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 |
|---|---|---|---|---|---|---|
case e_f.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply (set_integral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)).symm | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | contrapose! hy | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : (x, y) ∈ Function.support f.uncurry := hy | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact mem_image_of_mem _ (subset_tsupport _ this) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply (set_integral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)).symm | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : ∀ y, f x y = 0 := by
intro y
contrapose! hx
have : (x, y) ∈ Function.support f.uncurry := hx
exact mem_image_of_mem _ (subset_tsupport _ this) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro y | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | contrapose! hx | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : (x, y) ∈ Function.support f.uncurry := hx | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact mem_image_of_mem _ (subset_tsupport _ this) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp [this] | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply integral_integral_swap | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply (integrableOn_iff_integrable_of_support_subset (subset_tsupport f.uncurry)).mp | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | refine ⟨(h'f.stronglyMeasurable_of_prod hf).aestronglyMeasurable, ?_⟩ | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hf
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² :... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | obtain ⟨C, hC⟩ : ∃ C, ∀ p, ‖f.uncurry p‖ ≤ C := hf.bounded_above_of_compact_support h'f | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case hf.intro
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
ins... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact hasFiniteIntegral_of_bounded (C := C) (eventually_of_forall hC) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply set_integral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : ∀ x, f x y = 0 := by
intro x
contrapose! hy
have : (x, y) ∈ Function.support f.uncurry := hy
exact mem_image_of_mem _ (subset_tsupport _ this) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | intro x | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | contrapose! hy | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : (x, y) ∈ Function.support f.uncurry := hy | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact mem_image_of_mem _ (subset_tsupport _ this) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | simp [this] | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | congr 1 with y | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
case e_f.h
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | apply set_integral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | contrapose! hx | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | have : (x, y) ∈ Function.support f.uncurry := hx | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝¹⁸ : MeasurableSpace α
inst✝¹⁷ : MeasurableSpace α'
inst✝¹⁶ : MeasurableSpace β
inst✝¹⁵ : MeasurableSpace β'
inst✝¹⁴ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : SigmaFi... | /-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral ... | exact mem_image_of_mem _ (subset_tsupport _ this) | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib.MeasureTheory.Constructions.Prod.Integral.552_0.4tLli7hg1Kzhb8p | /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → ... | Mathlib_MeasureTheory_Constructions_Prod_Integral |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
⊢ Small.{max v w, u} (FullSubcategory (FilteredClosure f)) | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
| Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
⊢ ∀ (b : FullSubcategory (FilteredClosure f)),
∃ a, CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a = b.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rintro ⟨j, h⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
| Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
h : FilteredClosure f j
⊢ ∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j, property := h }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ hj₂ ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, FilteredClosureSmall.InductiveStep.max _ _ x y⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_righ... | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
| Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
h : FilteredClosure f j
⊢ ∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j, property := h }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ hj₂ ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, FilteredClosureSmall.InductiveStep.max _ _ x y⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_righ... | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
| Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.base
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
x : α
⊢ ∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := f x, property := (_ : FilteredClosure f (f x)) }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | | base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.base
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
x : α
⊢ ∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := f x, property := (_ : FilteredClosure f (f x)) }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | exact ⟨⟨0, ⟨x⟩⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : FilteredClosure f j✝
hj₂ : FilteredClosure f j'✝
ih :
∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j✝, property := hj₁ }.obj
ih' :
... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | | max hj₁ hj₂ ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, FilteredClosureSmall.InductiveStep.max _ _ x y⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : FilteredClosure f j✝
hj₂ : FilteredClosure f j'✝
ih :
∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j✝, property := hj₁ }.obj
ih' :
... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih with ⟨⟨n, x⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j j'✝ : C
hj₂ : FilteredClosure f j'✝
ih' :
∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j'✝, property := hj₂ }.obj
n : ℕ
x : (CategoryTheory.I... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih' with ⟨⟨m, y⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max.intro.mk.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosu... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | refine' ⟨⟨(Max.max n m).succ, FilteredClosureSmall.InductiveStep.max _ _ x y⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFil... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | all_goals apply Nat.lt_succ_of_le | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFil... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max.intro.mk.intro.mk.refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFil... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.max.intro.mk.intro.mk.refine'_1.a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractF... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : FilteredClosure f j✝
hj₂ : FilteredClosure f j'✝
g g' : j✝ ⟶ j'✝
ih :
∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j✝, property := h... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | | coeq hj₁ hj₂ g g' ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, FilteredClosureSmall.InductiveStep.coeq _ _ x y g g'⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : FilteredClosure f j✝
hj₂ : FilteredClosure f j'✝
g g' : j✝ ⟶ j'✝
ih :
∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j✝, property := h... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih with ⟨⟨n, x⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j j'✝ : C
hj₂ : FilteredClosure f j'✝
ih' :
∃ a,
CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClosureRealization f a =
{ obj := j'✝, property := hj₂ }.obj
n : ℕ
x : (CategoryTheory.... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih' with ⟨⟨m, y⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq.intro.mk.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFilteredClos... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | refine' ⟨⟨(Max.max n m).succ, FilteredClosureSmall.InductiveStep.coeq _ _ x y g g'⟩, rfl⟩ | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFi... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | all_goals apply Nat.lt_succ_of_le | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFi... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq.intro.mk.intro.mk.refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstractFi... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.coeq.intro.mk.intro.mk.refine'_1.a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsFiltered.FilteredClosureSmall.bundledAbstractFilteredClosure f n).fst
hj₁ :
FilteredClosure f
(CategoryTheory.IsFiltered.FilteredClosureSmall.abstract... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by
refine' small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ ... | Mathlib.CategoryTheory.Filtered.Small.94_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
⊢ Small.{max v w, u} (FullSubcategory (CofilteredClosure f)) | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
| Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
⊢ ∀ (b : FullSubcategory (CofilteredClosure f)),
∃ a, CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a = b.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rintro ⟨j, h⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
| Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
h : CofilteredClosure f j
⊢ ∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j, property := h }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| min hj₁ hj₂ ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, CofilteredClosureSmall.InductiveStep.min _ _ x y⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_ri... | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
| Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
h : CofilteredClosure f j
⊢ ∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j, property := h }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| min hj₁ hj₂ ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, CofilteredClosureSmall.InductiveStep.min _ _ x y⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_ri... | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
| Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.base
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
x : α
⊢ ∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := f x, property := (_ : CofilteredClosure f (f x)) }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | | base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.base
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
x : α
⊢ ∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := f x, property := (_ : CofilteredClosure f (f x)) }.obj | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | exact ⟨⟨0, ⟨x⟩⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : CofilteredClosure f j✝
hj₂ : CofilteredClosure f j'✝
ih :
∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j✝, property := hj₁ }.... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | | min hj₁ hj₂ ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, CofilteredClosureSmall.InductiveStep.min _ _ x y⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : CofilteredClosure f j✝
hj₂ : CofilteredClosure f j'✝
ih :
∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j✝, property := hj₁ }.... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih with ⟨⟨n, x⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j j'✝ : C
hj₂ : CofilteredClosure f j'✝
ih' :
∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j'✝, property := hj₂ }.obj
n : ℕ
x : (Catego... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih' with ⟨⟨m, y⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min.intro.mk.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstrac... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | refine' ⟨⟨(Max.max n m).succ, CofilteredClosureSmall.InductiveStep.min _ _ x y⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSma... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | all_goals apply Nat.lt_succ_of_le | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSma... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min.intro.mk.intro.mk.refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSma... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.min.intro.mk.intro.mk.refine'_1.a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureS... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : CofilteredClosure f j✝
hj₂ : CofilteredClosure f j'✝
g g' : j✝ ⟶ j'✝
ih :
∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j✝, pro... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | | eq hj₁ hj₂ g g' ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine' ⟨⟨(Max.max n m).succ, CofilteredClosureSmall.InductiveStep.eq _ _ x y g g'⟩, rfl⟩
all_goals apply Nat.lt_succ_of_le
exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j j✝ j'✝ : C
hj₁ : CofilteredClosure f j✝
hj₂ : CofilteredClosure f j'✝
g g' : j✝ ⟶ j'✝
ih :
∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j✝, pro... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih with ⟨⟨n, x⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j j'✝ : C
hj₂ : CofilteredClosure f j'✝
ih' :
∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := j'✝, property := hj₂ }.obj
n : ℕ
x : (Categor... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | rcases ih' with ⟨⟨m, y⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq.intro.mk.intro.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstract... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | refine' ⟨⟨(Max.max n m).succ, CofilteredClosureSmall.InductiveStep.eq _ _ x y g g'⟩, rfl⟩ | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSmal... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | all_goals apply Nat.lt_succ_of_le | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq.intro.mk.intro.mk.refine'_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSmal... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq.intro.mk.intro.mk.refine'_2
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSmal... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | apply Nat.lt_succ_of_le | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
case mk.eq.intro.mk.intro.mk.refine'_1.a
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
n : ℕ
x : (CategoryTheory.IsCofiltered.CofilteredClosureSmall.bundledAbstractCofilteredClosure f n).fst
hj₁ :
CofilteredClosure f
(CategoryTheory.IsCofiltered.CofilteredClosureSm... | /-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | exacts [Nat.le_max_left _ _, Nat.le_max_right _ _] | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) := by
refine' small_of_injective_of_exists
(CofilteredClosureSmall.abstractCofilteredClosureRealization f) FullSubcategory.ext _
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| ... | Mathlib.CategoryTheory.Filtered.Small.226_0.5hQB6DxvGqxowB3 | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | Mathlib_CategoryTheory_Filtered_Small |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
d : n → R
⊢ PosSemidef (diagonal d) ↔ ∀ (i : n), 0 ≤ d i | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine ⟨fun ⟨_, hP⟩ i ↦ by simpa using hP (Pi.single i 1), ?_⟩ | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) := by
| Mathlib.LinearAlgebra.Matrix.PosDef.51_0.RRbDg8T8pKv68Qo | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
d : n → R
x✝ : PosSemidef (diagonal d)
i : n
left✝ : IsHermitian (diagonal d)
hP : ∀ (x : n → R), 0 ≤ star x ⬝ᵥ mulVe... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa using hP (Pi.single i 1) | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) := by
refine ⟨fun ⟨_, hP⟩ i ↦ by | Mathlib.LinearAlgebra.Matrix.PosDef.51_0.RRbDg8T8pKv68Qo | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
d : n → R
⊢ (∀ (i : n), 0 ≤ d i) → PosSemidef (diagonal d) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine fun hd ↦ ⟨isHermitian_diagonal_iff.2 <| fun i ↦ IsSelfAdjoint.of_nonneg (hd i), ?_⟩ | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) := by
refine ⟨fun ⟨_, hP⟩ i ↦ by simpa using hP (Pi.single i 1), ?_⟩
| Mathlib.LinearAlgebra.Matrix.PosDef.51_0.RRbDg8T8pKv68Qo | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
d : n → R
hd : ∀ (i : n), 0 ≤ d i
⊢ ∀ (x : n → R), 0 ≤ star x ⬝ᵥ mulVec (diagonal d) x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine fun x ↦ Finset.sum_nonneg fun i _ ↦ ?_ | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) := by
refine ⟨fun ⟨_, hP⟩ i ↦ by simpa using hP (Pi.single i 1), ?_⟩
refine fun hd ↦ ⟨isHermitian_diagonal_iff.2 <|... | Mathlib.LinearAlgebra.Matrix.PosDef.51_0.RRbDg8T8pKv68Qo | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
d : n → R
hd : ∀ (i : n), 0 ≤ d i
x : n → R
i : n
x✝ : i ∈ Finset.univ
⊢ 0 ≤ star x i * mulVec (diagonal d) x i | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [mulVec_diagonal, mul_assoc] using conjugate_nonneg (hd i) _ | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) := by
refine ⟨fun ⟨_, hP⟩ i ↦ by simpa using hP (Pi.single i 1), ?_⟩
refine fun hd ↦ ⟨isHermitian_diagonal_iff.2 <|... | Mathlib.LinearAlgebra.Matrix.PosDef.51_0.RRbDg8T8pKv68Qo | /-- A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative. -/
lemma posSemidef_diagonal_iff [DecidableEq n] {d : n → R} :
PosSemidef (diagonal d) ↔ (∀ i : n, 0 ≤ d i) | Mathlib_LinearAlgebra_Matrix_PosDef |
m✝ : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m✝
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
A : Matrix n n R
hA : PosSemidef A
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
⊢ PosSemidef (Bᴴ * A * B) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | constructor | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) := by
| Mathlib.LinearAlgebra.Matrix.PosDef.68_0.RRbDg8T8pKv68Qo | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) | Mathlib_LinearAlgebra_Matrix_PosDef |
case left
m✝ : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m✝
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
A : Matrix n n R
hA : PosSemidef A
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
⊢ IsHermitian (Bᴴ * A * B) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact isHermitian_conjTranspose_mul_mul B hA.1 | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) := by
constructor
· | Mathlib.LinearAlgebra.Matrix.PosDef.68_0.RRbDg8T8pKv68Qo | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) | Mathlib_LinearAlgebra_Matrix_PosDef |
case right
m✝ : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m✝
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
A : Matrix n n R
hA : PosSemidef A
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
⊢ ∀ (x : m → R), 0 ≤ star x ⬝ᵥ mulVec (Bᴴ *... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | intro x | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) := by
constructor
· exact isHermitian_conjTranspose_mul_mul B hA.1
· | Mathlib.LinearAlgebra.Matrix.PosDef.68_0.RRbDg8T8pKv68Qo | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) | Mathlib_LinearAlgebra_Matrix_PosDef |
case right
m✝ : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m✝
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
A : Matrix n n R
hA : PosSemidef A
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
x : m → R
⊢ 0 ≤ star x ⬝ᵥ mulVec (Bᴴ * A * ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [star_mulVec, dotProduct_mulVec, vecMul_vecMul] using hA.2 (mulVec B x) | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) := by
constructor
· exact isHermitian_conjTranspose_mul_mul B hA.1
· intro x
| Mathlib.LinearAlgebra.Matrix.PosDef.68_0.RRbDg8T8pKv68Qo | lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B) | Mathlib_LinearAlgebra_Matrix_PosDef |
m✝ : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m✝
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
A : Matrix n n R
hA : PosSemidef A
m : Type u_5
inst✝ : Fintype m
B : Matrix m n R
⊢ PosSemidef (B * A * Bᴴ) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [conjTranspose_conjTranspose] using hA.conjTranspose_mul_mul_same Bᴴ | lemma mul_mul_conjTranspose_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix m n R):
PosSemidef (B * A * Bᴴ) := by
| Mathlib.LinearAlgebra.Matrix.PosDef.76_0.RRbDg8T8pKv68Qo | lemma mul_mul_conjTranspose_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix m n R):
PosSemidef (B * A * Bᴴ) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosSemidef M
e : m → n
⊢ PosSemidef (Matrix.submatrix M e e) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | classical
rw [(by simp : M = 1 * M * 1), submatrix_mul (he₂ := Function.bijective_id),
submatrix_mul (he₂ := Function.bijective_id), submatrix_id_id]
simpa only [conjTranspose_submatrix, conjTranspose_one] using
conjTranspose_mul_mul_same hM (Matrix.submatrix 1 id e) | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.81_0.RRbDg8T8pKv68Qo | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosSemidef M
e : m → n
⊢ PosSemidef (Matrix.submatrix M e e) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [(by simp : M = 1 * M * 1), submatrix_mul (he₂ := Function.bijective_id),
submatrix_mul (he₂ := Function.bijective_id), submatrix_id_id] | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef := by
classical
| Mathlib.LinearAlgebra.Matrix.PosDef.81_0.RRbDg8T8pKv68Qo | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosSemidef M
e : m → n
⊢ M = 1 * M * 1 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef := by
classical
rw [(by | Mathlib.LinearAlgebra.Matrix.PosDef.81_0.RRbDg8T8pKv68Qo | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosSemidef M
e : m → n
⊢ PosSemidef (Matrix.submatrix 1 e id * M * Matrix.submatrix 1 id e) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [conjTranspose_submatrix, conjTranspose_one] using
conjTranspose_mul_mul_same hM (Matrix.submatrix 1 id e) | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef := by
classical
rw [(by simp : M = 1 * M * 1), submatrix_mul (he₂ := Function.bijective_id),
submatrix_mul (he₂ := Function.bijective_id), submatrix_id_id]
| Mathlib.LinearAlgebra.Matrix.PosDef.81_0.RRbDg8T8pKv68Qo | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosSemidef M
⊢ PosSemidef Mᵀ | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩ | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.90_0.RRbDg8T8pKv68Qo | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosSemidef M
x : n → R
⊢ 0 ≤ star x ⬝ᵥ mulVec Mᵀ x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | convert hM.2 (star x) using 1 | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by
refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩
| Mathlib.LinearAlgebra.Matrix.PosDef.90_0.RRbDg8T8pKv68Qo | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case h.e'_4
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosSemidef M
x : n → R
⊢ star x ⬝ᵥ mulVec Mᵀ x = star (star x) ⬝ᵥ mulVec M (star x) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [mulVec_transpose, Matrix.dotProduct_mulVec, star_star, dotProduct_comm] | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by
refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩
convert hM.2 (star x) using 1
| Mathlib.LinearAlgebra.Matrix.PosDef.90_0.RRbDg8T8pKv68Qo | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
⊢ ∀ (x : n → R), 0 ≤ star x ⬝ᵥ mulVec 0 x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp | protected lemma zero : PosSemidef (0 : Matrix n n R) :=
⟨isHermitian_zero, by | Mathlib.LinearAlgebra.Matrix.PosDef.97_0.RRbDg8T8pKv68Qo | protected lemma zero : PosSemidef (0 : Matrix n n R) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
x : n → R
⊢ 0 ≤ star x ⬝ᵥ mulVec 1 x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [one_mulVec] | protected lemma one [DecidableEq n] : PosSemidef (1 : Matrix n n R) :=
⟨isHermitian_one, fun x => by
| Mathlib.LinearAlgebra.Matrix.PosDef.100_0.RRbDg8T8pKv68Qo | protected lemma one [DecidableEq n] : PosSemidef (1 : Matrix n n R) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
x : n → R
⊢ 0 ≤ star x ⬝ᵥ x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact Fintype.sum_nonneg fun i => star_mul_self_nonneg _ | protected lemma one [DecidableEq n] : PosSemidef (1 : Matrix n n R) :=
⟨isHermitian_one, fun x => by
rw [one_mulVec]; | Mathlib.LinearAlgebra.Matrix.PosDef.100_0.RRbDg8T8pKv68Qo | protected lemma one [DecidableEq n] : PosSemidef (1 : Matrix n n R) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
k : ℕ
⊢ PosSemidef (M ^ 1) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa using hM | protected lemma pow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (k : ℕ) :
PosSemidef (M ^ k) :=
match k with
| 0 => .one
| 1 => by | Mathlib.LinearAlgebra.Matrix.PosDef.104_0.RRbDg8T8pKv68Qo | protected lemma pow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (k : ℕ) :
PosSemidef (M ^ k) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
k✝ k : ℕ
⊢ PosSemidef (M ^ (k + 2)) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [pow_succ', pow_succ] | protected lemma pow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (k : ℕ) :
PosSemidef (M ^ k) :=
match k with
| 0 => .one
| 1 => by simpa using hM
| (k + 2) => by
| Mathlib.LinearAlgebra.Matrix.PosDef.104_0.RRbDg8T8pKv68Qo | protected lemma pow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (k : ℕ) :
PosSemidef (M ^ k) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
k✝ k : ℕ
⊢ PosSemidef (M * M ^ k * M) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [hM.isHermitian.eq] using (hM.pow k).mul_mul_conjTranspose_same M | protected lemma pow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (k : ℕ) :
PosSemidef (M ^ k) :=
match k with
| 0 => .one
| 1 => by simpa using hM
| (k + 2) => by
rw [pow_succ', pow_succ]
| Mathlib.LinearAlgebra.Matrix.PosDef.104_0.RRbDg8T8pKv68Qo | protected lemma pow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (k : ℕ) :
PosSemidef (M ^ k) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
⊢ PosSemidef M⁻¹ | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | by_cases h : IsUnit M.det | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.113_0.RRbDg8T8pKv68Qo | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case pos
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
h : IsUnit (det M)
⊢ PosSemidef M⁻¹ | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | have := (conjTranspose_mul_mul_same hM M⁻¹).conjTranspose | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef := by
by_cases h : IsUnit M.det
· | Mathlib.LinearAlgebra.Matrix.PosDef.113_0.RRbDg8T8pKv68Qo | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case pos
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
h : IsUnit (det M)
this : PosSemidef (M⁻¹ᴴ * M * M⁻¹)ᴴ
⊢ PosSemidef M⁻¹ | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rwa [mul_nonsing_inv_cancel_right _ _ h, conjTranspose_conjTranspose] at this | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef := by
by_cases h : IsUnit M.det
· have := (conjTranspose_mul_mul_same hM M⁻¹).conjTranspose
| Mathlib.LinearAlgebra.Matrix.PosDef.113_0.RRbDg8T8pKv68Qo | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case neg
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
h : ¬IsUnit (det M)
⊢ PosSemidef M⁻¹ | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [nonsing_inv_apply_not_isUnit _ h] | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef := by
by_cases h : IsUnit M.det
· have := (conjTranspose_mul_mul_same hM M⁻¹).conjTranspose
rwa [mul_nonsing_inv_cancel_right _ _ h, conjTranspose_conjTranspose] at this
· | Mathlib.LinearAlgebra.Matrix.PosDef.113_0.RRbDg8T8pKv68Qo | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case neg
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
h : ¬IsUnit (det M)
⊢ PosSemidef 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact .zero | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef := by
by_cases h : IsUnit M.det
· have := (conjTranspose_mul_mul_same hM M⁻¹).conjTranspose
rwa [mul_nonsing_inv_cancel_right _ _ h, conjTranspose_conjTranspose] at this
· rw [nonsing_inv_apply_not_isUnit _ h]
| Mathlib.LinearAlgebra.Matrix.PosDef.113_0.RRbDg8T8pKv68Qo | protected lemma inv [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) : M⁻¹.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
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