@@ -44,7 +44,6 @@ section Submodule
4444variable [Semiring R] [PartialOrder R] [IsOrderedRing R] [AddCommMonoid E] [Module R E]
4545variable {C : PointedCone R E}
4646
47- set_option backward.isDefEq.respectTransparency false in
4847/-- A submodule is a pointed cone. -/
4948@[coe] abbrev ofSubmodule (S : Submodule R E) : PointedCone R E := S.restrictScalars _
5049
@@ -54,25 +53,20 @@ instance : Coe (Submodule R E) (PointedCone R E) := ⟨ofSubmodule⟩
5453
5554lemma mem_ofSubmodule_iff {S : Submodule R E} {x : E} : x ∈ (S : PointedCone R E) ↔ x ∈ S := by rfl
5655
57- set_option backward.isDefEq.respectTransparency false in
5856lemma ofSubmodule_inj {S T : Submodule R E} : ofSubmodule S = ofSubmodule T ↔ S = T :=
5957 restrictScalars_inj ..
6058
61- set_option backward.isDefEq.respectTransparency false in
6259/-- Coercion from submodules to pointed cones as an order embedding. -/
6360abbrev ofSubmoduleEmbedding : Submodule R E ↪o PointedCone R E :=
6461 restrictScalarsEmbedding ..
6562
66- set_option backward.isDefEq.respectTransparency false in
6763/-- Coercion from submodules to pointed cones as a lattice homomorphism. -/
6864abbrev ofSubmoduleLatticeHom : CompleteLatticeHom (Submodule R E) (PointedCone R E) :=
6965 restrictScalarsLatticeHom ..
7066
71- set_option backward.isDefEq.respectTransparency false in
7267lemma ofSubmodule_inf (S T : Submodule R E) : S ⊓ T = (S ⊓ T : PointedCone R E) :=
7368 restrictScalars_inf _ _ _
7469
75- set_option backward.isDefEq.respectTransparency false in
7670lemma ofSubmodule_sup (S T : Submodule R E) : S ⊔ T = (S ⊔ T : PointedCone R E) :=
7771 restrictScalars_sup _ _ _
7872
@@ -91,7 +85,6 @@ lemma ofSubmodule_iSup (s : Set (Submodule R E)) : ⨆ S ∈ s, S = ⨆ S ∈ s,
9185variable {R E : Type *}
9286variable [Semiring R] [PartialOrder R] [IsOrderedRing R] [AddCommGroup E] [Module R E]
9387
94- set_option backward.isDefEq.respectTransparency false in
9588lemma neg_ofSubmodule (S : Submodule R E) : -(ofSubmodule S) = ofSubmodule (-S) :=
9689 neg_restrictScalars S
9790
@@ -129,7 +122,6 @@ lemma convex (C : PointedCone R E) : Convex R (C : Set E) := C.toConvexCone.conv
129122nonrec lemma smul_mem (C : PointedCone R E) (hr : 0 ≤ r) (hx : x ∈ C) : r • x ∈ C :=
130123 C.smul_mem ⟨r, hr⟩ hx
131124
132- set_option backward.isDefEq.respectTransparency false in
133125/-- The `PointedCone` constructed from a pointed `ConvexCone`. -/
134126def _root_.ConvexCone.toPointedCone (C : ConvexCone R E) (hC : C.Pointed) : PointedCone R E where
135127 carrier := C
@@ -222,7 +214,6 @@ between pointed cones induced from linear maps between the ambient modules that
222214
223215-/
224216
225- set_option backward.isDefEq.respectTransparency false in
226217/-- The image of a pointed cone under an `R`-linear map is a pointed cone. -/
227218def map (f : E →ₗ[R] F) (C : PointedCone R E) : PointedCone R F :=
228219 Submodule.map (f : E →ₗ[R≥0 ] F) C
@@ -248,7 +239,6 @@ theorem map_map (g : F →ₗ[R] G) (f : E →ₗ[R] F) (C : PointedCone R E) :
248239theorem map_id (C : PointedCone R E) : C.map LinearMap.id = C :=
249240 SetLike.coe_injective <| Set.image_id _
250241
251- set_option backward.isDefEq.respectTransparency false in
252242/-- The preimage of a pointed cone under an `R`-linear map is a pointed cone. -/
253243def comap (f : E →ₗ[R] F) (C : PointedCone R F) : PointedCone R E :=
254244 Submodule.comap (f : E →ₗ[R≥0 ] F) C
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