feat: port Analysis.Convex.Cone.Proper (#5607)

The remaining errors are about `coe` and `norm_cast`. I do not understand these well enough in Lean 4. 
Please feel free to push changes. 



Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

Author: apurvnakade

Mathlib.lean

@@ -579,6 +579,7 @@ import Mathlib.Analysis.Convex.Combination
 import Mathlib.Analysis.Convex.Complex
 import Mathlib.Analysis.Convex.Cone.Basic
 import Mathlib.Analysis.Convex.Cone.Dual
+import Mathlib.Analysis.Convex.Cone.Proper
 import Mathlib.Analysis.Convex.Contractible
 import Mathlib.Analysis.Convex.Exposed
 import Mathlib.Analysis.Convex.Extrema

Mathlib/Analysis/Convex/Cone/Proper.lean

@@ -0,0 +1,229 @@
+/-
+Copyright (c) 2022 Apurva Nakade All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Apurva Nakade
+
+! This file was ported from Lean 3 source module analysis.convex.cone.proper
+! leanprover-community/mathlib commit 74f1d61944a5a793e8c939d47608178c0a0cb0c2
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Analysis.Convex.Cone.Dual
+
+/-!
+# Proper cones
+
+We define a proper cone as a nonempty, closed, convex cone. Proper cones are used in defining conic
+programs which generalize linear programs. A linear program is a conic program for the positive
+cone. We then prove Farkas' lemma for conic programs following the proof in the reference below.
+Farkas' lemma is equivalent to strong duality. So, once have the definitions of conic programs and
+linear programs, the results from this file can be used to prove duality theorems.
+
+## TODO
+
+The next steps are:
+- Prove the cone version of Farkas' lemma (2.3.4 in the reference).
+- Add comap, adjoint
+- Add convex_cone_class that extends set_like and replace the below instance
+- Define the positive cone as a proper cone.
+- Define primal and dual cone programs and prove weak duality.
+- Prove regular and strong duality for cone programs using Farkas' lemma (see reference).
+- Define linear programs and prove LP duality as a special case of cone duality.
+- Find a better reference (textbook instead of lecture notes).
+- Show submodules are (proper) cones.
+
+## References
+
+- [B. Gartner and J. Matousek, Cone Programming][gartnerMatousek]
+
+-/
+
+
+namespace ConvexCone
+
+variable {𝕜 : Type _} [OrderedSemiring 𝕜]
+
+variable {E : Type _} [AddCommMonoid E] [TopologicalSpace E] [ContinuousAdd E] [SMul 𝕜 E]
+  [ContinuousConstSMul 𝕜 E]
+
+/-- The closure of a convex cone inside a topological space as a convex cone. This
+construction is mainly used for defining maps between proper cones. -/
+protected def closure (K : ConvexCone 𝕜 E) : ConvexCone 𝕜 E where
+  carrier := closure ↑K
+  smul_mem' c hc _ h₁ :=
+    map_mem_closure (continuous_id'.const_smul c) h₁ fun _ h₂ => K.smul_mem hc h₂
+  add_mem' _ h₁ _ h₂ := map_mem_closure₂ continuous_add h₁ h₂ K.add_mem
+#align convex_cone.closure ConvexCone.closure
+
+@[simp, norm_cast]
+theorem coe_closure (K : ConvexCone 𝕜 E) : (K.closure : Set E) = closure K :=
+  rfl
+#align convex_cone.coe_closure ConvexCone.coe_closure
+
+@[simp]
+protected theorem mem_closure {K : ConvexCone 𝕜 E} {a : E} :
+    a ∈ K.closure ↔ a ∈ closure (K : Set E) :=
+  Iff.rfl
+#align convex_cone.mem_closure ConvexCone.mem_closure
+
+@[simp]
+theorem closure_eq {K L : ConvexCone 𝕜 E} : K.closure = L ↔ closure (K : Set E) = L :=
+  SetLike.ext'_iff
+#align convex_cone.closure_eq ConvexCone.closure_eq
+
+end ConvexCone
+
+/-- A proper cone is a convex cone `K` that is nonempty and closed. Proper cones have the nice
+property that the dual of the dual of a proper cone is itself. This makes them useful for defining
+cone programs and proving duality theorems. -/
+structure ProperCone (𝕜 : Type _) (E : Type _) [OrderedSemiring 𝕜] [AddCommMonoid E]
+    [TopologicalSpace E] [SMul 𝕜 E] extends ConvexCone 𝕜 E where
+  nonempty' : (carrier : Set E).Nonempty
+  is_closed' : IsClosed (carrier : Set E)
+#align proper_cone ProperCone
+
+namespace ProperCone
+
+section SMul
+
+variable {𝕜 : Type _} [OrderedSemiring 𝕜]
+
+variable {E : Type _} [AddCommMonoid E] [TopologicalSpace E] [SMul 𝕜 E]
+
+instance : Coe (ProperCone 𝕜 E) (ConvexCone 𝕜 E) :=
+  ⟨fun K => K.1⟩
+
+-- Porting note: now a syntactic tautology
+-- @[simp]
+-- theorem toConvexCone_eq_coe (K : ProperCone 𝕜 E) : K.toConvexCone = K :=
+--   rfl
+-- #align proper_cone.to_convex_cone_eq_coe ProperCone.toConvexCone_eq_coe
+
+theorem ext' : Function.Injective ((↑) : ProperCone 𝕜 E → ConvexCone 𝕜 E) := fun S T h => by
+  cases S; cases T; congr
+#align proper_cone.ext' ProperCone.ext'
+
+-- TODO: add `ConvexConeClass` that extends `SetLike` and replace the below instance
+instance : SetLike (ProperCone 𝕜 E) E where
+  coe K := K.carrier
+  coe_injective' _ _ h := ProperCone.ext' (SetLike.coe_injective h)
+
+@[ext]
+theorem ext {S T : ProperCone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=
+  SetLike.ext h
+#align proper_cone.ext ProperCone.ext
+
+@[simp]
+theorem mem_coe {x : E} {K : ProperCone 𝕜 E} : x ∈ (K : ConvexCone 𝕜 E) ↔ x ∈ K :=
+  Iff.rfl
+#align proper_cone.mem_coe ProperCone.mem_coe
+
+protected theorem nonempty (K : ProperCone 𝕜 E) : (K : Set E).Nonempty :=
+  K.nonempty'
+#align proper_cone.nonempty ProperCone.nonempty
+
+protected theorem isClosed (K : ProperCone 𝕜 E) : IsClosed (K : Set E) :=
+  K.is_closed'
+#align proper_cone.is_closed ProperCone.isClosed
+
+end SMul
+
+section Module
+
+variable {𝕜 : Type _} [OrderedSemiring 𝕜]
+
+variable {E : Type _} [AddCommMonoid E] [TopologicalSpace E] [T1Space E] [Module 𝕜 E]
+
+instance : Zero (ProperCone 𝕜 E) :=
+  ⟨{  toConvexCone := 0
+      nonempty' := ⟨0, rfl⟩
+      is_closed' := isClosed_singleton }⟩
+
+instance : Inhabited (ProperCone 𝕜 E) :=
+  ⟨0⟩
+
+@[simp]
+theorem mem_zero (x : E) : x ∈ (0 : ProperCone 𝕜 E) ↔ x = 0 :=
+  Iff.rfl
+#align proper_cone.mem_zero ProperCone.mem_zero
+
+@[simp] -- Porting note: removed `norm_cast` (new-style structures)
+theorem coe_zero : ↑(0 : ProperCone 𝕜 E) = (0 : ConvexCone 𝕜 E) :=
+  rfl
+#align proper_cone.coe_zero ProperCone.coe_zero
+
+theorem pointed_zero : (0 : ProperCone 𝕜 E).Pointed := by simp [ConvexCone.pointed_zero]
+#align proper_cone.pointed_zero ProperCone.pointed_zero
+
+end Module
+
+section InnerProductSpace
+
+variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+variable {F : Type _} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+
+protected theorem pointed (K : ProperCone ℝ E) : (K : ConvexCone ℝ E).Pointed :=
+  (K : ConvexCone ℝ E).pointed_of_nonempty_of_isClosed K.nonempty' K.isClosed
+#align proper_cone.pointed ProperCone.pointed
+
+/-- The closure of image of a proper cone under a continuous `ℝ`-linear map is a proper cone. We
+use continuous maps here so that the adjoint of f is also a map between proper cones. -/
+noncomputable def map (f : E →L[ℝ] F) (K : ProperCone ℝ E) : ProperCone ℝ F where
+  toConvexCone := ConvexCone.closure (ConvexCone.map (f : E →ₗ[ℝ] F) ↑K)
+  nonempty' :=
+    ⟨0, subset_closure <| SetLike.mem_coe.2 <| ConvexCone.mem_map.2 ⟨0, K.pointed, map_zero _⟩⟩
+  is_closed' := isClosed_closure
+#align proper_cone.map ProperCone.map
+
+@[simp] -- Porting note: removed `norm_cast` (new-style structures)
+theorem coe_map (f : E →L[ℝ] F) (K : ProperCone ℝ E) :
+    ↑(K.map f) = (ConvexCone.map (f : E →ₗ[ℝ] F) ↑K).closure :=
+  rfl
+#align proper_cone.coe_map ProperCone.coe_map
+
+@[simp]
+theorem mem_map {f : E →L[ℝ] F} {K : ProperCone ℝ E} {y : F} :
+    y ∈ K.map f ↔ y ∈ (ConvexCone.map (f : E →ₗ[ℝ] F) ↑K).closure :=
+  Iff.rfl
+#align proper_cone.mem_map ProperCone.mem_map
+
+@[simp]
+theorem map_id (K : ProperCone ℝ E) : K.map (ContinuousLinearMap.id ℝ E) = K :=
+  ProperCone.ext' <| by simpa using IsClosed.closure_eq K.isClosed
+#align proper_cone.map_id ProperCone.map_id
+
+/-- The inner dual cone of a proper cone is a proper cone. -/
+def dual (K : ProperCone ℝ E) : ProperCone ℝ E where
+  toConvexCone := (K : Set E).innerDualCone
+  nonempty' := ⟨0, pointed_innerDualCone _⟩
+  is_closed' := isClosed_innerDualCone _
+#align proper_cone.dual ProperCone.dual
+
+@[simp] -- Porting note: removed `norm_cast` (new-style structures)
+theorem coe_dual (K : ProperCone ℝ E) : ↑(dual K) = (K : Set E).innerDualCone :=
+  rfl
+#align proper_cone.coe_dual ProperCone.coe_dual
+
+@[simp]
+theorem mem_dual {K : ProperCone ℝ E} {y : E} : y ∈ dual K ↔ ∀ ⦃x⦄, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ := by
+  rw [← mem_coe, coe_dual, mem_innerDualCone _ _]; rfl
+#align proper_cone.mem_dual ProperCone.mem_dual
+
+-- TODO: add comap, adjoint
+end InnerProductSpace
+
+section CompleteSpace
+
+variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+
+/-- The dual of the dual of a proper cone is itself. -/
+@[simp]
+theorem dual_dual (K : ProperCone ℝ E) : K.dual.dual = K :=
+  ProperCone.ext' <|
+    (K : ConvexCone ℝ E).innerDualCone_of_innerDualCone_eq_self K.nonempty' K.isClosed
+#align proper_cone.dual_dual ProperCone.dual_dual
+
+end CompleteSpace
+
+end ProperCone