Bousfield

📁 Source: Mathlib/CategoryTheory/Localization/Bousfield.lean

Statistics

Metric	Count
Definitions `homEquiv`, `isColocal`, `homEquiv`, `isLocal`, `homEquiv`	5
Theorems `W_adj_unit_app`, `W_eq_inverseImage_isomorphisms`, `W_iff_isIso`, `W_iff_isIso_map`, `W_isoClosure`, `W_of_isIso`, `galoisConnection`, `isLocalization`, `le_W_iff`, `le_isColocal_isColocal`, `le_isLocal_isLocal`, `le_leftBousfieldW_isLocal`, `galoisConnection_isColocal`, `galoisConnection_isLocal`, `instHasTwoOutOfThreePropertyIsColocal`, `instHasTwoOutOfThreePropertyIsLocal`, `instIsMultiplicativeIsColocal`, `instIsMultiplicativeIsLocal`, `instRespectsIsoIsColocal`, `instRespectsIsoIsLocal`, `homEquiv_apply`, `isColocal_adj_counit_app`, `isColocal_eq_inverseImage_isomorphisms`, `isColocal_iff_isIso`, `isColocal_iff_isIso_map`, `isColocal_of_isIso`, `homEquiv_apply`, `isLocal_adj_unit_app`, `isLocal_eq_inverseImage_isomorphisms`, `isLocal_iff_isIso`, `isLocal_iff_isIso_map`, `isLocal_of_isIso`, `isLocalization_isColocal`, `isLocalization_isLocal`, `isoClosure_isColocal`, `isoClosure_isLocal`, `le_isColocal_iff`, `le_isColocal_isColocal`, `le_isLocal_W`, `le_isLocal_iff`, `le_isLocal_isLocal`	41
Total	46

CategoryTheory.Localization.LeftBousfield

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`W_adj_unit_app` 📖	mathematical	—	`CategoryTheory.ObjectProperty.isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.Functor.id` `CategoryTheory.Functor.comp` `CategoryTheory.NatTrans.app` `CategoryTheory.Adjunction.unit`	—	`CategoryTheory.ObjectProperty.isLocal_adj_unit_app`
`W_eq_inverseImage_isomorphisms` 📖	mathematical	—	`CategoryTheory.ObjectProperty.isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.MorphismProperty.inverseImage` `CategoryTheory.MorphismProperty.isomorphisms`	—	`CategoryTheory.ObjectProperty.isLocal_eq_inverseImage_isomorphisms`
`W_iff_isIso` 📖	mathematical	—	`CategoryTheory.ObjectProperty.isLocal` `CategoryTheory.IsIso`	—	`CategoryTheory.ObjectProperty.isLocal_iff_isIso`
`W_iff_isIso_map` 📖	mathematical	—	`CategoryTheory.ObjectProperty.isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.IsIso` `CategoryTheory.Functor.map`	—	`CategoryTheory.ObjectProperty.isLocal_iff_isIso_map`
`W_isoClosure` 📖	mathematical	—	`CategoryTheory.ObjectProperty.isLocal` `CategoryTheory.ObjectProperty.isoClosure`	—	`CategoryTheory.ObjectProperty.isoClosure_isLocal`
`W_of_isIso` 📖	mathematical	—	`CategoryTheory.ObjectProperty.isLocal`	—	`CategoryTheory.ObjectProperty.isLocal_of_isIso`
`galoisConnection` 📖	mathematical	—	`GaloisConnection` `CategoryTheory.ObjectProperty` `CategoryTheory.Category.toCategoryStruct` `OrderDual` `CategoryTheory.MorphismProperty` `Pi.preorder` `PartialOrder.toPreorder` `Prop.partialOrder` `OrderDual.instPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra` `DFunLike.coe` `Equiv` `EquivLike.toFunLike` `Equiv.instEquivLike` `OrderDual.toDual` `CategoryTheory.ObjectProperty.isLocal` `CategoryTheory.MorphismProperty.isLocal` `OrderDual.ofDual`	—	`CategoryTheory.ObjectProperty.galoisConnection_isLocal`
`isLocalization` 📖	mathematical	—	`CategoryTheory.Functor.IsLocalization` `CategoryTheory.ObjectProperty.isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj`	—	`CategoryTheory.ObjectProperty.isLocalization_isLocal`
`le_W_iff` 📖	mathematical	—	`CategoryTheory.MorphismProperty` `CategoryTheory.Category.toCategoryStruct` `Preorder.toLE` `PartialOrder.toPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra` `CategoryTheory.ObjectProperty.isLocal` `CategoryTheory.ObjectProperty` `Pi.hasLe` `Prop.le` `CategoryTheory.MorphismProperty.isLocal`	—	`CategoryTheory.ObjectProperty.le_isLocal_iff`

CategoryTheory.Localization.LeftBousfield.W

Definitions

Name	Category	Theorems
`homEquiv` 📖	CompOp	—

CategoryTheory.MorphismProperty

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`le_isColocal_isColocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty` `CategoryTheory.Category.toCategoryStruct` `Preorder.toLE` `PartialOrder.toPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `instCompleteBooleanAlgebra` `CategoryTheory.ObjectProperty.isColocal` `isColocal`	—	`CategoryTheory.ObjectProperty.le_isColocal_iff` `le_refl`
`le_isLocal_isLocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty` `CategoryTheory.Category.toCategoryStruct` `Preorder.toLE` `PartialOrder.toPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `instCompleteBooleanAlgebra` `CategoryTheory.ObjectProperty.isLocal` `isLocal`	—	`CategoryTheory.ObjectProperty.le_isLocal_iff` `le_refl`
`le_leftBousfieldW_isLocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty` `CategoryTheory.Category.toCategoryStruct` `Preorder.toLE` `PartialOrder.toPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `instCompleteBooleanAlgebra` `CategoryTheory.ObjectProperty.isLocal` `isLocal`	—	`le_isLocal_isLocal`

CategoryTheory.ObjectProperty

Definitions

Name	Category	Theorems
`isColocal` 📖	CompOp	14 math math: `instRespectsIsoIsColocal`, `isLocalization_isColocal`, `isColocal_eq_inverseImage_isomorphisms`, `isoClosure_isColocal`, `galoisConnection_isColocal`, `isColocal_iff_isIso`, `isColocal_adj_counit_app`, `le_isColocal_iff`, `instIsMultiplicativeIsColocal`, `isColocal_of_isIso`, `isColocal_iff_isIso_map`, `instHasTwoOutOfThreePropertyIsColocal`, `CategoryTheory.MorphismProperty.le_isColocal_isColocal`, `le_isColocal_isColocal`
`isLocal` 📖	CompOp	34 math math: `isLocal_of_isIso`, `CategoryTheory.Localization.LeftBousfield.galoisConnection`, `CategoryTheory.Abelian.isoModSerre_kernel_eq_isLocal_of_rightAdjoint`, `CategoryTheory.Localization.LeftBousfield.W_eq_inverseImage_isomorphisms`, `CategoryTheory.Localization.LeftBousfield.W_adj_unit_app`, `CategoryTheory.Localization.LeftBousfield.le_W_iff`, `instHasTwoOutOfThreePropertyIsLocal`, `CategoryTheory.OrthogonalReflection.isLocal_isLocal_reflection`, `CategoryTheory.OrthogonalReflection.leftBousfieldW_isLocal_toSucc`, `isLocalization_isLocal`, `isLocal_iff_isIso`, `CategoryTheory.MorphismProperty.le_isLocal_isLocal`, `instIsStableUnderTransfiniteCompositionIsLocal`, `CategoryTheory.OrthogonalReflection.isLocal_isLocal_toSucc`, `le_isLocal_iff`, `le_isLocal_isLocal`, `CategoryTheory.Abelian.isoModSerre_kernel_eq_leftBousfield_W_of_rightAdjoint`, `instIsMultiplicativeIsLocal`, `CategoryTheory.Localization.LeftBousfield.W_iff_isIso`, `CategoryTheory.MorphismProperty.le_leftBousfieldW_isLocal`, `isLocal_iff_isIso_map`, `CategoryTheory.Localization.LeftBousfield.W_isoClosure`, `instIsStableUnderTransfiniteCompositionOfShapeIsLocal`, `isLocal_adj_unit_app`, `CategoryTheory.Localization.LeftBousfield.W_of_isIso`, `CategoryTheory.GrothendieckTopology.W_eq_isLocal_range_sheafToPresheaf_obj`, `galoisConnection_isLocal`, `CategoryTheory.GrothendieckTopology.W_eq_W_range_sheafToPresheaf_obj`, `instRespectsIsoIsLocal`, `le_isLocal_W`, `CategoryTheory.Localization.LeftBousfield.isLocalization`, `CategoryTheory.Localization.LeftBousfield.W_iff_isIso_map`, `isoClosure_isLocal`, `isLocal_eq_inverseImage_isomorphisms`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`galoisConnection_isColocal` 📖	mathematical	—	`GaloisConnection` `CategoryTheory.ObjectProperty` `CategoryTheory.Category.toCategoryStruct` `OrderDual` `CategoryTheory.MorphismProperty` `Pi.preorder` `PartialOrder.toPreorder` `Prop.partialOrder` `OrderDual.instPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra` `DFunLike.coe` `Equiv` `EquivLike.toFunLike` `Equiv.instEquivLike` `OrderDual.toDual` `isColocal` `CategoryTheory.MorphismProperty.isColocal` `OrderDual.ofDual`	—	`le_isColocal_iff`
`galoisConnection_isLocal` 📖	mathematical	—	`GaloisConnection` `CategoryTheory.ObjectProperty` `CategoryTheory.Category.toCategoryStruct` `OrderDual` `CategoryTheory.MorphismProperty` `Pi.preorder` `PartialOrder.toPreorder` `Prop.partialOrder` `OrderDual.instPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra` `DFunLike.coe` `Equiv` `EquivLike.toFunLike` `Equiv.instEquivLike` `OrderDual.toDual` `isLocal` `CategoryTheory.MorphismProperty.isLocal` `OrderDual.ofDual`	—	`le_isLocal_iff`
`instHasTwoOutOfThreePropertyIsColocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty` `isColocal`	—	`CategoryTheory.MorphismProperty.IsMultiplicative.toIsStableUnderComposition` `instIsMultiplicativeIsColocal` `Function.Bijective.of_comp_iff'` `CategoryTheory.Category.assoc` `Function.Bijective.of_comp_iff`
`instHasTwoOutOfThreePropertyIsLocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty` `isLocal`	—	`CategoryTheory.MorphismProperty.IsMultiplicative.toIsStableUnderComposition` `instIsMultiplicativeIsLocal` `Function.Bijective.of_comp_iff` `CategoryTheory.Category.assoc` `Function.Bijective.of_comp_iff'`
`instIsMultiplicativeIsColocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty.IsMultiplicative` `isColocal`	—	`CategoryTheory.Category.comp_id` `Function.bijective_id` `CategoryTheory.Category.assoc` `Function.Bijective.comp`
`instIsMultiplicativeIsLocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty.IsMultiplicative` `isLocal`	—	`CategoryTheory.Category.id_comp` `Function.bijective_id` `CategoryTheory.Category.assoc` `Function.Bijective.comp`
`instRespectsIsoIsColocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty.RespectsIso` `isColocal`	—	`CategoryTheory.MorphismProperty.comp_mem` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toIsStableUnderComposition` `instHasTwoOutOfThreePropertyIsColocal` `isColocal_of_isIso`
`instRespectsIsoIsLocal` 📖	mathematical	—	`CategoryTheory.MorphismProperty.RespectsIso` `isLocal`	—	`CategoryTheory.MorphismProperty.comp_mem` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toIsStableUnderComposition` `instHasTwoOutOfThreePropertyIsLocal` `isLocal_of_isIso`
`isColocal_adj_counit_app` 📖	mathematical	—	`isColocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.Functor.comp` `CategoryTheory.Functor.id` `CategoryTheory.NatTrans.app` `CategoryTheory.Adjunction.counit`	—	`CategoryTheory.Functor.FullyFaithful.map_preimage` `Equiv.bijective`
`isColocal_eq_inverseImage_isomorphisms` 📖	mathematical	—	`isColocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.MorphismProperty.inverseImage` `CategoryTheory.MorphismProperty.isomorphisms`	—	`CategoryTheory.MorphismProperty.ext` `isColocal_iff_isIso_map`
`isColocal_iff_isIso` 📖	mathematical	—	`isColocal` `CategoryTheory.IsIso`	—	`CategoryTheory.Category.assoc` `CategoryTheory.Category.comp_id` `CategoryTheory.Category.id_comp` `isColocal_of_isIso`
`isColocal_iff_isIso_map` 📖	mathematical	—	`isColocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.IsIso` `CategoryTheory.Functor.map`	—	`CategoryTheory.NatTrans.naturality` `CategoryTheory.MorphismProperty.precomp_iff` `CategoryTheory.MorphismProperty.Respects.toRespectsLeft` `CategoryTheory.MorphismProperty.instRespectsOfIsStableUnderComposition` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toIsStableUnderComposition` `instHasTwoOutOfThreePropertyIsColocal` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toHasOfPrecompProperty` `isColocal_adj_counit_app` `CategoryTheory.MorphismProperty.postcomp_iff` `CategoryTheory.MorphismProperty.Respects.toRespectsRight` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toHasOfPostcompProperty` `isColocal_iff_isIso` `CategoryTheory.isIso_of_fully_faithful` `CategoryTheory.Functor.map_isIso`
`isColocal_of_isIso` 📖	mathematical	—	`isColocal`	—	`CategoryTheory.cancel_mono` `CategoryTheory.mono_of_strongMono` `CategoryTheory.strongMono_of_isIso` `CategoryTheory.Category.assoc` `CategoryTheory.IsIso.inv_hom_id` `CategoryTheory.Category.comp_id`
`isLocal_adj_unit_app` 📖	mathematical	—	`isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.Functor.id` `CategoryTheory.Functor.comp` `CategoryTheory.NatTrans.app` `CategoryTheory.Adjunction.unit`	—	`CategoryTheory.Functor.FullyFaithful.map_preimage` `Equiv.bijective`
`isLocal_eq_inverseImage_isomorphisms` 📖	mathematical	—	`isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.MorphismProperty.inverseImage` `CategoryTheory.MorphismProperty.isomorphisms`	—	`CategoryTheory.MorphismProperty.ext` `isLocal_iff_isIso_map`
`isLocal_iff_isIso` 📖	mathematical	—	`isLocal` `CategoryTheory.IsIso`	—	`CategoryTheory.Category.assoc` `CategoryTheory.Category.id_comp` `Mathlib.Tactic.Reassoc.eq_whisker'` `CategoryTheory.Category.comp_id` `isLocal_of_isIso`
`isLocal_iff_isIso_map` 📖	mathematical	—	`isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj` `CategoryTheory.IsIso` `CategoryTheory.Functor.map`	—	`CategoryTheory.NatTrans.naturality` `CategoryTheory.MorphismProperty.postcomp_iff` `CategoryTheory.MorphismProperty.Respects.toRespectsRight` `CategoryTheory.MorphismProperty.instRespectsOfIsStableUnderComposition` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toIsStableUnderComposition` `instHasTwoOutOfThreePropertyIsLocal` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toHasOfPostcompProperty` `isLocal_adj_unit_app` `CategoryTheory.MorphismProperty.precomp_iff` `CategoryTheory.MorphismProperty.Respects.toRespectsLeft` `CategoryTheory.MorphismProperty.HasTwoOutOfThreeProperty.toHasOfPrecompProperty` `isLocal_iff_isIso` `CategoryTheory.isIso_of_fully_faithful` `CategoryTheory.Functor.map_isIso`
`isLocal_of_isIso` 📖	mathematical	—	`isLocal`	—	`CategoryTheory.cancel_epi` `CategoryTheory.epi_of_strongEpi` `CategoryTheory.strongEpi_of_isIso` `CategoryTheory.IsIso.hom_inv_id_assoc`
`isLocalization_isColocal` 📖	mathematical	—	`CategoryTheory.Functor.IsLocalization` `isColocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj`	—	`isColocal_eq_inverseImage_isomorphisms` `CategoryTheory.Adjunction.isLocalization'`
`isLocalization_isLocal` 📖	mathematical	—	`CategoryTheory.Functor.IsLocalization` `isLocal` `Set` `Set.instMembership` `Set.range` `CategoryTheory.Functor.obj`	—	`isLocal_eq_inverseImage_isomorphisms` `CategoryTheory.Adjunction.isLocalization`
`isoClosure_isColocal` 📖	mathematical	—	`isColocal` `isoClosure`	—	`CategoryTheory.MorphismProperty.ext` `le_isoClosure` `CategoryTheory.cancel_epi` `CategoryTheory.epi_of_strongEpi` `CategoryTheory.strongEpi_of_isIso` `CategoryTheory.Iso.isIso_inv` `CategoryTheory.Category.assoc` `CategoryTheory.Iso.hom_inv_id_assoc`
`isoClosure_isLocal` 📖	mathematical	—	`isLocal` `isoClosure`	—	`CategoryTheory.MorphismProperty.ext` `le_isoClosure` `CategoryTheory.cancel_mono` `CategoryTheory.mono_of_strongMono` `CategoryTheory.strongMono_of_isIso` `CategoryTheory.Iso.isIso_hom` `CategoryTheory.Category.assoc` `Mathlib.Tactic.Reassoc.eq_whisker'` `CategoryTheory.Iso.hom_inv_id` `CategoryTheory.Category.comp_id`
`le_isColocal_iff` 📖	mathematical	—	`CategoryTheory.MorphismProperty` `CategoryTheory.Category.toCategoryStruct` `Preorder.toLE` `PartialOrder.toPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra` `isColocal` `CategoryTheory.ObjectProperty` `Pi.hasLe` `Prop.le` `CategoryTheory.MorphismProperty.isColocal`	—	—
`le_isColocal_isColocal` 📖	mathematical	—	`CategoryTheory.ObjectProperty` `CategoryTheory.Category.toCategoryStruct` `Pi.hasLe` `Prop.le` `CategoryTheory.MorphismProperty.isColocal` `isColocal`	—	`le_isColocal_iff` `le_refl`
`le_isLocal_W` 📖	mathematical	—	`CategoryTheory.ObjectProperty` `CategoryTheory.Category.toCategoryStruct` `Pi.hasLe` `Prop.le` `CategoryTheory.MorphismProperty.isLocal` `isLocal`	—	`le_isLocal_isLocal`
`le_isLocal_iff` 📖	mathematical	—	`CategoryTheory.MorphismProperty` `CategoryTheory.Category.toCategoryStruct` `Preorder.toLE` `PartialOrder.toPreorder` `CompleteSemilatticeInf.toPartialOrder` `CompleteLattice.toCompleteSemilatticeInf` `CompleteBooleanAlgebra.toCompleteLattice` `CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra` `isLocal` `CategoryTheory.ObjectProperty` `Pi.hasLe` `Prop.le` `CategoryTheory.MorphismProperty.isLocal`	—	—
`le_isLocal_isLocal` 📖	mathematical	—	`CategoryTheory.ObjectProperty` `CategoryTheory.Category.toCategoryStruct` `Pi.hasLe` `Prop.le` `CategoryTheory.MorphismProperty.isLocal` `isLocal`	—	`le_isLocal_iff` `le_refl`

CategoryTheory.ObjectProperty.isColocal

Definitions

Name	Category	Theorems
`homEquiv` 📖	CompOp	1 math math: `homEquiv_apply`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`homEquiv_apply` 📖	mathematical	`CategoryTheory.ObjectProperty.isColocal`	`DFunLike.coe` `Equiv` `Quiver.Hom` `CategoryTheory.CategoryStruct.toQuiver` `CategoryTheory.Category.toCategoryStruct` `EquivLike.toFunLike` `Equiv.instEquivLike` `homEquiv` `CategoryTheory.CategoryStruct.comp`	—	—

CategoryTheory.ObjectProperty.isLocal

Definitions

Name	Category	Theorems
`homEquiv` 📖	CompOp	2 math math: `PresheafOfModules.homEquivOfIsLocallyBijective_symm_apply`, `homEquiv_apply`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`homEquiv_apply` 📖	mathematical	`CategoryTheory.ObjectProperty.isLocal`	`DFunLike.coe` `Equiv` `Quiver.Hom` `CategoryTheory.CategoryStruct.toQuiver` `CategoryTheory.Category.toCategoryStruct` `EquivLike.toFunLike` `Equiv.instEquivLike` `homEquiv` `CategoryTheory.CategoryStruct.comp`	—	—

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