state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
C0 : 0 < C
⊢ (∀ (x : α), dist ((mkOfCompact f) x) ((mkOfCompact g) x) < C) ↔ ∀ (x : α), dist (f x) (g x) < C | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [mkOfCompact_apply] | theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
| Mathlib.Topology.ContinuousFunction.Compact.152_0.Mig2jTVnn2FLKEB | theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
src✝¹ : MetricSpace C(α, E) := metricSpace α E
src✝ : AddCommGroup C(α, E) := instAddCommGroupContinuousMap
x y : C(α, E)
⊢ dist x y = ‖x - y‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub] | instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
| Mathlib.Topology.ContinuousFunction.Compact.185_0.Mig2jTVnn2FLKEB | instance : NormedAddCommGroup C(α, E) | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁶ : TopologicalSpace α
inst✝⁵ : CompactSpace α
inst✝⁴ : MetricSpace β
inst✝³ : NormedAddCommGroup E
inst✝² : Nonempty α
inst✝¹ : One E
inst✝ : NormOneClass E
⊢ ‖1‖ = 1 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one] | instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by | Mathlib.Topology.ContinuousFunction.Compact.193_0.Mig2jTVnn2FLKEB | instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CompactSpace α
inst✝³ : MetricSpace β
inst✝² : NormedAddCommGroup E
𝕜 : Type u_4
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
src✝ : C(α, E) ≃+ (α →ᵇ E) := addEquivBoundedOfCompact α E
c : 𝕜
f : C(α, E)
⊢ AddHom.toFun
{ toFun := src... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | ext | /-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
| Mathlib.Topology.ContinuousFunction.Compact.277_0.Mig2jTVnn2FLKEB | /-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E | Mathlib_Topology_ContinuousFunction_Compact |
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CompactSpace α
inst✝³ : MetricSpace β
inst✝² : NormedAddCommGroup E
𝕜 : Type u_4
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
src✝ : C(α, E) ≃+ (α →ᵇ E) := addEquivBoundedOfCompact α E
c : 𝕜
f : C(α, E)
x✝ : α
⊢ (AddHom.toFun
... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | norm_cast | /-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
| Mathlib.Topology.ContinuousFunction.Compact.277_0.Mig2jTVnn2FLKEB | /-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
𝕜 : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁶ : TopologicalSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup β
inst✝² : NormedSpace 𝕜 β
inst✝¹ : NormedAddCommGroup γ
inst✝ : NormedSpace 𝕜 γ
g : β →L[𝕜] γ
⊢ ↑(ContinuousLinearMap.compLeftContinuousCompact X g... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | ext f | @[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
| Mathlib.Topology.ContinuousFunction.Compact.406_0.Mig2jTVnn2FLKEB | @[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X | Mathlib_Topology_ContinuousFunction_Compact |
case h.h
X : Type u_1
𝕜 : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁶ : TopologicalSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup β
inst✝² : NormedSpace 𝕜 β
inst✝¹ : NormedAddCommGroup γ
inst✝ : NormedSpace 𝕜 γ
g : β →L[𝕜] γ
f : C(X, β)
a✝ : X
⊢ (↑(ContinuousLinearMap.... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rfl | @[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
| Mathlib.Topology.ContinuousFunction.Compact.406_0.Mig2jTVnn2FLKEB | @[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
⊢ Continuous fun g => comp g f | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | refine' Metric.continuous_iff.mpr _ | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
c... | Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
⊢ ∀ (b : C(Y, T)), ∀ ε > 0, ∃ δ > 0, ∀ (a : C(Y, T)), dist a b < δ → dist (comp a f) (comp b f) < ε | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | intro g ε ε_pos | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
c... | Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
g : C(Y, T)
ε : ℝ
ε_pos : ε > 0
⊢ ∃ δ > 0, ∀ (a : C(Y, T)), dist a g < δ → dist (comp a f) (comp g f) < ε | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | refine' ⟨ε, ε_pos, fun g' h => _⟩ | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
c... | Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
g : C(Y, T)
ε : ℝ
ε_pos : ε > 0
g' : C(Y, T)
h : dist g' g < ε
⊢ dist (comp g' f) (comp g f) < ε | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢ | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
c... | Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
g : C(Y, T)
ε : ℝ
ε_pos : ε > 0
g' : C(Y, T)
h : ∀ (x : Y), dist (g' x) (g x) < ε
⊢ ∀ (x : X), dist ((comp g' f) x) ((comp g f) x) < ε | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | exact fun x => h (f x) | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
c... | Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB | /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
⊢ Summable F | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _ | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
| Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Set X
hK : IsCompact K
⊢ ∃ f, Tendsto (fun i => restrict K (∑ b ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | lift K to Compacts X using hK | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
| Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
case intro
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
⊢ ∃ f, Tendsto (fun i => restrict (↑K) (∑ ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]... | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
| Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
⊢ ∀ (s : Finset ι), restrict (↑K) (∑ i in s, F i) = ∑... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | intro s | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
⊢ restrict (↑K) (∑ i in s, F i) = ∑ i in... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | ext1 x | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (restrict (↑K) (∑ i in ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (restrict (↑K) (∑ i in ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply] | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (∑ i in s, F i) ↑x = ∑ ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp? says simp only [coe_sum, Finset.sum_apply] | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (∑ i in s, F i) ↑x = ∑ ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [coe_sum, Finset.sum_apply] | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ ∑ c in s, (F c) ↑x = ∑ ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | congr! | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
case intro
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
A : ∀ (s : Finset ι), restrict (↑K) (∑ i i... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simpa only [HasSum, A] using (hF K).of_norm | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in ... | Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB | theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖star f‖ = ‖f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact] | instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
| Mathlib.Topology.ContinuousFunction.Compact.528_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ‖star f * f‖ = ‖f‖ * ‖f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | refine' le_antisymm _ _ | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
| Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ‖star f * f‖ ≤ ‖f‖ * ‖f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)] | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ∀ (x : α), ‖(star f * f) x‖ ≤ ‖f‖ ^ 2 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | intro x | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
| Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖(star f * f) x‖ ≤ ‖f‖ ^ 2 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq] | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
| Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ ^ 2 ≤ ‖f‖ ^ 2 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | refine' sq_le_sq' _ _ | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
... | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_1.refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ -‖f‖ ≤ ‖f x‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | linarith [norm_nonneg (f x), norm_nonneg f] | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
... | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_1.refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ ≤ ‖f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | exact ContinuousMap.norm_coe_le_norm f x | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
... | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ‖f‖ * ‖f‖ ≤ ‖star f * f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)] | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
... | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ∀ (x : α), ‖f x‖ ≤ Real.sqrt ‖star f * f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | intro x | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
... | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ ≤ Real.sqrt ‖star f * f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rw [Real.le_sqrt (norm_nonneg _) (norm_nonneg _), sq, ← CstarRing.norm_star_mul_self] | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
... | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖star (f x) * f x‖ ≤ ‖star f * f‖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | exact ContinuousMap.norm_coe_le_norm (star f * f) x | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
... | Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB | instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} | Mathlib_Topology_ContinuousFunction_Compact |
⊢ Tendsto (fun n => ↑(numDerangements n) / ↑(Nat.factorial n)) atTop (𝓝 (Real.exp (-1))) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
⊢ Tendsto (fun n => ↑(numDerangements n) / ↑(Nat.factorial n)) atTop (𝓝 (Real.exp (-1))) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply... | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ Tendsto (fun n => ↑(numDerangements n) / ↑(Nat.factorial n)) atTop (𝓝 (Real.exp (-1))) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | simp_rw [this] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ Tendsto (fun n => ∑ k in Finset.range (n + 1), (-1) ^ k / ↑(Nat.factorial k)) atTop (𝓝 (Real.exp (-1))) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ Tendsto (fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)) atTop (𝓝 (Real.exp (-1))) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | apply HasSum.tendsto_sum_nat | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
case h
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ HasSum (fun i => (-1) ^ i / ↑(Nat.factorial i)) (Real.exp (-1)) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | rw [Real.exp_eq_exp_ℝ] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
case h
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ HasSum (fun i => (-1) ^ i / ↑(Nat.factorial i)) (exp ℝ (-1)) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | exact expSeries_div_hasSum_exp ℝ (-1 : ℝ) | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
⊢ ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | intro n | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | rw [← Int.cast_ofNat, numDerangements_sum] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ↑(∑ k in Finset.range (n + 1), (-1) ^ k * ↑(Nat.ascFactorial k (n - k))) / ↑(Nat.factorial n) = s (n + 1) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | push_cast | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ (∑ x in Finset.range (n + 1), (-1) ^ x * ↑(Nat.ascFactorial x (n - x))) / ↑(Nat.factorial n) =
∑ k in Finset.range (n + 1), (-1) ^ k / ↑(Nat.factorial k) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | rw [Finset.sum_div] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ∑ x in Finset.range (n + 1), (-1) ^ x * ↑(Nat.ascFactorial x (n - x)) / ↑(Nat.factorial n) =
∑ k in Finset.range (n + 1), (-1) ^ k / ↑(Nat.factorial k) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | refine' Finset.sum_congr (refl _) _ | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ∀ x ∈ Finset.range (n + 1),
(-1) ^ x * ↑(Nat.ascFactorial x (n - x)) / ↑(Nat.factorial n) = (-1) ^ x / ↑(Nat.factorial x) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | intro k hk | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
⊢ (-1) ^ k * ↑(Nat.ascFactorial k (n - k)) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * ↑(Nat.ascFactorial k (n - k)) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * ↑(Nat.factorial n / Nat.factorial k) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | push_cast [Nat.factorial_dvd_factorial h_le] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * (↑(Nat.factorial n) / ↑(Nat.factorial k)) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | field_simp [Nat.factorial_ne_zero] | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * ↑(Nat.factorial n) * ↑(Nat.factorial k) = (-1) ^ k * (↑(Nat.factorial k) * ↑(Nat.factorial n)) | /-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import c... | ring | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the ... | Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | Mathlib_Combinatorics_Derangements_Exponential |
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
⊢ (dite p a fun hp => dite q (b hp) (c hp)) = if hq : q then dite p a fun hp => b hp hq else dite p a fun hp => c hp hq | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | split_ifs | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
| Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq | Mathlib_Logic_Lemmas |
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : p
h✝ : q
⊢ a h✝¹ = a h✝¹ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> | Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq | Mathlib_Logic_Lemmas |
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : p
h✝ : ¬q
⊢ a h✝¹ = a h✝¹ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> | Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq | Mathlib_Logic_Lemmas |
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : ¬p
h✝ : q
⊢ b h✝¹ h✝ = b h✝¹ (_ : q) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> | Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq | Mathlib_Logic_Lemmas |
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : ¬p
h✝ : ¬q
⊢ c h✝¹ h✝ = c h✝¹ (_ : ¬q) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> | Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq | Mathlib_Logic_Lemmas |
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
⊢ dite p (fun hp => dite q (a hp) (b hp)) c =
if hq : q then dite p (fun hp => a hp hq) c else dite p (fun hp => b hp hq) c | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | split_ifs | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
| Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c | Mathlib_Logic_Lemmas |
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : p
h✝ : q
⊢ a h✝¹ h✝ = a h✝¹ (_ : q) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> | Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c | Mathlib_Logic_Lemmas |
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : p
h✝ : ¬q
⊢ b h✝¹ h✝ = b h✝¹ (_ : ¬q) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> | Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c | Mathlib_Logic_Lemmas |
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : ¬p
h✝ : q
⊢ c h✝¹ = c h✝¹ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> | Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c | Mathlib_Logic_Lemmas |
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : ¬p
h✝ : ¬q
⊢ c h✝¹ = c h✝¹ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rfl | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> | Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj | theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c | Mathlib_Logic_Lemmas |
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
⊢ f True ∧ f False → ∀ (p : Prop), f p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rintro ⟨h₁, h₀⟩ p | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by | Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False | Mathlib_Logic_Lemmas |
case intro
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
⊢ f p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | by_cases hp : p | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; | Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False | Mathlib_Logic_Lemmas |
case pos
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : p
⊢ f p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | simp only [hp] | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> | Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False | Mathlib_Logic_Lemmas |
case neg
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : ¬p
⊢ f p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | simp only [hp] | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> | Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False | Mathlib_Logic_Lemmas |
case pos
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : p
⊢ f True | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | assumption | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> | Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False | Mathlib_Logic_Lemmas |
case neg
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : ¬p
⊢ f False | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | assumption | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> | Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj | lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False | Mathlib_Logic_Lemmas |
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
⊢ f True ∨ f False | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | refine' (em p).imp _ _ | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case refine'_1
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
⊢ p → f True | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | intro H | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case refine'_2
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
⊢ ¬p → f False | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | intro H | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case refine'_1
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : p
⊢ f True | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | convert h | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case refine'_2
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : ¬p
⊢ f False | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | convert h | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case h.e'_1.a
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : p
⊢ True ↔ p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | simp [H] | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case h.e'_1.a
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : ¬p
⊢ False ↔ p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | simp [H] | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
⊢ f True ∨ f False → ∃ p, f p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | rintro (h | h) | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case inl
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h : f True
⊢ ∃ p, f p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | exact ⟨_, h⟩ | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by rintro (h | h) <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
case inr
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h : f False
⊢ ∃ p, f p | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a... | exact ⟨_, h⟩ | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by rintro (h | h) <;> | Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False | Mathlib_Logic_Lemmas |
C : Type u_1
inst✝¹ : Category.{?u.28, u_1} C
inst✝ : Abelian C
⊢ Abelian Cᵒᵖ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) } | instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
| Mathlib.CategoryTheory.Abelian.Opposite.31_0.3nBRs3fSYrCoEsT | instance : Abelian Cᵒᵖ | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.π f).op ≫ f.op = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [← op_comp] | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by | Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ f ≫ (kernel.ι f.op).unop = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | rw [← f.unop_op, ← unop_comp, f.unop_op] | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
| Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (kernel.ι f.op ≫ f.op).unop = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
| Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop ≫
cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0) =
𝟙 (kernel f.op).unop | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp] | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom... | Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0)).op ≫
kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop =
(𝟙 (kernel f.op)).unop | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | congr 1 | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom... | Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
case e_f
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0)).op ≫
kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0) =
𝟙 (kernel f.op) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | ext | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom... | Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
case e_f.h
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0)).op ≫
kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)) ≫
equalizer.ι f.op 0 =
𝟙 (kernel f.o... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [← op_comp] | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom... | Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0) ≫
(kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop =
𝟙 (cokernel f) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | ext | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom... | Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
case h
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ coequalizer.π f 0 ≫
cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0) ≫
(kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop =
coequalizer.π f 0 ≫... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [← unop_comp] | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom... | Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT | /-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.π f.op).unop ≫ f = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | rw [← f.unop_op, ← unop_comp, f.unop_op] | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
| Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (f.op ≫ cokernel.π f.op).unop = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
| Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ f.op ≫ (kernel.ι f).op = 0 | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [← op_comp] | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by | Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0) ≫
(cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0)).unop =
𝟙 (cokernel f.op).unop | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp] | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
h... | Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0) ≫
(kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)).op).unop =
(𝟙 (cokernel f.op)).unop | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | congr 1 | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
h... | Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
case e_f
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0) ≫
(kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)).op =
𝟙 (cokernel f.op) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | ext | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
h... | Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
case e_f.h
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ coequalizer.π f.op 0 ≫
cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0) ≫
(kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)).op =
coequalizer.π... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [← op_comp] | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
h... | Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0)).unop ≫
kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0) =
𝟙 (kernel f) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | ext | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
h... | Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
case h
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0)).unop ≫
kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)) ≫
equalizer.ι f 0 =
𝟙 (kernel f) ≫ eq... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [← unop_comp] | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
h... | Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT | /-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.π f.op).unop = (cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (_ : X = (Opposite.op X).unop) | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [cokernelOpUnop] | theorem cokernel.π_op :
(cokernel.π f.op).unop =
(cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (Opposite.unop_op _).symm :=
by | Mathlib.CategoryTheory.Abelian.Opposite.95_0.3nBRs3fSYrCoEsT | theorem cokernel.π_op :
(cokernel.π f.op).unop =
(cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (Opposite.unop_op _).symm | Mathlib_CategoryTheory_Abelian_Opposite |
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (kernel.ι f.op).unop = eqToHom (_ : (Opposite.op Y).unop = Y) ≫ cokernel.π f ≫ (kernelOpUnop f).inv | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | simp [kernelOpUnop] | theorem kernel.ι_op :
(kernel.ι f.op).unop = eqToHom (Opposite.unop_op _) ≫ cokernel.π f ≫ (kernelOpUnop f).inv := by
| Mathlib.CategoryTheory.Abelian.Opposite.101_0.3nBRs3fSYrCoEsT | theorem kernel.ι_op :
(kernel.ι f.op).unop = eqToHom (Opposite.unop_op _) ≫ cokernel.π f ≫ (kernelOpUnop f).inv | Mathlib_CategoryTheory_Abelian_Opposite |
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