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 : α) → Acc r a → Sort v}\n (intro : (x : α) → (h : ∀ (y : α), r y x → Acc r y) →\n ((y : α) → (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 α 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 → False) : ¬a :=",
"role": "user"
},
{
"content": "theorem Not.intro {a : Prop} (h : a → 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 : ¬b) (H1 : a → b) : ¬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 ↔ b) : ¬a ↔ ¬b :=",
"role": "user"
},
{
"content": "theorem not_congr (h : a ↔ b) : ¬a ↔ ¬b := ⟨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 ↔ ¬a :=",
"role": "user"
},
{
"content": "theorem not_not_not : ¬¬¬a ↔ ¬a := ⟨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 : ¬(a → b) → ¬¬a :=",
"role": "user"
},
{
"content": "theorem not_not_of_not_imp : ¬(a → b) → ¬¬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} : ¬(a → b) → ¬b :=",
"role": "user"
},
{
"content": "theorem not_of_not_imp {a : Prop} : ¬(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 → ¬a) ↔ ¬a :=",
"role": "user"
},
{
"content": "theorem imp_not_self : (a → ¬a) ↔ ¬a := ⟨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 ↔ b) ↔ (a → b) ∧ (b → a) :=",
"role": "user"
},
{
"content": "theorem iff_def : (a ↔ b) ↔ (a → b) ∧ (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_def' : (a ↔ b) ↔ (b → a) ∧ (a → b) :=",
"role": "user"
},
{
"content": "theorem iff_def' : (a ↔ b) ↔ (b → 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 Eq.to_iff : a = b → (a ↔ b)",
"role": "user"
},
{
"content": "theorem Eq.to_iff : a = b → (a ↔ 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 → (a ↔ b) :=",
"role": "user"
},
{
"content": "theorem iff_of_eq : a = b → (a ↔ 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 : ¬(a ↔ b) → a ≠ b :=",
"role": "user"
},
{
"content": "theorem neq_of_not_iff : ¬(a ↔ b) → a ≠ 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 ↔ b) ↔ a = b :=",
"role": "user"
},
{
"content": "theorem iff_iff_eq : (a ↔ b) ↔ a = b := ⟨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) ↔ (p ↔ 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 ↔ True) : a :=",
"role": "user"
},
{
"content": "theorem of_iff_true (h : a ↔ True) : a := h.2 ⟨⟩",... |
[
{
"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 ↔ False) → ¬a :=",
"role": "user"
},
{
"content": "theorem not_of_iff_false : (a ↔ False) → ¬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 ↔ b :=",
"role": "user"
},
{
"content": "theorem iff_of_true (ha : a) (hb : b) : 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_false (ha : ¬a) (hb : ¬b) : a ↔ b :=",
"role": "user"
},
{
"content": "theorem iff_of_false (ha : ¬a) (hb : ¬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 ↔ b) ↔ b :=",
"role": "user"
},
{
"content": "theorem iff_true_left (ha : a) : (a ↔ b) ↔ 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 ↔ a) ↔ b :=",
"role": "user"
},
{
"content": "theorem iff_true_right (ha : a) : (b ↔ 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_false_left (ha : ¬a) : (a ↔ b) ↔ ¬b :=",
"role": "user"
},
{
"content": "theorem iff_false_left (ha : ¬a) : (a ↔ 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 : ¬a) : (b ↔ a) ↔ ¬b :=",
"role": "user"
},
{
"content": "theorem iff_false_right (ha : ¬a) : (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_intro (h : a) : a ↔ True :=",
"role": "user"
},
{
"content": "theorem iff_true_intro (h : a) : a ↔ 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 : ¬a) : a ↔ False :=",
"role": "user"
},
{
"content": "theorem iff_false_intro (h : ¬a) : a ↔ 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) : ¬a ↔ False :=",
"role": "user"
},
{
"content": "theorem not_iff_false_intro (h : 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 iff_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a ↔ b) ↔ (c ↔ d) :=",
"role": "user"
},
{
"content": "theorem iff_congr (h₁ : 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 : (¬True) ↔ False :=",
"role": "user"
},
{
"content": "theorem not_true : (¬True) ↔ 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 : (¬False) ↔ True :=",
"role": "user"
},
{
"content": "theorem not_false_iff : (¬False) ↔ 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 : α) : a ≠ a ↔ False :=",
"role": "user"
},
{
"content": "theorem ne_self_iff_false (a : α) : 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_self_iff_true (a : α) : a = a ↔ True :=",
"role": "user"
},
{
"content": "theorem eq_self_iff_true (a : α) : 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 : α) : HEq a a ↔ True :=",
"role": "user"
},
{
"content": "theorem heq_self_iff_true (a : α) : 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 : ¬(a ↔ ¬a)",
"role": "user"
},
{
"content": "theorem iff_not_self : ¬(a ↔ ¬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 : ¬(¬a ↔ a)",
"role": "user"
},
{
"content": "theorem not_iff_self : ¬(¬a ↔ 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 ↔ False) ↔ False :=",
"role": "user"
},
{
"content": "theorem true_iff_false : (True ↔ 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_iff_true : (False ↔ True) ↔ False :=",
"role": "user"
},
{
"content": "theorem false_iff_true : (False ↔ 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 false_of_true_iff_false : (True ↔ False) → 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) → 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 → (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 : α} : a = b ↔ b = a :=",
"role": "user"
},
{
"content": "theorem eq_comm {a b : α} : a = b ↔ b = a := ⟨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 {α β : Prop} (h : α) : β → α :=",
"role": "user"
},
{
"content": "theorem imp_intro {α β : Prop} (h : α) : 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 imp_imp_imp {a b c d : Prop} (h₀ : c → a) (h₁ : b → d) : (a → b) → (c → 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 → b) ↔ 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 (α : Sort u) : (α → True) ↔ True :=",
"role": "user"
},
{
"content": "theorem imp_true_iff (α : 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 → a) ↔ 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 (α : Prop) : (True → α) ↔ α :=",
"role": "user"
},
{
"content": "theorem true_imp_iff (α : 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 → a) ↔ True :=",
"role": "user"
},
{
"content": "theorem imp_self : (a → a) ↔ 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 → False) ↔ ¬a :=",
"role": "user"
},
{
"content": "theorem imp_false : (a → False) ↔ ¬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 → b → c) ↔ (b → a → c) :=",
"role": "user"
},
{
"content": "theorem imp.swap : (a → b → c) ↔ (b → a → 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 → ¬b) ↔ (b → ¬a) :=",
"role": "user"
},
{
"content": "theorem imp_not_comm : (a → ¬b) ↔ (b → ¬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 ↔ b) : (a → c) ↔ (b → c) :=",
"role": "user"
},
{
"content": "theorem imp_congr_left (h : a ↔ 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 → (b ↔ c)) : (a → b) ↔ (a → 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 (h₁ : a ↔ c) (h₂ : c → (b ↔ d)) : (a → b) ↔ (c → 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 (h₁ : a ↔ c) (h₂ : b ↔ d) : (a → b) ↔ (c → d) :=",
"role": "user"
},
{
"content": "theorem imp_congr (h₁ : 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 : ¬b) : a → b ↔ ¬a :=",
"role": "user"
},
{
"content": "theorem imp_iff_not (hb : ¬b) : a → b ↔ ¬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 ∧ b → b ∧ a",
"role": "user"
},
{
"content": "theorem And.symm : a ∧ b → b ∧ a | ⟨ha, hb⟩ => ⟨hb, 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.imp (f : a → c) (g : b → d) (h : a ∧ b) : c ∧ d :=",
"role": "user"
},
{
"content": "theorem And.imp (f : a → 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 → b) : a ∧ c → b ∧ c :=",
"role": "user"
},
{
"content": "theorem And.imp_left (h : a → b) : a ∧ 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 → b) : c ∧ a → c ∧ b :=",
"role": "user"
},
{
"content": "theorem And.imp_right (h : a → b) : 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 (h₁ : a ↔ c) (h₂ : b ↔ d) : a ∧ b ↔ c ∧ d :=",
"role": "user"
},
{
"content": "theorem and_congr (h₁ : a ↔ 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 ∧ b ↔ b ∧ a :=",
"role": "user"
},
{
"content": "theorem and_comm : a ∧ b ↔ b ∧ 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 → (b ↔ c)) : a ∧ b ↔ a ∧ 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 → (a ↔ b)) : a ∧ c ↔ b ∧ c :=",
"role": "user"
},
{
"content": "theorem and_congr_left (h : 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_left' (h : a ↔ b) : a ∧ c ↔ b ∧ c :=",
"role": "user"
},
{
"content": "theorem and_congr_left' (h : a ↔ 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 ↔ c) : a ∧ b ↔ a ∧ c :=",
"role": "user"
},
{
"content": "theorem and_congr_right' (h : b ↔ 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 → b = c) : (a ∧ b) = (a ∧ 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 → a = b) : (a ∧ c) = (b ∧ 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 ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=",
"role": "user"
},
{
"content": "theorem and_assoc : (a ∧ b) ∧ c ↔ a ∧ (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_left_comm : a ∧ (b ∧ c) ↔ b ∧ (a ∧ c) :=",
"role": "user"
},
{
"content": "theorem and_left_comm : a ∧ (b ∧ c) ↔ 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 ∧ b) ∧ c ↔ (a ∧ c) ∧ b :=",
"role": "user"
},
{
"content": "theorem and_right_comm : (a ∧ b) ∧ 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_rotate : a ∧ b ∧ c ↔ b ∧ c ∧ a :=",
"role": "user"
},
{
"content": "theorem and_rotate : a ∧ b ∧ c ↔ b ∧ c ∧ 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 ∧ b) ∧ c ∧ d ↔ (a ∧ c) ∧ b ∧ 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 ∧ b ∧ c ↔ (a ∧ b) ∧ a ∧ c :=",
"role": "user"
},
{
"content": "theorem and_and_left : a ∧ b ∧ c ↔ (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 ∧ b) ∧ c ↔ (a ∧ c) ∧ b ∧ c :=",
"role": "user"
},
{
"content": "theorem and_and_right : (a ∧ b) ∧ 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 → b) : (a ∧ b) ↔ 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 → a) : (a ∧ b) ↔ 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 ∧ b ↔ a :=",
"role": "user"
},
{
"content": "theorem and_iff_left (hb : b) : a ∧ b ↔ 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 ∧ b ↔ b :=",
"role": "user"
},
{
"content": "theorem and_iff_right (ha : a) : a ∧ b ↔ 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 ∧ b) ↔ a) ↔ (a → 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 ∧ b) ↔ b) ↔ (b → 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 ↔ p ∧ q) ↔ (p → q) :=",
"role": "user"
},
{
"content": "theorem iff_self_and : (p ↔ p ∧ q) ↔ (p → 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 ↔ q ∧ p) ↔ (p → q) :=",
"role": "user"
},
{
"content": "theorem iff_and_self : (p ↔ q ∧ p) ↔ (p → 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 ∧ b ↔ a ∧ c) ↔ (a → (b ↔ 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 ∧ c ↔ b ∧ c) ↔ c → (a ↔ 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 ∧ a ∧ b ↔ a ∧ b :=",
"role": "user"
},
{
"content": "theorem and_self_left : a ∧ a ∧ b ↔ a ∧ 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 ∧ b) ∧ b ↔ a ∧ b :=",
"role": "user"
},
{
"content": "theorem and_self_right : (a ∧ b) ∧ b ↔ a ∧ 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) : ¬a → ¬(a ∧ 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} : ¬b → ¬(a ∧ 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 : ¬(a ∧ ¬a)",
"role": "user"
},
{
"content": "theorem and_not_self : ¬(a ∧ ¬a) | ⟨ha, hn⟩ => 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 : ¬(¬a ∧ a)",
"role": "user"
},
{
"content": "theorem not_and_self : ¬(¬a ∧ a) | ⟨hn, ha⟩ => 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 ∧ ¬a ↔ False :=",
"role": "user"
},
{
"content": "theorem and_not_self_iff (a : Prop) ... |
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