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 dvd_refl (a : Nat) : a ∣ a :="
},
{
"role": "assistant",
"content": "theorem dvd_refl (... |
[
{
"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 pred_top_le_iff : pred ⊤ ≤ a ↔ a = ⊤ ∨ a = pred ⊤ :="
},
{
"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 pred_lt_top (a : α) : pred a < ⊤ :="
},
{
"role": "assistant",
"content": "theorem pred... |
[
{
"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 pred_ne_top (a : α) : pred a ≠ ⊤ :="
},
{
"role": "assistant",
"content": "-- If `a` is... |
[
{
"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 pred_eq_iSup (a : α) : pred a = ⨆ (b) (_ : b < a), b :="
},
{
"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 succ_pred_of_not_isMin (h : ¬IsMin a) : succ (pred a) = a :="
},
{
"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 pred_succ_of_not_isMax (h : ¬IsMax a) : pred (succ a) = a :="
},
{
"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 succ_pred [NoMinOrder α] (a : α) : succ (pred a) = a :="
},
{
"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 pred_succ [NoMaxOrder α] (a : α) : pred (succ a) = a :="
},
{
"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 pred_succ_iterate_of_not_isMax (i : α) (n : ℕ) (hin : ¬IsMax (succ^[n - 1] i)) :\n pred^[n] (suc... |
[
{
"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 succ_pred_iterate_of_not_isMin (i : α) (n : ℕ) (hin : ¬IsMin (pred^[n - 1] i)) :\n succ^[n] (pre... |
[
{
"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 succ_coe_top : succ ↑(⊤ : α) = (⊤ : WithTop α) :="
},
{
"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 succ_coe_of_ne_top {a : α} (h : a ≠ ⊤) : succ (↑a : WithTop α) = ↑(succ a) :="
},
{
"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 pred_top : pred (⊤ : WithTop α) = ↑(⊤ : α) :="
},
{
"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 pred_coe (a : α) : pred (↑a : WithTop α) = ↑(pred a) :="
},
{
"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 pred_untop :\n ∀ (a : WithTop α) (ha : a ≠ ⊤),\n pred (a.untop ha) = (pred a).untop (by ind... |
[
{
"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 succ_coe (a : α) : succ (↑a : WithTop α) = ↑(succ a) :="
},
{
"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 succ_bot : succ (⊥ : WithBot α) = ↑(⊥ : α) :="
},
{
"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 succ_coe (a : α) : succ (↑a : WithBot α) = ↑(succ a) :="
},
{
"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 succ_unbot :\n ∀ (a : WithBot α) (ha : a ≠ ⊥),\n succ (a.unbot ha) = (succ a).unbot (by ind... |
[
{
"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 pred_coe_bot : pred ↑(⊥ : α) = (⊥ : WithBot α) :="
},
{
"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 pred_coe_of_ne_bot {a : α} (h : a ≠ ⊥) : pred (↑a : WithBot α) = ↑(pred a) :="
},
{
"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 pred_coe (a : α) : pred (↑a : WithBot α) = ↑(pred a) :="
},
{
"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 LE.le.exists_succ_iterate (h : a ≤ b) : ∃ n, succ^[n] 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 exists_succ_iterate_iff_le : (∃ n, succ^[n] a = 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 Succ.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, m ≤ n → P n → P (succ n)) ⦃n : α⦄\n (hmn :... |
[
{
"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 Succ.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) {a b : α} (h : a ≤ b) :\n p a ↔ p 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 LE.le.exists_pred_iterate (h : a ≤ b) : ∃ n, pred^[n] b = a :="
},
{
"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 exists_pred_iterate_iff_le : (∃ n, pred^[n] b = a) ↔ 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 Pred.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, n ≤ m → P n → P (pred n)) ⦃n : α⦄\n (hmn :... |
[
{
"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 Pred.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) {a b : α} (h : a ≤ b) :\n p a ↔ p 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 exists_succ_iterate_or : (∃ n, succ^[n] a = b) ∨ ∃ n, succ^[n] b = a :="
},
{
"role": "assi... |
[
{
"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 Succ.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) (a b : α) : p a ↔ p 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 exists_pred_iterate_or : (∃ n, pred^[n] b = a) ∨ ∃ n, pred^[n] a = b :="
},
{
"role": "assi... |
[
{
"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 Pred.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) (a b : α) : p a ↔ p 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": "lemma StrictMono.not_bddAbove_range [NoMaxOrder α] [SuccOrder β] [IsSuccArchimedean β]\n (hf : StrictMon... |
[
{
"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": "lemma StrictMono.not_bddBelow_range [NoMinOrder α] [PredOrder β] [IsPredArchimedean β]\n (hf : StrictMon... |
[
{
"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": "lemma StrictAnti.not_bddAbove_range [NoMinOrder α] [SuccOrder β] [IsSuccArchimedean β]\n (hf : StrictAnt... |
[
{
"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": "lemma StrictAnti.not_bddBelow_range [NoMaxOrder α] [PredOrder β] [IsPredArchimedean β]\n (hf : StrictAnt... |
[
{
"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 Succ.rec_bot (p : α → Prop) (hbot : p ⊥) (hsucc : ∀ a, p a → p (succ a)) (a : α) : p 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 Pred.rec_top (p : α → Prop) (htop : p ⊤) (hpred : ∀ a, p a → p (pred a)) (a : α) : p 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": "lemma SuccOrder.forall_ne_bot_iff\n [Nontrivial α] [PartialOrder α] [OrderBot α] [SuccOrder α] [IsSuccAr... |
[
{
"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 zero_zeroAtFilter [Zero β] [TopologicalSpace β] (l : Filter α) :\n ZeroAtFilter l (0 : α → 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 ZeroAtFilter.add [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β]\n {l : Filter α} {f g :... |
[
{
"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 ZeroAtFilter.neg [TopologicalSpace β] [AddGroup β] [ContinuousNeg β] {l : Filter α}\n {f : α → 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 ZeroAtFilter.smul {𝕜 : Type*} [TopologicalSpace 𝕜] [TopologicalSpace β] [Zero 𝕜] [Zero β]\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 ZeroAtFilter.boundedAtFilter [NormedAddCommGroup β] {l : Filter α} {f : α → β}\n (hf : ZeroAtFil... |
[
{
"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 const_boundedAtFilter [NormedField β] (l : Filter α) (c : β) :\n BoundedAtFilter l (Function.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 BoundedAtFilter.add [NormedAddCommGroup β] {l : Filter α} {f g : α → β}\n (hf : BoundedAtFilter ... |
[
{
"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 BoundedAtFilter.neg [NormedAddCommGroup β] {l : Filter α} {f : α → β}\n (hf : BoundedAtFilter l ... |
[
{
"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 BoundedAtFilter.smul {𝕜 : Type*} [NormedField 𝕜] [NormedAddCommGroup β] [NormedSpace 𝕜 β]\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 BoundedAtFilter.mul [NormedField β] {l : Filter α} {f g : α → β}\n (hf : BoundedAtFilter l f) (h... |
[
{
"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 HasBasis.subsingleton_iff {ι : Sort*} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) :\n l.... |
[
{
"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 Subsingleton.anti {l'} (hl : l.Subsingleton) (hl' : l' ≤ l) : l'.Subsingleton :="
},
{
"rol... |
[
{
"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 Subsingleton.of_subsingleton [Subsingleton α] : l.Subsingleton :="
},
{
"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 Subsingleton.map (hl : l.Subsingleton) (f : α → β) : (map f l).Subsingleton :="
},
{
"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 Subsingleton.prod (hl : l.Subsingleton) {l' : Filter β} (hl' : l'.Subsingleton) :\n (l ×ˢ l').Su... |
[
{
"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 subsingleton_pure {a : α} : Filter.Subsingleton (pure a) :="
},
{
"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 subsingleton_bot : Filter.Subsingleton (⊥ : Filter α) :="
},
{
"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 Subsingleton.exists_eq_pure [l.NeBot] (hl : l.Subsingleton) : ∃ a, l = pure a :="
},
{
"rol... |
[
{
"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 subsingleton_iff_bot_or_pure : l.Subsingleton ↔ l = ⊥ ∨ ∃ a, l = pure a :="
},
{
"role": "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 subsingleton_iff_exists_le_pure [Nonempty α] : l.Subsingleton ↔ ∃ a, l ≤ pure a :="
},
{
"r... |
[
{
"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 subsingleton_iff_exists_singleton_mem [Nonempty α] : l.Subsingleton ↔ ∃ a, {a} ∈ l :="
},
{
... |
[
{
"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": "lemma Subsingleton.isCountablyGenerated (hl : l.Subsingleton) : IsCountablyGenerated l :="
},
{
"ro... |
[
{
"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 Nat.comap_cast_atTop [StrictOrderedSemiring R] [Archimedean R] :\n comap ((↑) : ℕ → R) atTop = 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 tendsto_nat_cast_atTop_iff [StrictOrderedSemiring R] [Archimedean R] {f : α → ℕ}\n {l : Filter α... |
[
{
"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 tendsto_nat_cast_atTop_atTop [StrictOrderedSemiring R] [Archimedean R] :\n Tendsto ((↑) : ℕ → R)... |
[
{
"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 Filter.Eventually.nat_cast_atTop [StrictOrderedSemiring R] [Archimedean R] {p : R → Prop}\n (h :... |
[
{
"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 Int.comap_cast_atTop [StrictOrderedRing R] [Archimedean R] :\n comap ((↑) : ℤ → R) atTop = atTop... |
[
{
"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 Int.comap_cast_atBot [StrictOrderedRing R] [Archimedean R] :\n comap ((↑) : ℤ → R) atBot = atBot... |
[
{
"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 tendsto_int_cast_atTop_iff [StrictOrderedRing R] [Archimedean R] {f : α → ℤ}\n {l : Filter α} : ... |
[
{
"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 tendsto_int_cast_atBot_iff [StrictOrderedRing R] [Archimedean R] {f : α → ℤ}\n {l : Filter α} : ... |
[
{
"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 tendsto_int_cast_atTop_atTop [StrictOrderedRing R] [Archimedean R] :\n Tendsto ((↑) : ℤ → R) atT... |
[
{
"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 Filter.Eventually.int_cast_atTop [StrictOrderedRing R] [Archimedean R] {p : R → Prop}\n (h : ∀ᶠ ... |
[
{
"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 Filter.Eventually.int_cast_atBot [StrictOrderedRing R] [Archimedean R] {p : R → Prop}\n (h : ∀ᶠ ... |
[
{
"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 Rat.comap_cast_atTop [LinearOrderedField R] [Archimedean R] :\n comap ((↑) : ℚ → R) atTop = atTo... |
[
{
"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 Rat.comap_cast_atBot [LinearOrderedField R] [Archimedean R] :\n comap ((↑) : ℚ → R) atBot = atBo... |
[
{
"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 tendsto_rat_cast_atTop_iff [LinearOrderedField R] [Archimedean R] {f : α → ℚ}\n {l : Filter α} :... |
[
{
"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 tendsto_rat_cast_atBot_iff [LinearOrderedField R] [Archimedean R] {f : α → ℚ}\n {l : Filter α} :... |
[
{
"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 Filter.Eventually.rat_cast_atTop [LinearOrderedField R] [Archimedean R] {p : R → Prop}\n (h : ∀ᶠ... |
[
{
"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 Filter.Eventually.rat_cast_atBot [LinearOrderedField R] [Archimedean R] {p : R → Prop}\n (h : ∀ᶠ... |
[
{
"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 atTop_hasAntitoneBasis_of_archimedean [StrictOrderedSemiring R] [Archimedean R] :\n (atTop : Fil... |
[
{
"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 atTop_hasCountableBasis_of_archimedean [StrictOrderedSemiring R] [Archimedean R] :\n (atTop : Fi... |
[
{
"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 atBot_hasCountableBasis_of_archimedean [LinearOrderedRing R] [Archimedean R] :\n (atBot : Filter... |
[
{
"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 Tendsto.const_mul_atTop' (hr : 0 < r) (hf : Tendsto f l atTop) :\n Tendsto (fun x => r * f x) l ... |
[
{
"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 Tendsto.atTop_mul_const' (hr : 0 < r) (hf : Tendsto f l atTop) :\n Tendsto (fun x => f x * r) l ... |
[
{
"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 Tendsto.atTop_mul_neg_const' (hr : r < 0) (hf : Tendsto f l atTop) :\n Tendsto (fun x => f x * r... |
[
{
"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 Tendsto.atBot_mul_const' (hr : 0 < r) (hf : Tendsto f l atBot) :\n Tendsto (fun x => f x * r) l ... |
[
{
"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 Tendsto.atBot_mul_neg_const' (hr : r < 0) (hf : Tendsto f l atBot) :\n Tendsto (fun x => f x * r... |
[
{
"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 Tendsto.atTop_nsmul_const {f : α → ℕ} (hr : 0 < r) (hf : Tendsto f l atTop) :\n Tendsto (fun x =... |
[
{
"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 Tendsto.atTop_nsmul_neg_const {f : α → ℕ} (hr : r < 0) (hf : Tendsto f l atTop) :\n Tendsto (fun... |
[
{
"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 Tendsto.atTop_zsmul_const {f : α → ℤ} (hr : 0 < r) (hf : Tendsto f l atTop) :\n Tendsto (fun x =... |
[
{
"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 Tendsto.atTop_zsmul_neg_const {f : α → ℤ} (hr : r < 0) (hf : Tendsto f l atTop) :\n Tendsto (fun... |
[
{
"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 Tendsto.atBot_zsmul_const {f : α → ℤ} (hr : 0 < r) (hf : Tendsto f l atBot) :\n Tendsto (fun x =... |
[
{
"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 Tendsto.atBot_zsmul_neg_const {f : α → ℤ} (hr : r < 0) (hf : Tendsto f l atBot) :\n Tendsto (fun... |
[
{
"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 isUpperSet_empty : IsUpperSet (∅ : Set α) :="
},
{
"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 isLowerSet_empty : IsLowerSet (∅ : Set α) :="
},
{
"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 isUpperSet_univ : IsUpperSet (univ : Set α) :="
},
{
"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 isLowerSet_univ : IsLowerSet (univ : Set α) :="
},
{
"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 IsUpperSet.compl (hs : IsUpperSet s) : IsLowerSet sᶜ :="
},
{
"role": "assistant",
"con... |
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