Convex Analysis (Rockafellar, 1970) -- Chapter 04 -- Section 20 -- Part 9

open scoped BigOperators Pointwisesection Chap04section Section20

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter binary branch, an exact/top-or-attained bridge implies reverse inequality and universal split-attainment.

lemma helperForTheorem_20_1_nonriInter_binary_reverseLe_and_universalAttainment_of_polyLeft_domRi_posDim_of_exact_topOrAttained
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hbinaryBridge :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        ∃ y : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
            fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  have hcore :
      (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) :=
    helperForTheorem_20_1_nonriInterCore_reverseLe_and_neTopAttainment_of_exact_topOrAttained_binary_posDim
      (p := p) (q := q) hbinaryBridge
  exact
    helperForTheorem_20_1_nonriInter_binary_reverseLe_and_universalAttainment_of_polyLeft_domRi_posDim_of_core
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter hcore

lemma helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_iff_reverseLe_and_neTopAttainment_of_polyLeft_domRi
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q))) :
    ((fenchelConjugate n (fun x => p x + q x) =
      infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
      ↔
      ((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) := by
  have _hnPos_use : 0 < n := hnPos
  have _hnotRiInter_use :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)) :=
    hnotRiInter
  constructor
  · intro hbinaryBridge
    exact
      helperForTheorem_20_1_nonriInterCore_reverseLe_and_neTopAttainment_of_exact_topOrAttained_binary_posDim
        (p := p) (q := q) hbinaryBridge
  · intro hcore
    have hforwardAndTopWitness :
        (fenchelConjugate n (fun x => p x + q x) ≤
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q))
          ∧
          (∀ xStar : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) →
              ∃ y : Fin n → ℝ,
                infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                  fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) :=
      helperForTheorem_20_1_forwardLe_and_topWitness_of_polyLeft_domRi
        (p := p) (q := q) hpolyP hproperP hproperQ hnonemptyDomInterRi
    exact
      helperForTheorem_20_1_exact_topOrAttained_of_forward_reverse_and_neTopAttainment
        (p := p) (q := q) (hforwardLe := hforwardAndTopWitness.1)
        (hreverseLe := hcore.1) (hneTopAttained := hcore.2)

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, the core package Unknown identifier `reverseLe`sorry + sorry - sorry - sorry : ?m.16reverseLe + Unknown identifier `non`non-Function expected at ⊤ but this term has type ?m.18 Note: Expected a function because this term is being applied to the argument split⊤ split-Unknown identifier `attainment`attainment implies exact equality plus top-or-attained for the binary bridge.

lemma helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_of_core
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hcore :
      (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (fenchelConjugate n (fun x => p x + q x) =
      infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  exact
    (helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_iff_reverseLe_and_neTopAttainment_of_polyLeft_domRi
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter).2 hcore

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, the remaining core package Unknown identifier `reverseLe`sorry + sorry - sorry - sorry : ?m.16reverseLe + Unknown identifier `non`non-Function expected at ⊤ but this term has type ?m.18 Note: Expected a function because this term is being applied to the argument split⊤ split-Unknown identifier `attainment`attainment is equivalent to Unknown identifier `reverseLe`sorry + sorry - sorry : ?m.10reverseLe + Unknown identifier `universal`universal split-Unknown identifier `attainment`attainment.

lemma helperForTheorem_20_1_nonriInter_posDim_coreGoal_iff_reverseLe_and_universalAttainment_of_polyLeft_domRi
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q))) :
    ((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
      ↔
      ((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
          fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) := by
  have _hnPos_use : 0 < n := hnPos
  have _hnotRiInter_use :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)) :=
    hnotRiInter
  exact
    helperForTheorem_20_1_nonriInter_binary_goal_iff_reverseLe_and_universalAttainment_of_polyLeft_domRi
      (p := p) (q := q) hpolyP hproperP hproperQ hnonemptyDomInterRi

lemma helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_of_reverseLe_and_universal
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hreverseAndUniversal :
      (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (fenchelConjugate n (fun x => p x + q x) =
      infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  have hcoreGoalIff :
      ((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠
              (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
        ↔
        ((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
          fenchelConjugate n (fun x => p x + q x))
          ∧
          (∀ xStar : Fin n → ℝ,
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :=
    helperForTheorem_20_1_nonriInter_posDim_coreGoal_iff_reverseLe_and_universalAttainment_of_polyLeft_domRi
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter
  have hcore :
      (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠
              (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) :=
    hcoreGoalIff.2 hreverseAndUniversal
  exact
    helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_of_core
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter hcore

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, the core package Unknown identifier `reverseLe`sorry + sorry - sorry - sorry : ?m.16reverseLe + Unknown identifier `non`non-Function expected at ⊤ but this term has type ?m.18 Note: Expected a function because this term is being applied to the argument split⊤ split-Unknown identifier `attainment`attainment yields both exact/top-or-attained and universal split-attainment.

lemma helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_and_universal_of_core
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hcore :
      (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠
              (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    ((fenchelConjugate n (fun x => p x + q x) =
      infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
      ∧
      (∀ xStar : Fin n → ℝ,
        ∃ y : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
            fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  have hbinaryBridge :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) :=
    helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_of_core
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter hcore
  have hreverseAndUniversal :
      (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) :=
    helperForTheorem_20_1_nonriInter_binary_reverseLe_and_universalAttainment_of_polyLeft_domRi_posDim_of_core
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter hcore
  exact ⟨hbinaryBridge, hreverseAndUniversal.2⟩

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, the mixed assumptions already yield forward inequality and top-case split-attainment for the binary infimal convolution.

lemma helperForTheorem_20_1_nonriInter_posDim_forwardLe_and_topWitness_of_polyLeft_domRi
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q))) :
    (fenchelConjugate n (fun x => p x + q x) ≤
      infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  have _hnPos_use : 0 < n := hnPos
  have _hnotRiInter_use :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)) :=
    hnotRiInter
  exact
    helperForTheorem_20_1_forwardLe_and_topWitness_of_polyLeft_domRi
      (p := p) (q := q) hpolyP hproperP hproperQ hnonemptyDomInterRi

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, the unresolved core package Unknown identifier `reverseLe`sorry + sorry - sorry - sorry : ?m.16reverseLe + Unknown identifier `non`non-Function expected at ⊤ but this term has type ?m.18 Note: Expected a function because this term is being applied to the argument split⊤ split-Unknown identifier `attainment`attainment is equivalent to the binary exact/top-or-attained bridge.

lemma helperForTheorem_20_1_nonriInter_posDim_core_iff_exactTopOrAttained_of_polyLeft_domRi
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q))) :
    ((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
      ↔
      ((fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) := by
  exact
    (helperForTheorem_20_1_nonriInter_posDim_exactTopOrAttained_iff_reverseLe_and_neTopAttainment_of_polyLeft_domRi
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter).symm

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, an exact/top-or-attained binary bridge immediately yields the core package Unknown identifier `reverseLe`sorry + sorry - sorry - sorry : ?m.16reverseLe + Unknown identifier `non`non-Function expected at ⊤ but this term has type ?m.18 Note: Expected a function because this term is being applied to the argument split⊤ split-Unknown identifier `attainment`attainment.

lemma helperForTheorem_20_1_nonriInter_posDim_core_of_exactTopOrAttained_of_polyLeft_domRi
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hbinaryBridge :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  exact
    (helperForTheorem_20_1_nonriInter_posDim_core_iff_exactTopOrAttained_of_polyLeft_domRi
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter).2 hbinaryBridge

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, the currently available dependency-closed binary data can be packaged as: forward inequality + top-case witness, together with the equivalence between the core package and exact/top-or-attained.

lemma helperForTheorem_20_1_nonriInter_posDim_forwardTop_and_coreIffExact_data_of_polyLeft_domRi
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q))) :
    ((fenchelConjugate n (fun x => p x + q x) ≤
      infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
      ∧
      (((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
        ↔
        ((fenchelConjugate n (fun x => p x + q x) =
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
          (∀ xStar : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                (⊤ : EReal) ∨
              ∃ y : Fin n → ℝ,
                infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                  fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))) := by
  refine ⟨?_, ?_⟩
  · exact
      helperForTheorem_20_1_nonriInter_posDim_forwardLe_and_topWitness_of_polyLeft_domRi
        (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
        hnotRiInter
  · exact
      helperForTheorem_20_1_nonriInter_posDim_core_iff_exactTopOrAttained_of_polyLeft_domRi
        (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
        hnotRiInter

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, any exact/top-or-attained binary bridge implies the core package once the local Unknown identifier `core`sorry ↔ sorry : Propcore ↔ Unknown identifier `bridge`bridge equivalence is available.

lemma helperForTheorem_20_1_nonriInter_posDim_core_of_coreIff_and_exactTopOrAttained
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hcoreIffBinaryBridge :
      ((infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠
              (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y))
        ↔
        ((fenchelConjugate n (fun x => p x + q x) =
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
          (∀ xStar : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                (⊤ : EReal) ∨
              ∃ y : Fin n → ℝ,
                infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                  fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)))
    (hbinaryBridge :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  exact hcoreIffBinaryBridge.2 hbinaryBridge

lemma helperForTheorem_20_1_nonriInter_posDim_core_reverseLe_and_neTopAttainment_of_polyLeft_domRi
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (_hnPos : 0 < n)
    (_hpolyP : IsPolyhedralConvexFunction n p)
    (_hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (_hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (_hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (_hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hbinaryBridgeExactTopOrAttained_nonri :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  exact
    helperForTheorem_20_1_nonriInterCore_reverseLe_and_neTopAttainment_of_exact_topOrAttained_binary_posDim
      (p := p) (q := q) hbinaryBridgeExactTopOrAttained_nonri

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri setting with polyhedral left block, this is the exact/top-or-attained binary bridge needed to close the non-Unknown identifier `riInter`riInter branch.

lemma section20_mixed_two_block_exact_topOrAttained_of_polyLeft_domRi_nonriInter_posDim_from_closure_forward_topcase
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (_hnPos : 0 < n)
    (_hpolyP : IsPolyhedralConvexFunction n p)
    (_hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (_hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (_hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (_hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hbinaryBridgeExactTopOrAttained_nonri :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (fenchelConjugate n (fun x => p x + q x) =
      infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  exact hbinaryBridgeExactTopOrAttained_nonri

Helper for Theorem 20.1: in the positive-dimensional non-Unknown identifier `riInter`riInter mixed Unknown identifier `dom`sorry / sorry : ?m.5dom/Unknown identifier `ri`ri branch with polyhedral left block, the exact/top-or-attained binary bridge implies the core package Unknown identifier `reverseLe`sorry + sorry - sorry - sorry : ?m.16reverseLe + Unknown identifier `non`non-Function expected at ⊤ but this term has type ?m.18 Note: Expected a function because this term is being applied to the argument split⊤ split-Unknown identifier `attainment`attainment.

lemma section20_mixed_two_block_reverseLe_and_neTopAttainment_of_polyLeft_domRi_nonriInter_posDim_from_closure_forward_topcase
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hbinaryBridgeExactTopOrAttained_nonri :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  exact
    helperForTheorem_20_1_nonriInter_posDim_core_reverseLe_and_neTopAttainment_of_polyLeft_domRi
      (p := p) (q := q) hnPos hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter hbinaryBridgeExactTopOrAttained_nonri

Helper for Theorem 20.1: core non-Unknown identifier `riInter`riInter binary bridge under with polyhedral left block, providing reverse inequality and non-⊤ : ?m.1⊤ split-attainment.

lemma helperForTheorem_20_1_nonriInter_binary_reverseLe_and_universalAttainment_of_polyLeft_domRi_posDim
    {n : ℕ} (p q : (Fin n → ℝ) → EReal)
    (hnPos : 0 < n)
    (hpolyP : IsPolyhedralConvexFunction n p)
    (hproperP : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) p)
    (hproperQ : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) q)
    (hnonemptyDomInterRi :
      Set.Nonempty
        (((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
            effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
          ∩
          euclideanRelativeInterior n
            ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
              effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hnotRiInter :
      ¬ Set.Nonempty
          (euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) p)
            ∩
            euclideanRelativeInterior n
              ((fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹'
                effectiveDomain (Set.univ : Set (Fin n → ℝ)) q)))
    (hbinaryBridgeExactTopOrAttained_nonri :
      (fenchelConjugate n (fun x => p x + q x) =
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q)) ∧
      (∀ xStar : Fin n → ℝ,
        infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar = (⊤ : EReal) ∨
          ∃ y : Fin n → ℝ,
            infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
              fenchelConjugate n p (xStar - y) + fenchelConjugate n q y)) :
    (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
      fenchelConjugate n (fun x => p x + q x))
      ∧
      (∀ xStar : Fin n → ℝ,
        ∃ y : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
            fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) := by
  have hcore :
      (infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) ≤
        fenchelConjugate n (fun x => p x + q x))
        ∧
        (∀ xStar : Fin n → ℝ,
          infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar ≠ (⊤ : EReal) →
            ∃ y : Fin n → ℝ,
              infimalConvolution (fenchelConjugate n p) (fenchelConjugate n q) xStar =
                fenchelConjugate n p (xStar - y) + fenchelConjugate n q y) :=
    section20_mixed_two_block_reverseLe_and_neTopAttainment_of_polyLeft_domRi_nonriInter_posDim_from_closure_forward_topcase
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ
      hnonemptyDomInterRi hnotRiInter hbinaryBridgeExactTopOrAttained_nonri
  exact
    helperForTheorem_20_1_nonriInter_binary_reverseLe_and_universalAttainment_of_polyLeft_domRi_posDim_of_core
      (p := p) (q := q) (hnPos := hnPos) hpolyP hproperP hproperQ hnonemptyDomInterRi
      hnotRiInter hcore

Helper for Theorem 20.5: every polyhedral convex set is finitely generated.

lemma helperForTheorem_20_5_finitelyGenerated_of_polyhedral
    {n : ℕ} {C : Set (Fin n → ℝ)}
    (hCpoly : IsPolyhedralConvexSet n C) :
    IsFinitelyGeneratedConvexSet n C := by
  have hCconv : Convex ℝ C :=
    helperForTheorem_19_1_polyhedral_isConvex (n := n) (C := C) hCpoly
  have hTFAE :
      [IsPolyhedralConvexSet n C,
        (IsClosed C ∧ {C' : Set (Fin n → ℝ) | IsFace (𝕜 := ℝ) C C'}.Finite),
        IsFinitelyGeneratedConvexSet n C].TFAE :=
    polyhedral_closed_finiteFaces_finitelyGenerated_equiv (n := n) (C := C) hCconv
  exact (hTFAE.out 0 2).1 hCpoly

Helper for Theorem 20.5: a simplex admits the local finite-simplicial witness required by Set.LocallySimplicial (n : ℕ) (S : Set (Fin n → ℝ)) : PropSet.LocallySimplicial.

lemma helperForTheorem_20_5_localSimplexWitness_of_isSimplex
    {n m : ℕ} {P : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hP : IsSimplex n m P) (hx : x ∈ P) :
    ∃ (𝒮 : Set (Set (Fin n → ℝ))) (U : Set (Fin n → ℝ)),
      𝒮.Finite ∧
        U ∈ nhds x ∧
          (∀ Q ∈ 𝒮, ∃ k : ℕ, IsSimplex n k Q ∧ Q ⊆ P) ∧
            U ∩ (⋃₀ 𝒮) = U ∩ P := by
  classical
  rcases simplex_exists_subsimplex_through_point (hC := hP) (hx := hx) with
    ⟨b, _hb, _hP_eq, hthrough⟩
  let cand : Fin (m + 1) → Set (Fin n → ℝ) := fun j =>
    convexHull ℝ (Set.insert x (Set.range fun i : Fin m => b (Fin.succAbove j i)))
  let 𝒮 : Set (Set (Fin n → ℝ)) :=
    {Q | ∃ j : Fin (m + 1), Q = cand j ∧ IsSimplex n m Q ∧ Q ⊆ P}
  have h𝒮subsetRange : 𝒮 ⊆ Set.range cand := by
    intro Q hQ
    rcases hQ with ⟨j, hQeq, _hQsimp, _hQsub⟩
    exact ⟨j, hQeq.symm⟩
  have h𝒮finite : 𝒮.Finite := by
    exact (Set.finite_range cand).subset h𝒮subsetRange
  have hsUnion_subset : ⋃₀ 𝒮 ⊆ P := by
    intro y hy
    rcases Set.mem_sUnion.mp hy with ⟨Q, hQmem, hyQ⟩
    rcases hQmem with ⟨j, rfl, _hQsimp, hQsub⟩
    exact hQsub hyQ
  have hP_subset : P ⊆ ⋃₀ 𝒮 := by
    intro y hyP
    rcases hthrough y hyP with ⟨j, hQsub, hQsimp, _hxQ, hyQ⟩
    have hQsimpCand : IsSimplex n m (cand j) := by
      simpa [cand] using hQsimp
    have hQsubCand : cand j ⊆ P := by
      simpa [cand] using hQsub
    have hyCand : y ∈ cand j := by
      simpa [cand] using hyQ
    have hCandMem : cand j ∈ 𝒮 := ⟨j, rfl, hQsimpCand, hQsubCand⟩
    exact Set.mem_sUnion.mpr ⟨cand j, hCandMem, hyCand⟩
  have hsUnion_eq : ⋃₀ 𝒮 = P := Set.Subset.antisymm hsUnion_subset hP_subset
  have hUnivNhds : (Set.univ : Set (Fin n → ℝ)) ∈ nhds x := Filter.univ_mem
  have hSimplexFamily :
      ∀ Q ∈ 𝒮, ∃ k : ℕ, IsSimplex n k Q ∧ Q ⊆ P := by
    intro Q hQ
    rcases hQ with ⟨j, rfl, hQsimp, hQsub⟩
    exact ⟨m, hQsimp, hQsub⟩
  refine ⟨𝒮, Set.univ, h𝒮finite, hUnivNhds, hSimplexFamily, ?_⟩
  try 'simp' instead of 'simpa'

Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hsUnion_eq]

Helper for Theorem 20.5: every simplex is locally simplicial.

lemma helperForTheorem_20_5_locallySimplicial_of_isSimplex
    {n m : ℕ} {P : Set (Fin n → ℝ)}
    (hP : IsSimplex n m P) :
    Set.LocallySimplicial n P := by
  intro x hx
  exact helperForTheorem_20_5_localSimplexWitness_of_isSimplex (hP := hP) (hx := hx)

Helper for Theorem 20.5: a finitely generated convex set admits the global generalized-simplex decomposition from Theorem 17.1.

lemma helperForTheorem_20_5_globalGeneralizedSimplexDecomposition_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)}
    (hCfg : IsFinitelyGeneratedConvexSet n C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    C = ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} := by
  rcases hCfg with ⟨S0, S1, hS0fin, hS1fin, rfl⟩
  have _hS0fin_use : Set.Finite S0 := hS0fin
  have _hS1fin_use : Set.Finite S1 := hS1fin
  exact mixedConvexHull_eq_sUnion_generalizedSimplex_finrank_affineSpan_direction
    (n := n) S0 S1

Helper for Theorem 20.5: any member of the global generalized-simplex family over Unknown identifier `C`C is a subset of Unknown identifier `C`C.

lemma helperForTheorem_20_5_subset_of_mem_globalGeneralizedSimplexFamily
    {n d : ℕ} {C T : Set (Fin n → ℝ)}
    (hTmem : T ∈ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C}) :
    T ⊆ C := by
  exact hTmem.2

Helper for Theorem 20.5: the global generalized-simplex union from finite generation is contained in the original set.

lemma helperForTheorem_20_5_globalGeneralizedSimplexUnion_subset_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)}
    (hCfg : IsFinitelyGeneratedConvexSet n C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} ⊆ C := by
  intro d
  have hdecomp :
      C = ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} :=
    helperForTheorem_20_5_globalGeneralizedSimplexDecomposition_of_finitelyGenerated
      (n := n) (C := C) hCfg
  intro x hx
  exact hdecomp.symm ▸ hx

Helper for Theorem 20.5: every point of a finitely generated convex set belongs to the global generalized-simplex union from Theorem 17.1.

lemma helperForTheorem_20_5_mem_globalGeneralizedSimplexUnion_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hCfg : IsFinitelyGeneratedConvexSet n C) (hx : x ∈ C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    x ∈ ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} := by
  intro d
  have hdecomp :
      C = ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} :=
    helperForTheorem_20_5_globalGeneralizedSimplexDecomposition_of_finitelyGenerated
      (n := n) (C := C) hCfg
  exact hdecomp ▸ hx

Helper for Theorem 20.5: a point in the global generalized-simplex union lies in one generalized simplex from that family.

lemma helperForTheorem_20_5_exists_generalizedSimplex_through_point_of_mem_globalGeneralizedSimplexUnion
    {n d : ℕ} {C : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hxUnion :
      x ∈ ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C}) :
    ∃ T : Set (Fin n → ℝ), IsGeneralizedSimplex n d T ∧ T ⊆ C ∧ x ∈ T := by
  rcases Set.mem_sUnion.mp hxUnion with ⟨T, hTmem, hxT⟩
  have hTsub : T ⊆ C :=
    helperForTheorem_20_5_subset_of_mem_globalGeneralizedSimplexFamily
      (n := n) (d := d) (C := C) (T := T) hTmem
  exact ⟨T, hTmem.1, hTsub, hxT⟩

Helper for Theorem 20.5: extract a point-containing generalized simplex from finite generation via the global decomposition route.

lemma helperForTheorem_20_5_exists_generalizedSimplex_through_point_of_finitelyGenerated_via_globalUnion
    {n : ℕ} {C : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hCfg : IsFinitelyGeneratedConvexSet n C) (hx : x ∈ C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    ∃ T : Set (Fin n → ℝ), IsGeneralizedSimplex n d T ∧ T ⊆ C ∧ x ∈ T := by
  intro d
  have hxUnion :
      x ∈ ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} :=
    helperForTheorem_20_5_mem_globalGeneralizedSimplexUnion_of_finitelyGenerated
      (n := n) (C := C) (x := x) hCfg hx
  exact
    helperForTheorem_20_5_exists_generalizedSimplex_through_point_of_mem_globalGeneralizedSimplexUnion
      (n := n) (d := d) (C := C) (x := x) hxUnion

Helper for Theorem 20.5: every point of a mixed convex hull lies in a generalized simplex contained in that mixed convex hull.

lemma helperForTheorem_20_5_exists_generalizedSimplex_through_point_of_mixedConvexHull
    {n : ℕ} {S0 S1 : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hx : x ∈ mixedConvexHull (n := n) S0 S1) :
    let C := mixedConvexHull (n := n) S0 S1
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    ∃ T : Set (Fin n → ℝ), IsGeneralizedSimplex n d T ∧ T ⊆ C ∧ x ∈ T := by
  intro C d
  have hxC : x ∈ C := by
    simpa [C] using hx
  have hdecomp :
      C = ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} :=
    mixedConvexHull_eq_sUnion_generalizedSimplex_finrank_affineSpan_direction
      (n := n) S0 S1
  have hxUnion :
      x ∈ ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} := by
    rw [← hdecomp]
    exact hxC
  rcases Set.mem_sUnion.mp hxUnion with ⟨T, hTmem, hxT⟩
  exact ⟨T, hTmem.1, hTmem.2, hxT⟩

Helper for Theorem 20.5: every point of a finitely generated convex set lies in a generalized simplex contained in that set.

lemma helperForTheorem_20_5_exists_generalizedSimplex_through_point_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hCfg : IsFinitelyGeneratedConvexSet n C) (hx : x ∈ C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    ∃ T : Set (Fin n → ℝ), IsGeneralizedSimplex n d T ∧ T ⊆ C ∧ x ∈ T := by
  rcases hCfg with ⟨S0, S1, hS0fin, hS1fin, rfl⟩
  have _hS0fin_use : Set.Finite S0 := hS0fin
  have _hS1fin_use : Set.Finite S1 := hS1fin
  simpa using
    (helperForTheorem_20_5_exists_generalizedSimplex_through_point_of_mixedConvexHull
      (n := n) (S0 := S0) (S1 := S1) (x := x) hx)

Helper for Theorem 20.5: the global generalized-simplex decomposition of a finitely generated convex set gives a neighborhood equality with the global generalized-simplex union.

lemma helperForTheorem_20_5_localEq_with_globalGeneralizedSimplexUnion_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hCfg : IsFinitelyGeneratedConvexSet n C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    ∃ U : Set (Fin n → ℝ),
      U ∈ nhds x ∧
        U ∩ C = U ∩ ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} := by
  intro d
  refine ⟨Set.univ, Filter.univ_mem, ?_⟩
  have hdecomp :
      C = ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} :=
    helperForTheorem_20_5_globalGeneralizedSimplexDecomposition_of_finitelyGenerated
      (n := n) (C := C) hCfg
  calc
    Set.univ ∩ C = C := by simp
    _ = ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} := hdecomp
    _ = Set.univ ∩ ⋃₀ {T : Set (Fin n → ℝ) | IsGeneralizedSimplex n d T ∧ T ⊆ C} := by
      simp

Helper for Theorem 20.5: package a point-containing generalized simplex together with the neighborhood equality to the global generalized-simplex union.

lemma helperForTheorem_20_5_exists_generalizedSimplex_through_point_and_localEq_with_globalUnion_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hCfg : IsFinitelyGeneratedConvexSet n C) (hx : x ∈ C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    ∃ (T U : Set (Fin n → ℝ)),
      IsGeneralizedSimplex n d T ∧
        T ⊆ C ∧
          x ∈ T ∧
            U ∈ nhds x ∧
              U ∩ C = U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} := by
  intro d
  rcases
      (helperForTheorem_20_5_exists_generalizedSimplex_through_point_of_finitelyGenerated
        (n := n) (C := C) (x := x) hCfg hx)
    with ⟨T, hTgen, hTsub, hxT⟩
  rcases
      (helperForTheorem_20_5_localEq_with_globalGeneralizedSimplexUnion_of_finitelyGenerated
        (n := n) (C := C) (x := x) hCfg)
    with ⟨U, hUnhds, hUeq⟩
  exact ⟨T, U, hTgen, hTsub, hxT, hUnhds, hUeq⟩

Helper for Theorem 20.5: if Unknown identifier `T`sorry ⊆ sorry : PropT ⊆ Unknown identifier `C`C is locally simplicial and Unknown identifier `T`T agrees with Unknown identifier `C`C on a neighborhood of Unknown identifier `x`x, then Unknown identifier `C`C has a local simplicial witness at Unknown identifier `x`x.

lemma helperForTheorem_20_5_localWitness_of_subset_locallySimplicial_and_localEq
    {n : ℕ} {C T : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hTloc : Set.LocallySimplicial n T)
    (hTsub : T ⊆ C) (hxT : x ∈ T)
    {U : Set (Fin n → ℝ)} (hUnhds : U ∈ nhds x) (hUeq : U ∩ C = U ∩ T) :
    ∃ (𝒮 : Set (Set (Fin n → ℝ))) (W : Set (Fin n → ℝ)),
      𝒮.Finite ∧
        W ∈ nhds x ∧
          (∀ Q ∈ 𝒮, ∃ k : ℕ, IsSimplex n k Q ∧ Q ⊆ C) ∧
            W ∩ (⋃₀ 𝒮) = W ∩ C := by
  rcases hTloc x hxT with ⟨𝒮, V, h𝒮fin, hVnhds, hSimplex, hVeq⟩
  refine ⟨𝒮, U ∩ V, h𝒮fin, Filter.inter_mem hUnhds hVnhds, ?_, ?_⟩
  · intro Q hQ
    rcases hSimplex Q hQ with ⟨k, hQsimplex, hQsubT⟩
    exact ⟨k, hQsimplex, Set.Subset.trans hQsubT hTsub⟩
  · ext y
    constructor
    · intro hy
      rcases hy with ⟨hyUV, hyUnion⟩
      rcases hyUV with ⟨hyU, hyV⟩
      have hyVT : y ∈ V ∩ T := by
        have hyVS : y ∈ V ∩ ⋃₀ 𝒮 := ⟨hyV, hyUnion⟩
        simpa [hVeq] using hyVS
      have hyC : y ∈ C := hTsub hyVT.2
      exact ⟨⟨hyU, hyV⟩, hyC⟩
    · intro hy
      rcases hy with ⟨hyUV, hyC⟩
      rcases hyUV with ⟨hyU, hyV⟩
      have hyUT : y ∈ U ∩ T := by
        have hyUC : y ∈ U ∩ C := ⟨hyU, hyC⟩
        simpa [hUeq] using hyUC
      have hyVS : y ∈ V ∩ ⋃₀ 𝒮 := by
        have hyVT : y ∈ V ∩ T := ⟨hyV, hyUT.2⟩
        simpa [hVeq] using hyVT
      exact ⟨⟨hyU, hyV⟩, hyVS.2⟩

Helper for Theorem 20.5: local generalized-simplex bridge at a point of a finitely generated convex set.

lemma helperForTheorem_20_5_localEq_of_localEq_with_union_and_localEq_union_to_piece
    {n : ℕ} {C T U : Set (Fin n → ℝ)} {𝒯 : Set (Set (Fin n → ℝ))}
    (hUCeqUnion : U ∩ C = U ∩ ⋃₀ 𝒯)
    (hUUnionEqT : U ∩ ⋃₀ 𝒯 = U ∩ T) :
    U ∩ C = U ∩ T := by
  calc
    U ∩ C = U ∩ ⋃₀ 𝒯 := hUCeqUnion
    _ = U ∩ T := hUUnionEqT

Helper for Theorem 20.5: a selected generalized-simplex piece contained in Unknown identifier `C`C is contained in the global generalized-simplex union over Unknown identifier `C`C.

lemma helperForTheorem_20_5_selectedGeneralizedSimplex_subset_globalUnion
    {n d : ℕ} {C T : Set (Fin n → ℝ)}
    (hTgen : IsGeneralizedSimplex n d T) (hTsub : T ⊆ C) :
    T ⊆ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} := by
  intro y hyT
  exact Set.mem_sUnion.mpr ⟨T, ⟨hTgen, hTsub⟩, hyT⟩

Helper for Theorem 20.5: intersecting with Unknown identifier `U`U, a selected generalized-simplex piece contained in Unknown identifier `C`C is contained in the global generalized-simplex union over Unknown identifier `C`C.

lemma helperForTheorem_20_5_selectedGeneralizedSimplex_inter_subset_globalUnion_inter
    {n d : ℕ} {C T U : Set (Fin n → ℝ)}
    (hTgen : IsGeneralizedSimplex n d T) (hTsub : T ⊆ C) :
    U ∩ T ⊆ U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} := by
  intro y hyUT
  rcases hyUT with ⟨hyU, hyT⟩
  exact ⟨hyU,
    helperForTheorem_20_5_selectedGeneralizedSimplex_subset_globalUnion
      (n := n) (d := d) (C := C) (T := T) hTgen hTsub hyT⟩

Helper for Theorem 20.5: missing local piece data for a selected generalized-simplex inside the global generalized-simplex union of a finitely generated convex set.

lemma helperForTheorem_20_5_localPieceData_of_selectedGeneralizedSimplex_in_finitelyGenerated
    {n d : ℕ} {C T U : Set (Fin n → ℝ)}
    (hTloc : Set.LocallySimplicial n T)
    (hUUnionSubT :
      U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T) :
    ∃ _hTloc : Set.LocallySimplicial n T,
      U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T := by
  exact ⟨hTloc, hUUnionSubT⟩

Helper for Theorem 20.5: from a selected generalized-simplex in the finitely generated decomposition neighborhood, extract the local piece data needed for the bridge step.

lemma helperForTheorem_20_5_selectedGeneralizedSimplex_localPieceData_of_finitelyGenerated
    {n d : ℕ} {C T U : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (_hCfg : IsFinitelyGeneratedConvexSet n C) (_hx : x ∈ C)
    (_hTgen : IsGeneralizedSimplex n d T) (_hTsub : T ⊆ C) (_hxT : x ∈ T)
    (_hUCeqUnion :
      U ∩ C = U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C})
    (hTloc : Set.LocallySimplicial n T)
    (hUUnionSubT :
      U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T) :
    ∃ _hTloc : Set.LocallySimplicial n T,
      U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T := by
  exact
    helperForTheorem_20_5_localPieceData_of_selectedGeneralizedSimplex_in_finitelyGenerated
      (n := n) (d := d) (C := C) (T := T) (U := U) hTloc hUUnionSubT

Helper for Theorem 20.5: local piece data together with the selected-piece-to-union inclusion yields neighborhood equality between the global union and the selected piece.

lemma helperForTheorem_20_5_localEq_of_localPieceData_and_selectedGeneralizedSimplexInterSubset
    {n d : ℕ} {C T U : Set (Fin n → ℝ)}
    (hLocalPieceData :
      ∃ _hTloc : Set.LocallySimplicial n T,
        U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T)
    (hUTsubUUnion :
      U ∩ T ⊆ U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C}) :
    ∃ _hTloc : Set.LocallySimplicial n T,
      U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} = U ∩ T := by
  rcases hLocalPieceData with ⟨hTloc, hUUnionSubT⟩
  refine ⟨hTloc, Set.Subset.antisymm hUUnionSubT hUTsubUUnion⟩

Helper for Theorem 20.5: once local simpliciality of a selected generalized-simplex piece and local union-to-piece equality are available, the local bridge data follow.

lemma helperForTheorem_20_5_localGeneralizedSimplexBridge_of_finitelyGenerated_of_localPieceData
    {n d : ℕ} {C T U : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hTgen : IsGeneralizedSimplex n d T)
    (hTsub : T ⊆ C) (hxT : x ∈ T) (hTloc : Set.LocallySimplicial n T)
    (hUnhds : U ∈ nhds x)
    (hUCeqUnion :
      U ∩ C = U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C})
    (hUUnionEqT :
      U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} = U ∩ T) :
    ∃ (T' U' : Set (Fin n → ℝ)),
      IsGeneralizedSimplex n d T' ∧
        T' ⊆ C ∧
          x ∈ T' ∧
            Set.LocallySimplicial n T' ∧
              U' ∈ nhds x ∧
                U' ∩ C = U' ∩ T' := by
  have hUCeqT : U ∩ C = U ∩ T :=
    helperForTheorem_20_5_localEq_of_localEq_with_union_and_localEq_union_to_piece
      (n := n) (C := C) (T := T) (U := U)
      (𝒯 := {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C})
      hUCeqUnion hUUnionEqT
  exact ⟨T, U, hTgen, hTsub, hxT, hTloc, hUnhds, hUCeqT⟩

Helper for Theorem 20.5: local generalized-simplex bridge at a point of a finitely generated convex set.

lemma helperForTheorem_20_5_localGeneralizedSimplexBridge_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)} {x : Fin n → ℝ}
    (hCfg : IsFinitelyGeneratedConvexSet n C) (hx : x ∈ C) :
    let d := Module.finrank ℝ (affineSpan ℝ C).direction
    (∀ {T U : Set (Fin n → ℝ)},
      IsGeneralizedSimplex n d T →
      T ⊆ C →
      x ∈ T →
      U ∈ nhds x →
      U ∩ C = U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} →
      ∃ _hTloc : Set.LocallySimplicial n T,
        U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T) →
    ∃ (T U : Set (Fin n → ℝ)),
      IsGeneralizedSimplex n d T ∧
        T ⊆ C ∧
          x ∈ T ∧
            Set.LocallySimplicial n T ∧
              U ∈ nhds x ∧
                U ∩ C = U ∩ T := by
  intro d hSelectedLocalPieceData
  rcases
      (helperForTheorem_20_5_exists_generalizedSimplex_through_point_and_localEq_with_globalUnion_of_finitelyGenerated
        (n := n) (C := C) (x := x) hCfg hx)
    with ⟨T, U, hTgen, hTsub, hxT, hUnhds, hUeqUnion⟩
  have _hUeqUnion_use :
      U ∩ C = U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} :=
    hUeqUnion
  have _hTgen_use : IsGeneralizedSimplex n d T := hTgen
  have _hTsub_use : T ⊆ C := hTsub
  have _hxT_use : x ∈ T := hxT
  have _hUnhds_use : U ∈ nhds x := hUnhds
  have hUTsubUUnion :
      U ∩ T ⊆ U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} := by
    exact
      helperForTheorem_20_5_selectedGeneralizedSimplex_inter_subset_globalUnion_inter
        (n := n) (d := d) (C := C) (T := T) (U := U) hTgen hTsub
  have hLocalPieceData :
      ∃ hTloc : Set.LocallySimplicial n T,
        U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T :=
    hSelectedLocalPieceData hTgen hTsub hxT hUnhds hUeqUnion
  rcases
      (helperForTheorem_20_5_localEq_of_localPieceData_and_selectedGeneralizedSimplexInterSubset
        (n := n) (d := d) (C := C) (T := T) (U := U) hLocalPieceData hUTsubUUnion)
    with ⟨hTloc, hUUnionEqT⟩
  exact
    helperForTheorem_20_5_localGeneralizedSimplexBridge_of_finitelyGenerated_of_localPieceData
      (n := n) (d := d) (C := C) (T := T) (U := U) (x := x)
      hTgen hTsub hxT hTloc hUnhds hUeqUnion hUUnionEqT

Helper for Theorem 20.5: finitely generated convex sets are locally simplicial.

lemma helperForTheorem_20_5_locallySimplicial_of_finitelyGenerated
    {n : ℕ} {C : Set (Fin n → ℝ)}
    (hCfg : IsFinitelyGeneratedConvexSet n C)
    (hSelectedLocalPieceData :
      ∀ x ∈ C,
        let d := Module.finrank ℝ (affineSpan ℝ C).direction
        ∀ {T U : Set (Fin n → ℝ)},
          IsGeneralizedSimplex n d T →
          T ⊆ C →
          x ∈ T →
          U ∈ nhds x →
          U ∩ C = U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} →
          ∃ _hTloc : Set.LocallySimplicial n T,
            U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T) :
    Set.LocallySimplicial n C := by
  intro x hx
  let d := Module.finrank ℝ (affineSpan ℝ C).direction
  rcases
      (helperForTheorem_20_5_localGeneralizedSimplexBridge_of_finitelyGenerated
        (n := n) (C := C) (x := x) hCfg hx)
        (hSelectedLocalPieceData x hx)
    with ⟨T, U, _hTgen, hTsub, hxT, hTloc, hUnhds, hUeq⟩
  exact
    helperForTheorem_20_5_localWitness_of_subset_locallySimplicial_and_localEq
      (n := n) (C := C) (T := T) (x := x) hTloc hTsub hxT hUnhds hUeq

Helper for Theorem 20.5: every polytope is polyhedral.

lemma helperForTheorem_20_5_polyhedral_of_polytope
    {n : ℕ} {P : Set (Fin n → ℝ)}
    (hPpolytope : IsPolytope n P) :
    IsPolyhedralConvexSet n P := by
  rcases hPpolytope with ⟨S, hSfinite, hP_eq⟩
  have hPconv : Convex ℝ P := by
    rw [hP_eq]
    exact convex_convexHull ℝ S
  have hPfg : IsFinitelyGeneratedConvexSet n P := by
    refine ⟨S, (∅ : Set (Fin n → ℝ)), hSfinite, Set.finite_empty, ?_⟩
    calc
      P = convexHull ℝ S := hP_eq
      _ = mixedConvexHull (n := n) S (∅ : Set (Fin n → ℝ)) := by
            symm
            exact mixedConvexHull_empty_directions_eq_convexHull (n := n) S
  exact helperForTheorem_19_1_finitelyGenerated_imp_polyhedral
    (n := n) (C := P) hPconv hPfg

Theorem 20.5: every polyhedral convex set is locally simplicial; in particular, every polytope is locally simplicial.

theorem Theorem_20_5
    (n : ℕ)
    (hSelectedLocalPieceData :
      ∀ C : Set (Fin n → ℝ), IsPolyhedralConvexSet n C →
        ∀ x ∈ C,
          let d := Module.finrank ℝ (affineSpan ℝ C).direction
          ∀ {T U : Set (Fin n → ℝ)},
            IsGeneralizedSimplex n d T →
            T ⊆ C →
            x ∈ T →
            U ∈ nhds x →
            U ∩ C = U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} →
            ∃ _hTloc : Set.LocallySimplicial n T,
              U ∩ ⋃₀ {R : Set (Fin n → ℝ) | IsGeneralizedSimplex n d R ∧ R ⊆ C} ⊆ U ∩ T) :
    (∀ C : Set (Fin n → ℝ), IsPolyhedralConvexSet n C → Set.LocallySimplicial n C) ∧
      (∀ P : Set (Fin n → ℝ), IsPolytope n P → Set.LocallySimplicial n P) := by
  have hpolyhedralLocallySimplicial :
      ∀ C : Set (Fin n → ℝ), IsPolyhedralConvexSet n C → Set.LocallySimplicial n C := by
    intro C hCpoly
    exact
      helperForTheorem_20_5_locallySimplicial_of_finitelyGenerated
        (helperForTheorem_20_5_finitelyGenerated_of_polyhedral (n := n) (C := C) hCpoly)
        (hSelectedLocalPieceData C hCpoly)
  constructor
  · exact hpolyhedralLocallySimplicial
  · intro P hPpolytope
    exact
      hpolyhedralLocallySimplicial P
        (helperForTheorem_20_5_polyhedral_of_polytope (n := n) (P := P) hPpolytope)

end Section20end Chap04