mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ ∀ x ∈ ↑I, ((Algebra.lsmul R R K) (I'.divMod I J')) x ∈ ↑I'

pp: case refine_2
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Submodule.comap (↑I).subtype ↑J ≤
    Submodule.comap (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ?refine_1)
      (Submodule.comap (↑I').subtype ↑J')

pp: case refine_3
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Injective
    ⇑((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ?refine_1) ?refine_2)

pp: case refine_4
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Surjective
    ⇑((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ?refine_1) ?refine_2)

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ ∀ x ∈ ↑I, ((Algebra.lsmul R R K) (I'.divMod I J')) x ∈ ↑I'
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: case inl
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : Iio x = ∅
⊢ Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x)))

pp: case inr
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
⊢ Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x)))

Tactic:
·

Goal state after:
case inl
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : Iio x = ∅
⊢ Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x)))
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: case inr.refine_1
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
l : β
hl : l < sSup (f '' Iio x)
⊢ ∀ᶠ (b : α) in 𝓝[<] x, l < f b

pp: case inr.refine_2
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
m : β
hm : m > sSup (f '' Iio x)
⊢ ∀ᶠ (b : α) in 𝓝[<] x, f b < m

Tactic:
·

Goal state after:
case inr.refine_1
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
l : β
hl : l < sSup (f '' Iio x)
⊢ ∀ᶠ (b : α) in 𝓝[<] x, l < f b
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Submodule.comap (↑I).subtype ↑J ≤
    Submodule.comap (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) (Submodule.comap (↑I').subtype ↑J')

pp: case refine_3
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Injective
    ⇑((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) ?refine_2)

pp: case refine_4
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Surjective
    ⇑((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) ?refine_2)

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Submodule.comap (↑I).subtype ↑J ≤
    Submodule.comap (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) (Submodule.comap (↑I').subtype ↑J')
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: case inr.refine_2
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
m : β
hm : m > sSup (f '' Iio x)
⊢ ∀ᶠ (b : α) in 𝓝[<] x, f b < m

Tactic:
·

Goal state after:
case inr.refine_2
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
m : β
hm : m > sSup (f '' Iio x)
⊢ ∀ᶠ (b : α) in 𝓝[<] x, f b < m
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: α✝ : Type u_1
α : Type u_2
inst✝² : CompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
⊢ IsCompact univ

Tactic:
simp only [← Icc_bot_top, isCompact_Icc]

Goal state after:
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case refine_1
α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : CompactIccSpace α
s : Set α
hs : s ∈ atBot ⊔ atTop
h✝ : IsEmpty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s

pp: case refine_2
α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : CompactIccSpace α
s : Set α
hs : s ∈ atBot ⊔ atTop
h✝ : Nonempty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s

Tactic:
·

Goal state after:
case refine_1
α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : CompactIccSpace α
s : Set α
hs : s ∈ atBot ⊔ atTop
h✝ : IsEmpty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case refine_2
α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : CompactIccSpace α
s : Set α
hs : s ∈ atBot ⊔ atTop
h✝ : Nonempty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s

Tactic:
·

Goal state after:
case refine_2
α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : CompactIccSpace α
s : Set α
hs : s ∈ atBot ⊔ atTop
h✝ : Nonempty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case refine_3
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Injective
    ⇑((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) ⋯)

pp: case refine_4
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Surjective
    ⇑((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) ⋯)

Tactic:
·

Goal state after:
case refine_3
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ Injective
    ⇑((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) ⋯)
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case refine_1
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTop α
inst✝ : CompactIccSpace α
x✝ : Set α
hs : x✝ ∈ atBot
h✝ : IsEmpty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ x✝

pp: case refine_2
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTop α
inst✝ : CompactIccSpace α
x✝ : Set α
hs : x✝ ∈ atBot
h✝ : Nonempty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ x✝

Tactic:
·

Goal state after:
case refine_1
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTop α
inst✝ : CompactIccSpace α
x✝ : Set α
hs : x✝ ∈ atBot
h✝ : IsEmpty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ x✝
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case refine_2
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTop α
inst✝ : CompactIccSpace α
x✝ : Set α
hs : x✝ ∈ atBot
h✝ : Nonempty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ x✝

Tactic:
·

Goal state after:
case refine_2
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTop α
inst✝ : CompactIccSpace α
x✝ : Set α
hs : x✝ ∈ atBot
h✝ : Nonempty α
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ x✝
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case refine_1
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : NoMinOrder α
inst✝ : ClosedIicTopology α
s : Set α
hs : s ∈ cocompact α
t : Set α
ht : IsCompact t
hts : tᶜ ⊆ s
h_empty : t = ∅
⊢ s ∈ atBot

pp: case refine_2
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : NoMinOrder α
inst✝ : ClosedIicTopology α
s : Set α
hs : s ∈ cocompact α
t : Set α
ht : IsCompact t
hts : tᶜ ⊆ s
h_nonempty : t.Nonempty
⊢ s ∈ atBot

Tactic:
·

Goal state after:
case refine_1
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : NoMinOrder α
inst✝ : ClosedIicTopology α
s : Set α
hs : s ∈ cocompact α
t : Set α
ht : IsCompact t
hts : tᶜ ⊆ s
h_empty : t = ∅
⊢ s ∈ atBot
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case inl
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
hs : IsPreconnected ∅
⊢ ∅ ∈
    {Icc (sInf ∅) (sSup ∅), Ico (sInf ∅) (sSup ∅), Ioc (sInf ∅) (sSup ∅), Ioo (sInf ∅) (sSup ∅), Ici (sInf ∅),
      Ioi (sInf ∅), Iic (sSup ∅), Iio (sSup ∅), univ, ∅}

pp: case inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
·

Goal state after:
case inl
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
hs : IsPreconnected ∅
⊢ ∅ ∈
    {Icc (sInf ∅) (sSup ∅), Ico (sInf ∅) (sSup ∅), Ioc (sInf ∅) (sSup ∅), Ioo (sInf ∅) (sSup ∅), Ici (sInf ∅),
      Ioi (sInf ∅), Iic (sSup ∅), Iio (sSup ∅), univ, ∅}
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case refine_2
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : NoMinOrder α
inst✝ : ClosedIicTopology α
s : Set α
hs : s ∈ cocompact α
t : Set α
ht : IsCompact t
hts : tᶜ ⊆ s
h_nonempty : t.Nonempty
⊢ s ∈ atBot

Tactic:
·

Goal state after:
case refine_2
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : NoMinOrder α
inst✝ : ClosedIicTopology α
s : Set α
hs : s ∈ cocompact α
t : Set α
ht : IsCompact t
hts : tᶜ ⊆ s
h_nonempty : t.Nonempty
⊢ s ∈ atBot
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inl
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
a : α
hs : IsCompact ∅
hf : ContinuousOn f ∅
hf' : ∀ b ∈ ∅, a < f b
⊢ ∃ a', a < a' ∧ ∀ b ∈ ∅, a' ≤ f b

pp: case inr
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
s : Set β
hs : IsCompact s
hf : ContinuousOn f s
a : α
hf' : ∀ b ∈ s, a < f b
hs' : s.Nonempty
⊢ ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b

Tactic:
·

Goal state after:
case inl
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
a : α
hs : IsCompact ∅
hf : ContinuousOn f ∅
hf' : ∀ b ∈ ∅, a < f b
⊢ ∃ a', a < a' ∧ ∀ b ∈ ∅, a' ≤ f b
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inr
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
s : Set β
hs : IsCompact s
hf : ContinuousOn f s
a : α
hf' : ∀ b ∈ s, a < f b
hs' : s.Nonempty
⊢ ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b

Tactic:
·

Goal state after:
case inr
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
s : Set β
hs : IsCompact s
hf : ContinuousOn f s
a : α
hf' : ∀ b ∈ s, a < f b
hs' : s.Nonempty
⊢ ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case pos
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case neg
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case pos
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : ¬BddBelow s
ha : BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case neg
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : ¬BddBelow s
ha : ¬BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
·

Goal state after:
case pos
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inl
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : ClosedIicTopology α
inst✝ : Nonempty α
hs : IsCompact ∅
⊢ BddBelow ∅

pp: case inr
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : ClosedIicTopology α
inst✝ : Nonempty α
s : Set α
hs : IsCompact s
hne : s.Nonempty
⊢ BddBelow s

Tactic:
·

Goal state after:
case inl
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : ClosedIicTopology α
inst✝ : Nonempty α
hs : IsCompact ∅
⊢ BddBelow ∅
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case neg
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case pos
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : ¬BddBelow s
ha : BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case neg
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : ¬BddBelow s
ha : ¬BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
·

Goal state after:
case neg
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
⊢ s ∈
    {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s),
      Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inr
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : ClosedIicTopology α
inst✝ : Nonempty α
s : Set α
hs : IsCompact s
hne : s.Nonempty
⊢ BddBelow s

Tactic:
·

Goal state after:
case inr
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : ClosedIicTopology α
inst✝ : Nonempty α
s : Set α
hs : IsCompact s
hne : s.Nonempty
⊢ BddBelow s
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case neg.inl
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s = Ici (sInf s)
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case neg.inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s ∈ {Ioi (sInf s)}
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
·

Goal state after:
case neg.inl
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s = Ici (sInf s)
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case neg.inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s ∈ {Ioi (sInf s)}
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
·

Goal state after:
case neg.inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs✝ : IsPreconnected s
hne : s.Nonempty
hs' : IsConnected s
hb : BddBelow s
ha : ¬BddAbove s
hs : s ∈ {Ioi (sInf s)}
⊢ s ∈ {Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅}
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inl
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : γ → β → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f
hK : IsCompact ∅
⊢ Continuous[inst✝, inst✝³] fun x ↦ sSup (f x '' ∅)

pp: case inr
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : γ → β → α
K : Set β
hK : IsCompact K
hf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f
h0K : K.Nonempty
⊢ Continuous[inst✝, inst✝³] fun x ↦ sSup (f x '' K)

Tactic:
·

Goal state after:
case inl
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : γ → β → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f
hK : IsCompact ∅
⊢ Continuous[inst✝, inst✝³] fun x ↦ sSup (f x '' ∅)