awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
hb : b ∈ s
hgt : ∀ x ∈ s ∩ Ioc a b, ∀ y ∈ Iio x, (s ∩ Ico y x).Nonempty
y : α
hy : y ∈ Icc a b
⊢ IsClosed[inst] (s ∩ Icc y b)
have suffices : s ∩ Icc y b = s ∩ Icc a b ∩ Icc y b
Tactic: suffices s ∩ Icc y b = s ∩ Icc a b ∩ Icc y b from this ▸ hs.inter isClosed_Icc
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
hb : b ∈ s
hgt : ∀ x ∈ s ∩ Ioc a b, ∀ y ∈ Iio x, (s ∩ Ico y x).Nonempty
y : α
hy : y ∈ Icc a b
⊢ y ∈ s
have this : IsClosed[inst] (s ∩ Icc y b)
Tactic: have : IsClosed (s ∩ Icc y b) := by
    suffices s ∩ Icc y b = s ∩ Icc a b ∩ Icc y b from this ▸ hs.inter isClosed_Icc
    grind [Icc_subset_Icc_left, inter_eq_self_of_subset_right, inter_assoc]
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
⊢ Icc a b ⊆ s
have inl : ∀ (hab : b < a), Icc a b ⊆ s
Tactic: rcases lt_or_ge b a with hab | hab
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
inl : ∀ (hab : b < a), Icc a b ⊆ s
⊢ Icc a b ⊆ s
have inr : ∀ (hab : a ≤ b), Icc a b ⊆ s
Tactic: rcases lt_or_ge b a with hab | hab
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
⊢ Icc a b ⊆ s
have a_mem : a ∈ A
Tactic: have a_mem : a ∈ A := ⟨left_mem_Icc.mpr hab, by simp [ha]⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
⊢ Icc a b ⊆ s
have bdd_A : BddAbove A
Tactic: have bdd_A : BddAbove A := ⟨b, fun t ht ↦ ht.1.2⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
⊢ Icc a b ⊆ s
have t₁_mem : t₁ ∈ Icc a b
Tactic: have t₁_mem : t₁ ∈ Icc a b := ⟨le_csSup bdd_A a_mem, csSup_le ⟨a, a_mem⟩ (fun t ht ↦ ht.1.2)⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
⊢ t₁ ∈ A
have refine : ∀ (t : α) (ht : t ∈ Icc a t₁), t ∈ s
Tactic: refine ⟨t₁_mem, fun t ht ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
t : α
ht : t ∈ Icc a t₁
⊢ t ∈ s
have inl : ∀ (ht : t₁ ∈ Icc a t₁), t₁ ∈ s
Tactic: rcases ht.2.eq_or_lt with rfl | h
Cleared hypotheses: t, ht
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
t : α
ht : t ∈ Icc a t₁
inl : ∀ (ht : t₁ ∈ Icc a t₁), t₁ ∈ s
⊢ t ∈ s
have inr : ∀ (h : t < t₁), t ∈ s
Tactic: rcases ht.2.eq_or_lt with rfl | h
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht : t₁ ∈ Icc a t₁
⊢ closure[inst] A ⊆ s ∩ Icc a b
have apply : ∀ (t : α) (ht : t ∈ A), t ∈ s
Tactic: apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩)
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht_1 : t₁ ∈ Icc a t₁
t : α
ht : t ∈ A
⊢ t ∈ s
have this : t ∈ Icc a t
Tactic: have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht : t₁ ∈ Icc a t₁
⊢ t₁ ∈ s
have this : closure[inst] A ⊆ s ∩ Icc a b
Tactic: have : closure A ⊆ s ∩ Icc a b := by
        apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩)
        have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩
        exact ht.2 this
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht : t₁ ∈ Icc a t₁
this : closure[inst] A ⊆ s ∩ Icc a b
⊢ t₁ ∈ s
have inl : t₁ ∈ closure[inst] A
Tactic: apply this.trans inter_subset_left
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h_1 : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
t : α
ht : t ∈ Icc a t₁
h : t < t₁
⊢ t ∈ s
have inr : ∀ (c : α) (cA : c ∈ A) (tc : t < c), t ∈ s
Tactic: obtain ⟨c, cA, tc⟩ : ∃ c ∈ A, t < c := (lt_csSup_iff bdd_A ⟨a, a_mem⟩).1 h
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h_1 : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
t : α
ht : t ∈ Icc a t₁
h : t < t₁
c : α
cA : c ∈ A
tc : t < c
⊢ t ∈ s
have inr : t ∈ Icc a c
Tactic: apply cA.2
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
⊢ Icc a b ⊆ s
have ht₁ : Icc a t₁ ⊆ s
Tactic: obtain ⟨⟨t₁a, t₁b⟩, ht₁⟩ : t₁ ∈ A := by
    refine ⟨t₁_mem, fun t ht ↦ ?_⟩
    rcases ht.2.eq_or_lt with rfl | h
    · have : closure A ⊆ s ∩ Icc a b := by
        apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩)
        have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩
        exact ht.2 this
      apply this.trans inter_subset_left
      exact csSup_mem_closure ⟨a, a_mem⟩ bdd_A
    · obtain ⟨c, cA, tc⟩ : ∃ c ∈ A, t < c := (lt_csSup_iff bdd_A ⟨a, a_mem⟩).1 h
      apply cA.2
      exact ⟨ht.1, tc.le⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
⊢ Icc a b ⊆ s
have t₁a : a ≤ t₁
Tactic: obtain ⟨⟨t₁a, t₁b⟩, ht₁⟩ : t₁ ∈ A := by
    refine ⟨t₁_mem, fun t ht ↦ ?_⟩
    rcases ht.2.eq_or_lt with rfl | h
    · have : closure A ⊆ s ∩ Icc a b := by
        apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩)
        have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩
        exact ht.2 this
      apply this.trans inter_subset_left
      exact csSup_mem_closure ⟨a, a_mem⟩ bdd_A
    · obtain ⟨c, cA, tc⟩ : ∃ c ∈ A, t < c := (lt_csSup_iff bdd_A ⟨a, a_mem⟩).1 h
      apply cA.2
      exact ⟨ht.1, tc.le⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
⊢ Icc a b ⊆ s
have t₁b : t₁ ≤ b
Tactic: obtain ⟨⟨t₁a, t₁b⟩, ht₁⟩ : t₁ ∈ A := by
    refine ⟨t₁_mem, fun t ht ↦ ?_⟩
    rcases ht.2.eq_or_lt with rfl | h
    · have : closure A ⊆ s ∩ Icc a b := by
        apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩)
        have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩
        exact ht.2 this
      apply this.trans inter_subset_left
      exact csSup_mem_closure ⟨a, a_mem⟩ bdd_A
    · obtain ⟨c, cA, tc⟩ : ∃ c ∈ A, t < c := (lt_csSup_iff bdd_A ⟨a, a_mem⟩).1 h
      apply cA.2
      exact ⟨ht.1, tc.le⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
⊢ Icc a b ⊆ s
have inr : t₁ = b
Tactic: suffices t₁ = b by simpa [this] using ht₁
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
⊢ t₁ = b
have apply : ∀ (t₁b' : t₁ < b), False
Tactic: apply eq_of_le_of_not_lt t₁b fun t₁b' ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
⊢ False
have obtain : ∀ (m : α) (t₁m : m > t₁) (H : Ioo t₁ m ⊆ s), False
Tactic: obtain ⟨m, t₁m, H⟩ : ∃ m > t₁, Ioo t₁ m ⊆ s :=
    (mem_nhdsGT_iff_exists_Ioo_subset' t₁b').mp (h t₁ ⟨t₁a, t₁b'⟩ (fun s hs ↦ ht₁ hs))
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
⊢ False
have obtain : ∀ (t : α) (hat : t₁ < t) (ht : t < min m b), False
Tactic: obtain ⟨t, hat, ht⟩ : ∃ t, t₁ < t ∧ t < min m b := exists_between (lt_min t₁m t₁b')
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
⊢ a ≤ t
have order : ¬a ≤ t → False
Tactic: order
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
order order₁ order₂ order₃ : ¬a ≤ t → False
⊢ a ≤ t
have order₄ : t < a → False
Tactic: order
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
⊢ t ∈ A
have refine : ∀ (t' : α) (ht' : t' ∈ Icc a t), t' ∈ s
Tactic: refine ⟨⟨by order, ht.le.trans (min_le_right _ _)⟩, fun t' ht' ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
t' : α
ht' : t' ∈ Icc a t
⊢ t' ∈ s
have inl : ∀ (h't' : t' ≤ t₁), t' ∈ s
Tactic: rcases le_or_gt t' t₁ with h't' | h't'
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
t' : α
ht' : t' ∈ Icc a t
inl : ∀ (h't' : t' ≤ t₁), t' ∈ s
⊢ t' ∈ s
have inr : ∀ (h't' : t₁ < t'), t' ∈ s
Tactic: rcases le_or_gt t' t₁ with h't' | h't'
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
⊢ False
have this : t ∈ A
Tactic: have : t ∈ A := by
    refine ⟨⟨by order, ht.le.trans (min_le_right _ _)⟩, fun t' ht' ↦ ?_⟩
    rcases le_or_gt t' t₁ with h't' | h't'
    · exact ht₁ ⟨ht'.1, h't'⟩
    · exact H ⟨h't', ht'.2.trans_lt <| ht.trans_le <| min_le_left ..⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
this : t ∈ A
⊢ False
have this₁ : t ≤ t₁
Tactic: have : t ≤ t₁ := le_csSup bdd_A this
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
this_1 : t ∈ A
this : t ≤ t₁
⊢ False
have order : ¬False → False
Tactic: order
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t : α | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
this_1 : t ∈ A
this : t ≤ t₁
order order₁ order₂ order₃ : ¬False → False
⊢ False
have order₄ : True → False
Tactic: order
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst] s
ht : IsClosed[inst] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
⊢ (Icc a b ∩ (s ∩ t)).Nonempty
have xyab : Icc x y ⊆ Icc a b
Tactic: have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst] s
ht : IsClosed[inst] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
⊢ (Icc a b ∩ (s ∩ t)).Nonempty
have by_contra : ∀ (hst : ¬(Icc a b ∩ (s ∩ t)).Nonempty), False
Tactic: by_contra hst
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst] s
ht : IsClosed[inst] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬(Icc a b ∩ (s ∩ t)).Nonempty
⊢ False
have suffices : Icc x y ⊆ s
Tactic: suffices Icc x y ⊆ s from
    hst ⟨y, xyab <| right_mem_Icc.2 hxy, this <| right_mem_Icc.2 hxy, hy.2⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst] s
ht : IsClosed[inst] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬(Icc a b ∩ (s ∩ t)).Nonempty
⊢ Icc x y ⊆ s
have apply : ∀ x_1 ∈ s ∩ Ico x y, s ∈ 𝓝[>] x_1
Tactic: apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (hypothesis)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst] s
ht : IsClosed[inst] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬(Icc a b ∩ (s ∩ t)).Nonempty
z : α
zs : z ∈ s
hz : z ∈ Ico x y
⊢ s ∈ 𝓝[>] z
have zt : z ∈ tᶜ
Tactic: have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <| Ico_subset_Icc_self hz, zs, zt⟩
-- Lean 4 proof step from Mathlib.Topology.Order.IntermediateValue (goal)
Goal:
α : Type u
inst : TopologicalSpace α
inst_1 : ConditionallyCompleteLinearOrder α
inst_2 : OrderTopology α
inst_3 : DenselyOrdered α
a b x y : α
s t : Set α
hxy : x ≤ y
hs : IsClosed[inst] s
ht : IsClosed[inst] t
hab : Icc a b ⊆ s ∪ t
hx : x ∈ Icc a b ∩ s
hy : y ∈ Icc a b ∩ t
xyab : Icc x y ⊆ Icc a b
hst : ¬(Icc a b ∩ (s ∩ t)).Nonempty
z : α
zs : z ∈ s
hz : z ∈ Ico x y
zt : z ∈ tᶜ
⊢ tᶜ ∩ Ioc z y ∈ 𝓝[>] z
have rw : tᶜ ∩ Ioc z y ∈ 𝓝[Ioc z y] z
Tactic: rw [← nhdsWithin_Ioc_eq_nhdsGT hz.2]