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_self_right : (a β§ b) β§ b β a β§ b :=",
"role": "user"
},
{
"content": "theorem and_self_right : (a β§ b) β§ b β a β§ b... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_and_of_not_left (b : Prop) : Β¬a β Β¬(a β§ b) :=",
"role": "user"
},
{
"content": "theorem not_and_of_not_left (b : P... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_and_of_not_right (a : Prop) {b : Prop} : Β¬b β Β¬(a β§ b) :=",
"role": "user"
},
{
"content": "theorem not_and_of_not... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_not_self : Β¬(a β§ Β¬a)",
"role": "user"
},
{
"content": "theorem and_not_self : Β¬(a β§ Β¬a) | β¨ha, hnβ© => hn ha",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem not_and_self : Β¬(Β¬a β§ a)",
"role": "user"
},
{
"content": "theorem not_and_self : Β¬(Β¬a β§ a) | β¨hn, haβ© => hn ha",
... |
[
{
"content": "You are a Lean4 expert. Your goal is to provide a complete proof in Lean4 for the given mathematical statement.",
"role": "system"
},
{
"content": "theorem and_not_self_iff (a : Prop) : a β§ Β¬a β False :=",
"role": "user"
},
{
"content": "theorem and_not_self_iff (a : Prop) ... |
End of preview. Expand
in Data Studio
README.md exists but content is empty.
- Downloads last month
- 21