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

Goal state before:
pp: case pos
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
H' : I'.divMod I J' = 0
⊢ ↑I ⊓ Submodule.comap ((Algebra.lsmul R R K) (I'.divMod I J')) ↑J' ≤ ↑J

pp: case neg
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
H' : ¬I'.divMod I J' = 0
⊢ ↑I ⊓ Submodule.comap ((Algebra.lsmul R R K) (I'.divMod I J')) ↑J' ≤ ↑J

Tactic:
·

Goal state after:
case pos
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
H' : I'.divMod I J' = 0
⊢ ↑I ⊓ Submodule.comap ((Algebra.lsmul R R K) (I'.divMod I J')) ↑J' ≤ ↑J
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, ∅}

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 inr.refine_1
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
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z < f x y
⊢ ∀ᶠ (b : γ) in 𝓝 x, z < sSup (f b '' K)

pp: case inr.refine_2
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
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z > f x y
⊢ ∀ᶠ (b : γ) in 𝓝 x, sSup (f b '' K) < z

Tactic:
·

Goal state after:
case inr.refine_1
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
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z < f x y
⊢ ∀ᶠ (b : γ) in 𝓝 x, z < sSup (f b '' K)
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case pos.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 = Iic (sSup s)
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}

pp: case pos.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 ∈ {Iio (sSup s)}
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
·

Goal state after:
case pos.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 = Iic (sSup s)
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case pos.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 ∈ {Iio (sSup s)}
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}

Tactic:
·

Goal state after:
case pos.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 ∈ {Iio (sSup s)}
⊢ s ∈ {Iic (sSup s), Iio (sSup s), univ, ∅}
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inr.refine_2
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
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z > f x y
⊢ ∀ᶠ (b : γ) in 𝓝 x, sSup (f b '' K) < z

Tactic:
·

Goal state after:
case inr.refine_2
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
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z > f x y
⊢ ∀ᶠ (b : γ) in 𝓝 x, sSup (f b '' K) < z
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, ∅}

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.IntermediateValue

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
s : Set α
hs : IsPreconnected 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:
rcases s.eq_empty_or_nonempty with (rfl | hne)
  · apply_rules [Or.inr, mem_singleton]
  have hs' : IsConnected s := ⟨hne, hs⟩
  by_cases hb : BddBelow s <;> by_cases ha : BddAbove s
  · refine mem_of_subset_of_mem ?_ <| mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset
      (hs'.Ioo_csInf_csSup_subset hb ha) (subset_Icc_csInf_csSup hb ha)
    simp only [insert_subset_iff, mem_insert_iff, mem_singleton_iff, true_or, or_true,
      singleton_subset_iff, and_self]
  · refine Or.inr <| Or.inr <| Or.inr <| Or.inr ?_
    rcases mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_csInf_subset hb ha) fun x hx ↦
      csInf_le hb hx with hs | hs
    · exact Or.inl hs
    · exact Or.inr (Or.inl hs)
  · iterate 6 apply Or.inr
    rcases mem_Iic_Iio_of_subset_of_subset (hs.Iio_csSup_subset hb ha) fun x hx ↦
      le_csSup ha hx with hs | hs
    · exact Or.inl hs
    · exact Or.inr (Or.inl hs)
  · iterate 8 apply Or.inr
    exact Or.inl (hs.eq_univ_of_unbounded hb ha)

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace

Goal state before:
pp: case inl
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
⊢ SigmaCompactSpace α

pp: case inr
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
hn : ¬Nonempty α
⊢ SigmaCompactSpace α

Tactic:
·

Goal state after:
case inl
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
⊢ SigmaCompactSpace α
Module: Mathlib.Topology.MetricSpace.ProperSpace

Goal state before:
pp: case inr
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
hn : ¬Nonempty α
⊢ SigmaCompactSpace α

Tactic:
·

Goal state after:
case inr
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
hn : ¬Nonempty α
⊢ SigmaCompactSpace α
Module: Mathlib.Topology.MetricSpace.Cauchy

Goal state before:
pp: case mp
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
⊢ UniformCauchySeqOn F atTop s → ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε

pp: case mpr
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
⊢ (∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε) → UniformCauchySeqOn F atTop s

Tactic:
·

Goal state after:
case mp
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
⊢ UniformCauchySeqOn F atTop s → ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε
Module: Mathlib.Topology.MetricSpace.Cauchy

Goal state before:
pp: case mpr
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
⊢ (∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε) → UniformCauchySeqOn F atTop s

Tactic:
·

Goal state after:
case mpr
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
γ : Type u_3
F : β → γ → α
s : Set γ
⊢ (∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε) → UniformCauchySeqOn F atTop s
Module: Mathlib.Topology.MetricSpace.Cauchy

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : ℝ
R0 : R > 0
H : ∀ (n : ℕ), dist (u n) (u N) < R
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R

pp: α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
⊢ ∃ R > 0, ∀ (n : ℕ), dist (u n) (u N) < R

Tactic:
·

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : ℝ
R0 : R > 0
H : ∀ (n : ℕ), dist (u n) (u N) < R
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R
Module: Mathlib.Topology.MetricSpace.Cauchy

Goal state before:
pp: case inl
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun n ↦ nndist (u n) (u N)
n : ℕ
h : N ≤ n
⊢ dist (u n) (u N) < ↑R + 1

pp: case inr
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun n ↦ nndist (u n) (u N)
n : ℕ
h : n < N
⊢ dist (u n) (u N) < ↑R + 1

Tactic:
·

Goal state after:
case inl
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun n ↦ nndist (u n) (u N)
n : ℕ
h : N ≤ n
⊢ dist (u n) (u N) < ↑R + 1
Module: Mathlib.Topology.MetricSpace.Cauchy

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun n ↦ nndist (u n) (u N)
n : ℕ
h : n < N
⊢ dist (u n) (u N) < ↑R + 1

Tactic:
·

Goal state after:
case inr
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
R : NNReal := (Finset.range N).sup fun n ↦ nndist (u n) (u N)
n : ℕ
h : n < N
⊢ dist (u n) (u N) < ↑R + 1
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖x‖ = 0
⊢ ‖x * y‖ ≤ ‖y‖

pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖x‖ = 0
⊢ ‖y‖ ≤ ‖x * y‖

Tactic:
·

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖x‖ = 0
⊢ ‖x * y‖ ≤ ‖y‖
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖x‖ = 0
⊢ ‖y‖ ≤ ‖x * y‖

Tactic:
·

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖x‖ = 0
⊢ ‖y‖ ≤ ‖x * y‖
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
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_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')) ⋯) ⋯)
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x * y‖ ≤ ‖x‖

pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x‖ ≤ ‖x * y‖

Tactic:
·

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x * y‖ ≤ ‖x‖
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x‖ ≤ ‖x * y‖

Tactic:
·

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x‖ ≤ ‖x * y‖
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
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty
S : Set α := s ∩ Icc a b
ha : a ∈ S
Sbd : BddAbove S
c : α := sSup (s ∩ Icc a b)
c_mem : c ∈ S
c_le : c ≤ b
hc : c = b
⊢ b ∈ s

pp: case inr
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty
S : Set α := s ∩ Icc a b
ha : a ∈ S
Sbd : BddAbove S
c : α := sSup (s ∩ Icc a b)
c_mem : c ∈ S
c_le : c ≤ b
hc : c < b
⊢ b ∈ s

Tactic:
·

Goal state after:
case inl
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty
S : Set α := s ∩ Icc a b
ha : a ∈ S
Sbd : BddAbove S
c : α := sSup (s ∩ Icc a b)
c_mem : c ∈ S
c_le : c ≤ b
hc : c = b
⊢ b ∈ s
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
r : ℝ
⊢ a ∈ closedBall 1 r ↔ ‖a‖ ≤ r

Tactic:
rw [mem_closedBall, dist_one_right]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
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
H : Submodule.map ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I = ↑(spanSingleton R⁰ (I'.divMod I J')) * ↑I
⊢ Submodule.map (Submodule.comap (↑I').subtype ↑J').mkQ
      (LinearMap.codRestrict (↑I') (LinearMap.domRestrict ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I) ⋯).range =
    ⊤

Tactic:
LinearMap.range_codRestrict,

Goal state after:
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
H : Submodule.map ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I = ↑(spanSingleton R⁰ (I'.divMod I J')) * ↑I
⊢ Submodule.map (Submodule.comap (↑I').subtype ↑J').mkQ
      (Submodule.comap (↑I').subtype (LinearMap.domRestrict ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I).range) =
    ⊤
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case inl
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : b < a
⊢ Icc a b ⊆ s

pp: case inr
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
⊢ Icc a b ⊆ s

Tactic:
·

Goal state after:
case inl
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : b < a
⊢ Icc a b ⊆ s
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case inl
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (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 | 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

pp: case inr
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (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 | 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

Tactic:
·

Goal state after:
case inl
α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (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 | 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
Module: Mathlib.Topology.EMetricSpace.Diam

Goal state before:
pp: case inl.inl
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ s
h'b : b ∈ s
⊢ edist a b ≤ ediam s + edist x y + ediam t

pp: case inl.inr
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ s
h'b : b ∈ t
⊢ edist a b ≤ ediam s + edist x y + ediam t

pp: case inr.inl
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ t
h'b : b ∈ s
⊢ edist a b ≤ ediam s + edist x y + ediam t

pp: case inr.inr
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ t
h'b : b ∈ t
⊢ edist a b ≤ ediam s + edist x y + ediam t

Tactic:
·

Goal state after:
case inl.inl
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ s
h'b : b ∈ s
⊢ edist a b ≤ ediam s + edist x y + ediam t
Module: Mathlib.Topology.EMetricSpace.Diam

Goal state before:
pp: case inl.inr
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ s
h'b : b ∈ t
⊢ edist a b ≤ ediam s + edist x y + ediam t

pp: case inr.inl
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ t
h'b : b ∈ s
⊢ edist a b ≤ ediam s + edist x y + ediam t

pp: case inr.inr
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ t
h'b : b ∈ t
⊢ edist a b ≤ ediam s + edist x y + ediam t

Tactic:
·

Goal state after:
case inl.inr
X : Type u_2
s t : Set X
x y : X
inst✝ : PseudoEMetricSpace X
xs : x ∈ s
yt : y ∈ t
A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ ediam s + edist x y + ediam t
a b : X
h'a : a ∈ s
h'b : b ∈ t
⊢ edist a b ≤ ediam s + edist x y + ediam t