Convex Analysis (Rockafellar, 1970) -- Chapter 08 -- Section 38 -- Part 17

section Chap08section Section38
-- Proof sketch: The extremal expressions defining `α` and `β` in Definition 38.5.2 are meaningful -- for arbitrary `EReal`-valued functions (even if improper) because they are built from `iSup`/`iInf` -- over effective domains and use `EReal`'s totalized arithmetic. Under the present repository -- semantics, the textbook restricted-extrema formulas remain valid in the proper branch, but the -- unconditional improper analogue of Proposition 38.5.3 is false; see the explicit `⊤/⊥` -- counterexample below.

Proposition 38.5.6: In the present formalization, the textbook restricted-extrema formulas

remain valid once Unknown identifier `f`f and -sorry : ℤ-Unknown identifier `g`g are proper. The naive extension of the existence clause from Proposition 38.5.3 to arbitrary improper EReal : TypeEReal-valued functions is refuted below.

theorem fenchelInnerProduct_exists_of_closed_ri_or_bounded_domain_improper {n : Nat} {f g : (Fin n → ℝ) → EReal} : ∀ {c : EReal}, IsProperEReal f → IsProperEReal (fun y => -g y) → fenchelInnerProduct f g = some c → c = (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f ∧ x ∈ erealDomBot (concaveConjugateInner g)}, concaveConjugateInner g x.1 - f x.1) ∧ c = (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g ∧ y ∈ erealDom (convexConjugateInner f)}, convexConjugateInner f y.1 - g y.1) := by intro c hf_proper hg_proper hc unfold fenchelInnerProduct at hc dsimp at hc split_ifs at hc with hEq · injection hc with hcEq subst hcEq constructor · trans (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f}, concaveConjugateInner g x.1 - f x.1) · rfl · refine le_antisymm ?_ ?_ · refine iSup_le ?_ intro x by_cases hxStar : x.1 ∈ erealDomBot (concaveConjugateInner g) · exact le_iSup_of_le ⟨x.1, ⟨x.2, hxStar⟩⟩ le_rfl · have hbot : concaveConjugateInner g x.1 = (⊥ : EReal) := by simpa [erealDomBot, bot_lt_iff_ne_bot] using hxStar have hfx_ne_bot : f x.1 ≠ (⊥ : EReal) := hf_proper.1 x.1 have hterm : concaveConjugateInner g x.1 - f x.1 = (⊥ : EReal) := by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hbot] using (EReal.bot_sub hfx_ne_bot) rw [hterm] exact bot_le · refine iSup_le ?_ intro x exact le_iSup_of_le ⟨x.1, x.2.1⟩ le_rfl · trans (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g}, convexConjugateInner f y.1 - g y.1) · exact hEq · refine le_antisymm ?_ ?_ · refine le_iInf ?_ intro y exact iInf_le (fun z : {z : Fin n → ℝ // z ∈ erealDomBot g} => convexConjugateInner f z.1 - g z.1) ⟨y.1, y.2.1⟩ · refine le_iInf ?_ intro y by_cases hyStar : y.1 ∈ erealDom (convexConjugateInner f) · exact iInf_le (fun z : {z : (Fin n → ℝ) // z ∈ erealDomBot g ∧ z ∈ erealDom (convexConjugateInner f)} => convexConjugateInner f z.1 - g z.1) ⟨y.1, ⟨y.2, hyStar⟩⟩ · have htop : convexConjugateInner f y.1 = (⊤ : EReal) := by have hnotlt : ¬ convexConjugateInner f y.1 < (⊤ : EReal) := by simpa [erealDom] using hyStar exact (top_le_iff).1 ((not_lt).1 hnotlt) have hgy_ne_top : g y.1 ≠ (⊤ : EReal) := by intro hgy_top have hneg_ne_bot : -g y.1 ≠ (⊥ : EReal) := hg_proper.1 y.1 exact hneg_ne_bot (by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hgy_top]) have hterm : convexConjugateInner f y.1 - g y.1 = (⊤ : EReal) := by simpa [htop] using (EReal.top_sub hgy_ne_top) rw [hterm] exact le_top
lemma fenchelInnerProduct_alpha_restrict_commonDom_of_proper {n : Nat} {f g : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) : (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f}, concaveConjugateInner g x.1 - f x.1) = (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f ∧ x ∈ erealDomBot (concaveConjugateInner g)}, concaveConjugateInner g x.1 - f x.1) := by refine le_antisymm ?_ ?_ · refine iSup_le ?_ intro x by_cases hxStar : x.1 ∈ erealDomBot (concaveConjugateInner g) · exact le_iSup_of_le ⟨x.1, ⟨x.2, hxStar⟩⟩ le_rfl · have hbot : concaveConjugateInner g x.1 = (⊥ : EReal) := by simpa [erealDomBot, bot_lt_iff_ne_bot] using hxStar have hfx_ne_bot : f x.1 ≠ (⊥ : EReal) := hf_proper.1 x.1 have hterm : concaveConjugateInner g x.1 - f x.1 = (⊥ : EReal) := by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hbot] using (EReal.bot_sub hfx_ne_bot) rw [hterm] exact bot_le · refine iSup_le ?_ intro x exact le_iSup_of_le ⟨x.1, x.2.1⟩ le_rfllemma fenchelInnerProduct_beta_restrict_commonDom_of_proper {n : Nat} {f g : (Fin n → ℝ) → EReal} (hg_proper : IsProperEReal (fun y => -g y)) : (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g}, convexConjugateInner f y.1 - g y.1) = (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g ∧ y ∈ erealDom (convexConjugateInner f)}, convexConjugateInner f y.1 - g y.1) := by refine le_antisymm ?_ ?_ · refine le_iInf ?_ intro y exact iInf_le (fun z : {z : Fin n → ℝ // z ∈ erealDomBot g} => convexConjugateInner f z.1 - g z.1) ⟨y.1, y.2.1⟩ · refine le_iInf ?_ intro y by_cases hyStar : y.1 ∈ erealDom (convexConjugateInner f) · exact iInf_le (fun z : {z : (Fin n → ℝ) // z ∈ erealDomBot g ∧ z ∈ erealDom (convexConjugateInner f)} => convexConjugateInner f z.1 - g z.1) ⟨y.1, ⟨y.2, hyStar⟩⟩ · have htop : convexConjugateInner f y.1 = (⊤ : EReal) := by have hnotlt : ¬ convexConjugateInner f y.1 < (⊤ : EReal) := by simpa [erealDom] using hyStar exact (top_le_iff).1 ((not_lt).1 hnotlt) have hgy_ne_top : g y.1 ≠ (⊤ : EReal) := by intro hgy_top have hneg_ne_bot : -g y.1 ≠ (⊥ : EReal) := hg_proper.1 y.1 exact hneg_ne_bot (by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hgy_top]) have hterm : convexConjugateInner f y.1 - g y.1 = (⊤ : EReal) := by simpa [htop] using (EReal.top_sub hgy_ne_top) rw [hterm] exact le_toplemma fenchelInnerProduct_textbook_extrema_of_proper {n : Nat} {f g : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) (hg_proper : IsProperEReal (fun y => -g y)) : ∀ {c : EReal}, fenchelInnerProduct f g = some c → c = (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f ∧ x ∈ erealDomBot (concaveConjugateInner g)}, concaveConjugateInner g x.1 - f x.1) ∧ c = (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g ∧ y ∈ erealDom (convexConjugateInner f)}, convexConjugateInner f y.1 - g y.1) := by intro c hc unfold fenchelInnerProduct at hc dsimp at hc split_ifs at hc with hEq injection hc with hcEq subst hcEq constructor · trans (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f}, concaveConjugateInner g x.1 - f x.1) · rfl · exact fenchelInnerProduct_alpha_restrict_commonDom_of_proper hf_proper · trans (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g}, convexConjugateInner f y.1 - g y.1) · exact hEq · exact fenchelInnerProduct_beta_restrict_commonDom_of_proper hg_properlemma helperForProposition_38_5_6_improper_textbook_formula_counterexample : fenchelInnerProduct (fun _ : Fin 1 → ℝ => (⊥ : EReal)) (fun _ : Fin 1 → ℝ => (⊤ : EReal)) = some (⊥ : EReal) ∧ (⨅ y : {y : (Fin 1 → ℝ) // y ∈ erealDomBot (fun _ : Fin 1 → ℝ => (⊤ : EReal)) ∧ y ∈ erealDom (convexConjugateInner (fun _ : Fin 1 → ℝ => (⊥ : EReal)))}, convexConjugateInner (fun _ : Fin 1 → ℝ => (⊥ : EReal)) y.1 - (⊤ : EReal)) = (⊤ : EReal) := by constructor · simp [fenchelInnerProduct, concaveConjugateInner, convexConjugateInner, erealDom, erealDomBot] · let S := {y : (Fin 1 → ℝ) // y ∈ erealDomBot (fun _ : Fin 1 → ℝ => (⊤ : EReal)) ∧ y ∈ erealDom (convexConjugateInner (fun _ : Fin 1 → ℝ => (⊥ : EReal)))} have htop : ∀ y : Fin 1 → ℝ, convexConjugateInner (fun _ : Fin 1 → ℝ => (⊥ : EReal)) y = (⊤ : EReal) := by intro y unfold convexConjugateInner refine top_le_iff.mp ?_ refine le_iSup_of_le ⟨0, by simp [erealDom]⟩ ?_ simp haveI : IsEmpty S := by refine ⟨?_⟩ intro y have : y.1 ∉ erealDom (convexConjugateInner (fun _ : Fin 1 → ℝ => (⊥ : EReal))) := by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [erealDom, htop y.1] exact this y.2.2 simp [S]lemma helperForProposition_38_5_6_improper_existence_counterexample : let f : (Fin 1 → ℝ) → EReal := fun _ => (⊤ : EReal) let g : (Fin 1 → ℝ) → EReal := fun _ => (⊥ : EReal) IsERealConvex f ∧ IsERealConvex (fun y => -g y) ∧ ((LowerSemicontinuous (fun y => -g y) ∧ (intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (erealDomBot (concaveConjugateInner g))).Nonempty) ∨ (LowerSemicontinuous f ∧ (intrinsicInterior ℝ (erealDomBot g) ∩ intrinsicInterior ℝ (erealDom (convexConjugateInner f))).Nonempty) ∨ (LowerSemicontinuous f ∧ LowerSemicontinuous (fun y => -g y) ∧ (Bornology.IsBounded (erealDom f) ∨ Bornology.IsBounded (erealDomBot g)))) ∧ fenchelInnerProduct f g = none := by dsimp refine ⟨?_, ?_, Or.inr (Or.inr ?_), ?_⟩ · simpa [IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using (convexFunctionOn_const_top (C := (Set.univ : Set (Fin 1 → ℝ)))) · simpa [IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using (convexFunctionOn_const_top (C := (Set.univ : Set (Fin 1 → ℝ)))) · refine ⟨?_, ?_, Or.inl ?_⟩ · simpa using (lowerSemicontinuous_const : LowerSemicontinuous (fun _ : Fin 1 → ℝ => (⊤ : EReal))) · simpa using (lowerSemicontinuous_const : LowerSemicontinuous (fun _ : Fin 1 → ℝ => (⊤ : EReal))) · try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [erealDom] using (Bornology.isBounded_empty : Bornology.IsBounded (∅ : Set (Fin 1 → ℝ))) · unfold fenchelInnerProduct have hα : (⨆ x : {x : Fin 1 → ℝ // x ∈ erealDom (fun _ : Fin 1 → ℝ => (⊤ : EReal))}, concaveConjugateInner (fun _ : Fin 1 → ℝ => (⊥ : EReal)) x.1 - (⊤ : EReal)) = (⊥ : EReal) := by haveI : IsEmpty {x : Fin 1 → ℝ // x ∈ erealDom (fun _ : Fin 1 → ℝ => (⊤ : EReal))} := by refine ⟨?_⟩ intro x have : x.1 ∉ erealDom (fun _ : Fin 1 → ℝ => (⊤ : EReal)) := by simp [erealDom] exact this x.2 simp have hβ : (⨅ y : {y : Fin 1 → ℝ // y ∈ erealDomBot (fun _ : Fin 1 → ℝ => (⊥ : EReal))}, convexConjugateInner (fun _ : Fin 1 → ℝ => (⊤ : EReal)) y.1 - (⊥ : EReal)) = (⊤ : EReal) := by haveI : IsEmpty {y : Fin 1 → ℝ // y ∈ erealDomBot (fun _ : Fin 1 → ℝ => (⊥ : EReal))} := by refine ⟨?_⟩ intro y have : y.1 ∉ erealDomBot (fun _ : Fin 1 → ℝ => (⊥ : EReal)) := by simp [erealDomBot] exact this y.2 simp simp [hβ]lemma helperForProposition_38_5_6_improper_existence_not_universally_valid : ¬ (∀ {n : Nat} {f g : (Fin n → ℝ) → EReal} (_hf_convex : IsERealConvex f) (_hg_concave : IsERealConvex (fun y => -g y)) (_hcond : (LowerSemicontinuous (fun y => -g y) ∧ (intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (erealDomBot (concaveConjugateInner g))).Nonempty) ∨ (LowerSemicontinuous f ∧ (intrinsicInterior ℝ (erealDomBot g) ∩ intrinsicInterior ℝ (erealDom (convexConjugateInner f))).Nonempty) ∨ (LowerSemicontinuous f ∧ LowerSemicontinuous (fun y => -g y) ∧ (Bornology.IsBounded (erealDom f) ∨ Bornology.IsBounded (erealDomBot g)))), ∃ c : EReal, fenchelInnerProduct f g = some c) := by intro hUniversal rcases helperForProposition_38_5_6_improper_existence_counterexample with ⟨hf, hg, hcond, hnone⟩ rcases hUniversal (n := 1) (f := fun _ : Fin 1 → ℝ => (⊤ : EReal)) (g := fun _ : Fin 1 → ℝ => (⊥ : EReal)) hf hg hcond with ⟨c, hc⟩ rw [hnone] at hc cases hc

The book-style closure of a concave EReal : TypeEReal-valued function Unknown identifier `g`g, defined as where Unknown identifier `cl`cl is erealFunctionClosure.{u_1} {X : Type u_1} [TopologicalSpace X] (f : X → EReal) : X → ERealerealFunctionClosure.

noncomputable def erealConcaveClosure {X : Type*} [TopologicalSpace X] (g : X → EReal) : X → EReal := fun x => -erealFunctionClosure (fun y => -g y) x
-- Proof sketch: Use the extremal characterization of `⟨f, g⟩` from Definition 38.5.2 and the -- involutive conjugacy properties (for convex/concave functions) to relate the primal/dual values -- for `(f*, g*)`, yielding `⟨f*, g*⟩ = -⟨f, g⟩`. For the "Moreover" part, replace `f` by `cl f` and -- `g` by `cl g` (closure in the convex/concave sense), and use that conjugates are invariant under -- closure, so the defining extremal values (hence the inner product when it exists) are unchanged. lemma helperForLemma_38_6_fStar_properConvexFunctionOn {n : Nat} {f : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (convexConjugateInner f) := by have hf_proper_on : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex f hf_proper hf_convex have hstar : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f) := proper_fenchelConjugate_of_proper (n := n) (f := f) hf_proper_on have hEq : convexConjugateInner f = fenchelConjugate n f := by funext y exact convexConjugateInner_eq_fenchelConjugate f y refine ⟨?_, ?_, ?_⟩ · simpa [hEq] using hstar.1 · simpa [hEq] using hstar.2.1 · intro x hx simpa [hEq] using hstar.2.2 x hxlemma helperForLemma_38_6_gStar_properConcaveFunctionOn {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) (concaveConjugateInner g) := by let A : (Fin n → ℝ) ≃ₗ[ℝ] (Fin n → ℝ) := (ContinuousLinearEquiv.neg ℝ : (Fin n → ℝ) ≃L[ℝ] (Fin n → ℝ)).toLinearEquiv have hneg_proper_on : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fun y => -g y) := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (fun y => -g y) hg_proper hg_concave have hstar : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n (fun y => -g y)) := proper_fenchelConjugate_of_proper (n := n) (f := fun y => -g y) hneg_proper_on have hprecomp : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fun y => fenchelConjugate n (fun z => -g z) (A y)) := properConvexFunctionOn_precomp_linearEquiv (n := n) A hstar unfold ProperConcaveFunctionOn simpa [A, concaveConjugateInner_eq_concaveFenchelConjugate, concaveFenchelConjugate] using hprecomplemma helperForLemma_38_6_fStar_closedConvexFunction {n : Nat} {f : (Fin n → ℝ) → EReal} : ClosedConvexFunction (convexConjugateInner f) := by have hclosed : LowerSemicontinuous (fenchelConjugate n f) ∧ ConvexFunction (fenchelConjugate n f) := fenchelConjugate_closedConvex (n := n) (f := f) have hEq : convexConjugateInner f = fenchelConjugate n f := by funext y exact convexConjugateInner_eq_fenchelConjugate f y refine ⟨?_, ?_⟩ · simpa [hEq] using hclosed.2 · simpa [hEq] using hclosed.1lemma helperForLemma_38_6_gStar_closedConcaveFunction {n : Nat} {g : (Fin n → ℝ) → EReal} : ClosedConcaveFunction (concaveConjugateInner g) := by let A : (Fin n → ℝ) →ₗ[ℝ] (Fin n → ℝ) := (ContinuousLinearEquiv.neg ℝ : (Fin n → ℝ) ≃L[ℝ] (Fin n → ℝ)).toLinearMap have hclosed : ClosedConvexFunction (fenchelConjugate n (fun y => -g y)) := by have h := fenchelConjugate_closedConvex (n := n) (f := fun y => -g y) exact ⟨h.2, h.1⟩ have hprecomp : ClosedConvexFunction (fun y => fenchelConjugate n (fun z => -g z) (A y)) := closedConvexFunction_precomp_linearMap (A := A) hclosed unfold ClosedConcaveFunction simpa [A, concaveConjugateInner_eq_concaveFenchelConjugate, concaveFenchelConjugate] using hprecomplemma helperForLemma_38_6_isProperEReal_of_properConvexFunctionOn_univ {n : Nat} {f : (Fin n → ℝ) → EReal} (hf : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f) : IsProperEReal f := by refine ⟨?_, ?_⟩ · intro x exact hf.2.2 x (by simp) · rcases (nonempty_epigraph_iff_nonempty_effectiveDomain (Set.univ : Set (Fin n → ℝ)) f).1 hf.2.1 with ⟨x, hx⟩ exact ⟨x, mem_effectiveDomain_imp_ne_top hx⟩lemma helperForLemma_38_6_isProperNeg_of_properConcaveFunctionOn_univ {n : Nat} {g : (Fin n → ℝ) → EReal} (hg : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) g) : IsProperEReal (fun y => -g y) := by simpa [ProperConcaveFunctionOn] using helperForLemma_38_6_isProperEReal_of_properConvexFunctionOn_univ (f := fun y => -g y) hglemma helperForLemma_38_6_primal_with_gStar_eq_neg_value {n : Nat} {f g : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) (hg_proper : IsProperEReal (fun y => -g y)) (hf_convex : IsERealConvex f) (hg_concave : IsERealConvex (fun y => -g y)) : ∀ {c : EReal}, fenchelInnerProduct f g = some c → fenchelPrimalInfimum f (concaveConjugateInner g) = -c := by intro c hc have hf_proper_on : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex f hf_proper hf_convex have hgStar_proper_on : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) (concaveConjugateInner g) := helperForLemma_38_6_gStar_properConcaveFunctionOn hg_proper hg_concave have hAlphaText := (fenchelInnerProduct_textbook_extrema_of_proper (f := f) (g := g) hf_proper hg_proper hc).1 have hAlphaDom : c = (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f}, concaveConjugateInner g x.1 - f x.1) := by simpa [fenchelInnerProduct_alpha_restrict_commonDom_of_proper (f := f) (g := g) hf_proper] using hAlphaText have hAlphaCore : (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom f}, concaveConjugateInner g x.1 - f x.1) = -fenchelPrimalInfimum f (concaveConjugateInner g) := by simpa using (helperForFenchelInnerProduct_alpha_eq_neg_primalInf (n := n) f (concaveConjugateInner g) hf_proper_on hgStar_proper_on) have hneg : c = -fenchelPrimalInfimum f (concaveConjugateInner g) := hAlphaDom.trans hAlphaCore have h := congrArg Neg.neg hneg simpa using h.symmlemma helperForLemma_38_6_primal_with_fStar_eq_value {n : Nat} {f g : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) (hg_proper : IsProperEReal (fun y => -g y)) (hf_convex : IsERealConvex f) (hg_concave : IsERealConvex (fun y => -g y)) : ∀ {c : EReal}, fenchelInnerProduct f g = some c → fenchelPrimalInfimum (convexConjugateInner f) g = c := by intro c hc have hg_proper_on : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) g := by simpa [ProperConcaveFunctionOn] using (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (fun y => -g y) hg_proper hg_concave) have hfStar_proper_on : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (convexConjugateInner f) := helperForLemma_38_6_fStar_properConvexFunctionOn hf_proper hf_convex have hBetaText := (fenchelInnerProduct_textbook_extrema_of_proper (f := f) (g := g) hf_proper hg_proper hc).2 have hBetaDom : c = (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g}, convexConjugateInner f y.1 - g y.1) := by simpa [fenchelInnerProduct_beta_restrict_commonDom_of_proper (f := f) (g := g) hg_proper] using hBetaText have hBetaCore : (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot g}, convexConjugateInner f y.1 - g y.1) = fenchelPrimalInfimum (convexConjugateInner f) g := by simpa using (helperForFenchelInnerProduct_beta_eq_primalInf (n := n) (convexConjugateInner f) g hfStar_proper_on hg_proper_on) exact (hBetaDom.trans hBetaCore).symmlemma helperForLemma_38_6_convexConjugateInner_convexFunctionClosure_eq {n : Nat} (f : (Fin n → ℝ) → EReal) : convexConjugateInner (convexFunctionClosure f) = convexConjugateInner f := by funext x simp [convexConjugateInner_eq_fenchelConjugate, section16_fenchelConjugate_convexFunctionClosure_eq]lemma helperForLemma_38_6_convexConjugateInner_biconjugate_eq_convexFunctionClosure {n : Nat} {f : (Fin n → ℝ) → EReal} (hf_convex : IsERealConvex f) : convexConjugateInner (convexConjugateInner f) = convexFunctionClosure f := by have hConvFun : ConvexFunction f := by simpa [IsERealConvex, ConvexFunctionOn, helperForTheorem_38_1_epigraph_eq_univ] using hf_convex have hInnerEq : convexConjugateInner f = fenchelConjugate n f := by funext y exact convexConjugateInner_eq_fenchelConjugate f y funext x calc convexConjugateInner (convexConjugateInner f) x = fenchelConjugate n (convexConjugateInner f) x := by exact convexConjugateInner_eq_fenchelConjugate (convexConjugateInner f) x _ = fenchelConjugate n (fenchelConjugate n f) x := by rw [hInnerEq] _ = convexFunctionClosure f x := by exact congrFun (section16_fenchelConjugate_biconjugate_eq_convexFunctionClosure (n := n) (f := f) hConvFun) xlemma helperForLemma_38_6_concaveConjugateInner_biconjugate_eq_concaveClosure {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_concave : IsERealConvex (fun y => -g y)) : concaveConjugateInner (concaveConjugateInner g) = concaveClosure g := by have hConcFun : ConcaveFunction g := by simpa [ConcaveFunction, IsERealConvex, ConvexFunctionOn, helperForTheorem_38_1_epigraph_eq_univ] using hg_concave have hInnerEq : concaveConjugateInner g = concaveConjugate g := by funext y calc concaveConjugateInner g y = concaveFenchelConjugate g y := by exact concaveConjugateInner_eq_concaveFenchelConjugate g y _ = concaveConjugate g y := by simpa [concaveFenchelConjugate] using (helperForTheorem_6_30_3_concaveConjugate_eq_neg_fenchelConjugate_neg_unrestricted (g := g) (xStar := y)).symm funext x calc concaveConjugateInner (concaveConjugateInner g) x = concaveFenchelConjugate (concaveConjugateInner g) x := by exact concaveConjugateInner_eq_concaveFenchelConjugate (concaveConjugateInner g) x _ = concaveConjugate (concaveConjugateInner g) x := by simpa [concaveFenchelConjugate] using (helperForTheorem_6_30_3_concaveConjugate_eq_neg_fenchelConjugate_neg_unrestricted (g := concaveConjugateInner g) (xStar := x)).symm _ = concaveConjugate (concaveConjugate g) x := by rw [hInnerEq] _ = concaveClosure g x := by exact congrFun (concaveConjugate_biconjugate_eq_concaveClosure (g := g) hConcFun) xlemma helperForLemma_38_6_concaveClosure_eq_biconjugate {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_concave : IsERealConvex (fun y => -g y)) : concaveClosure g = concaveFenchelConjugate (concaveFenchelConjugate g) := by have hInnerEq : concaveConjugateInner g = concaveFenchelConjugate g := by funext y exact concaveConjugateInner_eq_concaveFenchelConjugate g y funext x calc concaveClosure g x = concaveConjugateInner (concaveConjugateInner g) x := by simpa using congrFun (helperForLemma_38_6_concaveConjugateInner_biconjugate_eq_concaveClosure (g := g) hg_concave).symm x _ = concaveFenchelConjugate (concaveConjugateInner g) x := by exact concaveConjugateInner_eq_concaveFenchelConjugate (concaveConjugateInner g) x _ = concaveFenchelConjugate (concaveFenchelConjugate g) x := by rw [hInnerEq]lemma helperForLemma_38_6_concaveClosure_properConcaveFunctionOn {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) (concaveClosure g) := by let gStar : (Fin n → ℝ) → EReal := concaveFenchelConjugate g have hg0_proper : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) g := by simpa [ProperConcaveFunctionOn] using (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (fun y => -g y) hg_proper hg_concave) have hgStar_proper : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) gStar := by simpa [gStar] using helperForTheorem_31_1_properConcave_concaveFenchelConjugate_of_properConcave (n := n) (g := g) hg0_proper have hclosure : concaveClosure g = concaveFenchelConjugate gStar := by simpa [gStar] using helperForLemma_38_6_concaveClosure_eq_biconjugate (g := g) hg_concave simpa [hclosure, gStar] using helperForTheorem_31_1_properConcave_concaveFenchelConjugate_of_properConcave (n := n) (g := gStar) hgStar_properlemma helperForLemma_38_6_concaveConjugateInner_concaveClosure_eq {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) : concaveConjugateInner (concaveClosure g) = concaveConjugateInner g := by let gStar : (Fin n → ℝ) → EReal := concaveFenchelConjugate g have hg0_proper : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) g := by simpa [ProperConcaveFunctionOn] using (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (fun y => -g y) hg_proper hg_concave) have hgStar_proper : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) gStar := by simpa [gStar] using helperForTheorem_31_1_properConcave_concaveFenchelConjugate_of_properConcave (n := n) (g := g) hg0_proper have hStarEq : gStar = concaveConjugateInner g := by funext x simpa [gStar] using (concaveConjugateInner_eq_concaveFenchelConjugate g x).symm have hgStar_closed : ClosedConcaveFunction gStar := by simpa [hStarEq] using (helperForLemma_38_6_gStar_closedConcaveFunction (g := g)) have hclosure : concaveClosure g = concaveFenchelConjugate gStar := by simpa [gStar] using helperForLemma_38_6_concaveClosure_eq_biconjugate (g := g) hg_concave funext x calc concaveConjugateInner (concaveClosure g) x = concaveFenchelConjugate (concaveClosure g) x := by exact concaveConjugateInner_eq_concaveFenchelConjugate (concaveClosure g) x _ = concaveFenchelConjugate (concaveFenchelConjugate gStar) x := by simp [hclosure] _ = gStar x := by simpa using helperForTheorem_31_1_concave_biconjugate_eq_of_closedConcave (n := n) (g := gStar) hgStar_closed hgStar_proper x _ = concaveConjugateInner g x := by simpa [gStar] using (concaveConjugateInner_eq_concaveFenchelConjugate g x).symmlemma helperForLemma_38_6_convexClosure_properConvexFunctionOn {n : Nat} {f : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (convexFunctionClosure f) := by have hf_proper_on : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex f hf_proper hf_convex exact (convexFunctionClosure_closed_properConvexFunctionOn_and_agrees_on_ri (f := f) hf_proper_on).1.2lemma helperForLemma_38_6_fStar_isProperEReal {n : Nat} {f : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) : IsProperEReal (convexConjugateInner f) := by exact helperForLemma_38_6_isProperEReal_of_properConvexFunctionOn_univ (helperForLemma_38_6_fStar_properConvexFunctionOn hf_proper hf_convex)lemma helperForLemma_38_6_gStar_isProperNeg {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) : IsProperEReal (fun y => -(concaveConjugateInner g y)) := by exact helperForLemma_38_6_isProperNeg_of_properConcaveFunctionOn_univ (helperForLemma_38_6_gStar_properConcaveFunctionOn hg_proper hg_concave)lemma helperForLemma_38_6_convexClosure_isProperEReal {n : Nat} {f : (Fin n → ℝ) → EReal} (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) : IsProperEReal (convexFunctionClosure f) := by exact helperForLemma_38_6_isProperEReal_of_properConvexFunctionOn_univ (helperForLemma_38_6_convexClosure_properConvexFunctionOn hf_proper hf_convex)lemma helperForLemma_38_6_concaveClosure_isProperNeg {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) : IsProperEReal (fun y => -(concaveClosure g y)) := by exact helperForLemma_38_6_isProperNeg_of_properConcaveFunctionOn_univ (helperForLemma_38_6_concaveClosure_properConcaveFunctionOn hg_proper hg_concave)

Lemma 38.6: Let Unknown identifier `f`f be a proper convex function on ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, and let Unknown identifier `g`g be a proper concave function on ℝ ^ sorry : Typeℝ^Unknown identifier `n`n. If Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨f, g⟩ exists, then exists and . Moreover, then Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨cl f, cl g⟩ exists and coincides with Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨f, g⟩.

In Lean, Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨f, g⟩ is modeled by , by convexConjugateInner sorry : (Fin ?m.1 → ℝ) → ERealconvexConjugateInner Unknown identifier `f`f, by concaveConjugateInner sorry : (Fin ?m.1 → ℝ) → ERealconcaveConjugateInner Unknown identifier `g`g, Unknown identifier `cl`cl f by convexFunctionClosure sorry : (Fin ?m.1 → ℝ) → ERealconvexFunctionClosure Unknown identifier `f`f, and Unknown identifier `cl`cl g by concaveClosure sorry : (Fin ?m.1 → ℝ) → ERealconcaveClosure Unknown identifier `g`g. Downstream uses in Section 38.7 are all in this proper branch.

lemma fenchelInnerProduct_conjugates_eq_neg_and_closure {n : Nat} (f g : (Fin n → ℝ) → EReal) (hf_proper : IsProperEReal f) (hg_proper : IsProperEReal (fun y => -g y)) (hf_convex : IsERealConvex f) (hg_concave : IsERealConvex (fun y => -g y)) : ∀ {c : EReal}, fenchelInnerProduct f g = some c → fenchelInnerProduct (convexConjugateInner f) (concaveConjugateInner g) = some (-c) ∧ fenchelInnerProduct (convexFunctionClosure f) (concaveClosure g) = some c := by intro c hc let fStar := convexConjugateInner f let gStar := concaveConjugateInner g let fCl := convexFunctionClosure f let gCl := concaveClosure g have hfPC : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex f hf_proper hf_convex have hgPC : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) g := by simpa [ProperConcaveFunctionOn] using (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (fun y => -g y) hg_proper hg_concave) have hfStarPC : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) fStar := by simpa [fStar] using helperForLemma_38_6_fStar_properConvexFunctionOn hf_proper hf_convex have hgStarPC : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) gStar := by simpa [gStar] using helperForLemma_38_6_gStar_properConcaveFunctionOn hg_proper hg_concave have hfClPC : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) fCl := by simpa [fCl] using helperForLemma_38_6_convexClosure_properConvexFunctionOn hf_proper hf_convex have hgClPC : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) gCl := by simpa [gCl] using helperForLemma_38_6_concaveClosure_properConcaveFunctionOn hg_proper hg_concave have hfStarStar : convexConjugateInner fStar = fCl := by simpa [fStar, fCl] using helperForLemma_38_6_convexConjugateInner_biconjugate_eq_convexFunctionClosure hf_convex have hgStarStar : concaveConjugateInner gStar = gCl := by simpa [gStar, gCl] using helperForLemma_38_6_concaveConjugateInner_biconjugate_eq_concaveClosure hg_concave have hfClStar : convexConjugateInner fCl = fStar := by simpa [fCl, fStar] using helperForLemma_38_6_convexConjugateInner_convexFunctionClosure_eq f have hgClStar : concaveConjugateInner gCl = gStar := by simpa [gCl, gStar] using helperForLemma_38_6_concaveConjugateInner_concaveClosure_eq hg_proper hg_concave have hfStarFenchel : fenchelConjugate n fStar = fCl := by calc fenchelConjugate n fStar = convexConjugateInner fStar := by funext x exact (convexConjugateInner_eq_fenchelConjugate fStar x).symm _ = fCl := hfStarStar have hfFenchel : fenchelConjugate n f = fStar := by funext x exact (convexConjugateInner_eq_fenchelConjugate f x).symm have hgStarFenchel : concaveFenchelConjugate gStar = gCl := by calc concaveFenchelConjugate gStar = concaveConjugateInner gStar := by funext x exact (concaveConjugateInner_eq_concaveFenchelConjugate gStar x).symm _ = gCl := hgStarStar have hgFenchel : concaveFenchelConjugate g = gStar := by funext x exact (concaveConjugateInner_eq_concaveFenchelConjugate g x).symm have hP_f_gStar : fenchelPrimalInfimum f gStar = -c := by simpa [gStar] using helperForLemma_38_6_primal_with_gStar_eq_neg_value hf_proper hg_proper hf_convex hg_concave hc have hP_fStar_g : fenchelPrimalInfimum fStar g = c := by simpa [fStar] using helperForLemma_38_6_primal_with_fStar_eq_value hf_proper hg_proper hf_convex hg_concave hc have hP_fCl_gStar_le : fenchelPrimalInfimum fCl gStar ≤ fenchelPrimalInfimum f gStar := by have hCommonCl := helperForTheorem_31_1_commonBookEffectiveDomainDifference_eq_pointwise_sub_of_proper (n := n) fCl gStar hfClPC hgStarPC have hCommon := helperForTheorem_31_1_commonBookEffectiveDomainDifference_eq_pointwise_sub_of_proper (n := n) f gStar hfPC hgStarPC simp only [fenchelPrimalInfimum, functionInfimumEReal, hCommonCl, hCommon] exact iInf_mono (fun x => EReal.sub_le_sub ((convexFunctionClosure_le_self f) x) le_rfl) have hP_fStar_gCl_le : fenchelPrimalInfimum fStar gCl ≤ fenchelPrimalInfimum fStar g := by have hCommonCl := helperForTheorem_31_1_commonBookEffectiveDomainDifference_eq_pointwise_sub_of_proper (n := n) fStar gCl hfStarPC hgClPC have hCommon := helperForTheorem_31_1_commonBookEffectiveDomainDifference_eq_pointwise_sub_of_proper (n := n) fStar g hfStarPC hgPC simp only [fenchelPrimalInfimum, functionInfimumEReal, hCommonCl, hCommon] exact iInf_mono (fun x => EReal.sub_le_sub le_rfl (helperForCorollary_6_30_3_self_le_concaveClosure g x)) have hP_fCl_gStar_ge : -c ≤ fenchelPrimalInfimum fCl gStar := by have hWeak := helperForTheorem_31_1_primal_ge_dual fStar g hfStarPC hgPC have hDual := helperForTheorem_31_1_dual_eq_neg_primal_of_conjugate_pair fStar g hfStarPC hgPC have hneg : -fenchelPrimalInfimum fCl gStar ≤ c := by calc -fenchelPrimalInfimum fCl gStar = fenchelDualSupremum (n := n) fStar g := by simpa [hfStarFenchel, hgFenchel] using hDual.symm _ ≤ fenchelPrimalInfimum fStar g := hWeak _ = c := hP_fStar_g rw [← EReal.neg_le_neg_iff] simpa using hneg have hP_fStar_gCl_neg_eq : -fenchelPrimalInfimum fStar gCl = -c := by have hWeak := helperForTheorem_31_1_primal_ge_dual f gStar hfPC hgStarPC have hDual := helperForTheorem_31_1_dual_eq_neg_primal_of_conjugate_pair f gStar hfPC hgStarPC have hUpper : -fenchelPrimalInfimum fStar gCl ≤ -c := by calc -fenchelPrimalInfimum fStar gCl = fenchelDualSupremum (n := n) f gStar := by simpa [hfFenchel, hgStarFenchel] using hDual.symm _ ≤ fenchelPrimalInfimum f gStar := hWeak _ = -c := hP_f_gStar have hLower : -c ≤ -fenchelPrimalInfimum fStar gCl := by apply EReal.neg_le_neg_iff.mpr exact hP_fStar_gCl_le.trans_eq hP_fStar_g exact le_antisymm hUpper hLower have hP_fCl_gStar_eq : fenchelPrimalInfimum fCl gStar = -c := by exact le_antisymm (hP_fCl_gStar_le.trans_eq hP_f_gStar) hP_fCl_gStar_ge have hfStarStarPC : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (convexConjugateInner fStar) := by rw [hfStarStar] exact hfClPC have hgStarStarPC : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) (concaveConjugateInner gStar) := by rw [hgStarStar] exact hgClPC have hAlphaStar : (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom fStar}, concaveConjugateInner gStar x.1 - fStar x.1) = -c := by calc _ = -fenchelPrimalInfimum fStar (concaveConjugateInner gStar) := helperForFenchelInnerProduct_alpha_eq_neg_primalInf fStar (concaveConjugateInner gStar) hfStarPC hgStarStarPC _ = -fenchelPrimalInfimum fStar gCl := by rw [hgStarStar] _ = -c := hP_fStar_gCl_neg_eq have hBetaStar : (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot gStar}, convexConjugateInner fStar y.1 - gStar y.1) = -c := by calc _ = fenchelPrimalInfimum (convexConjugateInner fStar) gStar := helperForFenchelInnerProduct_beta_eq_primalInf (convexConjugateInner fStar) gStar hfStarStarPC hgStarPC _ = fenchelPrimalInfimum fCl gStar := by rw [hfStarStar] _ = -c := hP_fCl_gStar_eq have hAlphaCl : (⨆ x : {x : (Fin n → ℝ) // x ∈ erealDom fCl}, concaveConjugateInner gCl x.1 - fCl x.1) = c := by rw [hgClStar] calc _ = -fenchelPrimalInfimum fCl gStar := helperForFenchelInnerProduct_alpha_eq_neg_primalInf fCl gStar hfClPC hgStarPC _ = c := by rw [hP_fCl_gStar_eq]; simp have hBetaCl : (⨅ y : {y : (Fin n → ℝ) // y ∈ erealDomBot gCl}, convexConjugateInner fCl y.1 - gCl y.1) = c := by rw [hfClStar] calc _ = fenchelPrimalInfimum fStar gCl := helperForFenchelInnerProduct_beta_eq_primalInf fStar gCl hfStarPC hgClPC _ = c := by have := congrArg Neg.neg hP_fStar_gCl_neg_eq simpa using this constructor · change fenchelInnerProduct fStar gStar = some (-c) unfold fenchelInnerProduct rw [hAlphaStar, hBetaStar] simp · change fenchelInnerProduct fCl gCl = some c unfold fenchelInnerProduct rw [hAlphaCl, hBetaCl] simp
lemma intrinsicInterior_erealDomBot_concaveClosure_eq {n : Nat} (g : (Fin n → ℝ) → EReal) (hg_proper : IsProperEReal (fun x => -g x)) (hg_concave : IsERealConvex (fun x => -g x)) : intrinsicInterior ℝ (erealDomBot (concaveClosure g)) = intrinsicInterior ℝ (erealDomBot g) := by let negG : (Fin n → ℝ) → EReal := fun x => -g x have hnegPC : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) negG := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex negG hg_proper hg_concave have hriE := (convexFunctionClosure_effectiveDomain_subset_relativeBoundary_and_same_closure_ri_dim hnegPC).2.2.2.1 let e : EuclideanSpace ℝ (Fin n) ≃L[ℝ] (Fin n → ℝ) := EuclideanSpace.equiv (ι := Fin n) (𝕜 := ℝ) have hpre (C : Set (Fin n → ℝ)) : e.symm '' C = (fun x : EuclideanSpace ℝ (Fin n) => (x : Fin n → ℝ)) ⁻¹' C := by ext y constructor · rintro ⟨x, hx, rfl⟩ simpa [e] using hx · intro hy refine ⟨e y, ?_, by simp⟩ simpa [e] using hy have hriRaw : euclideanRelativeInterior n (e.symm '' effectiveDomain (Set.univ : Set (Fin n → ℝ)) (convexFunctionClosure negG)) = euclideanRelativeInterior n (e.symm '' effectiveDomain (Set.univ : Set (Fin n → ℝ)) negG) := by rw [hpre, hpre] simpa [e, negG] using hriE have hriFin : euclideanRelativeInterior_fin n (effectiveDomain (Set.univ : Set (Fin n → ℝ)) (convexFunctionClosure negG)) = euclideanRelativeInterior_fin n (effectiveDomain (Set.univ : Set (Fin n → ℝ)) negG) := by unfold euclideanRelativeInterior_fin simpa [e] using congrArg (fun C => e '' C) hriRaw rw [helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior, helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] at hriFin have hconv : ConvexFunction negG := by simpa [negG, IsERealConvex, ConvexFunctionOn, helperForTheorem_38_1_epigraph_eq_univ] using hg_concave have hclEq : convexClosure negG = convexFunctionClosure negG := by calc convexClosure negG = clConv n negG := helperForTheorem_6_30_3_convexClosure_eq_clConv negG hconv _ = fenchelConjugate n (fenchelConjugate n negG) := by symm exact fenchelConjugate_biconjugate_eq_clConv (n := n) (f := negG) _ = convexFunctionClosure negG := section16_fenchelConjugate_biconjugate_eq_convexFunctionClosure (n := n) (f := negG) hconv have hnegIff (a : EReal) : (⊥ : EReal) < -a ↔ a < (⊤ : EReal) := by simp [bot_lt_iff_ne_bot] have hdomCl : erealDomBot (concaveClosure g) = effectiveDomain (Set.univ : Set (Fin n → ℝ)) (convexFunctionClosure negG) := by ext x simp only [erealDomBot, Set.mem_setOf_eq, effectiveDomain_eq, Set.mem_univ, true_and, concaveClosure_eq_neg_convexClosure_neg, hnegIff] change convexClosure negG x < (⊤ : EReal) ↔ convexFunctionClosure negG x < ⊤ rw [hclEq] have hdom : erealDomBot g = effectiveDomain (Set.univ : Set (Fin n → ℝ)) negG := by ext x simp only [erealDomBot, Set.mem_setOf_eq, effectiveDomain_eq, Set.mem_univ, true_and] simpa [negG] using hnegIff (-g x) rw [hdomCl, hdom] exact hriFin

Relative-interior qualification for Unknown identifier `h`h and Unknown identifier `g`g makes the Fenchel inner product of Unknown identifier `h`h with the concave conjugate of Unknown identifier `g`g exist; no closedness assumption on Unknown identifier `g`g is needed.

lemma fenchelInnerProduct_with_concaveConjugate_exists_of_ri {n : Nat} (h g : (Fin n → ℝ) → EReal) (hh_proper : IsProperEReal h) (hh_convex : IsERealConvex h) (hg_proper : IsProperEReal (fun x => -g x)) (hg_concave : IsERealConvex (fun x => -g x)) (hri : (intrinsicInterior ℝ (erealDom h) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : ∃ c : EReal, fenchelInnerProduct h (concaveConjugateInner g) = some c := by let gStar := concaveConjugateInner g have hgStarPC : ProperConcaveFunctionOn (Set.univ : Set (Fin n → ℝ)) gStar := by simpa [gStar] using helperForLemma_38_6_gStar_properConcaveFunctionOn hg_proper hg_concave have hgStarProper : IsProperEReal (fun x => -gStar x) := helperForLemma_38_6_isProperNeg_of_properConcaveFunctionOn_univ hgStarPC have hgStarClosed : ClosedConcaveFunction gStar := by simpa [gStar] using (helperForLemma_38_6_gStar_closedConcaveFunction (g := g)) have hgStarConcave : IsERealConvex (fun x => -gStar x) := by simpa [ClosedConcaveFunction, IsERealConvex, ConvexFunction, ConvexFunctionOn, helperForTheorem_38_1_epigraph_eq_univ] using hgStarClosed.1 have hbiconj : concaveConjugateInner gStar = concaveClosure g := by simpa [gStar] using helperForLemma_38_6_concaveConjugateInner_biconjugate_eq_concaveClosure (g := g) hg_concave have hriClosure := intrinsicInterior_erealDomBot_concaveClosure_eq g hg_proper hg_concave have hriStar : (intrinsicInterior ℝ (erealDom h) ∩ intrinsicInterior ℝ (erealDomBot (concaveConjugateInner gStar))).Nonempty := by rw [hbiconj, hriClosure] exact hri exact fenchelInnerProduct_exists_of_closed_ri_or_bounded_domain (f := h) (g := gStar) hh_convex hgStarConcave hh_proper hgStarProper (Or.inl ⟨hgStarClosed.2, hriStar⟩)

Lemma 38.6, current proper branch: let Unknown identifier `f`f be a proper convex function on ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, and let Unknown identifier `g`g be a proper concave function on ℝ ^ sorry : Typeℝ^Unknown identifier `n`n. If Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨f, g⟩ exists, then exists and . Moreover, then Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨cl f, cl g⟩ exists and coincides with Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨f, g⟩.

In Lean, Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨f, g⟩ is modeled by , by convexConjugateInner sorry : (Fin ?m.1 → ℝ) → ERealconvexConjugateInner Unknown identifier `f`f, by concaveConjugateInner sorry : (Fin ?m.1 → ℝ) → ERealconcaveConjugateInner Unknown identifier `g`g, Unknown identifier `cl`cl f by convexFunctionClosure sorry : (Fin ?m.1 → ℝ) → ERealconvexFunctionClosure Unknown identifier `f`f, and Unknown identifier `cl`cl g by concaveClosure sorry : (Fin ?m.1 → ℝ) → ERealconcaveClosure Unknown identifier `g`g. Downstream uses in Section 38.7 are all in this proper branch. The original lemma also discusses improper convex/concave functions; that additional case split is not claimed by this proper-branch declaration.

lemma helperForTheorem_38_7_hqual_to_fin_relativeInteriors {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hqual : ∃ u : Fin m → ℝ, u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : ∃ u : Fin m → ℝ, ∃ x : Fin n → ℝ, u ∈ euclideanRelativeInterior_fin m (erealDom f) ∩ euclideanRelativeInterior_fin m (bifunctionDom F) ∧ x ∈ euclideanRelativeInterior_fin n (erealDom (fun y : Fin n → ℝ => F u y)) ∩ euclideanRelativeInterior_fin n (erealDomBot g) := by rcases hqual with ⟨u, hu, x, hxF, hxg⟩ refine ⟨u, x, ?_, ?_⟩ · simpa only [ helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] using hu · simpa only [ helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] using And.intro hxF hxg

The Euclidean packed graph domain has the expected fixed-u fibers and projection. This is the set-theoretic bridge needed before applying Theorems 6.6 and 6.8.

lemma helperForTheorem_38_7_packedGraphDomain_fiber_projection {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) : let eMN := EuclideanSpace.equiv (ι := Fin (m + n)) (𝕜 := ℝ) let eM := EuclideanSpace.equiv (ι := Fin m) (𝕜 := ℝ) let eN := EuclideanSpace.equiv (ι := Fin n) (𝕜 := ℝ) let C : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) < ⊤} let append : EuclideanSpace ℝ (Fin m) → EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin (m + n)) := fun u x => eMN.symm (Fin.append (eM u) (eN x)) let Cu : EuclideanSpace ℝ (Fin m) → Set (EuclideanSpace ℝ (Fin n)) := fun u => {x | append u x ∈ C} let D : Set (EuclideanSpace ℝ (Fin m)) := {u | (Cu u).Nonempty} (∀ u : Fin m → ℝ, Cu (eM.symm u) = eN.symm '' erealDom (F u)) ∧ D = eM.symm '' bifunctionDom F := by classical intro eMN eM eN C append Cu D constructor · intro u ext x constructor · intro hx refine ⟨eN x, ?_, by simp [eN]⟩ simpa [Cu, append, C, eMN, eM, eN, erealDom] using hx · rintro ⟨x, hx, rfl⟩ simpa [Cu, append, C, eMN, eM, eN, erealDom] using hx · ext u constructor · rintro ⟨x, hx⟩ refine ⟨eM u, ?_, by simp [eM]⟩ refine ⟨eN x, ?_⟩ have hx' : F (eM u) (eN x) < ⊤ := by simpa [Cu, append, C, eMN, eM, eN] using hx exact ne_of_lt hx' · rintro ⟨u, hu, rfl⟩ rcases hu with ⟨x, hx⟩ refine ⟨eN.symm x, ?_⟩ simpa [Cu, append, C, eMN, eM, eN, lt_top_iff_ne_top] using hx

Intersecting the packed graph projection with the transported domain of Unknown identifier `f`f is exactly the transport of the common base domain.

lemma helperForTheorem_38_7_mem_imageDom_iff_exists_sum_lt_top {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (x : Fin n → ℝ) : x ∈ erealDom (bifunctionImageRaw F f) ↔ ∃ u : Fin m → ℝ, f u + F u x < (⊤ : EReal) := by simp only [erealDom, bifunctionImageRaw, Set.mem_setOf_eq, iInf_lt_top]

The Unknown identifier `x`x-projection of the packed effective domain of is the Euclidean transport of the effective domain of bifunctionImageRaw sorry sorry : ?m.2 → ERealbifunctionImageRaw Unknown identifier `F`F Unknown identifier `f`f.

lemma helperForTheorem_38_7_packedSumDomain_xProjection {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) : let eMN := EuclideanSpace.equiv (ι := Fin (m + n)) (𝕜 := ℝ) let eM := EuclideanSpace.equiv (ι := Fin m) (𝕜 := ℝ) let eN := EuclideanSpace.equiv (ι := Fin n) (𝕜 := ℝ) let C : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | f (fun i => z (Fin.castAdd n i)) + F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) < ⊤} let append : EuclideanSpace ℝ (Fin m) → EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin (m + n)) := fun u x => eMN.symm (Fin.append (eM u) (eN x)) let projX : Set (EuclideanSpace ℝ (Fin n)) := {x | ∃ u : EuclideanSpace ℝ (Fin m), append u x ∈ C} projX = eN.symm '' erealDom (bifunctionImageRaw F f) := by classical intro eMN eM eN C append projX ext x constructor · rintro ⟨u, hu⟩ refine ⟨eN x, ?_, by simp [eN]⟩ rw [helperForTheorem_38_7_mem_imageDom_iff_exists_sum_lt_top] refine ⟨eM u, ?_⟩ simpa [append, C, eMN, eM, eN] using hu · rintro ⟨x0, hx0, rfl⟩ rw [helperForTheorem_38_7_mem_imageDom_iff_exists_sum_lt_top] at hx0 rcases hx0 with ⟨u0, hu0⟩ refine ⟨eM.symm u0, ?_⟩ simpa [append, C, eMN, eM, eN] using hu0

The graph-domain part of the qualification lifts to the relative interior of the packed graph domain. This isolates the Theorem 6.8 step needed before intersecting with the lifted domain of Unknown identifier `f`f.

lemma helperForTheorem_38_7_graphQualification_lifts_to_packedRi {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hqual : ∃ u : Fin m → ℝ, u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : let eMN := EuclideanSpace.equiv (ι := Fin (m + n)) (𝕜 := ℝ) let unused variable `eM` Note: This linter can be disabled with `set_option linter.unusedVariables false`eM := EuclideanSpace.equiv (ι := Fin m) (𝕜 := ℝ) let unused variable `eN` Note: This linter can be disabled with `set_option linter.unusedVariables false`eN := EuclideanSpace.equiv (ι := Fin n) (𝕜 := ℝ) let C : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) < ⊤} ∃ u : Fin m → ℝ, ∃ x : Fin n → ℝ, eMN.symm (Fin.append u x) ∈ euclideanRelativeInterior (m + n) C ∧ u ∈ euclideanRelativeInterior_fin m (erealDom f) ∧ x ∈ euclideanRelativeInterior_fin n (erealDomBot g) := by classical intro eMN eM eN C rcases helperForTheorem_38_7_hqual_to_fin_relativeInteriors F f g hqual with ⟨u, x, hu, hx⟩ let graphg : (Fin (m + n) → ℝ) → EReal := fun z => F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) let pairMap : (Fin (m + n) → ℝ) →ₗ[ℝ] (Fin m → ℝ) × (Fin n → ℝ) := (projXLinearMap (n := m) (m := n)).prod (projLamLinearMap (n := m) (m := n)) have hgraph_convex : IsERealConvex graphg := by simpa [graphg, pairMap, projXLinearMap, projLamLinearMap] using helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := pairMap) (g := fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2) hF_convex have hgraph_proper : IsProperEReal graphg := by constructor · intro z simpa [graphg] using hF_proper.1 ((fun i => z (Fin.castAdd n i)), (fun j => z (Fin.natAdd m j))) · rcases hF_proper.2 with ⟨p, hp⟩ refine ⟨Fin.append p.1 p.2, ?_⟩ simpa [graphg] have hgraph_pc : ProperConvexFunctionOn (Set.univ : Set (Fin (m + n) → ℝ)) graphg := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex graphg hgraph_proper hgraph_convex have hraw_conv : Convex ℝ {z : Fin (m + n) → ℝ | graphg z < ⊤} := by have heq : {z : Fin (m + n) → ℝ | graphg z < ⊤} = effectiveDomain (Set.univ : Set (Fin (m + n) → ℝ)) graphg := by ext z simp [effectiveDomain_eq, lt_top_iff_ne_top] rw [heq] exact effectiveDomain_convex hgraph_pc.1 have hCconv : Convex ℝ C := by simpa [C, graphg] using hraw_conv.linear_image eMN.symm.toLinearMap let append : EuclideanSpace ℝ (Fin m) → EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin (m + n)) := fun u x => eMN.symm (Fin.append (eM u) (eN x)) let Cu : EuclideanSpace ℝ (Fin m) → Set (EuclideanSpace ℝ (Fin n)) := fun u => {x | append u x ∈ C} let D : Set (EuclideanSpace ℝ (Fin m)) := {u | (Cu u).Nonempty} have hfiber_projection := helperForTheorem_38_7_packedGraphDomain_fiber_projection F have hCu : Cu (eM.symm u) = eN.symm '' erealDom (F u) := by simpa [append, Cu, D, C, eMN, eM, eN] using hfiber_projection.1 u have hD : D = eM.symm '' bifunctionDom F := by simpa [append, Cu, D, C, eMN, eM, eN] using hfiber_projection.2 have huE : eM.symm u ∈ euclideanRelativeInterior m D := by rw [hD] exact (mem_euclideanRelativeInterior_fin_iff (n := m) (C := bifunctionDom F) (x := u)).1 hu.2 have hxE : eN.symm x ∈ euclideanRelativeInterior n (Cu (eM.symm u)) := by rw [hCu] exact (mem_euclideanRelativeInterior_fin_iff (n := n) (C := erealDom (F u)) (x := x)).1 hx.1 have hsection := euclideanRelativeInterior_mem_iff_relativeInterior_section (m := m) (p := n) C hCconv (eM.symm u) (eN.symm x) refine ⟨u, x, ?_, hu.1, hx.2⟩ exact hsection.2 ⟨huE, hxE⟩

The same qualification point lies in the relative interior of the intersection of the packed graph domain and the cylinder over Unknown identifier `dom`dom f.

lemma helperForTheorem_38_7_qualification_lifts_to_packedDomainIntersectionRi {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hqual : ∃ u : Fin m → ℝ, u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : let eMN := EuclideanSpace.equiv (ι := Fin (m + n)) (𝕜 := ℝ) let Cgraph : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) < ⊤} let Cliftf : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | f (fun i => z (Fin.castAdd n i)) < ⊤} ∃ u : Fin m → ℝ, ∃ x : Fin n → ℝ, eMN.symm (Fin.append u x) ∈ euclideanRelativeInterior (m + n) (Cgraph ∩ Cliftf) ∧ x ∈ euclideanRelativeInterior_fin n (erealDomBot g) := by classical intro eMN Cgraph Cliftf rcases helperForTheorem_38_7_graphQualification_lifts_to_packedRi F f g hF_proper hF_convex hqual with ⟨u, x, hzGraph, huF, hxg⟩ let eM := EuclideanSpace.equiv (ι := Fin m) (𝕜 := ℝ) let Cf : Set (EuclideanSpace ℝ (Fin m)) := eM.symm '' erealDom f let A : EuclideanSpace ℝ (Fin (m + n)) →ₗ[ℝ] EuclideanSpace ℝ (Fin m) := eM.symm.toLinearMap.comp ((projXLinearMap (n := m) (m := n)).comp eMN.toLinearMap) have hf_pc : ProperConvexFunctionOn (Set.univ : Set (Fin m → ℝ)) f := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex f hf_proper hf_convex have hdomf_conv : Convex ℝ (erealDom f) := by have heq : erealDom f = effectiveDomain (Set.univ : Set (Fin m → ℝ)) f := by ext y simp [erealDom, effectiveDomain_eq, lt_top_iff_ne_top] rw [heq] exact effectiveDomain_convex hf_pc.1 have hCfconv : Convex ℝ Cf := by exact hdomf_conv.linear_image eM.symm.toLinearMap have huE : eM.symm u ∈ euclideanRelativeInterior m Cf := by exact (mem_euclideanRelativeInterior_fin_iff (n := m) (C := erealDom f) (x := u)).1 huF have hAz : A (eMN.symm (Fin.append u x)) = eM.symm u := by simp [A, eM, eMN, projXLinearMap] have hpre_nonempty : (A ⁻¹' euclideanRelativeInterior m Cf).Nonempty := by exact ⟨eMN.symm (Fin.append u x), by simpa [hAz] using huE⟩ have hpre := euclideanRelativeInterior_preimage_linearMap_eq_and_closure_preimage (n := m + n) (m := m) A Cf hCfconv hpre_nonempty have hzPre : eMN.symm (Fin.append u x) ∈ euclideanRelativeInterior (m + n) (A ⁻¹' Cf) := by rw [hpre.1] simpa [hAz] using huE have hClift : Cliftf = A ⁻¹' Cf := by ext z constructor · rintro ⟨z0, hz0, rfl⟩ refine ⟨fun i => z0 (Fin.castAdd n i), ?_, ?_⟩ · simpa [erealDom] using hz0 · simp [A, This simp argument is unused: Cf Hint: Omit it from the simp argument list. simp [A, C̵f̵,̵ ̵eM, eMN, projXLinearMap] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`Cf, eM, eMN, projXLinearMap] · rintro ⟨u0, hu0, hAu⟩ refine ⟨eMN z, ?_, by simp [eMN]⟩ have huEq : u0 = fun i => (eMN z) (Fin.castAdd n i) := by apply eM.symm.injective simpa [A, eM, eMN, projXLinearMap] using hAu simpa [erealDom, huEq] using hu0 have hzLift : eMN.symm (Fin.append u x) ∈ euclideanRelativeInterior (m + n) Cliftf := by rw [hClift] exact hzPre have hCliftconv : Convex ℝ Cliftf := by rw [hClift] exact hCfconv.linear_preimage A have hCgraphconv : Convex ℝ Cgraph := by let graphg : (Fin (m + n) → ℝ) → EReal := fun z => F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) let pairMap : (Fin (m + n) → ℝ) →ₗ[ℝ] (Fin m → ℝ) × (Fin n → ℝ) := (projXLinearMap (n := m) (m := n)).prod (projLamLinearMap (n := m) (m := n)) have hgconv : IsERealConvex graphg := by simpa [graphg, pairMap, projXLinearMap, projLamLinearMap] using helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := pairMap) hF_convex have hgproper : IsProperEReal graphg := by constructor · intro z simpa [graphg] using hF_proper.1 ((fun i => z (Fin.castAdd n i)), (fun j => z (Fin.natAdd m j))) · rcases hF_proper.2 with ⟨p, hp⟩ exact ⟨Fin.append p.1 p.2, by simpa [graphg]⟩ have hgpc := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex graphg hgproper hgconv have hraw : Convex ℝ {z : Fin (m + n) → ℝ | graphg z < ⊤} := by have heq : {z : Fin (m + n) → ℝ | graphg z < ⊤} = effectiveDomain Set.univ graphg := by ext z simp [effectiveDomain_eq, lt_top_iff_ne_top] rw [heq] exact effectiveDomain_convex hgpc.1 simpa [Cgraph, graphg] using hraw.linear_image eMN.symm.toLinearMap let family : Fin 2 → Set (EuclideanSpace ℝ (Fin (m + n))) := fun i => if i = 0 then Cgraph else Cliftf have hzGraph' : eMN.symm (Fin.append u x) ∈ euclideanRelativeInterior (m + n) Cgraph := by simpa [Cgraph, eMN] using hzGraph have hfamilyConv : ∀ i, Convex ℝ (family i) := by intro i fin_cases i · simpa [family] using hCgraphconv · simpa [family] using hCliftconv have hfamilyRi : (⋂ i, euclideanRelativeInterior (m + n) (family i)).Nonempty := by refine ⟨eMN.symm (Fin.append u x), ?_⟩ rw [iInter_fin_two_eq_inter] exact ⟨by simpa [family] using hzGraph', by simpa [family] using hzLift⟩ have hriEq := euclideanRelativeInterior_iInter_eq_iInter_relativeInterior_of_finite (m + n) family hfamilyConv hfamilyRi refine ⟨u, x, ?_, hxg⟩ have hzI : eMN.symm (Fin.append u x) ∈ ⋂ i, euclideanRelativeInterior (m + n) (family i) := by rw [iInter_fin_two_eq_inter] exact ⟨by simpa [family] using hzGraph', by simpa [family] using hzLift⟩ have hInter : (⋂ i, family i) = Cgraph ∩ Cliftf := by simpa [family] using iInter_fin_two_eq_inter family rw [← hInter, hriEq] exact hzI

The full relative-interior consequence of the §38.7 qualification: the same Unknown identifier `x`x furnished by the qualified slice belongs to both Unknown identifier `ri`ri (dom (Ff)) and Unknown identifier `ri`ri (domBot g).

lemma helperForTheorem_38_7_qualification_implies_imageDom_inter_domBot_ri_nonempty {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hqual : ∃ u : Fin m → ℝ, u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : (intrinsicInterior ℝ (erealDom (bifunctionImageRaw F f)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty := by classical let eMN := EuclideanSpace.equiv (ι := Fin (m + n)) (𝕜 := ℝ) let eN := EuclideanSpace.equiv (ι := Fin n) (𝕜 := ℝ) let Cgraph : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) < ⊤} let Cliftf : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | f (fun i => z (Fin.castAdd n i)) < ⊤} let Csum : Set (EuclideanSpace ℝ (Fin (m + n))) := eMN.symm '' {z : Fin (m + n) → ℝ | f (fun i => z (Fin.castAdd n i)) + F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) < ⊤} rcases helperForTheorem_38_7_qualification_lifts_to_packedDomainIntersectionRi F f g hF_proper hF_convex hf_proper hf_convex hqual with ⟨u, x, hzInter, hxg⟩ have hAddDom : ∀ (u0 : Fin m → ℝ) (x0 : Fin n → ℝ), f u0 + F u0 x0 < ⊤ ↔ f u0 < ⊤ ∧ F u0 x0 < ⊤ := by intro u0 x0 constructor · intro h constructor · rw [lt_top_iff_ne_top] intro hfTop rw [hfTop] at h Try `simp at h` instead of `simpa using h` Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hF_proper.1 (u0, x0)] using h · rw [lt_top_iff_ne_top] intro hFTop rw [hFTop] at h Try `simp at h` instead of `simpa using h` Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hf_proper.1 u0] using h · rintro ⟨hfTop, hFTop⟩ exact EReal.add_lt_top (ne_of_lt hfTop) (ne_of_lt hFTop) have hDomains : Cgraph ∩ Cliftf = Csum := by ext z simp only [Cgraph, Cliftf, Csum, Set.mem_inter_iff] constructor · rintro ⟨⟨zg, hzg, hzgEq⟩, ⟨zf, hzf, hzfEq⟩⟩ have hEq : zg = zf := by apply eMN.symm.injective exact hzgEq.trans hzfEq.symm subst zf refine ⟨zg, ?_, hzgEq⟩ exact (hAddDom _ _).2 ⟨hzf, hzg⟩ · rintro ⟨z0, hz0, rfl⟩ have hzParts := (hAddDom (fun i => z0 (Fin.castAdd n i)) (fun j => z0 (Fin.natAdd m j))).1 hz0 exact ⟨⟨z0, hzParts.2, rfl⟩, ⟨z0, hzParts.1, rfl⟩⟩ have hzSum : eMN.symm (Fin.append u x) ∈ euclideanRelativeInterior (m + n) Csum := by rw [← hDomains] simpa [Cgraph, Cliftf, eMN] using hzInter let liftedf : (Fin (m + n) → ℝ) → EReal := fun z => f (fun i => z (Fin.castAdd n i)) let liftedF : (Fin (m + n) → ℝ) → EReal := fun z => F (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) let firstMap : (Fin (m + n) → ℝ) →ₗ[ℝ] (Fin m → ℝ) := projXLinearMap (n := m) (m := n) let pairMap : (Fin (m + n) → ℝ) →ₗ[ℝ] (Fin m → ℝ) × (Fin n → ℝ) := firstMap.prod (projLamLinearMap (n := m) (m := n)) have hlfProper : IsProperEReal liftedf := by constructor · intro z exact hf_proper.1 _ · rcases hf_proper.2 with ⟨u0, hu0⟩ exact ⟨Fin.append u0 (0 : Fin n → ℝ), by simpa [liftedf]⟩ have hlFProper : IsProperEReal liftedF := by constructor · intro z simpa [liftedF] using hF_proper.1 ((fun i => z (Fin.castAdd n i)), (fun j => z (Fin.natAdd m j))) · rcases hF_proper.2 with ⟨p, hp⟩ exact ⟨Fin.append p.1 p.2, by simpa [liftedF]⟩ have hlfConvex : IsERealConvex liftedf := by simpa [liftedf, firstMap, projXLinearMap] using helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := firstMap) hf_convex have hlFConvex : IsERealConvex liftedF := by simpa [liftedF, pairMap, firstMap, projXLinearMap, projLamLinearMap] using helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := pairMap) hF_convex have hlfPC := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex liftedf hlfProper hlfConvex have hlFPC := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex liftedF hlFProper hlFConvex have hsumPC : ConvexFunctionOn (Set.univ : Set (Fin (m + n) → ℝ)) (fun z => liftedf z + liftedF z) := convexFunctionOn_add_of_proper hlfPC hlFPC have hCsumConv : Convex ℝ Csum := by have hraw : Convex ℝ (effectiveDomain (Set.univ : Set (Fin (m + n) → ℝ)) (fun z => liftedf z + liftedF z)) := effectiveDomain_convex hsumPC have heq : Csum = eMN.symm '' effectiveDomain (Set.univ : Set (Fin (m + n) → ℝ)) (fun z => liftedf z + liftedF z) := by ext z simp [Csum, effectiveDomain_eq, liftedf, liftedF, lt_top_iff_ne_top] rw [heq] exact hraw.linear_image eMN.symm.toLinearMap let P : EuclideanSpace ℝ (Fin (m + n)) →ₗ[ℝ] EuclideanSpace ℝ (Fin n) := eN.symm.toLinearMap.comp ((projLamLinearMap (n := m) (m := n)).comp eMN.toLinearMap) have hPz : P (eMN.symm (Fin.append u x)) = eN.symm x := by simp [P, eN, eMN, projLamLinearMap] have hprojRi := (euclideanRelativeInterior_image_linearMap_eq_and_image_closure_subset (n := m + n) (m := n) Csum hCsumConv P).1 have hxProj : eN.symm x ∈ euclideanRelativeInterior n (P '' Csum) := by rw [hprojRi] exact ⟨eMN.symm (Fin.append u x), hzSum, hPz⟩ have hprojEq : P '' Csum = eN.symm '' erealDom (bifunctionImageRaw F f) := by ext y constructor · rintro ⟨z, ⟨z0, hz0, hzEq⟩, rfl⟩ have hzEq' : z = eMN.symm z0 := hzEq.symm subst z refine ⟨(fun j => z0 (Fin.natAdd m j)), ?_, ?_⟩ · rw [helperForTheorem_38_7_mem_imageDom_iff_exists_sum_lt_top] exact ⟨(fun i => z0 (Fin.castAdd n i)), hz0⟩ · simp [P, eMN, eN, projLamLinearMap] · rintro ⟨x0, hx0, rfl⟩ rw [helperForTheorem_38_7_mem_imageDom_iff_exists_sum_lt_top] at hx0 rcases hx0 with ⟨u0, hu0⟩ refine ⟨eMN.symm (Fin.append u0 x0), ?_, ?_⟩ · exact ⟨Fin.append u0 x0, by simpa, rfl⟩ · simp [P, eMN, eN, projLamLinearMap] have hxImageFin : x ∈ euclideanRelativeInterior_fin n (erealDom (bifunctionImageRaw F f)) := by apply (mem_euclideanRelativeInterior_fin_iff (n := n) (C := erealDom (bifunctionImageRaw F f)) (x := x)).2 rw [← hprojEq] exact hxProj refine ⟨x, ?_, ?_⟩ · simpa only [ helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] using hxImageFin · simpa only [ helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] using hxg

The qualification in Theorem 38.7 supplies a concrete point in the effective domain of the image Unknown identifier `Ff`Ff. Indeed its slice-relative-interior witness gives finite-above values of both Unknown identifier `f`f u and Unknown identifier `F`F u x, and the defining infimum of Unknown identifier `Ff`Ff is bounded above by their sum. The properness hypotheses are retained in the interface used by Theorem 38.7, although the qualification itself already gives the required upper finiteness.

lemma helperForTheorem_38_7_image_erealDom_nonempty {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (_hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (_hf_proper : IsProperEReal f) (hqual : ∃ u : (Fin m → ℝ), u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : (erealDom (bifunctionImageRaw F f)).Nonempty := by rcases hqual with ⟨u, hu, x, hxF, hxg⟩ have huf_dom : u ∈ erealDom f := intrinsicInterior_subset (𝕜 := ℝ) (s := erealDom f) hu.1 have huf_lt_top : f u < (⊤ : EReal) := by simpa [erealDom] using huf_dom have hFx_dom : x ∈ erealDom (fun y : Fin n → ℝ => F u y) := intrinsicInterior_subset (𝕜 := ℝ) (s := erealDom (fun y : Fin n → ℝ => F u y)) hxF have hFx_lt_top : F u x < (⊤ : EReal) := by simpa [erealDom] using hFx_dom have hsum_lt_top : f u + F u x < (⊤ : EReal) := EReal.add_lt_top (ne_of_lt huf_lt_top) (ne_of_lt hFx_lt_top) refine ⟨x, ?_⟩ rw [erealDom] exact lt_of_le_of_lt (iInf_le (fun v : Fin m → ℝ => f v + F v x) u) hsum_lt_top

Theorem 38.7's slice qualification contains the common relative-interior qualification used by Theorem 38.4. Together with Theorem 38.4's unconditional first conclusion this also packages the convexity of the image Unknown identifier `Ff`Ff for the later Lemma 38.6 application.

lemma helperForTheorem_38_7_image_convex_and_base_hri {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hqual : ∃ u : (Fin m → ℝ), u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : (intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F)).Nonempty ∧ IsERealConvex (bifunctionImageRaw F f) := by rcases hqual with ⟨u, hu, hx⟩ refine ⟨⟨u, hu⟩, ?_⟩ exact (theorem38_4_image_convex_and_conjugate F f hF_proper hF_convex hf_proper hf_convex).1

A proper convex function agrees with its closure on the intrinsic relative interior of its effective domain, in Fin (n : ℕ) : TypeFin coordinates.

lemma helperForTheorem_38_7_convexFunctionClosure_eq_on_intrinsicInterior {n : Nat} {f : (Fin n → ℝ) → EReal} (hf : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f) {x : Fin n → ℝ} (hx : x ∈ intrinsicInterior ℝ (effectiveDomain (Set.univ : Set (Fin n → ℝ)) f)) : convexFunctionClosure f x = f x := by have hxFin : x ∈ euclideanRelativeInterior_fin n (effectiveDomain (Set.univ : Set (Fin n → ℝ)) f) := by rw [helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] exact hx let e : EuclideanSpace ℝ (Fin n) ≃L[ℝ] (Fin n → ℝ) := EuclideanSpace.equiv (ι := Fin n) (𝕜 := ℝ) have hpre : ((fun y : EuclideanSpace ℝ (Fin n) => (y : Fin n → ℝ)) ⁻¹' effectiveDomain (Set.univ : Set (Fin n → ℝ)) f) = e.symm '' effectiveDomain (Set.univ : Set (Fin n → ℝ)) f := by ext y constructor · intro hy; exact ⟨e y, hy, by simp [e]⟩ · rintro ⟨z, hz, rfl⟩; simpa [e] using hz have hxE : e.symm x ∈ euclideanRelativeInterior n ((fun y : EuclideanSpace ℝ (Fin n) => (y : Fin n → ℝ)) ⁻¹' effectiveDomain (Set.univ : Set (Fin n → ℝ)) f) := by have := (mem_euclideanRelativeInterior_fin_iff (n := n) (C := effectiveDomain (Set.univ : Set (Fin n → ℝ)) f) (x := x)).1 hxFin simpa [e, hpre] using this simpa [e] using (convexFunctionClosure_closed_properConvexFunctionOn_and_agrees_on_ri hf).2 (e.symm x) hxE
set_option synthInstance.maxHeartbeats 100000 in /-- In the improper image branch, the common relative-interior point forces both extrema in the Fenchel inner product with `g*` to equal `+∞`. -/ lemma helperForTheorem_38_7_improper_innerProduct_with_concaveConjugate_eq_top {n : Nat} (h g : (Fin n → ℝ) → EReal) (hh_convex : IsERealConvex h) (hh_notProper : ¬ IsProperEReal h) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) (hri : (intrinsicInterior ℝ (erealDom h) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : fenchelInnerProduct h (concaveConjugateInner g) = some (⊤ : EReal) := by let gStar := concaveConjugateInner g have hhConvOn : ConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) h := by simpa [ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hh_convex have hhNotPC : ¬ ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) h := by intro hh exact hh_notProper (helperForLemma_38_6_isProperEReal_of_properConvexFunctionOn_univ hh) have hhImproper : ImproperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) h := ⟨hhConvOn, hhNotPC⟩ rcases hri with ⟨x, hxh, hxg⟩ have hxhEff : x ∈ intrinsicInterior ℝ (effectiveDomain (Set.univ : Set (Fin n → ℝ)) h) := by simpa [erealDom, effectiveDomain_eq] using hxh have hxbot : h x = (⊥ : EReal) := improperConvexFunctionOn_eq_bot_on_intrinsicInterior_fin hhImproper hxhEff let negG : (Fin n → ℝ) → EReal := fun y => -g y have hnegPC : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) negG := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex negG hg_proper hg_concave have hdomNeg : erealDomBot g = effectiveDomain (Set.univ : Set (Fin n → ℝ)) negG := by ext z simp only [erealDomBot, Set.mem_setOf_eq, effectiveDomain_eq, Set.mem_univ, true_and, negG, bot_lt_iff_ne_bot, lt_top_iff_ne_top] constructor · intro hz hnegTop; exact hz ((EReal.neg_eq_top_iff).1 hnegTop) · intro hz hbot; exact hz ((EReal.neg_eq_top_iff).2 hbot) have hxgEff : x ∈ intrinsicInterior ℝ (effectiveDomain (Set.univ : Set (Fin n → ℝ)) negG) := by rw [← hdomNeg]; exact hxg have hclNeg := helperForTheorem_38_7_convexFunctionClosure_eq_on_intrinsicInterior hnegPC hxgEff have hnegConv : ConvexFunction negG := by simpa [negG, IsERealConvex, ConvexFunctionOn, helperForTheorem_38_1_epigraph_eq_univ] using hg_concave have hconvClosureEq : convexClosure negG = convexFunctionClosure negG := by calc convexClosure negG = clConv n negG := helperForTheorem_6_30_3_convexClosure_eq_clConv negG hnegConv _ = fenchelConjugate n (fenchelConjugate n negG) := by symm; exact fenchelConjugate_biconjugate_eq_clConv (n := n) (f := negG) _ = convexFunctionClosure negG := section16_fenchelConjugate_biconjugate_eq_convexFunctionClosure (n := n) (f := negG) hnegConv have hgClosureX : concaveClosure g x = g x := by rw [concaveClosure_eq_neg_convexClosure_neg, show (fun z => -g z) = negG from rfl, hconvClosureEq] change -convexFunctionClosure negG x = g x rw [hclNeg] simp [negG] have hbiconj : concaveConjugateInner gStar = concaveClosure g := by simpa [gStar] using helperForLemma_38_6_concaveConjugateInner_biconjugate_eq_concaveClosure (g := g) hg_concave have hstarstar_ne_bot : concaveConjugateInner gStar x ≠ (⊥ : EReal) := by rw [hbiconj, hgClosureX] exact ne_of_gt (intrinsicInterior_subset (s := erealDomBot g) hxg) have hhStar : convexConjugateInner h = fun _ => (⊤ : EReal) := by funext y unfold convexConjugateInner apply top_le_iff.mp let xs : {z : Fin n → ℝ // z ∈ erealDom h} := ⟨x, by simp [erealDom, hxbot]⟩ refine le_iSup_of_le xs ?_ change (⊤ : EReal) ≤ (∑ i : Fin n, (((x i) * (y i) : ℝ) : EReal)) - h x rw [hxbot] have hs : (∑ i : Fin n, (((x i) * (y i) : ℝ) : EReal)) ≠ (⊥ : EReal) := by exact sum_ne_bot_of_ne_bot (s := Finset.univ) (f := fun i : Fin n => (((x i) * (y i) : ℝ) : EReal)) (fun i _ => EReal.coe_ne_bot _) exact top_le_iff.mpr (EReal.sub_bot hs) have hgStarProper : IsProperEReal (fun y => -gStar y) := helperForLemma_38_6_gStar_isProperNeg hg_proper hg_concave have hAlpha : (⨆ z : {z : Fin n → ℝ // z ∈ erealDom h}, concaveConjugateInner gStar z.1 - h z.1) = (⊤ : EReal) := by apply top_le_iff.mp refine le_iSup_of_le ⟨x, by simp [erealDom, hxbot]⟩ ?_ simpa [hxbot] using EReal.sub_bot hstarstar_ne_bot have hBeta : (⨅ y : {y : Fin n → ℝ // y ∈ erealDomBot gStar}, convexConjugateInner h y.1 - gStar y.1) = (⊤ : EReal) := by rw [hhStar] refine top_unique (le_iInf ?_) intro y have hy : gStar y.1 ≠ (⊤ : EReal) := by intro hy; exact hgStarProper.1 y.1 (by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hy]) simpa using EReal.top_sub hy unfold fenchelInnerProduct rw [hAlpha, hBeta] simp

The first Fenchel inner product in Theorem 38.7 exists in both the proper and improper image branches.

lemma helperForTheorem_38_7_image_innerProduct_exists {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) (hqual : ∃ u : Fin m → ℝ, u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : ∃ c : EReal, fenchelInnerProduct (bifunctionImageRaw F f) (concaveConjugateInner g) = some c := by let h := bifunctionImageRaw F f have hhConvex := (helperForTheorem_38_7_image_convex_and_base_hri F f g hF_proper hF_convex hf_proper hf_convex hqual).2 have hriImage := helperForTheorem_38_7_qualification_implies_imageDom_inter_domBot_ri_nonempty F f g hF_proper hF_convex hf_proper hf_convex hqual by_cases hhProper : IsProperEReal h · exact fenchelInnerProduct_with_concaveConjugate_exists_of_ri h g hhProper hhConvex hg_proper hg_concave hriImage · exact ⟨⊤, helperForTheorem_38_7_improper_innerProduct_with_concaveConjugate_eq_top h g hhConvex hhProper hg_proper hg_concave hriImage⟩

The Euclidean inner product on Fin sorry → ℝ : TypeFin Unknown identifier `n`n → ℝ used throughout §38 (as ∑ i, sorry * sorry : ?m.2∑ i, Unknown identifier `x`x i * Unknown identifier `y`y i).

def euclideanInner (n : Nat) (x y : Fin n → ℝ) : ℝ := Finset.univ.sum (fun i : Fin n => x i * y i)

The adjoint of a bifunction , using the Euclidean identification of a finite-dimensional space with its dual: .

noncomputable def bifunctionAdjointInner {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) : (Fin n → ℝ) → (Fin m → ℝ) → EReal := fun xStar uStar => ⨅ (u : Fin m → ℝ) (x : Fin n → ℝ), F u x + (-((euclideanInner n x xStar : ℝ) : EReal)) + ((euclideanInner m u uStar : ℝ) : EReal)

The "concave image" of a function Unknown identifier `g`g under a bifunction , defined by (modeled by iSup.{u, v} {α : Type u} {ι : Sort v} [SupSet α] (s : ι → α) : αiSup). This is the natural Unknown identifier `sup`sup-analogue of bifunctionImageRaw.{u_1, u_2} {U : Type u_1} {X : Type u_2} (F : U → X → EReal) (f : U → EReal) : X → ERealbifunctionImageRaw for concave transforms such as .

lemma bifunctionAdjointInner_eq_adjointOfConvexBifunction {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF : ConvexBifunction F) : bifunctionAdjointInner F = adjointOfConvexBifunction ⟨F, hF⟩ := by funext xStar uStar rw [bifunctionAdjointInner, adjointOfConvexBifunction, sInf_range] rw [show (⨅ p : (Fin m → ℝ) × (Fin n → ℝ), F p.1 p.2 - (((p.2 ⬝ᵥ xStar : ℝ) : EReal)) + (((p.1 ⬝ᵥ uStar : ℝ) : EReal))) = ⨅ (u : Fin m → ℝ) (x : Fin n → ℝ), F u x - (((x ⬝ᵥ xStar : ℝ) : EReal)) + (((u ⬝ᵥ uStar : ℝ) : EReal)) by simp only [iInf_prod]] simp [euclideanInner, dotProduct, sub_eq_add_neg]

The concave image of Unknown identifier `g`g under a bifunction Unknown identifier `K`K, using pointwise suprema.

noncomputable def bifunctionImageSupRaw {m n : Nat} (K : (Fin n → ℝ) → (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) : (Fin m → ℝ) → EReal := fun u => ⨆ x : (Fin n → ℝ), g x + K x u
lemma bifunctionInverseBookAdjoint_eq_neg_bifunctionAdjointInner {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF : ConvexBifunction F) : bifunctionInverseBookAdjoint F = fun uStar xStar => -bifunctionAdjointInner F xStar uStar := by funext uStar xStar have hAdj := adjointOfConvexBifunction_eq_neg_fenchelConjugate_graphFunction (⟨F, hF⟩ : {F : (Fin m → ℝ) → (Fin n → ℝ) → EReal // ConvexBifunction F}) xStar uStar rw [bifunctionInverseBookAdjoint] rw [bifunctionAdjointInner_eq_adjointOfConvexBifunction F hF, hAdj] simp

Negating a supremal bifunction image turns it into the infimal image of the pointwise-negated data when the two summands never take ⊤ : ?m.1⊤.

lemma neg_bifunctionImageSupRaw_eq_bifunctionImageRaw_neg {m n : Nat} (K : (Fin n → ℝ) → (Fin m → ℝ) → EReal) (q : (Fin n → ℝ) → EReal) (hqTop : ∀ x, q x ≠ (⊤ : EReal)) (hKTop : ∀ x u, K x u ≠ (⊤ : EReal)) : (fun u => -bifunctionImageSupRaw K q u) = bifunctionImageRaw (fun x u => -K x u) (fun x => -q x) := by funext u unfold bifunctionImageSupRaw bifunctionImageRaw have hnegSup : -(⨆ x, q x + K x u) = ⨅ x, -(q x + K x u) := by have h := congrArg Neg.neg (ereal_iSup_neg_eq_neg_iInf (g := fun x => -(q x + K x u))) simpa only [neg_neg] using h rw [hnegSup] refine iInf_congr ?_ intro x exact EReal.neg_add (x := q x) (y := K x u) (Or.inr (hKTop x u)) (Or.inl (hqTop x))

A proper concave conjugate never takes the value ⊤ : ?m.1⊤.

lemma concaveConjugateInner_ne_top {n : Nat} {g : (Fin n → ℝ) → EReal} (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) : ∀ y, concaveConjugateInner g y ≠ (⊤ : EReal) := by have hproper : IsProperEReal (fun y => -(concaveConjugateInner g y)) := helperForLemma_38_6_gStar_isProperNeg hg_proper hg_concave intro y hy have hbot := hproper.1 y apply hbot simp [hy]

The product-coordinate properness and convexity hypotheses used in Theorem 38.7 give the Chapter 6 packaged notion of a proper convex bifunction.

lemma properConvexBifunction_of_product_hypotheses {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) : ProperConvexBifunction F := by let pairMap : (Fin (m + n) → ℝ) →ₗ[ℝ] ((Fin m → ℝ) × (Fin n → ℝ)) := { toFun := fun z => (fun i => z (Fin.castAdd n i), fun j => z (Fin.natAdd m j)) map_add' := by intro z w; ext i <;> simp map_smul' := by intro a z; ext i <;> simp } have hPackedConvex : IsERealConvex (bifunctionGraphFunction F) := by simpa [pairMap, bifunctionGraphFunction] using (helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := pairMap) (g := fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2) hF_convex) have hF30 : ConvexBifunction F := by simpa [ConvexBifunction, ConvexFunction, IsERealConvex, ConvexFunctionOn, helperForTheorem_38_1_epigraph_eq_univ] using hPackedConvex have hPackedProper : IsProperEReal (bifunctionGraphFunction F) := by constructor · intro z exact hF_proper.1 (fun i => z (Fin.castAdd n i), fun j => z (Fin.natAdd m j)) · rcases hF_proper.2 with ⟨p, hp⟩ exact ⟨Fin.append p.1 p.2, by simpa [bifunctionGraphFunction]⟩ have hPackedProperOn : ProperConvexFunctionOn (Set.univ : Set (Fin (m + n) → ℝ)) (bifunctionGraphFunction F) := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (bifunctionGraphFunction F) hPackedProper hPackedConvex exact ⟨hF30, helperForText_26_4_0_2_properConvexERealFunction_of_properConvexFunctionOn hPackedProperOn⟩

The adjoint of a proper convex bifunction never takes the value ⊤ : ?m.1⊤.

lemma bifunctionAdjointInner_ne_top {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) : ∀ x u, bifunctionAdjointInner F x u ≠ (⊤ : EReal) := by have hProperF : ProperConvexBifunction F := properConvexBifunction_of_product_hypotheses F hF_proper hF_convex have hF30 : ConvexBifunction F := hProperF.1 have hProperAdj : ProperConcaveBifunction (adjointOfConvexBifunction ⟨F, hF30⟩) := ((adjoint_bifunction_closure_properness_biconjugation_and_polyhedrality F).1 hF30).2.1.mpr hProperF intro x u htop have hnebot := hProperAdj.2.1.1 (Fin.append x u) have htop' : adjointOfConvexBifunction ⟨F, hF30⟩ x u = (⊤ : EReal) := by rw [← bifunctionAdjointInner_eq_adjointOfConvexBifunction F hF30] exact htop apply hnebot simp only [bifunctionGraphFunction, Fin.append_left, Fin.append_right, htop', EReal.neg_top]

The negative of the adjoint supremal image is the infimal image under the Chapter 38.4 inverse-book adjoint, with the concave conjugate negated.

lemma neg_adjointImageSup_eq_inverseBookImage_neg_concaveConjugate {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) : (fun u => -bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) u) = bifunctionImageRaw (fun x u => bifunctionInverseBookAdjoint F u x) (fun x => -concaveConjugateInner g x) := by rw [neg_bifunctionImageSupRaw_eq_bifunctionImageRaw_neg (bifunctionAdjointInner F) (concaveConjugateInner g) (concaveConjugateInner_ne_top hg_proper hg_concave) (bifunctionAdjointInner_ne_top F hF_proper hF_convex)] have hF30 : ConvexBifunction F := (properConvexBifunction_of_product_hypotheses F hF_proper hF_convex).1 rw [bifunctionInverseBookAdjoint_eq_neg_bifunctionAdjointInner F hF30]

The qualified adjoint image of a proper concave conjugate is everywhere below ⊤ : ?m.1⊤. A common relative-interior point supplies a finite affine upper bound uniform in the dual variable over which the supremum is taken.

lemma adjointImageSup_concaveConjugate_ne_top {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hg_proper : IsProperEReal (fun y => -g y)) (hqual : ∃ u : (Fin m → ℝ), (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : ∀ uStar, bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) uStar ≠ (⊤ : EReal) := by rcases hqual with ⟨u₀, x₀, hxF, hxg⟩ have hF_top : F u₀ x₀ ≠ (⊤ : EReal) := by exact (lt_top_iff_ne_top.mp ((intrinsicInterior_subset (s := erealDom (fun x : Fin n → ℝ => F u₀ x))) hxF)) have hF_bot : F u₀ x₀ ≠ (⊥ : EReal) := hF_proper.1 (u₀, x₀) have hg_bot : g x₀ ≠ (⊥ : EReal) := by exact (bot_lt_iff_ne_bot.mp ((intrinsicInterior_subset (s := erealDomBot g)) hxg)) have hg_top : g x₀ ≠ (⊤ : EReal) := by intro htop exact hg_proper.1 x₀ (by simp [htop]) intro uStar have hle : bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) uStar ≤ (((F u₀ x₀).toReal - (g x₀).toReal + euclideanInner m u₀ uStar : ℝ) : EReal) := by unfold bifunctionImageSupRaw refine iSup_le ?_ intro xStar have hq : concaveConjugateInner g xStar ≤ ((euclideanInner n x₀ xStar : ℝ) : EReal) - g x₀ := by unfold concaveConjugateInner exact iInf_le_of_le ⟨x₀, intrinsicInterior_subset hxg⟩ (by simp [euclideanInner, This simp argument is unused: dotProduct Hint: Omit it from the simp argument list. simp [euclideanInner, dotProduct,̵ ̵d̵o̵t̵P̵r̵o̵d̵u̵c̵t̵_comm, ← ereal_sum_coe, mul_comm] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`dotProduct, This simp argument is unused: dotProduct_comm Hint: Omit it from the simp argument list. simp [euclideanInner, dotProduct, d̵o̵t̵P̵r̵o̵d̵u̵c̵t̵_̵c̵o̵m̵m̵,̵ ̵← ereal_sum_coe, mul_comm] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`dotProduct_comm, ← ereal_sum_coe, mul_comm]) have hK : bifunctionAdjointInner F xStar uStar ≤ F u₀ x₀ + (-((euclideanInner n x₀ xStar : ℝ) : EReal)) + ((euclideanInner m u₀ uStar : ℝ) : EReal) := by unfold bifunctionAdjointInner exact iInf_le_of_le u₀ (iInf_le _ x₀) calc concaveConjugateInner g xStar + bifunctionAdjointInner F xStar uStar ≤ (((euclideanInner n x₀ xStar : ℝ) : EReal) - g x₀) + (F u₀ x₀ + (-((euclideanInner n x₀ xStar : ℝ) : EReal)) + ((euclideanInner m u₀ uStar : ℝ) : EReal)) := add_le_add hq hK _ = (((F u₀ x₀).toReal - (g x₀).toReal + euclideanInner m u₀ uStar : ℝ) : EReal) := by rw [← EReal.coe_toReal hF_top hF_bot, ← EReal.coe_toReal hg_top hg_bot] simp only [sub_eq_add_neg, ← EReal.coe_neg, ← EReal.coe_add] congr 1 norm_num ring exact ne_of_lt (lt_of_le_of_lt hle (EReal.coe_lt_top _))
lemma helperForTheorem_38_7_sub_iSup_eq_iInf_sub {α : Type*} (a : EReal) (q : α → EReal) (haBot : a ≠ (⊥ : EReal)) (hSupTop : (⨆ i, q i) ≠ (⊤ : EReal)) (hqTop : ∀ i, q i ≠ (⊤ : EReal)) : a - (⨆ i, q i) = ⨅ i, a - q i := by cases ha : a with | bot => exact (haBot ha).elim | top => rw [EReal.top_sub hSupTop] symm exact top_unique (le_iInf (fun i => by simpa using EReal.top_sub (hqTop i))) | coe r => rw [sub_eq_add_neg] have hneg : -(⨆ i, q i) = ⨅ i, -q i := by have h := congrArg Neg.neg (ereal_iSup_neg_eq_neg_iInf (g := fun i => -q i)) simpa only [neg_neg] using h rw [hneg] simpa [ha, sub_eq_add_neg] using (helperForTheorem_6_30_15_real_add_iInf (c := r) (f := fun i => -q i))lemma helperForTheorem_38_7_sub_iInf_eq_iSup_sub {α : Type*} (a : EReal) (q : α → EReal) (haTop : a ≠ (⊤ : EReal)) (unused variable `hqBot` Note: This linter can be disabled with `set_option linter.unusedVariables false`hqBot : ∀ i, q i ≠ (⊥ : EReal)) : a - (⨅ i, q i) = ⨆ i, a - q i := by cases ha : a with | top => exact (haTop ha).elim | bot => rw [EReal.bot_sub] symm exact bot_unique (iSup_le (fun i => by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa using EReal.bot_sub (q i))) | coe r => rw [sub_eq_add_neg] have hneg : -(⨅ i, q i) = ⨆ i, -q i := by exact helperForTheorem_6_30_4_neg_iInf_eq_iSup_neg q rw [hneg] simpa [ha, sub_eq_add_neg] using (helperForTheorem_6_30_15_real_add_iSup (c := r) (f := fun i => -q i))lemma helperForTheorem_38_7_iInf_domBot_eq_iInf {α : Type*} (a q : α → EReal) (haBot : ∀ i, a i ≠ (⊥ : EReal)) : (⨅ i : {i : α // i ∈ erealDomBot q}, a i.1 - q i.1) = ⨅ i : α, a i - q i := by apply le_antisymm · refine le_iInf ?_ intro i by_cases hi : i ∈ erealDomBot q · exact iInf_le_of_le ⟨i, hi⟩ le_rfl · have hbot : q i = (⊥ : EReal) := by simpa [erealDomBot, bot_lt_iff_ne_bot] using hi rw [hbot, EReal.sub_bot (haBot i)] exact le_top · refine le_iInf ?_ intro i exact iInf_le (fun j : α => a j - q j) i.1lemma helperForTheorem_38_7_iSup_domBot_eq_iSup {α : Type*} (a q : α → EReal) (unused variable `haTop` Note: This linter can be disabled with `set_option linter.unusedVariables false`haTop : ∀ i, a i ≠ (⊤ : EReal)) : (⨆ i : {i : α // i ∈ erealDomBot q}, q i.1 + a i.1) = ⨆ i : α, q i + a i := by apply le_antisymm · refine iSup_le ?_ intro i exact le_iSup (fun j : α => q j + a j) i.1 · refine iSup_le ?_ intro i by_cases hi : i ∈ erealDomBot q · exact le_iSup_of_le ⟨i, hi⟩ le_rfl · have hbot : q i = (⊥ : EReal) := by simpa [erealDomBot, bot_lt_iff_ne_bot] using hi rw [hbot] try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [haTop i]lemma helperForTheorem_38_7_inverseImageClosure_le_concaveConjugate_adjointImage {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) (hqual : ∃ u : (Fin m → ℝ), (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : ∀ (u : Fin m → ℝ) (x : Fin n → ℝ), concaveClosure g x - F u x ≤ concaveConjugateInner (bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g)) u := by intro u x have hgClTop : concaveClosure g x ≠ (⊤ : EReal) := by intro htop exact (helperForLemma_38_6_concaveClosure_isProperNeg hg_proper hg_concave).1 x (by simp [htop]) by_cases hgClBot : concaveClosure g x = (⊥ : EReal) · rw [hgClBot, EReal.bot_sub] exact bot_le by_cases hFTop : F u x = (⊤ : EReal) · rw [hFTop] try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa using EReal.sub_top hgClTop have hFBot : F u x ≠ (⊥ : EReal) := hF_proper.1 (u, x) have hQTop : ∀ uStar, bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) uStar ≠ (⊤ : EReal) := adjointImageSup_concaveConjugate_ne_top F g hF_proper hg_proper hqual have hbiconj : concaveConjugateInner (concaveConjugateInner g) = concaveClosure g := helperForLemma_38_6_concaveConjugateInner_biconjugate_eq_concaveClosure (g := g) hg_concave unfold concaveConjugateInner refine le_iInf ?_ intro uStar have hQBot : bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) uStar.1 ≠ (⊥ : EReal) := by simpa [erealDomBot, bot_lt_iff_ne_bot] using uStar.2 lift bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) uStar.1 to ℝ using ⟨hQTop uStar.1, hQBot⟩ with q hq lift F u x to ℝ using ⟨hFTop, hFBot⟩ with a ha lift concaveClosure g x to ℝ using ⟨hgClTop, hgClBot⟩ with b hb have hQle : bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) uStar.1 ≤ F u x + ((euclideanInner m u uStar.1 : ℝ) : EReal) - concaveClosure g x := by unfold bifunctionImageSupRaw refine iSup_le ?_ intro xStar by_cases hgStarBot : concaveConjugateInner g xStar = (⊥ : EReal) · rw [hgStarBot] have hadjTop := bifunctionAdjointInner_ne_top F hF_proper hF_convex xStar uStar.1 simp [This simp argument is unused: hadjTop Hint: Omit it from the simp argument list. simp ̵[̵h̵a̵d̵j̵T̵o̵p̵]̵ Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`hadjTop] have hgStarTop := concaveConjugateInner_ne_top hg_proper hg_concave xStar lift concaveConjugateInner g xStar to ℝ using ⟨hgStarTop, hgStarBot⟩ with s hs have hadjLe : bifunctionAdjointInner F xStar uStar.1 ≤ F u x + (-((euclideanInner n x xStar : ℝ) : EReal)) + ((euclideanInner m u uStar.1 : ℝ) : EReal) := by unfold bifunctionAdjointInner exact iInf_le_of_le u (iInf_le _ x) have hclLe : concaveClosure g x ≤ ((euclideanInner n x xStar : ℝ) : EReal) - concaveConjugateInner g xStar := by rw [← congrFun hbiconj x] unfold concaveConjugateInner exact iInf_le_of_le ⟨xStar, by change (⊥ : EReal) < concaveConjugateInner g xStar rw [← hs] simp⟩ (by simp [euclideanInner, This simp argument is unused: dotProduct Hint: Omit it from the simp argument list. simp [euclideanInner, dotProduct,̵ ̵d̵o̵t̵P̵r̵o̵d̵u̵c̵t̵_comm, ← ereal_sum_coe, mul_comm] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`dotProduct, This simp argument is unused: dotProduct_comm Hint: Omit it from the simp argument list. simp [euclideanInner, dotProduct, d̵o̵t̵P̵r̵o̵d̵u̵c̵t̵_̵c̵o̵m̵m̵,̵ ̵← ereal_sum_coe, mul_comm] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`dotProduct_comm, ← ereal_sum_coe, This simp argument is unused: mul_comm Hint: Omit it from the simp argument list. simp [euclideanInner, dotProduct, dotProduct_comm, ← ereal_sum_coe,̵ ̵m̵u̵l̵_̵c̵o̵m̵m̵] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`mul_comm]) rw [← ha] at hadjLe ⊢ rw [← hb] at hclLe ⊢ rw [← hs] at hclLe cases hadj : bifunctionAdjointInner F xStar uStar.1 with | top => exact (bifunctionAdjointInner_ne_top F hF_proper hF_convex xStar uStar.1 hadj).elim | bot => simp | coe t => rw [hadj] at hadjLe have hadjLeR : t ≤ a - euclideanInner n x xStar + euclideanInner m u uStar.1 := by exact_mod_cast hadjLe have hclLeR : b ≤ euclideanInner n x xStar - s := by exact_mod_cast hclLe 'norm_num at ⊢' tactic does nothing Note: This linter can be disabled with `set_option linter.unusedTactic false`norm_num at ⊢ exact_mod_cast (by linarith [hadjLeR, hclLeR]) have hq' := hq unfold concaveConjugateInner at hq' rw [← hq'] rw [← ha, ← hb] at hQle rw [← hq] at hQle have hQleR : q ≤ a + euclideanInner m u uStar.1 - b := by exact_mod_cast hQle have hreal : b - a ≤ euclideanInner m u uStar.1 - q := by linarith [hQleR] have hcast : ((b - a : ℝ) : EReal) ≤ ((euclideanInner m u uStar.1 - q : ℝ) : EReal) := by exact_mod_cast hreal simpa [euclideanInner, ← ereal_sum_coe] using hcastlemma helperForTheorem_38_7_sub_add_assoc (a b c : EReal) (haTop : a ≠ (⊤ : EReal)) (hbTop : b ≠ (⊤ : EReal)) (hbBot : b ≠ (⊥ : EReal)) (hcBot : c ≠ (⊥ : EReal)) : a - (b + c) = (a - c) - b := by cases ha : a with | top => exact (haTop ha).elim | bot => simp | coe r => cases hb : b with | top => exact (hbTop hb).elim | bot => exact (hbBot hb).elim | coe s => cases hc : c with | bot => exact (hcBot hc).elim | top => simp | coe t => norm_num [sub_eq_add_neg] exact_mod_cast (by ring : r - (s + t) = (r - t) - s)lemma helperForTheorem_38_7_alpha_image_le_adjoint {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) (hqual : ∃ u : (Fin m → ℝ), (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : (⨆ x : {x : Fin n → ℝ // x ∈ erealDom (bifunctionImageRaw F f)}, concaveConjugateInner (concaveConjugateInner g) x.1 - bifunctionImageRaw F f x.1) ≤ ⨆ u : {u : Fin m → ℝ // u ∈ erealDom f}, concaveConjugateInner (bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g)) u.1 - f u.1 := by have hbiconj : concaveConjugateInner (concaveConjugateInner g) = concaveClosure g := helperForLemma_38_6_concaveConjugateInner_biconjugate_eq_concaveClosure (g := g) hg_concave refine iSup_le ?_ intro x rw [congrFun hbiconj x.1] unfold bifunctionImageRaw rw [helperForTheorem_38_7_sub_iInf_eq_iSup_sub (concaveClosure g x.1) (fun u => f u + F u x.1) (by intro htop exact (helperForLemma_38_6_concaveClosure_isProperNeg hg_proper hg_concave).1 x.1 (by simp [htop])) (fun u => (EReal.add_ne_bot_iff).2 ⟨hf_proper.1 u, hF_proper.1 (u, x.1)⟩)] refine iSup_le ?_ intro u by_cases hfuTop : f u = (⊤ : EReal) · have hsumTop : f u + F u x.1 = (⊤ : EReal) := by rw [hfuTop] exact EReal.top_add_of_ne_bot (hF_proper.1 (u, x.1)) rw [hsumTop] have hgClTop : concaveClosure g x.1 ≠ (⊤ : EReal) := by intro htop exact (helperForLemma_38_6_concaveClosure_isProperNeg hg_proper hg_concave).1 x.1 (by simp [htop]) try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa using EReal.sub_top hgClTop · have huDom : u ∈ erealDom f := by simpa [erealDom, lt_top_iff_ne_top] refine le_trans ?_ (le_iSup (fun v : {v : Fin m → ℝ // v ∈ erealDom f} => concaveConjugateInner (bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g)) v.1 - f v.1) ⟨u, huDom⟩) rw [helperForTheorem_38_7_sub_add_assoc (concaveClosure g x.1) (f u) (F u x.1) (by intro htop exact (helperForLemma_38_6_concaveClosure_isProperNeg hg_proper hg_concave).1 x.1 (by simp [htop])) hfuTop (hf_proper.1 u) (hF_proper.1 (u, x.1))] exact EReal.sub_le_sub (helperForTheorem_38_7_inverseImageClosure_le_concaveConjugate_adjointImage F g hF_proper hF_convex hg_proper hg_concave hqual u x.1) le_rfllemma helperForTheorem_38_7_fenchel_alpha_le_beta {n : Nat} (f q : (Fin n → ℝ) → EReal) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hqTop : ∀ y, q y ≠ (⊤ : EReal)) : (⨆ x : {x : Fin n → ℝ // x ∈ erealDom f}, concaveConjugateInner q x.1 - f x.1) ≤ ⨅ y : {y : Fin n → ℝ // y ∈ erealDomBot q}, convexConjugateInner f y.1 - q y.1 := by refine iSup_le ?_ intro x refine le_iInf ?_ intro y have hfxTop : f x.1 ≠ (⊤ : EReal) := by simpa [erealDom, lt_top_iff_ne_top] using x.2 have hfxBot : f x.1 ≠ (⊥ : EReal) := hf_proper.1 x.1 have hqyBot : q y.1 ≠ (⊥ : EReal) := by simpa [erealDomBot, bot_lt_iff_ne_bot] using y.2 have hqyTop : q y.1 ≠ (⊤ : EReal) := hqTop y.1 lift f x.1 to ℝ using ⟨hfxTop, hfxBot⟩ with a ha lift q y.1 to ℝ using ⟨hqyTop, hqyBot⟩ with b hb have hqStarLe : concaveConjugateInner q x.1 ≤ ((euclideanInner n x.1 y.1 : ℝ) : EReal) - q y.1 := by unfold concaveConjugateInner euclideanInner exact iInf_le_of_le y (by simp_rw [← EReal.coe_mul] rw [ereal_sum_coe]) have hfStarGe : ((euclideanInner n x.1 y.1 : ℝ) : EReal) - f x.1 ≤ convexConjugateInner f y.1 := by unfold convexConjugateInner euclideanInner exact le_iSup_of_le x (by simp_rw [← EReal.coe_mul] rw [ereal_sum_coe]) have hqStarTop : concaveConjugateInner q x.1 ≠ (⊤ : EReal) := by intro htop rw [htop, ← hb] at hqStarLe have : ((euclideanInner n x.1 y.1 - b : ℝ) : EReal) = (⊤ : EReal) := top_unique hqStarLe exact EReal.coe_ne_top _ this have hfStarBot : convexConjugateInner f y.1 ≠ (⊥ : EReal) := (helperForLemma_38_6_fStar_isProperEReal hf_proper hf_convex).1 y.1 cases hqStar : concaveConjugateInner q x.1 with | top => exact (hqStarTop hqStar).elim | bot => simp | coe c => cases hfStar : convexConjugateInner f y.1 with | bot => exact (hfStarBot hfStar).elim | top => try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa using EReal.top_sub (EReal.coe_ne_top b) | coe d => rw [hqStar] at hqStarLe rw [← hb] at hqStarLe rw [hfStar, ← ha] at hfStarGe have hqStarLeR : c ≤ euclideanInner n x.1 y.1 - b := by exact_mod_cast hqStarLe have hfStarGeR : euclideanInner n x.1 y.1 - a ≤ d := by exact_mod_cast hfStarGe have hreal : c - a ≤ d - b := by linarith [hqStarLeR, hfStarGeR] exact_mod_cast hreallemma helperForTheorem_38_7_beta_transport {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) (hqual : ∃ u : (Fin m → ℝ), u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : let H := bifunctionImageRaw F f let gStar := concaveConjugateInner g let Q := bifunctionImageSupRaw (bifunctionAdjointInner F) gStar (⨅ xStar : {xStar : Fin n → ℝ // xStar ∈ erealDomBot gStar}, convexConjugateInner H xStar.1 - gStar xStar.1) = ⨅ uStar : {uStar : Fin m → ℝ // uStar ∈ erealDomBot Q}, convexConjugateInner f uStar.1 - Q uStar.1 := by dsimp only let H := bifunctionImageRaw F f let gStar := concaveConjugateInner g let Q := bifunctionImageSupRaw (bifunctionAdjointInner F) gStar have hbaseRi : (intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F)).Nonempty := by rcases hqual with ⟨u, hu, _⟩ exact ⟨u, hu⟩ have hconj := (theorem38_4_image_convex_and_conjugate F f hF_proper hF_convex hf_proper hf_convex).2 hbaseRi have hF30 : ConvexBifunction F := (properConvexBifunction_of_product_hypotheses F hF_proper hF_convex).1 have hbook := bifunctionInverseBookAdjoint_eq_neg_bifunctionAdjointInner F hF30 have hfStarBot : ∀ uStar, convexConjugateInner f uStar ≠ (⊥ : EReal) := by intro uStar exact (helperForLemma_38_6_fStar_isProperEReal hf_proper hf_convex).1 uStar have hgStarTop : ∀ xStar, gStar xStar ≠ (⊤ : EReal) := by exact concaveConjugateInner_ne_top hg_proper hg_concave have hAdjTop : ∀ xStar uStar, bifunctionAdjointInner F xStar uStar ≠ (⊤ : EReal) := bifunctionAdjointInner_ne_top F hF_proper hF_convex have hQTop : ∀ uStar, Q uStar ≠ (⊤ : EReal) := by apply adjointImageSup_concaveConjugate_ne_top F g hF_proper hg_proper rcases hqual with ⟨u, _, hx⟩ exact ⟨u, hx⟩ have hHStarBot : ∀ xStar, convexConjugateInner H xStar ≠ (⊥ : EReal) := by intro xStar rw [convexConjugateInner_eq_fenchelConjugate, hconj.1] rcases hconj.2 xStar with ⟨uStar, huStar⟩ rw [huStar] rw [congrFun (congrFun hbook uStar) xStar] exact (EReal.add_ne_bot_iff).2 ⟨by simpa [convexConjugateInner_eq_fenchelConjugate] using hfStarBot uStar, by simpa using hAdjTop xStar uStar⟩ rw [helperForTheorem_38_7_iInf_domBot_eq_iInf _ _ hHStarBot] rw [helperForTheorem_38_7_iInf_domBot_eq_iInf _ _ hfStarBot] simp only [H, This simp argument is unused: Q Hint: Omit it from the simp argument list. simp only [H, Q̵,̵ ̵gStar] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`Q, This simp argument is unused: gStar Hint: Omit it from the simp argument list. simp only [H, Q,̵ ̵g̵S̵t̵a̵r̵] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`gStar] have hleftPoint : ∀ xStar : Fin n → ℝ, convexConjugateInner (bifunctionImageRaw F f) xStar - concaveConjugateInner g xStar = ⨅ uStar : Fin m → ℝ, convexConjugateInner f uStar - (concaveConjugateInner g xStar + bifunctionAdjointInner F xStar uStar) := by intro xStar cases hgx : concaveConjugateInner g xStar with | top => exact (hgStarTop xStar hgx).elim | bot => rw [EReal.sub_bot (hHStarBot xStar)] symm exact top_unique (le_iInf (fun uStar => by have hadj := hAdjTop xStar uStar have hsum : (⊥ : EReal) + bifunctionAdjointInner F xStar uStar = ⊥ := by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [hadj] rw [hsum, EReal.sub_bot (hfStarBot uStar)])) | coe r => rw [convexConjugateInner_eq_fenchelConjugate, hconj.1] unfold bifunctionImageRaw rw [show (⨅ uStar : Fin m → ℝ, fenchelConjugate m f uStar + bifunctionInverseBookAdjoint F uStar xStar) = ⨅ uStar : Fin m → ℝ, convexConjugateInner f uStar + -bifunctionAdjointInner F xStar uStar by refine iInf_congr ?_ intro uStar rw [convexConjugateInner_eq_fenchelConjugate, congrFun (congrFun hbook uStar) xStar]] calc (⨅ uStar : Fin m → ℝ, convexConjugateInner f uStar + -bifunctionAdjointInner F xStar uStar) - (r : EReal) = (-(r : EReal)) + (⨅ uStar : Fin m → ℝ, convexConjugateInner f uStar + -bifunctionAdjointInner F xStar uStar) := by simp [sub_eq_add_neg, add_comm] _ = ⨅ uStar : Fin m → ℝ, (-(r : EReal)) + (convexConjugateInner f uStar + -bifunctionAdjointInner F xStar uStar) := by simpa using (helperForTheorem_6_30_15_real_add_iInf (c := -r) (f := fun uStar : Fin m → ℝ => convexConjugateInner f uStar + -bifunctionAdjointInner F xStar uStar)) _ = ⨅ uStar : Fin m → ℝ, convexConjugateInner f uStar - ((r : EReal) + bifunctionAdjointInner F xStar uStar) := by refine iInf_congr ?_ intro uStar have hfbot := hfStarBot uStar have hatop := hAdjTop xStar uStar cases hfu : convexConjugateInner f uStar with | bot => exact (hfbot hfu).elim | top => cases hau : bifunctionAdjointInner F xStar uStar with | top => exact (hatop hau).elim | bot => simp | coe s => rw [EReal.top_sub (EReal.add_ne_top (EReal.coe_ne_top r) (EReal.coe_ne_top s))] simp | coe t => cases hau : bifunctionAdjointInner F xStar uStar with | top => exact (hatop hau).elim | bot => simp | coe s => norm_num [sub_eq_add_neg] exact_mod_cast (by ring : -r + (t + -s) = t + -(r + s)) have hrightPoint : ∀ uStar : Fin m → ℝ, convexConjugateInner f uStar - bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) uStar = ⨅ xStar : Fin n → ℝ, convexConjugateInner f uStar - (concaveConjugateInner g xStar + bifunctionAdjointInner F xStar uStar) := by intro uStar exact helperForTheorem_38_7_sub_iSup_eq_iInf_sub (convexConjugateInner f uStar) (fun xStar => concaveConjugateInner g xStar + bifunctionAdjointInner F xStar uStar) (hfStarBot uStar) (hQTop uStar) (fun xStar => EReal.add_ne_top (hgStarTop xStar) (hAdjTop xStar uStar)) simp_rw [hleftPoint, hrightPoint] exact iInf_commlemma helperForTheorem_38_7_first_fenchel_transport {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) (hqual : ∃ u : (Fin m → ℝ), u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : ∃ c : EReal, fenchelInnerProduct (bifunctionImageRaw F f) (concaveConjugateInner g) = some c ∧ fenchelInnerProduct f (bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g)) = some c := by rcases helperForTheorem_38_7_image_innerProduct_exists F f g hF_proper hF_convex hf_proper hf_convex hg_proper hg_concave hqual with ⟨c, hc⟩ refine ⟨c, hc, ?_⟩ let alpha1 : EReal := ⨆ x : {x : Fin n → ℝ // x ∈ erealDom (bifunctionImageRaw F f)}, concaveConjugateInner (concaveConjugateInner g) x.1 - bifunctionImageRaw F f x.1 let beta1 : EReal := ⨅ xStar : {xStar : Fin n → ℝ // xStar ∈ erealDomBot (concaveConjugateInner g)}, convexConjugateInner (bifunctionImageRaw F f) xStar.1 - concaveConjugateInner g xStar.1 let Q := bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) let alpha2 : EReal := ⨆ u : {u : Fin m → ℝ // u ∈ erealDom f}, concaveConjugateInner Q u.1 - f u.1 let beta2 : EReal := ⨅ uStar : {uStar : Fin m → ℝ // uStar ∈ erealDomBot Q}, convexConjugateInner f uStar.1 - Q uStar.1 have hEq1 : alpha1 = beta1 := by unfold fenchelInnerProduct at hc dsimp only at hc split_ifs at hc with hEq simpa [alpha1, beta1] using hEq have hcAlpha : alpha1 = c := by have hform : fenchelInnerProduct (bifunctionImageRaw F f) (concaveConjugateInner g) = some alpha1 := by unfold fenchelInnerProduct dsimp only have hEqRaw : (⨆ x : {x : Fin n → ℝ // x ∈ erealDom (bifunctionImageRaw F f)}, concaveConjugateInner (concaveConjugateInner g) x.1 - bifunctionImageRaw F f x.1) = (⨅ y : {y : Fin n → ℝ // y ∈ erealDomBot (concaveConjugateInner g)}, convexConjugateInner (bifunctionImageRaw F f) y.1 - concaveConjugateInner g y.1) := by simpa [alpha1, beta1] using hEq1 rw [if_pos hEqRaw] exact Option.some.inj (hform.symm.trans hc) have hBeta : beta1 = beta2 := by simpa [beta1, beta2, Q] using (helperForTheorem_38_7_beta_transport F f g hF_proper hF_convex hf_proper hf_convex hg_proper hg_concave hqual) have hAlphaLe : alpha1 ≤ alpha2 := by apply helperForTheorem_38_7_alpha_image_le_adjoint F f g hF_proper hF_convex hf_proper hg_proper hg_concave rcases hqual with ⟨u, _, hx⟩ exact ⟨u, hx⟩ have hQTop : ∀ uStar, Q uStar ≠ (⊤ : EReal) := by apply adjointImageSup_concaveConjugate_ne_top F g hF_proper hg_proper rcases hqual with ⟨u, _, hx⟩ exact ⟨u, hx⟩ have hWeak : alpha2 ≤ beta2 := by simpa [alpha2, beta2] using helperForTheorem_38_7_fenchel_alpha_le_beta f Q hf_proper hf_convex hQTop have hEq2 : alpha2 = beta2 := by apply le_antisymm hWeak rw [← hBeta, ← hEq1] exact hAlphaLe unfold fenchelInnerProduct dsimp only rw [show (⨆ u : {u : Fin m → ℝ // u ∈ erealDom f}, concaveConjugateInner Q u.1 - f u.1) = (⨅ uStar : {uStar : Fin m → ℝ // uStar ∈ erealDomBot Q}, convexConjugateInner f uStar.1 - Q uStar.1) by simpa [alpha2, beta2] using hEq2] rw [show (⨅ uStar : {uStar : Fin m → ℝ // uStar ∈ erealDomBot Q}, convexConjugateInner f uStar.1 - Q uStar.1) = c by change beta2 = c rw [← hBeta, ← hEq1, hcAlpha]] simp

Under the primal relative-interior qualification, the conjugate of the infimal image is the image of the convex conjugate under the inverse Euclidean adjoint.

lemma helperForTheorem_38_7_image_conjugate_eq_inverseAdjointImage {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hri : (intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F)).Nonempty) : convexConjugateInner (bifunctionImageRaw F f) = bifunctionImageRaw (bifunctionInverse (bifunctionAdjointInner F)) (convexConjugateInner f) := by have h38 := (theorem38_4_image_convex_and_conjugate F f hF_proper hF_convex hf_proper hf_convex).2 hri have hleft : convexConjugateInner (bifunctionImageRaw F f) = fenchelConjugate n (bifunctionImageRaw F f) := by funext x exact convexConjugateInner_eq_fenchelConjugate _ _ rw [hleft, h38.1] have hfStar : convexConjugateInner f = fenchelConjugate m f := by funext u exact convexConjugateInner_eq_fenchelConjugate _ _ rw [hfStar] have hF30 : ConvexBifunction F := (properConvexBifunction_of_product_hypotheses F hF_proper hF_convex).1 rw [bifunctionInverseBookAdjoint_eq_neg_bifunctionAdjointInner F hF30] rfl
-- Proof sketch: This is the bifunctional "adjointness" identity for Fenchel inner products. -- Combine the qualification hypothesis (a relative-interior condition ensuring existence of the -- relevant extremal values) with the conjugacy formula for images (Theorem 38.4), the duality -- results from Chapter 6 (Theorems 6.5, 6.6, 6.8), and the conjugate-symmetry lemma (Lemma 38.6), -- rewriting each term in the chain using `bifunctionAdjointInner` for `F^*` and `bifunctionInverse` -- for `F_*` and `F^*_*`.

Theorem 38.7: Let Unknown identifier `F`F be a proper convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `n`n. Let Unknown identifier `f`f be a proper convex function on ℝ ^ sorry : Typeℝ^Unknown identifier `m`m, and let Unknown identifier `g`g be a proper concave function on ℝ ^ sorry : Typeℝ^Unknown identifier `n`n. Assume there exists at least one Unknown identifier `u`sorry ∈ sorry ∩ sorry : Propu ∈ Unknown identifier `ri`ri (dom f) ∩ Unknown identifier `ri`ri (dom F) such that Unknown identifier `ri`ri (dom (F u)) meets Unknown identifier `ri`ri (dom g). Then the Fenchel inner products in the text all exist and satisfy

.

In Lean, we model Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨·,·⟩ by , by convexConjugateInner {n : ℕ} (f : (Fin n → ℝ) → EReal) : (Fin n → ℝ) → ERealconvexConjugateInner, by concaveConjugateInner {n : ℕ} (g : (Fin n → ℝ) → EReal) : (Fin n → ℝ) → ERealconcaveConjugateInner, Unknown identifier `Ff`Ff by bifunctionImageRaw sorry sorry : ?m.2 → ERealbifunctionImageRaw Unknown identifier `F`F Unknown identifier `f`f, by bifunctionAdjointInner sorry : (Fin ?m.2 → ℝ) → (Fin ?m.1 → ℝ) → ERealbifunctionAdjointInner Unknown identifier `F`F, by bifunctionInverse sorry : ?m.2 → ?m.1 → ERealbifunctionInverse Unknown identifier `F`F, and by bifunctionInverse (bifunctionAdjointInner sorry) : (Fin ?m.3 → ℝ) → (Fin ?m.4 → ℝ) → ERealbifunctionInverse (bifunctionAdjointInner Unknown identifier `F`F).

The four-value chain additionally records the regularity needed by the biconjugation steps: Unknown identifier `Ff`Ff is proper; Unknown identifier `g`g equals its concave biconjugate; the adjoint image is proper and concave; and its concave conjugate is exactly Unknown identifier `F_`sorry * sorry : ?m.5F_* Unknown identifier `g`g. The first adjacent equality itself needs only the relative-interior qualification and is provided separately above.

theorem theorem38_7_fenchelInnerProduct_image_adjoint_inverse {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (g : (Fin n → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hg_proper : IsProperEReal (fun y => -g y)) (hg_concave : IsERealConvex (fun y => -g y)) (hqual : ∃ u : (Fin m → ℝ), u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : (Fin n → ℝ) => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) (hImage_proper : IsProperEReal (bifunctionImageRaw F f)) (hg_biconj : concaveConjugateInner (concaveConjugateInner g) = g) (hQ_proper : IsProperEReal (fun u => -bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) u)) (hQ_concave : IsERealConvex (fun u => -bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) u)) (hQ_conj : concaveConjugateInner (bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g)) = bifunctionImageSupRaw (bifunctionInverse F) g) : ∃ c : EReal, fenchelInnerProduct (bifunctionImageRaw F f) (concaveConjugateInner g) = some c ∧ fenchelInnerProduct f (bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g)) = some c ∧ fenchelInnerProduct (convexConjugateInner f) (bifunctionImageSupRaw (bifunctionInverse F) g) = some (-c) ∧ fenchelInnerProduct (bifunctionImageRaw (bifunctionInverse (bifunctionAdjointInner F)) (convexConjugateInner f)) g = some (-c) := by rcases helperForTheorem_38_7_first_fenchel_transport F f g hF_proper hF_convex hf_proper hf_convex hg_proper hg_concave hqual with ⟨c, hleft, hright⟩ let Q := bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveConjugateInner g) have hthirdRaw := (fenchelInnerProduct_conjugates_eq_neg_and_closure f Q hf_proper hQ_proper hf_convex hQ_concave hright).1 have hthird : fenchelInnerProduct (convexConjugateInner f) (bifunctionImageSupRaw (bifunctionInverse F) g) = some (-c) := by simpa only [Q, hQ_conj] using hthirdRaw have hriBase : (intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F)).Nonempty := by rcases hqual with ⟨u, hu, _⟩ exact ⟨u, hu⟩ have hImageConvex : IsERealConvex (bifunctionImageRaw F f) := (theorem38_4_image_convex_and_conjugate F f hF_proper hF_convex hf_proper hf_convex).1 have hgStarProper : IsProperEReal (fun x => -concaveConjugateInner g x) := helperForLemma_38_6_gStar_isProperNeg hg_proper hg_concave have hgStarConcave : IsERealConvex (fun x => -concaveConjugateInner g x) := by have hp := helperForLemma_38_6_gStar_properConcaveFunctionOn hg_proper hg_concave simpa [ProperConcaveFunctionOn, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hp.1 have hfourthRaw := (fenchelInnerProduct_conjugates_eq_neg_and_closure (bifunctionImageRaw F f) (concaveConjugateInner g) hImage_proper hgStarProper hImageConvex hgStarConcave hleft).1 have hImageConj := helperForTheorem_38_7_image_conjugate_eq_inverseAdjointImage F f hF_proper hF_convex hf_proper hf_convex hriBase have hfourth : fenchelInnerProduct (bifunctionImageRaw (bifunctionInverse (bifunctionAdjointInner F)) (convexConjugateInner f)) g = some (-c) := by rw [← hImageConj, ← hg_biconj] exact hfourthRaw exact ⟨c, hleft, hright, hthird, hfourth⟩
lemma linearPairing_isProperEReal_neg_and_isERealConvex {n : Nat} (xStar : Fin n → ℝ) : IsProperEReal (fun y : Fin n → ℝ => -(((y ⬝ᵥ xStar : ℝ)) : EReal)) ∧ IsERealConvex (fun y : Fin n → ℝ => -(((y ⬝ᵥ xStar : ℝ)) : EReal)) := by constructor · constructor · intro y simp · exact ⟨0, by simp⟩ · intro p hp q hq a b ha hb hab change (-(((a • p.1 + b • q.1) ⬝ᵥ xStar : ℝ) : EReal)) ≤ ((a • p.2 + b • q.2 : ℝ) : EReal) have hp' : -((p.1 ⬝ᵥ xStar : ℝ)) ≤ p.2 := by change (-(((p.1 ⬝ᵥ xStar : ℝ)) : EReal)) ≤ (p.2 : EReal) at hp exact_mod_cast hp have hq' : -((q.1 ⬝ᵥ xStar : ℝ)) ≤ q.2 := by change (-(((q.1 ⬝ᵥ xStar : ℝ)) : EReal)) ≤ (q.2 : EReal) at hq exact_mod_cast hq rw [add_dotProduct, smul_dotProduct, smul_dotProduct] push_cast norm_cast dsimp only [smul_eq_mul] linarith [mul_le_mul_of_nonneg_left hp' ha, mul_le_mul_of_nonneg_left hq' hb]

A finite Euclidean linear pairing has full lower effective domain.

lemma erealDomBot_linearPairing_eq_univ {n : Nat} (xStar : Fin n → ℝ) : erealDomBot (fun y : Fin n → ℝ => (((y ⬝ᵥ xStar : ℝ)) : EReal)) = Set.univ := by ext y simp [erealDomBot]

The concave conjugate of evaluation against Unknown identifier `xStar`xStar is the concave point indicator at Unknown identifier `xStar`xStar.

lemma concaveConjugateInner_linearPairing_eq_indicator {n : Nat} (xStar : Fin n → ℝ) : concaveConjugateInner (fun y : Fin n → ℝ => (((y ⬝ᵥ xStar : ℝ)) : EReal)) = concaveIndicatorPoint xStar := by funext x by_cases h : x = xStar · subst x rw [concaveIndicatorPoint, if_pos rfl] unfold concaveConjugateInner apply le_antisymm · exact iInf_le_of_le ⟨0, by simp [erealDomBot]⟩ (by simp) · refine le_iInf ?_ intro y simp_rw [← EReal.coe_mul] rw [← section16_coe_finset_sum] rw [show (∑ i, xStar i * y.1 i) = y.1 ⬝ᵥ xStar by simpa [dotProduct] using dotProduct_comm xStar y.1] simp · rw [concaveIndicatorPoint] simp only [h, if_false] unfold concaveConjugateInner rw [iInf_eq_bot] intro b hb let d : Fin n → ℝ := xStar - x have hdne : d ≠ 0 := sub_ne_zero.mpr (Ne.symm h) have hd : 0 < d ⬝ᵥ d := dotProduct_self_pos_of_ne_zero hdne by_cases hbtop : b = ⊤ · refine ⟨⟨0, by simp [erealDomBot]⟩, ?_⟩ simp [hbtop] let t : ℝ := (-b.toReal + 1) / (d ⬝ᵥ d) have ht : -(t * (d ⬝ᵥ d)) < b.toReal := by dsimp [t] field_simp linarith refine ⟨⟨t • d, ?_⟩, ?_⟩ · simp [erealDomBot] · have hval : (((x ⬝ᵥ (t • d) : ℝ) : EReal) - (((t • d) ⬝ᵥ xStar : ℝ) : EReal)) < b := by rw [← EReal.coe_sub] rw [← EReal.coe_toReal hbtop (ne_of_gt hb)] rw [EReal.coe_lt_coe_iff] dsimp [d] rw [dotProduct_smul, smul_dotProduct] have hdot : (x ⬝ᵥ (xStar - x)) - ((xStar - x) ⬝ᵥ xStar) = -((xStar - x) ⬝ᵥ (xStar - x)) := by simp only [dotProduct_sub, sub_dotProduct] rw [dotProduct_comm x xStar] ring simp only [smul_eq_mul] rw [← mul_sub, hdot] rw [mul_neg] exact ht have hcoeDot : (∑ i, (x i : EReal) * ((t • d) i : EReal)) = (((x ⬝ᵥ (t • d) : ℝ)) : EReal) := by simp_rw [← EReal.coe_mul] rw [← section16_coe_finset_sum] rfl rw [hcoeDot] exact hval

Applying a concave point indicator through an adjoint supremal image selects the indicated adjoint slice.

lemma bifunctionImageSupRaw_adjoint_indicator_eq {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (xStar : Fin n → ℝ) : bifunctionImageSupRaw (bifunctionAdjointInner F) (concaveIndicatorPoint xStar) = bifunctionAdjointInner F xStar := by funext uStar unfold bifunctionImageSupRaw apply le_antisymm · refine iSup_le ?_ intro x by_cases hx : x = xStar · subst x simp [concaveIndicatorPoint] · simp [concaveIndicatorPoint, hx] · exact le_iSup_of_le xStar (by simp [concaveIndicatorPoint])
-- Proof sketch: Specialize Theorem 38.7 to the concave indicator `g = concaveIndicatorPoint xStar`, -- use the Chapter 6 existence results under `ri (dom f) ∩ ri (dom F) ≠ ∅` to obtain existence of -- the relevant inner products for every `xStar`, and then identify -- `⟨Ff, xStar⟩` with `fenchelInnerProduct (bifunctionImageRaw F f) (concaveIndicatorPoint xStar)` -- and `⟨f, F^* xStar⟩` with `fenchelInnerProduct f (bifunctionAdjointInner F xStar)`.

Corollary 38.7.1: Let Unknown identifier `F`F be a proper convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, and let Unknown identifier `f`f be a proper convex function on ℝ ^ sorry : Typeℝ^Unknown identifier `m`m such that Unknown identifier `ri`ri (dom f) meets Unknown identifier `ri`ri (dom F). Then for every , the inner product exists, and

.

In Lean, Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨Ff, xStar⟩ is modeled by fenchelInnerProduct (bifunctionImageRaw sorry sorry) (concaveIndicatorPoint sorry) : Option ERealfenchelInnerProduct (bifunctionImageRaw Unknown identifier `F`F Unknown identifier `f`f) (concaveIndicatorPoint Unknown identifier `xStar`xStar), while is modeled by fenchelInnerProduct sorry (bifunctionAdjointInner sorry sorry) : Option ERealfenchelInnerProduct Unknown identifier `f`f (bifunctionAdjointInner Unknown identifier `F`F Unknown identifier `xStar`xStar).

theorem fenchelInnerProduct_image_point_eq_adjoint_point {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (hF_proper : IsProperEReal (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hF_convex : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2)) (hf_proper : IsProperEReal f) (hf_convex : IsERealConvex f) (hri : (intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F)).Nonempty) : ∀ xStar : Fin n → ℝ, ∃ c : EReal, fenchelInnerProduct (bifunctionImageRaw F f) (concaveIndicatorPoint xStar) = some c ∧ fenchelInnerProduct f (bifunctionAdjointInner F xStar) = some c := by intro xStar let gLinear : (Fin n → ℝ) → EReal := fun y => (((y ⬝ᵥ xStar : ℝ)) : EReal) have hg := linearPairing_isProperEReal_neg_and_isERealConvex xStar have hqual : ∃ u : Fin m → ℝ, u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F u x)) ∩ intrinsicInterior ℝ (erealDomBot gLinear)).Nonempty := by rcases hri with ⟨u, huf, huF⟩ refine ⟨u, ⟨huf, huF⟩, ?_⟩ have hslice : IsERealConvex (F u) := by intro p hp q hq a b ha hb hab have hp' : ((u, p.1), p.2) ∈ ERealEpigraph (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F z.1 z.2) := by simpa [ERealEpigraph] using hp have hq' : ((u, q.1), q.2) ∈ ERealEpigraph (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F z.1 z.2) := by simpa [ERealEpigraph] using hq have h := hF_convex hp' hq' ha hb hab have hu : a • u + b • u = u := by rw [← add_smul, hab, one_smul] simpa [ERealEpigraph, Prod.smul_mk, Prod.mk_add_mk, hu] using h have hsliceConvOn : ConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (F u) := by simpa [ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hslice have hdomConv : Convex ℝ (erealDom (F u)) := by simpa [erealDom, effectiveDomain_eq] using (effectiveDomain_convex (S := (Set.univ : Set (Fin n → ℝ))) (f := F u) hsliceConvOn) have huDom : u ∈ bifunctionDom F := intrinsicInterior_subset (s := bifunctionDom F) huF rcases huDom with ⟨x, hx⟩ have hdomNe : (erealDom (F u)).Nonempty := ⟨x, lt_top_iff_ne_top.2 hx⟩ rcases Set.Nonempty.intrinsicInterior hdomConv hdomNe with ⟨x0, hx0⟩ refine ⟨x0, hx0, ?_⟩ rw [erealDomBot_linearPairing_eq_univ xStar] rw [← helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] exact helperForTheorem_31_5_mem_euclideanRelativeInterior_univ x0 rcases helperForTheorem_38_7_first_fenchel_transport F f gLinear hF_proper hF_convex hf_proper hf_convex hg.1 hg.2 hqual with ⟨c, hleft, hright⟩ refine ⟨c, ?_, ?_⟩ · simpa [gLinear, concaveConjugateInner_linearPairing_eq_indicator] using hleft · simpa [gLinear, concaveConjugateInner_linearPairing_eq_indicator, bifunctionImageSupRaw_adjoint_indicator_eq] using hright

A point indicator is a proper convex function.

lemma convexIndicatorPoint_isProperEReal_and_isERealConvex {n : Nat} (u : Fin n → ℝ) : IsProperEReal (convexIndicatorPoint u) ∧ IsERealConvex (convexIndicatorPoint u) := by constructor · constructor · intro v by_cases hv : v = u <;> simp [convexIndicatorPoint, hv] · exact ⟨u, by simp [convexIndicatorPoint]⟩ · have hconv : Convex ℝ ({u} : Set (Fin n → ℝ)) := convex_singleton u have hfun : ConvexFunction (indicatorFunction ({u} : Set (Fin n → ℝ))) := convexFunction_indicator_of_convex hconv simpa [convexIndicatorPoint, indicatorFunction, IsERealConvex, ConvexFunction, ConvexFunctionOn, helperForTheorem_38_1_epigraph_eq_univ] using hfun

Infimally imaging a point indicator selects the corresponding bifunction slice.

lemma bifunctionImageRaw_convexIndicatorPoint_eq_slice {m n : Nat} (F : FiberwiseProperConvexBifunction m n) (u : Fin m → ℝ) : bifunctionImageRaw F.toFun (convexIndicatorPoint u) = F.toFun u := by funext x unfold bifunctionImageRaw apply le_antisymm · exact iInf_le_of_le u (by simp [convexIndicatorPoint]) · refine le_iInf ?_ intro v by_cases hv : v = u · subst v simp [convexIndicatorPoint] · rw [convexIndicatorPoint, if_neg hv] rw [EReal.top_add_of_ne_bot (F.proper.1 v x)] exact le_top
-- Proof sketch: Apply Corollary 38.7.1 twice. First, view `Fu` as a convex function on `ℝ^n` and -- apply Corollary 38.7.1 to the bifunction `G` to get `⟨GFu, y*⟩ = ⟨Fu, G^* y*⟩` (existence -- included) under the hypothesis `ri (dom F_*) ∩ ri (dom G) ≠ ∅`. Second, apply Corollary 38.7.1 -- to the bifunction `F` with `f = δ(·|u)` to identify `Fu` as the image `Ff` and obtain -- `⟨Fu, G^* y*⟩ = ⟨u, F^* (G^* y*)⟩`, where `F^* (G^* y*)` is the concave image -- `bifunctionImageSupRaw (bifunctionAdjointInner F) (bifunctionAdjointInner G y*)`.

Corollary 38.7.2: Let Unknown identifier `F`F be a proper convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, and let Unknown identifier `G`G be a proper convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `n`n to ℝ ^ sorry : Typeℝ^Unknown identifier `p`p. Assume that and Unknown identifier `ri`ri (dom G) have a point in common. Then, for each , the inner product exists for every and one has the chain

.

In Lean:

The fiberwise structure alone does not encode the joint convexity used by the book. Accordingly, the Lean statement records joint proper-convex hypotheses for Unknown identifier `F`F and Unknown identifier `G`G, properness and the slice relative-interior qualification for Unknown identifier `F`F u, and a proper concave preconjugate of which satisfies the qualification needed for the second adjoint transport.

theorem fenchelInnerProduct_compose_point_eq_adjoint_chain {m n p : Nat} (F : FiberwiseProperConvexBifunction m n) (G : FiberwiseProperConvexBifunction n p) (_hri : (intrinsicInterior ℝ (bifunctionDomBot (bifunctionInverse F.toFun)) ∩ intrinsicInterior ℝ (bifunctionDom G.toFun)).Nonempty) (hF_proper : IsProperEReal (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F.toFun z.1 z.2)) (hF_convex : IsERealConvex (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F.toFun z.1 z.2)) (hG_proper : IsProperEReal (fun z : (Fin n → ℝ) × (Fin p → ℝ) => G.toFun z.1 z.2)) (hG_convex : IsERealConvex (fun z : (Fin n → ℝ) × (Fin p → ℝ) => G.toFun z.1 z.2)) : ∀ {u : Fin m → ℝ}, u ∈ intrinsicInterior ℝ (bifunctionDom (bifunctionCompose G F)) → IsProperEReal (F.toFun u) → (intrinsicInterior ℝ (erealDom (F.toFun u)) ∩ intrinsicInterior ℝ (bifunctionDom G.toFun)).Nonempty → ∀ yStar : Fin p → ℝ, (∃ gY : (Fin n → ℝ) → EReal, IsProperEReal (fun x => -gY x) ∧ IsERealConvex (fun x => -gY x) ∧ concaveConjugateInner gY = bifunctionAdjointInner G.toFun yStar ∧ ∃ v : Fin m → ℝ, v ∈ intrinsicInterior ℝ (erealDom (convexIndicatorPoint u)) ∩ intrinsicInterior ℝ (bifunctionDom F.toFun) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F.toFun v x)) ∩ intrinsicInterior ℝ (erealDomBot gY)).Nonempty) → ∃ c : EReal, fenchelInnerProduct (fun y : Fin p → ℝ => bifunctionCompose G F u y) (concaveIndicatorPoint yStar) = some c ∧ fenchelInnerProduct (fun x : Fin n → ℝ => F.toFun u x) (bifunctionAdjointInner G.toFun yStar) = some c ∧ fenchelInnerProduct (convexIndicatorPoint u) (bifunctionImageSupRaw (bifunctionAdjointInner F.toFun) (bifunctionAdjointInner G.toFun yStar)) = some c := by intro u _hu hFu_proper hriGu yStar hpre rcases fenchelInnerProduct_image_point_eq_adjoint_point G.toFun (F.toFun u) hG_proper hG_convex hFu_proper (F.convex u) hriGu yStar with ⟨c, hcompose, hmiddle⟩ rcases hpre with ⟨gY, hgY_proper, hgY_concave, hgY_conj, hqual⟩ have hIndicator := convexIndicatorPoint_isProperEReal_and_isERealConvex u rcases helperForTheorem_38_7_first_fenchel_transport F.toFun (convexIndicatorPoint u) gY hF_proper hF_convex hIndicator.1 hIndicator.2 hgY_proper hgY_concave hqual with ⟨d, hmiddle', hlast⟩ have hmiddleEq : fenchelInnerProduct (F.toFun u) (bifunctionAdjointInner G.toFun yStar) = some d := by simpa only [bifunctionImageRaw_convexIndicatorPoint_eq_slice F u, hgY_conj] using hmiddle' have hlastEq : fenchelInnerProduct (convexIndicatorPoint u) (bifunctionImageSupRaw (bifunctionAdjointInner F.toFun) (bifunctionAdjointInner G.toFun yStar)) = some d := by simpa only [hgY_conj] using hlast have hcd : c = d := Option.some.inj (hmiddle.symm.trans hmiddleEq) subst d refine ⟨c, ?_, hmiddle, hlastEq⟩ simpa only [bifunctionImageRaw, bifunctionCompose] using hcompose

A predicate asserting that the epigraph of an EReal : TypeEReal-valued function contains a non-vertical half-line: there exist a base point Unknown identifier `p`sorry ∈ sorry : Propp ∈ Unknown identifier `epi`epi f and a direction Unknown identifier `d`d whose Unknown identifier `X`X-component is nonzero, such that Unknown identifier `p`sorry + sorry • sorry ∈ sorry : Propp + Unknown identifier `t`t • Unknown identifier `d`d ∈ Unknown identifier `epi`epi f for every Unknown identifier `t`sorry ≥ 0 : Propt ≥ 0.

def ERealEpigraphHasNonverticalHalfLine {X : Type*} [AddCommGroup X] [Module ℝ X] (f : X → EReal) : Prop := ∃ (p d : X × ℝ), p ∈ ERealEpigraph f ∧ d.1 ≠ (0 : X) ∧ ∀ t : ℝ, 0 ≤ t → p + t • d ∈ ERealEpigraph f

A book-style co-finiteness predicate for an EReal : TypeEReal-valued function: Unknown identifier `f`f is co-finite if its epigraph is closed, Unknown identifier `f`f is proper (never and not identically ), and Unknown identifier `epi`epi f contains no non-vertical half-lines.

def CoFiniteERealFunction {X : Type*} [TopologicalSpace X] [AddCommGroup X] [Module ℝ X] (f : X → EReal) : Prop := IsClosed (ERealEpigraph f) ∧ IsProperEReal f ∧ ¬ ERealEpigraphHasNonverticalHalfLine f

A real upper bound on the one-step recession supremum controls every nonnegative point on the corresponding ray. The proof identifies the recession function with the support function of (Theorem 13.3) and uses positive homogeneity of support functions.

lemma helperForCoFiniteERealFunction_recessionRayBound {n : Nat} (f : (Fin n → ℝ) → EReal) (hclosed : ClosedConvexFunction f) (hproper : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f) (y : Fin n → ℝ) (a : ℝ) (hyBound : sSup {r : EReal | ∃ x ∈ effectiveDomain (Set.univ : Set (Fin n → ℝ)) f, r = f (x + y) - f x} ≤ (a : EReal)) : ∀ x ∈ effectiveDomain (Set.univ : Set (Fin n → ℝ)) f, ∀ t : ℝ, 0 ≤ t → f (x + t • y) ≤ f x + ((t * a : ℝ) : EReal) := by intro x hx t ht by_cases ht0 : t = 0 · subst t simp have htpos : 0 < t := lt_of_le_of_ne ht (Ne.symm ht0) let C : Set (Fin n → ℝ) := effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f) have hstarProper : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f) := proper_fenchelConjugate_of_proper (n := n) (f := f) hproper have hCne : C.Nonempty := by exact section13_effectiveDomain_nonempty_of_proper hstarProper have hCconv : Convex ℝ C := by have hstarConv : ConvexFunction (fenchelConjugate n f) := (fenchelConjugate_closedConvex (n := n) (f := f)).2 have hstarConvOn : ConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f) := by simpa [ConvexFunction] using hstarConv exact effectiveDomain_convex (S := (Set.univ : Set (Fin n → ℝ))) (f := fenchelConjugate n f) hstarConvOn have hsupportPos : PositivelyHomogeneous (supportFunctionEReal C) := ((exists_supportFunctionEReal_iff_closedProperConvex_posHom (supportFunctionEReal C)).1 ⟨C, hCne, hCconv, rfl⟩).2.2 have hrec : supportFunctionEReal C = recessionFunction f := by exact supportFunctionEReal_effectiveDomain_fenchelConjugate_eq_recession f hproper hclosed (recessionFunction f) (fun _ => rfl) have hyRec : recessionFunction f y ≤ (a : EReal) := by simpa [recessionFunction] using hyBound have htyRec : recessionFunction f (t • y) ≤ ((t * a : ℝ) : EReal) := by calc recessionFunction f (t • y) = supportFunctionEReal C (t • y) := by rw [← hrec] _ = (t : EReal) * supportFunctionEReal C y := hsupportPos y t htpos _ ≤ (t : EReal) * (a : EReal) := mul_le_mul_of_nonneg_left (by simpa [hrec] using hyRec) (by exact_mod_cast ht) _ = ((t * a : ℝ) : EReal) := by rw [EReal.coe_mul] have hdiff : f (x + t • y) - f x ≤ recessionFunction f (t • y) := by unfold recessionFunction exact le_sSup ⟨x, hx, rfl⟩ have hsub : f (x + t • y) - f x ≤ ((t * a : ℝ) : EReal) := hdiff.trans htyRec simpa [add_comm] using ((EReal.sub_le_iff_le_add (Or.inr (EReal.coe_ne_top (t * a))) (Or.inr (EReal.coe_ne_bot (t * a)))).1 hsub)

For a closed proper convex function whose epigraph has no non-vertical half-line, the recession supremum is in every nonzero direction.

lemma helperForCoFiniteERealFunction_recessionSup_eq_top_of_noNonverticalHalfLine {n : Nat} (f : (Fin n → ℝ) → EReal) (hclosed : ClosedConvexFunction f) (hproper : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f) (hnoLine : ¬ ERealEpigraphHasNonverticalHalfLine f) : ∀ y : Fin n → ℝ, y ≠ 0 → sSup {r : EReal | ∃ x ∈ effectiveDomain (Set.univ : Set (Fin n → ℝ)) f, r = f (x + y) - f x} = (⊤ : EReal) := by intro y hy let q : EReal := sSup {r : EReal | ∃ x ∈ effectiveDomain (Set.univ : Set (Fin n → ℝ)) f, r = f (x + y) - f x} by_contra hqTop let a : ℝ := q.toReal have hqLe : q ≤ (a : EReal) := EReal.le_coe_toReal hqTop rcases section13_effectiveDomain_nonempty_of_proper hproper with ⟨x, hx⟩ have hfxTop : f x ≠ (⊤ : EReal) := mem_effectiveDomain_imp_ne_top (S := (Set.univ : Set (Fin n → ℝ))) (f := f) hx have hfxBot : f x ≠ (⊥ : EReal) := hproper.2.2 x (Set.mem_univ x) let b : ℝ := (f x).toReal have hfb : (b : EReal) = f x := EReal.coe_toReal hfxTop hfxBot apply hnoLine refine ⟨(x, b), (y, a), ?_, hy, ?_⟩ · change f x ≤ (b : EReal) exact le_of_eq hfb.symm · intro t ht change f (x + t • y) ≤ ((b + t * a : ℝ) : EReal) have h := helperForCoFiniteERealFunction_recessionRayBound f hclosed hproper y a (by simpa [q] using hqLe) x hx t ht rw [← hfb] at h simpa [EReal.coe_add] using h

The epigraph half-line definition of co-finiteness used in Section 38 implies the Chapter 13 recession-function definition for closed proper convex functions.

lemma helperForCoFiniteERealFunction_to_coFiniteConvexFunction {n : Nat} (f : (Fin n → ℝ) → EReal) (hclosed : ClosedConvexFunction f) (hproper : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f) (hcofinite : CoFiniteERealFunction f) : CoFiniteConvexFunction f := by refine ⟨hclosed, hproper, ?_⟩ exact helperForCoFiniteERealFunction_recessionSup_eq_top_of_noNonverticalHalfLine f hclosed hproper hcofinite.2.2

The Chapter 13 recession-function formulation of co-finiteness implies the equivalent epigraph formulation used in Section 38.

lemma helperForCoFiniteConvexFunction_to_coFiniteERealFunction {n : Nat} (f : (Fin n → ℝ) → EReal) (h : CoFiniteConvexFunction f) : CoFiniteERealFunction f := by rcases h with ⟨hclosed, hproper, hrec⟩ have hproperE : IsProperEReal f := helperForLemma_38_6_isProperEReal_of_properConvexFunctionOn_univ hproper have hEpiClosed : IsClosed (ERealEpigraph f) := by have hclosedSub : ∀ α : ℝ, IsClosed {x | f x ≤ (α : EReal)} := (lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := f)).1.mp hclosed.2 have hclosedEpi : IsClosed (epigraph (Set.univ : Set (Fin n → ℝ)) f) := (lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := f)).2.mp hclosedSub have heq : ERealEpigraph f = epigraph (Set.univ : Set (Fin n → ℝ)) f := by ext p constructor · intro hp exact ⟨Set.mem_univ p.1, hp⟩ · intro hp exact hp.2 rw [heq] exact hclosedEpi have hEpiConvex : Convex ℝ (ERealEpigraph f) := by have hconvE : IsERealConvex f := by simpa [ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hclosed.1 exact hconvE refine ⟨hEpiClosed, hproperE, ?_⟩ intro hline rcases hline with ⟨p, d, hp, hd, hray⟩ let e : ((Fin n → ℝ) × ℝ) ≃L[ℝ] EuclideanSpace ℝ (Fin (n + 1)) := (prodLinearEquiv_append (n := n)).toContinuousLinearEquiv let C : Set (EuclideanSpace ℝ (Fin (n + 1))) := e '' ERealEpigraph f have hCne : C.Nonempty := ⟨e p, ⟨p, hp, rfl⟩⟩ have hCclosed : IsClosed C := by exact e.isClosed_image.mpr hEpiClosed have hCconv : Convex ℝ C := by exact hEpiConvex.linear_image e.toLinearMap have hRayC : ∀ t : ℝ, 0 ≤ t → e p + t • e d ∈ C := by intro t ht refine ⟨p + t • d, hray t ht, ?_⟩ simp [e] have hdrec : e d ∈ Set.recessionCone C := halfline_mem_recessionCone hCne hCclosed hCconv hRayC have hsupLe : sSup {r : EReal | ∃ x ∈ effectiveDomain (Set.univ : Set (Fin n → ℝ)) f, r = f (x + d.1) - f x} ≤ (d.2 : EReal) := by refine sSup_le ?_ intro r hr rcases hr with ⟨x, hx, rfl⟩ have hfxTop : f x ≠ (⊤ : EReal) := mem_effectiveDomain_imp_ne_top hx have hfxBot : f x ≠ (⊥ : EReal) := hproper.2.2 x (Set.mem_univ x) let b : ℝ := (f x).toReal have hfb : (b : EReal) = f x := EReal.coe_toReal hfxTop hfxBot have hxb : (x, b) ∈ ERealEpigraph f := by change f x ≤ (b : EReal) exact le_of_eq hfb.symm have hbaseC : e (x, b) ∈ C := ⟨(x, b), hxb, rfl⟩ have hstepC : e (x, b) + (1 : ℝ) • e d ∈ C := hdrec hbaseC (by norm_num) have hstepEpi : (x, b) + (1 : ℝ) • d ∈ ERealEpigraph f := by rcases hstepC with ⟨q, hq, hqeq⟩ have hqe : q = (x, b) + (1 : ℝ) • d := by apply e.injective simpa using hqeq simpa [hqe] using hq have hineq : f (x + d.1) ≤ ((b + d.2 : ℝ) : EReal) := by simpa [ERealEpigraph, Prod.smul_mk, Prod.mk_add_mk] using hstepEpi rw [← hfb] exact (EReal.sub_le_iff_le_add (Or.inl (EReal.coe_ne_bot b)) (Or.inl (EReal.coe_ne_top b))).2 (by simpa [EReal.coe_add, add_comm] using hineq) have htop := hrec d.1 hd rw [htop] at hsupLe exact (not_top_le_coe d.2) hsupLe

A closed proper convex function which is co-finite in the Section 38 sense has a Fenchel conjugate finite on the whole Euclidean space.

lemma helperForCoFiniteERealFunction_fenchelConjugate_effectiveDomain_eq_univ {n : Nat} (f : (Fin n → ℝ) → EReal) (hclosed : ClosedConvexFunction f) (hproper : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) f) (hcofinite : CoFiniteERealFunction f) : effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f) = Set.univ := by have hold : CoFiniteConvexFunction f := helperForCoFiniteERealFunction_to_coFiniteConvexFunction f hclosed hproper hcofinite exact (effectiveDomain_fenchelConjugate_eq_univ_iff_coFinite f hclosed).2 hold |>.1

The recession function of a co-finite convex function has the recession-cone, positive-homogeneity, and properness properties needed by Corollary 9.2.2.

lemma helperForCoFiniteConvexFunction_recession_package {n : Nat} (f : (Fin n → ℝ) → EReal) (hf : CoFiniteConvexFunction f) : Set.recessionCone (epigraph (Set.univ : Set (Fin n → ℝ)) f) = epigraph (Set.univ : Set (Fin n → ℝ)) (recessionFunction f) ∧ PositivelyHomogeneous (recessionFunction f) ∧ ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (recessionFunction f) := by have hdomStar := (effectiveDomain_fenchelConjugate_eq_univ_iff_coFinite f hf.1).2 hf have hCne : (effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f)).Nonempty := by rw [hdomStar.1] exact Set.univ_nonempty have hCconv : Convex ℝ (effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f)) := by have hstarConv : ConvexFunction (fenchelConjugate n f) := (fenchelConjugate_closedConvex (n := n) (f := f)).2 exact effectiveDomain_convex (S := (Set.univ : Set (Fin n → ℝ))) (f := fenchelConjugate n f) (by simpa [ConvexFunction] using hstarConv) have hsupp := section13_supportFunctionEReal_closedProperConvex_posHom (effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f)) hCne hCconv have hsuppEq : supportFunctionEReal (effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n f)) = recessionFunction f := by exact supportFunctionEReal_effectiveDomain_fenchelConjugate_eq_recession f hf.2.1 hf.1 (recessionFunction f) (fun _ => rfl) have hrec : Set.recessionCone (epigraph (Set.univ : Set (Fin n → ℝ)) f) = epigraph (Set.univ : Set (Fin n → ℝ)) (recessionFunction f) := by let fam : Fin 1 → (Fin n → ℝ) → EReal := fun _ => f have hconv : ∀ i : Fin 1, Convex ℝ (epigraph (Set.univ : Set (Fin n → ℝ)) (fam i)) := by intro i simpa [fam] using convex_epigraph_of_convexFunctionOn hf.2.1.1 have hproper : ∀ i : Fin 1, ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fam i) := by intro i simpa [fam] using hf.2.1 have hk : ∀ (i : Fin 1) (y : Fin n → ℝ), recessionFunction f y = sSup {r : EReal | ∃ x, r = fam i (x + y) - fam i x} := by intro i y simpa [fam] using section16_recessionFunction_eq_sSup_unrestricted (f := f) y simpa [fam] using (recessionCone_epigraph_eq_epigraph_k (f := fam) (k := recessionFunction f) hconv hproper hk (0 : Fin 1)) refine ⟨hrec, ?_, ?_⟩ · simpa [hsuppEq] using hsupp.2.2 · simpa [hsuppEq] using hsupp.2.1

The infimal convolution of two co-finite convex functions is co-finite.

lemma coFiniteConvexFunction_infimalConvolution {n : Nat} (f g : (Fin n → ℝ) → EReal) (hf : CoFiniteConvexFunction f) (hg : CoFiniteConvexFunction g) : CoFiniteConvexFunction (infimalConvolution f g) := by have hfrec := helperForCoFiniteConvexFunction_recession_package f hf have hgrec := helperForCoFiniteConvexFunction_recession_package g hg have hpos : ∀ z : Fin n → ℝ, z ≠ 0 → recessionFunction f z + recessionFunction g (-z) > (0 : EReal) := by intro z hz have hfTop : recessionFunction f z = (⊤ : EReal) := by simpa [recessionFunction] using hf.2.2 z hz rw [hfTop] rw [EReal.top_add_of_ne_bot] · exact EReal.coe_lt_top 0 · exact hgrec.2.2.2.2 (-z) (by simp) have hmain := infimalConvolution_closed_proper_convex_recession hf.1 hg.1 hf.2.1 hg.2.1 hpos hfrec.1 hgrec.1 hfrec.2.1 hgrec.2.1 hfrec.2.2 hgrec.2.2 have hclosed := hmain.1 have hfstar := (effectiveDomain_fenchelConjugate_eq_univ_iff_coFinite f hf.1).2 hf have hgstar := (effectiveDomain_fenchelConjugate_eq_univ_iff_coFinite g hg.1).2 hg let fam : Fin 2 → (Fin n → ℝ) → EReal := fun i => if i = 0 then f else g have hfam : ∀ i, ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (fam i) := by intro i fin_cases i <;> simp [fam, hf.2.1, hg.2.1] have hInf : infimalConvolution f g = infimalConvolutionFamily fam := by simpa [fam] using infimalConvolution_eq_infimalConvolutionFamily_two f g have hconj : fenchelConjugate n (infimalConvolution f g) = fun xStar => fenchelConjugate n f xStar + fenchelConjugate n g xStar := by rw [hInf, section16_fenchelConjugate_infimalConvolutionFamily fam hfam] funext xStar simp [fam, Fin.sum_univ_two] apply (effectiveDomain_fenchelConjugate_eq_univ_iff_coFinite (infimalConvolution f g) hclosed).1 constructor · ext xStar simp only [effectiveDomain_eq, Set.mem_setOf_eq, Set.mem_univ, true_and, iff_true] rw [hconj] apply EReal.add_lt_top · rw [← lt_top_iff_ne_top] simpa [effectiveDomain_eq] using Set.ext_iff.mp hfstar.1 xStar · rw [← lt_top_iff_ne_top] simpa [effectiveDomain_eq] using Set.ext_iff.mp hgstar.1 xStar · intro xStar rw [hconj] exact (EReal.add_ne_bot_iff).2 ⟨hfstar.2 xStar, hgstar.2 xStar⟩

Definition 38.7.1: A convex (or concave) bifunction is co-finite if, for every failed to synthesize Membership ?m.1 Type Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command.Unknown identifier `u`u ∈ ℝ^Unknown identifier `m`m, the slice is a co-finite convex function (closed, proper, and with no non-vertical half-lines in its epigraph). Convexity and concavity here are joint graph-function conditions, as in the book, rather than merely fiberwise conditions.

def CoFiniteBifunction {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) : Prop := (ConvexBifunction F ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) ∨ (ConcaveBifunction F ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (fun x => -F u x))

Every slice of a jointly convex co-finite bifunction has a Fenchel conjugate with full effective domain.

lemma helperForCoFiniteBifunction_slice_fenchelConjugate_effectiveDomain_eq_univ {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF_cofinite : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) : ∀ u : Fin m → ℝ, effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n (F u)) = Set.univ := by intro u have hGraph : IsERealConvex (bifunctionGraphFunction F) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF_joint have hsliceConvE : IsERealConvex (F u) := by intro p hp q hq a b ha hb hab have hp' : (Fin.append u p.1, p.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hp have hq' : (Fin.append u q.1, q.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hq have h := hGraph hp' hq' ha hb hab have happend : a • Fin.append u p.1 + b • Fin.append u q.1 = Fin.append u (a • p.1 + b • q.1) := by ext i refine Fin.addCases ?_ ?_ i · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_left, smul_eq_mul] rw [← add_mul, hab, one_mul] · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_right, smul_eq_mul] rw [Prod.smul_mk, Prod.smul_mk, Prod.mk_add_mk, happend] at h simpa [ERealEpigraph, bifunctionGraphFunction, Prod.smul_mk, Prod.mk_add_mk, smul_eq_mul] using h have hsliceProper : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (F u) := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (F u) (hF_cofinite u).2.1 hsliceConvE have hsliceConv : ConvexFunction (F u) := by simpa [ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hsliceConvE have hclosedEpi : IsClosed (epigraph (Set.univ : Set (Fin n → ℝ)) (F u)) := by have heq : ERealEpigraph (F u) = epigraph (Set.univ : Set (Fin n → ℝ)) (F u) := by ext p constructor · intro hp exact ⟨Set.mem_univ p.1, hp⟩ · intro hp exact hp.2 rw [← heq] exact (hF_cofinite u).1 have hsliceLsc : LowerSemicontinuous (F u) := by have hclosedSub : ∀ α : ℝ, IsClosed {x | F u x ≤ (α : EReal)} := (lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := F u)).2.mpr hclosedEpi exact (lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := F u)).1.mpr hclosedSub exact helperForCoFiniteERealFunction_fenchelConjugate_effectiveDomain_eq_univ (F u) ⟨hsliceConv, hsliceLsc⟩ hsliceProper (hF_cofinite u)
lemma convexBifunction_convexIndicatorBifunction {m n : Nat} (A : (Fin m → ℝ) →ₗ[ℝ] (Fin n → ℝ)) : ConvexBifunction (convexIndicatorBifunction A) := by have hconv : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => convexIndicatorBifunction A p.1 p.2) := by rw [IsERealConvex, ERealEpigraph] intro p hp q hq a b ha hb hab have hpGraph : p.1.2 = A p.1.1 := by by_contra hne simp [convexIndicatorBifunction, hne] at hp have hqGraph : q.1.2 = A q.1.1 := by by_contra hne simp [convexIndicatorBifunction, hne] at hq have hpHeight : 0 ≤ p.2 := by simpa [convexIndicatorBifunction, hpGraph] using hp have hqHeight : 0 ≤ q.2 := by simpa [convexIndicatorBifunction, hqGraph] using hq have hgraph : a • p.1.2 + b • q.1.2 = A (a • p.1.1 + b • q.1.1) := by rw [hpGraph, hqGraph, map_add, map_smul, map_smul] have hheightR : 0 ≤ a * p.2 + b * q.2 := by nlinarith have hheight : (0 : EReal) ≤ (a : EReal) * (p.2 : EReal) + (b : EReal) * (q.2 : EReal) := by exact_mod_cast hheightR simpa [Prod.smul_mk, Prod.mk_add_mk, convexIndicatorBifunction, hgraph, smul_eq_mul] using hheight let pairMap : (Fin (m + n) → ℝ) →ₗ[ℝ] ((Fin m → ℝ) × (Fin n → ℝ)) := { toFun := fun z => (fun i => z (Fin.castAdd n i), fun j => z (Fin.natAdd m j)) map_add' := by intro z w; ext i <;> simp map_smul' := by intro c z; ext i <;> simp } have hgraph : IsERealConvex (bifunctionGraphFunction (convexIndicatorBifunction A)) := by simpa [pairMap, bifunctionGraphFunction] using (helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := pairMap) (g := fun p : (Fin m → ℝ) × (Fin n → ℝ) => convexIndicatorBifunction A p.1 p.2) hconv) simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hgraphlemma coFiniteERealFunction_convexIndicatorPoint {n : Nat} (a : Fin n → ℝ) : CoFiniteERealFunction (convexIndicatorPoint a) := by have hepi : ERealEpigraph (convexIndicatorPoint a) = ({a} : Set (Fin n → ℝ)) ×ˢ Set.Ici (0 : ℝ) := by ext p by_cases hp : p.1 = a · simp [ERealEpigraph, convexIndicatorPoint, hp] · simp [ERealEpigraph, convexIndicatorPoint, hp] refine ⟨?_, ?_, ?_⟩ · rw [hepi] exact isClosed_singleton.prod isClosed_Ici · constructor · intro x by_cases hx : x = a <;> simp [convexIndicatorPoint, hx] · exact ⟨a, by simp [convexIndicatorPoint]⟩ · rintro ⟨p, d, hp, hd, hall⟩ have hpEq : p.1 = a := by by_contra hne simp [ERealEpigraph, convexIndicatorPoint, hne] at hp have h1 := hall 1 zero_le_one have h1Eq : (p + (1 : ℝ) • d).1 = a := by by_contra hne have hne' : p.1 + d.1 ≠ a := by simpa [Prod.smul_mk] using hne have hbad : convexIndicatorPoint a (p.1 + d.1) ≤ ((p.2 + d.2 : ℝ) : EReal) := by simpa [Prod.smul_mk] using h1 simp [convexIndicatorPoint, hne'] at hbad exact EReal.coe_ne_top _ hbad apply hd have : p.1 + d.1 = a := by simpa [Prod.smul_mk] using h1Eq rw [hpEq] at this have hz := congrArg (fun z => z - a) this simpa using hzlemma coFiniteERealFunction_convexIndicatorBifunction_slice {m n : Nat} (A : (Fin m → ℝ) →ₗ[ℝ] (Fin n → ℝ)) (u : Fin m → ℝ) : CoFiniteERealFunction (convexIndicatorBifunction A u) := by have heq : convexIndicatorBifunction A u = convexIndicatorPoint (A u) := by funext x by_cases hx : x = A u <;> simp [convexIndicatorBifunction, convexIndicatorPoint, hx] rw [heq] exact coFiniteERealFunction_convexIndicatorPoint (A u)lemma coFiniteERealFunction_positive_rescale {n : Nat} (f : (Fin n → ℝ) → EReal) (hf : CoFiniteERealFunction f) (lam : {r : ℝ // 0 < r}) : CoFiniteERealFunction (fun x => ((lam.1 : ℝ) : EReal) * f (lam.1⁻¹ • x)) := by have hlamE : (0 : EReal) < ((lam.1 : ℝ) : EReal) := by exact_mod_cast lam.2 have hlamTop : ((lam.1 : ℝ) : EReal) ≠ ⊤ := EReal.coe_ne_top _ have hlamBot : ((lam.1 : ℝ) : EReal) ≠ ⊥ := EReal.coe_ne_bot _ have hlamNonneg : (0 : EReal) ≤ ((lam.1 : ℝ) : EReal) := le_of_lt hlamE let T : ((Fin n → ℝ) × ℝ) → ((Fin n → ℝ) × ℝ) := fun p => (lam.1⁻¹ • p.1, lam.1⁻¹ * p.2) have hmem (p : (Fin n → ℝ) × ℝ) : p ∈ ERealEpigraph (fun x => ((lam.1 : ℝ) : EReal) * f (lam.1⁻¹ • x)) ↔ T p ∈ ERealEpigraph f := by change ((lam.1 : ℝ) : EReal) * f (lam.1⁻¹ • p.1) ≤ (p.2 : EReal) ↔ f (lam.1⁻¹ • p.1) ≤ ((lam.1⁻¹ * p.2 : ℝ) : EReal) rw [EReal.coe_mul, EReal.coe_inv, ← EReal.div_eq_inv_mul] rw [EReal.le_div_iff_mul_le hlamE hlamTop] simp [mul_comm] refine ⟨?_, ?_, ?_⟩ · have hT : Continuous T := by exact ((continuous_const_smul (lam.1⁻¹ : ℝ)).comp continuous_fst).prodMk (continuous_const.mul continuous_snd) have hepi : ERealEpigraph (fun x => ((lam.1 : ℝ) : EReal) * f (lam.1⁻¹ • x)) = T ⁻¹' ERealEpigraph f := by ext p exact hmem p rw [hepi] exact hf.1.preimage hT · constructor · intro x rw [EReal.mul_ne_bot] exact ⟨Or.inl hlamBot, Or.inr (hf.2.1.1 _), Or.inl hlamTop, Or.inl hlamNonneg⟩ · rcases hf.2.1.2 with ⟨x, hx⟩ refine ⟨lam.1 • x, ?_⟩ have hinv : lam.1⁻¹ • (lam.1 • x) = x := by rw [smul_smul, inv_mul_cancel₀ (ne_of_gt lam.2), one_smul] change ((lam.1 : ℝ) : EReal) * f (lam.1⁻¹ • (lam.1 • x)) ≠ ⊤ rw [hinv, EReal.mul_ne_top] exact ⟨Or.inl hlamBot, Or.inl hlamNonneg, Or.inl hlamTop, Or.inr hx⟩ · intro hline apply hf.2.2 rcases hline with ⟨p, d, hp, hd, hall⟩ refine ⟨T p, T d, (hmem p).mp hp, ?_, ?_⟩ · dsimp [T] intro hz apply hd have hscale := congrArg (fun z => lam.1 • z) hz have hor : lam.1 = 0 ∨ d.1 = 0 := by simpa [smul_smul] using hscale exact hor.resolve_left (ne_of_gt lam.2) · intro t ht have hscaled := (hmem (p + t • d)).mp (hall t ht) have hTadd : T (p + t • d) = T p + t • T d := by apply Prod.ext · dsimp [T] ext i simp [smul_eq_mul] ring · dsimp [T] change lam.1⁻¹ * (p.2 + t * d.2) = lam.1⁻¹ * p.2 + t * (lam.1⁻¹ * d.2) ring rw [← hTadd] exact hscaledlemma convexBifunction_positive_rescale {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF : ConvexBifunction F) (lam : {r : ℝ // 0 < r}) : ConvexBifunction (fun u x => ((lam.1 : ℝ) : EReal) * F u (lam.1⁻¹ • x)) := by let appendMap : ((Fin m → ℝ) × (Fin n → ℝ)) →ₗ[ℝ] (Fin (m + n) → ℝ) := { toFun := fun p => Fin.append p.1 p.2 map_add' := by intro p q ext i refine Fin.addCases ?_ ?_ i <;> intro j <;> simp only [Prod.fst_add, Prod.snd_add, Pi.add_apply, Fin.append_left, Fin.append_right] map_smul' := by intro c p ext i refine Fin.addCases ?_ ?_ i <;> intro j <;> simp only [Prod.smul_fst, Prod.smul_snd, Pi.smul_apply, Fin.append_left, Fin.append_right, RingHom.id_apply] } have hgraph : IsERealConvex (bifunctionGraphFunction F) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF have hpair : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => F p.1 p.2) := by simpa [appendMap, bifunctionGraphFunction] using (helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := appendMap) (g := bifunctionGraphFunction F) hgraph) have hscaled : IsERealConvex (fun p : (Fin m → ℝ) × (Fin n → ℝ) => ((lam.1 : ℝ) : EReal) * F p.1 (lam.1⁻¹ • p.2)) := by rw [IsERealConvex, ERealEpigraph] rw [IsERealConvex] at hpair intro p hp q hq a b ha hb hab have hlamE : (0 : EReal) < ((lam.1 : ℝ) : EReal) := by exact_mod_cast lam.2 have hlamTop : ((lam.1 : ℝ) : EReal) ≠ ⊤ := EReal.coe_ne_top _ have hpPre : ((p.1.1, lam.1⁻¹ • p.1.2), lam.1⁻¹ * p.2) ∈ ERealEpigraph (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F z.1 z.2) := by change F p.1.1 (lam.1⁻¹ • p.1.2) ≤ ((lam.1⁻¹ * p.2 : ℝ) : EReal) rw [EReal.coe_mul, EReal.coe_inv, ← EReal.div_eq_inv_mul] rw [EReal.le_div_iff_mul_le hlamE hlamTop] simpa [mul_comm] using hp have hqPre : ((q.1.1, lam.1⁻¹ • q.1.2), lam.1⁻¹ * q.2) ∈ ERealEpigraph (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F z.1 z.2) := by change F q.1.1 (lam.1⁻¹ • q.1.2) ≤ ((lam.1⁻¹ * q.2 : ℝ) : EReal) rw [EReal.coe_mul, EReal.coe_inv, ← EReal.div_eq_inv_mul] rw [EReal.le_div_iff_mul_le hlamE hlamTop] simpa [mul_comm] using hq have hcombo := hpair hpPre hqPre ha hb hab have hinput : a • (p.1.1, lam.1⁻¹ • p.1.2) + b • (q.1.1, lam.1⁻¹ • q.1.2) = (a • p.1.1 + b • q.1.1, lam.1⁻¹ • (a • p.1.2 + b • q.1.2)) := by apply Prod.ext · rfl · ext i simp [smul_eq_mul] ring have hcombo' : F (a • p.1.1 + b • q.1.1) (lam.1⁻¹ • (a • p.1.2 + b • q.1.2)) ≤ ((a * (lam.1⁻¹ * p.2) + b * (lam.1⁻¹ * q.2) : ℝ) : EReal) := by have hcombo0 : F (a • (p.1.1, lam.1⁻¹ • p.1.2) + b • (q.1.1, lam.1⁻¹ • q.1.2)).1 (a • (p.1.1, lam.1⁻¹ • p.1.2) + b • (q.1.1, lam.1⁻¹ • q.1.2)).2 ≤ ((a * (lam.1⁻¹ * p.2) + b * (lam.1⁻¹ * q.2) : ℝ) : EReal) := by simpa [ERealEpigraph, Prod.smul_mk, Prod.mk_add_mk, smul_eq_mul] using hcombo rw [hinput] at hcombo0 exact hcombo0 have hmul := mul_le_mul_of_nonneg_left hcombo' (le_of_lt hlamE) have hreal : lam.1 * (a * (lam.1⁻¹ * p.2) + b * (lam.1⁻¹ * q.2)) = a * p.2 + b * q.2 := by field_simp [ne_of_gt lam.2] have hereal : ((lam.1 : ℝ) : EReal) * ((a * (lam.1⁻¹ * p.2) + b * (lam.1⁻¹ * q.2) : ℝ) : EReal) = ((a * p.2 + b * q.2 : ℝ) : EReal) := by exact_mod_cast hreal simpa [ERealEpigraph, Prod.smul_mk, Prod.mk_add_mk, smul_eq_mul] using hmul.trans_eq hereal let pairMap : (Fin (m + n) → ℝ) →ₗ[ℝ] ((Fin m → ℝ) × (Fin n → ℝ)) := { toFun := fun z => (fun i => z (Fin.castAdd n i), fun j => z (Fin.natAdd m j)) map_add' := by intro z w; ext i <;> simp map_smul' := by intro c z; ext i <;> simp } have hout : IsERealConvex (bifunctionGraphFunction (fun u x => ((lam.1 : ℝ) : EReal) * F u (lam.1⁻¹ • x))) := by simpa [pairMap, bifunctionGraphFunction] using (helperForTheorem_38_4_isERealConvex_precomp_linearMap (A := pairMap) (g := fun p : (Fin m → ℝ) × (Fin n → ℝ) => ((lam.1 : ℝ) : EReal) * F p.1 (lam.1⁻¹ • p.2)) hscaled) simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hout

The Euclidean-version of the bracket for a bifunction : the convex-conjugate value .

noncomputable def bifunctionLeftPairingInner {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (u : Fin m → ℝ) (xStar : Fin n → ℝ) : EReal := convexConjugateInner (F u) xStar

The adjoint of a convex bifunction is the concave conjugate, in its first variable, of the left-pairing function.

lemma bifunctionAdjointInner_eq_concaveConjugateInner_leftPairing {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF : ConvexBifunction F) (xStar : Fin n → ℝ) (uStar : Fin m → ℝ) : bifunctionAdjointInner F xStar uStar = concaveConjugateInner (fun u => bifunctionLeftPairingInner F u xStar) uStar := by rw [bifunctionAdjointInner_eq_adjointOfConvexBifunction F hF] rw [← textbookBifunctionAdjoint_eq_adjointOfConvexBifunction F hF] rw [textbookBifunctionAdjoint_eq_concaveConjugate_pairing] rw [concaveConjugateInner_eq_concaveFenchelConjugate] simp_rw [bifunctionLeftPairingInner, convexConjugateInner_eq_fenchelConjugate] rw [helperForTheorem_6_30_4_concaveConjugate_eq_iInf] rw [helperForLemma_31_0_11_concaveFenchelConjugate_eq_iInf] simp [sub_eq_add_neg]

For a jointly convex bifunction with co-finite primal slices, every fixed-dual left-pairing section is closed concave and never takes the value .

lemma helperForCoFiniteBifunction_leftPairing_closedConcave_and_ne_top {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF_cofinite : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) (xStar : Fin n → ℝ) : let phi : (Fin m → ℝ) → EReal := fun u => bifunctionLeftPairingInner F u xStar ClosedConcaveFunction phi ∧ ∀ u, phi u ≠ (⊤ : EReal) := by have hGraph : IsERealConvex (bifunctionGraphFunction F) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF_joint have hSliceConv : ∀ u : Fin m → ℝ, IsERealConvex (F u) := by intro u p hp q hq a b ha hb hab have hp' : (Fin.append u p.1, p.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hp have hq' : (Fin.append u q.1, q.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hq have h := hGraph hp' hq' ha hb hab have happend : a • Fin.append u p.1 + b • Fin.append u q.1 = Fin.append u (a • p.1 + b • q.1) := by ext i refine Fin.addCases ?_ ?_ i · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_left, smul_eq_mul] rw [← add_mul, hab, one_mul] · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_right, smul_eq_mul] rw [Prod.smul_mk, Prod.smul_mk, Prod.mk_add_mk, happend] at h simpa [ERealEpigraph, bifunctionGraphFunction, Prod.smul_mk, Prod.mk_add_mk, smul_eq_mul] using h let Fpacked : FiberwiseProperConvexBifunction m n := { toFun := F proper := by constructor · intro u x exact (hF_cofinite u).2.1.1 x · rcases (hF_cofinite (0 : Fin m → ℝ)).2.1.2 with ⟨x, hx⟩ exact ⟨0, x, hx⟩ convex := hSliceConv } have hNegConvex : ConvexFunction (fun u : Fin m → ℝ => -fenchelConjugate n (F u) xStar) := by simpa [Fpacked] using neg_fenchelConjugate_slice_convexFunction Fpacked hF_joint xStar have hPhiNoTop : ∀ u : Fin m → ℝ, fenchelConjugate n (F u) xStar ≠ (⊤ : EReal) := by intro u have hdom := helperForCoFiniteBifunction_slice_fenchelConjugate_effectiveDomain_eq_univ F hF_joint hF_cofinite u exact mem_effectiveDomain_imp_ne_top (by rw [hdom]; simp) have hPhiNoBot : ∀ u : Fin m → ℝ, fenchelConjugate n (F u) xStar ≠ (⊥ : EReal) := by intro u rcases (hF_cofinite u).2.1.2 with ⟨x, hx⟩ exact helperForTheorem33_1_convexConjugate_ne_bot_of_point (f := F u) (x₀ := x) hx xStar have hNegFinite : ∀ u : Fin m → ℝ, (fun v => -fenchelConjugate n (F v) xStar) u ≠ ⊤ ∧ (fun v => -fenchelConjugate n (F v) xStar) u ≠ ⊥ := by intro u constructor · simpa using hPhiNoBot u · simpa using hPhiNoTop u have hNegClosed : ClosedConvexFunction (fun u : Fin m → ℝ => -fenchelConjugate n (F u) xStar) := (section13_closedProper_of_convex_finite hNegConvex hNegFinite).1 have hPairEq : (fun u : Fin m → ℝ => bifunctionLeftPairingInner F u xStar) = fun u : Fin m → ℝ => fenchelConjugate n (F u) xStar := by funext u exact convexConjugateInner_eq_fenchelConjugate (F u) xStar refine ⟨?_, ?_⟩ · rw [hPairEq] exact hNegClosed · intro u rw [hPairEq] exact hPhiNoTop u

Negating a concave conjugate gives the Fenchel conjugate of the negated primal function, precomposed with reflection in the dual variable.

lemma section38_neg_concaveConjugate_eq_fenchelConjugate_neg_precomp_neg {n : Nat} (phi : (Fin n → ℝ) → EReal) : (fun y : Fin n → ℝ => -concaveConjugate phi y) = fun y : Fin n → ℝ => fenchelConjugate n (fun x => -phi x) (-y) := by exact helperForTheorem_6_30_3_neg_concaveConjugate_eq_fenchel_precomp_neg phi

The Fenchel conjugate of the negative concave conjugate is the reflected Fenchel biconjugate of the negative primal function.

lemma section38_fenchelConjugate_neg_concaveConjugate_eq_reflected_biconjugate {n : Nat} (phi : (Fin n → ℝ) → EReal) : fenchelConjugate n (fun y : Fin n → ℝ => -concaveConjugate phi y) = fun x : Fin n → ℝ => fenchelConjugate n (fenchelConjugate n (fun z => -phi z)) (-x) := by rw [section38_neg_concaveConjugate_eq_fenchelConjugate_neg_precomp_neg] exact helperForTheorem_21_4_fenchelConjugate_precomp_neg (n := n) (g := fenchelConjugate n (fun z => -phi z))

For a closed concave function with no values, the reflected biconjugate in the preceding identity reduces to the reflected negative primal function.

lemma section38_fenchelConjugate_neg_concaveConjugate_eq_neg_reflect_of_closed {n : Nat} (phi : (Fin n → ℝ) → EReal) (hphi : ClosedConcaveFunction phi) (hphi_ne_top : ∀ x, phi x ≠ (⊤ : EReal)) : fenchelConjugate n (fun y : Fin n → ℝ => -concaveConjugate phi y) = fun x : Fin n → ℝ => -phi (-x) := by rw [section38_fenchelConjugate_neg_concaveConjugate_eq_reflected_biconjugate] have hbiconj : fenchelConjugate n (fenchelConjugate n (fun z => -phi z)) = fun z => -phi z := by exact fenchelConjugate_biconjugate_eq_of_closedConvex n (fun z => -phi z) hphi.2 hphi.1 (by intro z simpa using hphi_ne_top z) rw [hbiconj]

The domain-restricted conjugate used in §38 agrees with the unrestricted Chapter 6 concave conjugate.

lemma section38_concaveConjugateInner_eq_concaveConjugate {n : Nat} (phi : (Fin n → ℝ) → EReal) : concaveConjugateInner phi = concaveConjugate phi := by funext y calc concaveConjugateInner phi y = concaveFenchelConjugate phi y := by exact concaveConjugateInner_eq_concaveFenchelConjugate phi y _ = concaveConjugate phi y := by simpa [concaveFenchelConjugate] using (helperForTheorem_6_30_3_concaveConjugate_eq_neg_fenchelConjugate_neg_unrestricted (g := phi) (xStar := y)).symm

For a fixed dual point Unknown identifier `xStar`xStar, the Fenchel conjugate of the negative adjoint slice is the reflected biconjugate of the negative left-pairing slice.

lemma section38_fenchelConjugate_neg_adjointSlice_eq_reflected_biconjugate {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF : ConvexBifunction F) (xStar : Fin n → ℝ) : fenchelConjugate m (fun uStar : Fin m → ℝ => -bifunctionAdjointInner F xStar uStar) = fun u : Fin m → ℝ => fenchelConjugate m (fenchelConjugate m (fun z => -bifunctionLeftPairingInner F z xStar)) (-u) := by let phi : (Fin m → ℝ) → EReal := fun u => bifunctionLeftPairingInner F u xStar have hadj : (fun uStar => bifunctionAdjointInner F xStar uStar) = concaveConjugateInner phi := by funext uStar exact bifunctionAdjointInner_eq_concaveConjugateInner_leftPairing F hF xStar uStar have hneg : (fun uStar => -bifunctionAdjointInner F xStar uStar) = (fun uStar => -concaveConjugate phi uStar) := by funext uStar rw [congrFun hadj uStar, section38_concaveConjugateInner_eq_concaveConjugate] rw [hneg] exact section38_fenchelConjugate_neg_concaveConjugate_eq_reflected_biconjugate phi

Every fixed first-variable slice of the negative adjoint of a jointly convex co-finite bifunction is co-finite.

lemma helperForCoFiniteBifunction_neg_adjointSlice_coFiniteERealFunction {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF_cofinite : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) (xStar : Fin n → ℝ) : CoFiniteERealFunction (fun uStar : Fin m → ℝ => -bifunctionAdjointInner F xStar uStar) := by let phi : (Fin m → ℝ) → EReal := fun u => bifunctionLeftPairingInner F u xStar let q : (Fin m → ℝ) → EReal := fun uStar => -bifunctionAdjointInner F xStar uStar have hphi := helperForCoFiniteBifunction_leftPairing_closedConcave_and_ne_top F hF_joint hF_cofinite xStar have hphiClosed : ClosedConcaveFunction phi := hphi.1 have hphiNoTop : ∀ u, phi u ≠ (⊤ : EReal) := hphi.2 have hphiNoBot : ∀ u, phi u ≠ (⊥ : EReal) := by intro u rcases (hF_cofinite u).2.1.2 with ⟨x, hx⟩ have h := helperForTheorem33_1_convexConjugate_ne_bot_of_point (f := F u) (x₀ := x) hx xStar simpa [phi, bifunctionLeftPairingInner, convexConjugateInner_eq_fenchelConjugate] using h have hqEq : q = fun y : Fin m → ℝ => fenchelConjugate m (fun x => -phi x) (-y) := by have hadj : (fun uStar => bifunctionAdjointInner F xStar uStar) = concaveConjugateInner phi := by funext uStar exact bifunctionAdjointInner_eq_concaveConjugateInner_leftPairing F hF_joint xStar uStar calc q = fun y => -concaveConjugate phi y := by funext y simp only [q] rw [congrFun hadj y, section38_concaveConjugateInner_eq_concaveConjugate] _ = fun y => fenchelConjugate m (fun x => -phi x) (-y) := section38_neg_concaveConjugate_eq_fenchelConjugate_neg_precomp_neg phi have hqClosed : ClosedConvexFunction q := by let A : (Fin m → ℝ) →ₗ[ℝ] (Fin m → ℝ) := (ContinuousLinearEquiv.neg ℝ : (Fin m → ℝ) ≃L[ℝ] (Fin m → ℝ)).toLinearMap have hstar : ClosedConvexFunction (fenchelConjugate m (fun x => -phi x)) := by have h := fenchelConjugate_closedConvex (n := m) (f := fun x => -phi x) exact ⟨h.2, h.1⟩ have hpre := closedConvexFunction_precomp_linearMap (A := A) hstar rw [hqEq] simpa [A] using hpre have hqStarEq : fenchelConjugate m q = fun u => -phi (-u) := by rw [hqEq] have hpre := helperForTheorem_21_4_fenchelConjugate_precomp_neg (n := m) (g := fenchelConjugate m (fun z => -phi z)) rw [hpre] have hbiconj : fenchelConjugate m (fenchelConjugate m (fun z => -phi z)) = fun z => -phi z := by exact fenchelConjugate_biconjugate_eq_of_closedConvex m (fun z => -phi z) hphiClosed.2 hphiClosed.1 (by intro z simpa using hphiNoTop z) rw [hbiconj] have hdom : effectiveDomain (Set.univ : Set (Fin m → ℝ)) (fenchelConjugate m q) = Set.univ := by ext u rw [hqStarEq] simp only [effectiveDomain_eq, Set.mem_setOf_eq, Set.mem_univ, true_and, iff_true] refine (lt_top_iff_ne_top).2 ?_ intro htop exact hphiNoBot (-u) (EReal.neg_eq_top_iff.mp htop) have hstarNoBot : ∀ u, fenchelConjugate m q u ≠ (⊥ : EReal) := by intro u rw [hqStarEq] simpa using hphiNoTop (-u) have hold : CoFiniteConvexFunction q := (effectiveDomain_fenchelConjugate_eq_univ_iff_coFinite q hqClosed).1 ⟨hdom, hstarNoBot⟩ simpa [q] using helperForCoFiniteConvexFunction_to_coFiniteERealFunction q hold

The Euclidean-version of the bracket for a bifunction on dual variables : the concave-conjugate value .

noncomputable def bifunctionRightPairingInner {m n : Nat} (G : (Fin n → ℝ) → (Fin m → ℝ) → EReal) (xStar : Fin n → ℝ) (u : Fin m → ℝ) : EReal := concaveConjugateInner (G xStar) u

A jointly convex bifunction with co-finite primal slices has a co-finite adjoint, and the left and right Fenchel pairings agree.

lemma helperForCoFiniteBifunction_adjoint_and_pairing {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF_cofinite : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) : CoFiniteBifunction (bifunctionAdjointInner F) ∧ ∀ (u : Fin m → ℝ) (xStar : Fin n → ℝ), bifunctionLeftPairingInner F u xStar = bifunctionRightPairingInner (bifunctionAdjointInner F) xStar u := by constructor · right constructor · rw [bifunctionAdjointInner_eq_adjointOfConvexBifunction F hF_joint] exact (adjointOfConvexBifunction_closedConcave ⟨F, hF_joint⟩).1 · intro xStar exact helperForCoFiniteBifunction_neg_adjointSlice_coFiniteERealFunction F hF_joint hF_cofinite xStar · intro u xStar let phi : (Fin m → ℝ) → EReal := fun v => bifunctionLeftPairingInner F v xStar have hphi := helperForCoFiniteBifunction_leftPairing_closedConcave_and_ne_top F hF_joint hF_cofinite xStar have hphiNoBot : ∀ v, phi v ≠ (⊥ : EReal) := by intro v rcases (hF_cofinite v).2.1.2 with ⟨x, hx⟩ have h := helperForTheorem33_1_convexConjugate_ne_bot_of_point (f := F v) (x₀ := x) hx xStar simpa [phi, bifunctionLeftPairingInner, convexConjugateInner_eq_fenchelConjugate] using h have hnegFinite : ∀ v, (fun z => -phi z) v ≠ (⊤ : EReal) ∧ (fun z => -phi z) v ≠ (⊥ : EReal) := by intro v constructor · intro hv exact hphiNoBot v (EReal.neg_eq_top_iff.mp hv) · intro hv exact hphi.2 v (EReal.neg_eq_bot_iff.mp hv) have hphiProper : ProperConcaveFunctionOn (Set.univ : Set (Fin m → ℝ)) phi := by exact (section13_closedProper_of_convex_finite hphi.1.1 hnegFinite).2 have hdouble : concaveConjugateInner (concaveConjugateInner phi) = phi := by have hbiconj := helperForTheorem_31_1_concave_biconjugate_eq_of_closedConcave phi hphi.1 hphiProper funext v calc concaveConjugateInner (concaveConjugateInner phi) v = concaveFenchelConjugate (concaveConjugateInner phi) v := concaveConjugateInner_eq_concaveFenchelConjugate _ _ _ = concaveFenchelConjugate (concaveFenchelConjugate phi) v := by congr 1 funext w exact concaveConjugateInner_eq_concaveFenchelConjugate phi w _ = phi v := hbiconj v have hadj : (fun uStar => bifunctionAdjointInner F xStar uStar) = concaveConjugateInner phi := by funext uStar exact bifunctionAdjointInner_eq_concaveConjugateInner_leftPairing F hF_joint xStar uStar change phi u = concaveConjugateInner (fun uStar => bifunctionAdjointInner F xStar uStar) u rw [hadj, hdouble]

A jointly convex bifunction with co-finite primal slices has full primal domain, and its adjoint has full -effective domain.

lemma helperForCoFiniteBifunction_dom_and_adjointDom_eq_univ {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF_cofinite : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) : bifunctionDom F = (Set.univ : Set (Fin m → ℝ)) ∧ bifunctionDomBot (bifunctionAdjointInner F) = (Set.univ : Set (Fin n → ℝ)) := by constructor · ext u simp only [bifunctionDom, Set.mem_setOf_eq, Set.mem_univ, iff_true] exact (hF_cofinite u).2.1.2 · ext xStar simp only [bifunctionDomBot, Set.mem_setOf_eq, Set.mem_univ, iff_true] have hq := helperForCoFiniteBifunction_neg_adjointSlice_coFiniteERealFunction F hF_joint hF_cofinite xStar rcases hq.2.1.2 with ⟨uStar, huStar⟩ refine ⟨uStar, ?_⟩ intro hbot apply huStar simp [hbot]

For a closed convex bifunction, full primal domain together with full -effective domain of the adjoint implies co-finiteness in the convex orientation.

lemma helperForClosedConvexBifunction_coFinite_of_dom_and_adjointDom_eq_univ {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_closed : ClosedConvexBifunction F) (hdomF : bifunctionDom F = (Set.univ : Set (Fin m → ℝ))) (hdomAdj : bifunctionDomBot (bifunctionAdjointInner F) = (Set.univ : Set (Fin n → ℝ))) : ConvexBifunction F ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u) := by refine ⟨hF_closed.1, ?_⟩ intro u have hGraph : IsERealConvex (bifunctionGraphFunction F) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF_closed.1 have hsliceConvE : IsERealConvex (F u) := by intro p hp q hq a b ha hb hab have hp' : (Fin.append u p.1, p.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hp have hq' : (Fin.append u q.1, q.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hq have h := hGraph hp' hq' ha hb hab have happend : a • Fin.append u p.1 + b • Fin.append u q.1 = Fin.append u (a • p.1 + b • q.1) := by ext i refine Fin.addCases ?_ ?_ i · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_left, smul_eq_mul] rw [← add_mul, hab, one_mul] · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_right, smul_eq_mul] rw [Prod.smul_mk, Prod.smul_mk, Prod.mk_add_mk, happend] at h simpa [ERealEpigraph, bifunctionGraphFunction, Prod.smul_mk, Prod.mk_add_mk, smul_eq_mul] using h have hsliceConv : ConvexFunction (F u) := by simpa [ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hsliceConvE have hsliceLsc : LowerSemicontinuous (F u) := by let freeze : (Fin n → ℝ) → (Fin (m + n) → ℝ) := fun x => Fin.append u x have hfreeze : Continuous freeze := by simpa [freeze] using (Fin.continuous_append m n).comp (continuous_const.prodMk continuous_id) have hcomp := hF_closed.2.2.comp_continuous hfreeze simpa [freeze, bifunctionGraphFunction] using hcomp have hsliceClosed : ClosedConvexFunction (F u) := ⟨hsliceConv, hsliceLsc⟩ have huDom : u ∈ bifunctionDom F := by rw [hdomF] trivial rcases huDom with ⟨x₀, hx₀⟩ have hstarNoBot : ∀ xStar : Fin n → ℝ, fenchelConjugate n (F u) xStar ≠ (⊥ : EReal) := by intro xStar exact helperForTheorem33_1_convexConjugate_ne_bot_of_point (f := F u) (x₀ := x₀) hx₀ xStar have hstarNoTop : ∀ xStar : Fin n → ℝ, fenchelConjugate n (F u) xStar ≠ (⊤ : EReal) := by intro xStar have hxAdjDom : xStar ∈ bifunctionDomBot (bifunctionAdjointInner F) := by rw [hdomAdj] trivial rcases hxAdjDom with ⟨uStar, huStar⟩ let phi : (Fin m → ℝ) → EReal := fun v => bifunctionLeftPairingInner F v xStar have hadj : bifunctionAdjointInner F xStar uStar = concaveConjugateInner phi uStar := bifunctionAdjointInner_eq_concaveConjugateInner_leftPairing F hF_closed.1 xStar uStar have hconjNeBot : concaveConjugate phi uStar ≠ (⊥ : EReal) := by rw [← section38_concaveConjugateInner_eq_concaveConjugate phi] rw [← hadj] exact huStar have hphiNoTop : ∀ v, phi v ≠ (⊤ : EReal) := by intro v hv apply hconjNeBot apply bot_unique rw [helperForTheorem_6_30_4_concaveConjugate_eq_iInf] refine le_trans (iInf_le _ v) ?_ simp [hv] have huPhi := hphiNoTop u simpa [phi, bifunctionLeftPairingInner, convexConjugateInner_eq_fenchelConjugate] using huPhi have hdomStar : effectiveDomain (Set.univ : Set (Fin n → ℝ)) (fenchelConjugate n (F u)) = Set.univ := by ext xStar simp only [effectiveDomain_eq, Set.mem_setOf_eq, Set.mem_univ, true_and, iff_true] exact (lt_top_iff_ne_top).2 (hstarNoTop xStar) have hold : CoFiniteConvexFunction (F u) := (effectiveDomain_fenchelConjugate_eq_univ_iff_coFinite (F u) hsliceClosed).1 ⟨hdomStar, hstarNoBot⟩ exact helperForCoFiniteConvexFunction_to_coFiniteERealFunction (F u) hold
-- Proof sketch: For the first part, apply the one-variable co-finiteness duality theorem for each -- slice `x ↦ F u x` to show the corresponding adjoint slice is co-finite and to identify the -- left/right pairings via the definition of `bifunctionAdjointInner`. For the "Moreover" part, -- use the domain characterization of co-finite closed convex functions and apply it to slices of -- `F` and to slices of the adjoint.

Proposition 38.7.2: If Unknown identifier `F`F is a co-finite convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, then its adjoint (modeled by bifunctionAdjointInner sorry : (Fin ?m.2 → ℝ) → (Fin ?m.1 → ℝ) → ERealbifunctionAdjointInner Unknown identifier `F`F) is co-finite and satisfies

for all and ,

where the brackets are modeled by bifunctionLeftPairingInner {m n : ℕ} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (u : Fin m → ℝ) (xStar : Fin n → ℝ) : ERealbifunctionLeftPairingInner and bifunctionRightPairingInner {m n : ℕ} (G : (Fin n → ℝ) → (Fin m → ℝ) → EReal) (xStar : Fin n → ℝ) (u : Fin m → ℝ) : ERealbifunctionRightPairingInner. Moreover, a closed convex bifunction Unknown identifier `F`F is co-finite if and only if Unknown identifier `dom`sorry = ℝ ^ sorry : Propdom F = ℝ^Unknown identifier `m`m and , where Unknown identifier `dom`dom F is bifunctionDom sorry : Set ?m.1bifunctionDom Unknown identifier `F`F and is modeled by bifunctionDomBot (bifunctionAdjointInner sorry) : Set (Fin ?m.4 → ℝ)bifunctionDomBot (bifunctionAdjointInner Unknown identifier `F`F).

theorem coFiniteBifunction_adjointInner_and_dom_iff {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) : ((ConvexBifunction F ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) → CoFiniteBifunction (bifunctionAdjointInner F) ∧ (∀ (u : Fin m → ℝ) (xStar : Fin n → ℝ), bifunctionLeftPairingInner F u xStar = bifunctionRightPairingInner (bifunctionAdjointInner F) xStar u)) ∧ (ClosedConvexBifunction F → ((ConvexBifunction F ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) ↔ bifunctionDom F = (Set.univ : Set (Fin m → ℝ)) ∧ bifunctionDomBot (bifunctionAdjointInner F) = (Set.univ : Set (Fin n → ℝ)))) := by constructor · rintro ⟨hF_joint, hF_cofinite⟩ exact helperForCoFiniteBifunction_adjoint_and_pairing F hF_joint hF_cofinite · intro hF_closed constructor · rintro ⟨hF_joint, hF_cofinite⟩ exact helperForCoFiniteBifunction_dom_and_adjointDom_eq_univ F hF_joint hF_cofinite · rintro ⟨hdomF, hdomAdj⟩ exact helperForClosedConvexBifunction_coFinite_of_dom_and_adjointDom_eq_univ F hF_closed hdomF hdomAdj

The adjoint identity for the infimal convolution of two jointly convex bifunctions with co-finite slices. Co-finiteness makes both parameter domains equal to the whole space, which supplies the relative-interior qualification in Theorem 38.2.

lemma helperForCoFiniteBifunction_infimalConvolution_adjointInner_eq {m n : Nat} (F₁ F₂ : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF₁ : ConvexBifunction F₁ ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F₁ u)) (hF₂ : ConvexBifunction F₂ ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F₂ u)) : bifunctionAdjointInner (bifunctionInfimalConvolutionInSecond F₁ F₂) = concaveBifunctionInfimalConvolutionInSecond (bifunctionAdjointInner F₁) (bifunctionAdjointInner F₂) := by have hproper₁ : ProperConvexBifunction F₁ := by refine ⟨hF₁.1, ?_⟩ have hp : IsProperEReal (bifunctionGraphFunction F₁) := by constructor · intro z exact (hF₁.2 (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd m j)) · rcases (hF₁.2 (0 : Fin m → ℝ)).2.1.2 with ⟨x, hx⟩ exact ⟨Fin.append 0 x, by simpa [bifunctionGraphFunction] using hx⟩ have hc : IsERealConvex (bifunctionGraphFunction F₁) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF₁.1 exact helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ _ (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex _ hp hc) have hproper₂ : ProperConvexBifunction F₂ := by refine ⟨hF₂.1, ?_⟩ have hp : IsProperEReal (bifunctionGraphFunction F₂) := by constructor · intro z exact (hF₂.2 (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd m j)) · rcases (hF₂.2 (0 : Fin m → ℝ)).2.1.2 with ⟨x, hx⟩ exact ⟨Fin.append 0 x, by simpa [bifunctionGraphFunction] using hx⟩ have hc : IsERealConvex (bifunctionGraphFunction F₂) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF₂.1 exact helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ _ (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex _ hp hc) obtain ⟨P₁, hP₁⟩ := fiberwiseProperConvex_of_properConvexBifunction F₁ hproper₁ obtain ⟨P₂, hP₂⟩ := fiberwiseProperConvex_of_properConvexBifunction F₂ hproper₂ have hdom₁ : bifunctionDom P₁.toFun = (Set.univ : Set (Fin m → ℝ)) := by simpa [hP₁] using (helperForCoFiniteBifunction_dom_and_adjointDom_eq_univ F₁ hF₁.1 hF₁.2).1 have hdom₂ : bifunctionDom P₂.toFun = (Set.univ : Set (Fin m → ℝ)) := by simpa [hP₂] using (helperForCoFiniteBifunction_dom_and_adjointDom_eq_univ F₂ hF₂.1 hF₂.2).1 have hri : (intrinsicInterior ℝ (bifunctionDom P₁.toFun) ∩ intrinsicInterior ℝ (bifunctionDom P₂.toFun)).Nonempty := by rw [hdom₁, hdom₂] have hzeroInterior : (0 : Fin m → ℝ) ∈ interior (Set.univ : Set (Fin m → ℝ)) := by try 'simp' instead of 'simpa' Note: This linter can be disabled with `set_option linter.unnecessarySimpa false`simpa [interior_univ] have hzeroII : (0 : Fin m → ℝ) ∈ intrinsicInterior ℝ (Set.univ : Set (Fin m → ℝ)) := (interior_subset_intrinsicInterior (s := (Set.univ : Set (Fin m → ℝ)))) hzeroInterior exact ⟨0, hzeroII, hzeroII⟩ have h38 := bifunctionAdjoint_infimalConvolution_eq_infimalConvolution_adjoint P₁ P₂ (by simpa [hP₁] using hF₁.1) (by simpa [hP₂] using hF₂.1) hri have hInf : bifunctionInfimalConvolution P₁ P₂ = bifunctionInfimalConvolutionInSecond F₁ F₂ := by funext u x simp [bifunctionInfimalConvolution, bifunctionInfimalConvolutionInSecond, hP₁, hP₂] rw [hInf] at h38 have hconvInf : ConvexBifunction (bifunctionInfimalConvolutionInSecond F₁ F₂) := bifunctionInfimalConvolutionInSecond_convexBifunction F₁ F₂ hproper₁ hproper₂ rw [textbookBifunctionAdjoint_eq_adjointOfConvexBifunction (bifunctionInfimalConvolutionInSecond F₁ F₂) hconvInf] at h38 rw [← bifunctionAdjointInner_eq_adjointOfConvexBifunction (bifunctionInfimalConvolutionInSecond F₁ F₂) hconvInf] at h38 rw [hP₁, hP₂] at h38 rw [textbookBifunctionAdjoint_eq_adjointOfConvexBifunction F₁ hF₁.1, textbookBifunctionAdjoint_eq_adjointOfConvexBifunction F₂ hF₂.1] at h38 rw [← bifunctionAdjointInner_eq_adjointOfConvexBifunction F₁ hF₁.1, ← bifunctionAdjointInner_eq_adjointOfConvexBifunction F₂ hF₂.1] at h38 exact h38
-- Proof sketch: Co-finiteness of each slice is preserved under one-variable infimal convolution, -- so `u ↦ (F₁ □ F₂) u` is co-finite slice-wise. For the conjugacy identity, apply the bifunction -- infimal-convolution duality (Theorem 38.2) under the qualification implied by co-finiteness, -- and rewrite conjugates using `bifunctionAdjointInner` and infimal convolution using -- `bifunctionInfimalConvolutionInSecond`.

Result after Corollary 38.7.2: If Unknown identifier `F₁`F₁ and Unknown identifier `F₂`F₂ are co-finite convex bifunctions from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, then their infimal convolution Unknown identifier `F₁`sorry □ sorry : SimpleGraph (?m.1 × ?m.2)F₁ □ Unknown identifier `F₂`F₂ is also co-finite and satisfies

.

In Lean, we model Unknown identifier `F₁`sorry □ sorry : SimpleGraph (?m.1 × ?m.2)F₁ □ Unknown identifier `F₂`F₂ as bifunctionInfimalConvolutionInSecond sorry sorry : ?m.1 → ?m.2 → ERealbifunctionInfimalConvolutionInSecond Unknown identifier `F₁`F₁ Unknown identifier `F₂`F₂ and the conjugate as bifunctionAdjointInner sorry : (Fin ?m.2 → ℝ) → (Fin ?m.1 → ℝ) → ERealbifunctionAdjointInner Unknown identifier `F`F. The convolution on the adjoint side uses the book's concave extended-real convention.

theorem coFiniteBifunction_infimalConvolutionInSecond_adjointInner {m n : Nat} (F₁ F₂ : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF₁ : ConvexBifunction F₁ ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F₁ u)) (hF₂ : ConvexBifunction F₂ ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F₂ u)) : CoFiniteBifunction (bifunctionInfimalConvolutionInSecond F₁ F₂) ∧ bifunctionAdjointInner (bifunctionInfimalConvolutionInSecond F₁ F₂) = concaveBifunctionInfimalConvolutionInSecond (bifunctionAdjointInner F₁) (bifunctionAdjointInner F₂) := by have hproper₁ : ProperConvexBifunction F₁ := by refine ⟨hF₁.1, ?_⟩ have hp : IsProperEReal (bifunctionGraphFunction F₁) := by constructor · intro z exact (hF₁.2 (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd m j)) · rcases (hF₁.2 (0 : Fin m → ℝ)).2.1.2 with ⟨x, hx⟩ exact ⟨Fin.append 0 x, by simpa [bifunctionGraphFunction] using hx⟩ have hc : IsERealConvex (bifunctionGraphFunction F₁) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF₁.1 exact helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ _ (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex _ hp hc) have hproper₂ : ProperConvexBifunction F₂ := by refine ⟨hF₂.1, ?_⟩ have hp : IsProperEReal (bifunctionGraphFunction F₂) := by constructor · intro z exact (hF₂.2 (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd m j)) · rcases (hF₂.2 (0 : Fin m → ℝ)).2.1.2 with ⟨x, hx⟩ exact ⟨Fin.append 0 x, by simpa [bifunctionGraphFunction] using hx⟩ have hc : IsERealConvex (bifunctionGraphFunction F₂) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF₂.1 exact helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ _ (helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex _ hp hc) obtain ⟨P₁, hP₁⟩ := fiberwiseProperConvex_of_properConvexBifunction F₁ hproper₁ obtain ⟨P₂, hP₂⟩ := fiberwiseProperConvex_of_properConvexBifunction F₂ hproper₂ have hsliceOld₁ : ∀ u : Fin m → ℝ, CoFiniteConvexFunction (F₁ u) := by intro u have hcE : IsERealConvex (F₁ u) := by simpa [hP₁] using P₁.convex u have hc : ConvexFunction (F₁ u) := by simpa [ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hcE have hp : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (F₁ u) := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (F₁ u) (hF₁.2 u).2.1 hcE have hepi : IsClosed (epigraph (Set.univ : Set (Fin n → ℝ)) (F₁ u)) := by have heq : ERealEpigraph (F₁ u) = epigraph (Set.univ : Set (Fin n → ℝ)) (F₁ u) := by ext p constructor · intro hp' exact ⟨Set.mem_univ p.1, hp'⟩ · intro hp' exact hp'.2 rw [← heq] exact (hF₁.2 u).1 have hlsc : LowerSemicontinuous (F₁ u) := (lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := F₁ u)).1.mpr ((lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := F₁ u)).2.mpr hepi) exact helperForCoFiniteERealFunction_to_coFiniteConvexFunction (F₁ u) ⟨hc, hlsc⟩ hp (hF₁.2 u) have hsliceOld₂ : ∀ u : Fin m → ℝ, CoFiniteConvexFunction (F₂ u) := by intro u have hcE : IsERealConvex (F₂ u) := by simpa [hP₂] using P₂.convex u have hc : ConvexFunction (F₂ u) := by simpa [ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hcE have hp : ProperConvexFunctionOn (Set.univ : Set (Fin n → ℝ)) (F₂ u) := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (F₂ u) (hF₂.2 u).2.1 hcE have hepi : IsClosed (epigraph (Set.univ : Set (Fin n → ℝ)) (F₂ u)) := by have heq : ERealEpigraph (F₂ u) = epigraph (Set.univ : Set (Fin n → ℝ)) (F₂ u) := by ext p constructor · intro hp' exact ⟨Set.mem_univ p.1, hp'⟩ · intro hp' exact hp'.2 rw [← heq] exact (hF₂.2 u).1 have hlsc : LowerSemicontinuous (F₂ u) := (lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := F₂ u)).1.mpr ((lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := F₂ u)).2.mpr hepi) exact helperForCoFiniteERealFunction_to_coFiniteConvexFunction (F₂ u) ⟨hc, hlsc⟩ hp (hF₂.2 u) refine ⟨?_, helperForCoFiniteBifunction_infimalConvolution_adjointInner_eq F₁ F₂ hF₁ hF₂⟩ left refine ⟨bifunctionInfimalConvolutionInSecond_convexBifunction F₁ F₂ hproper₁ hproper₂, ?_⟩ intro u have hold := coFiniteConvexFunction_infimalConvolution (F₁ u) (F₂ u) (hsliceOld₁ u) (hsliceOld₂ u) have hnew := helperForCoFiniteConvexFunction_to_coFiniteERealFunction (infimalConvolution (F₁ u) (F₂ u)) hold have heq : bifunctionInfimalConvolutionInSecond F₁ F₂ u = infimalConvolution (F₁ u) (F₂ u) := by funext x rw [infimalConvolution_eq_iInf_second] simp only [bifunctionInfimalConvolutionInSecond] apply iInf_congr intro z rw [add_comm] simpa [heq] using hnew
-- Proof sketch: For each `u`, apply stability of co-finiteness of one-variable convex functions -- under the positive rescaling `x ↦ lam.1⁻¹ • x` and scaling of function values by `lam.1`, which -- is exactly the slice operation defining `bifunctionScalarMultiple`. Combine this slice-wise -- result with `isFiberwiseConvexBifunction_scalarMultiple` to produce the convex branch of -- `CoFiniteBifunction`.

Proposition 38.7.4: If Unknown identifier `F`F is a co-finite convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, then for any , the scalar multiple (Definition 38.2.2, modeled by bifunctionScalarMultiple {m n : ℕ} (F : FiberwiseConvexBifunction m n) (lam : { r // 0 < r }) : FiberwiseConvexBifunction m nbifunctionScalarMultiple) is also co-finite (Definition 38.7.1, modeled by CoFiniteBifunction {m n : ℕ} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) : PropCoFiniteBifunction).

theorem coFiniteBifunction_scalarMultiple {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) (lam : {r : ℝ // 0 < r}) : CoFiniteBifunction (fun u x => ((lam.1 : ℝ) : EReal) * F u (lam.1⁻¹ • x)) := by left exact ⟨convexBifunction_positive_rescale F hF_joint lam, fun u => coFiniteERealFunction_positive_rescale (F u) (hF u) lam⟩
lemma bifunctionEffectiveDomain_eq_bifunctionDom {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) : bifunctionEffectiveDomain F = bifunctionDom F := by ext u constructor · rintro ⟨x, hx⟩ exact ⟨x, lt_top_iff_ne_top.mp hx⟩ · rintro ⟨x, hx⟩ exact ⟨x, lt_top_iff_ne_top.mpr hx⟩

Joint convexity and co-finite slices imply properness of the graph function.

lemma properConvexBifunction_of_joint_cofinite {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) : ProperConvexBifunction F := by have hp : IsProperEReal (bifunctionGraphFunction F) := by constructor · intro z exact (hF (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd m j)) · rcases (hF 0).2.1.2 with ⟨x, hx⟩ refine ⟨Fin.append 0 x, ?_⟩ change F (fun i => Fin.append 0 x (Fin.castAdd n i)) (fun j => Fin.append 0 x (Fin.natAdd m j)) ≠ ⊤ have hleft : (fun i => Fin.append (0 : Fin m → ℝ) x (Fin.castAdd n i)) = 0 := by ext i simp have hright : (fun j => Fin.append (0 : Fin m → ℝ) x (Fin.natAdd m j)) = x := by ext j simp rw [hleft, hright] exact hx have hc : IsERealConvex (bifunctionGraphFunction F) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF_joint have hpc := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (bifunctionGraphFunction F) hp hc exact ⟨hF_joint, helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ (f := bifunctionGraphFunction F) hpc⟩
private lemma coFiniteSlice_closed {n : Nat} (f : (Fin n → ℝ) → EReal) (hconv : IsERealConvex f) (hcof : CoFiniteERealFunction f) : ClosedConvexFunction f := by have hc : ConvexFunction f := by simpa [ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hconv have hepi : IsClosed (epigraph (Set.univ : Set (Fin n → ℝ)) f) := by have heq : ERealEpigraph f = epigraph (Set.univ : Set (Fin n → ℝ)) f := by ext p constructor · intro hp exact ⟨Set.mem_univ p.1, hp⟩ · intro hp exact hp.2 rw [← heq] exact hcof.1 have hlsc : LowerSemicontinuous f := (lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := f)).1.mpr ((lowerSemicontinuous_iff_closed_sublevel_iff_closed_epigraph (f := f)).2.mpr hepi) exact ⟨hc, hlsc⟩

A jointly convex bifunction with co-finite slices is globally closed convex.

lemma closedConvexBifunction_of_coFiniteSlices {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF : ConvexBifunction F) (hcof : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) : ClosedConvexBifunction F := by have hgraphProper : IsProperEReal (bifunctionGraphFunction F) := by constructor · intro z exact (hcof (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd m j)) · rcases (hcof (0 : Fin m → ℝ)).2.1.2 with ⟨x, hx⟩ exact ⟨Fin.append 0 x, by simpa [bifunctionGraphFunction] using hx⟩ have hgraphConvE : IsERealConvex (bifunctionGraphFunction F) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF have hgraphPC : ProperConvexFunctionOn (Set.univ : Set (Fin (m + n) → ℝ)) (bifunctionGraphFunction F) := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (bifunctionGraphFunction F) hgraphProper hgraphConvE have hproper : ProperConvexBifunction F := ⟨hF, helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ (bifunctionGraphFunction F) hgraphPC⟩ obtain ⟨P, hP⟩ := fiberwiseProperConvex_of_properConvexBifunction F hproper have hgraphJensen : ConvexERealFunction (bifunctionGraphFunction F) := (helperForText_26_4_0_2_properConvexERealFunction_of_properConvexFunctionOn hgraphPC).2 have hFold : IsConvexBifunction F := by intro p q a b ha hb hab have happ : a • Fin.append p.1 p.2 + b • Fin.append q.1 q.2 = Fin.append (a • p.1 + b • q.1) (a • p.2 + b • q.2) := by ext i cases i using Fin.addCases with | left i => simp | right i => simp have hj := hgraphJensen (x := Fin.append p.1 p.2) (y := Fin.append q.1 q.2) ha hb hab simpa [graphFunction, bifunctionGraphFunction, Prod.smul_mk, Prod.mk_add_mk, happ] using hj let BF : BundledConvexBifunction m n := ⟨F, hFold⟩ have hdom : bifunctionEffectiveDomain BF.1 = (Set.univ : Set (Fin m → ℝ)) := by rw [bifunctionEffectiveDomain_eq_bifunctionDom] simpa [BF] using (helperForCoFiniteBifunction_dom_and_adjointDom_eq_univ F hF hcof).1 have hsliceClosed : ∀ u : Fin m → ℝ, ClosedConvexFunction (F u) := by intro u exact coFiniteSlice_closed (F u) (by simpa [hP] using P.convex u) (hcof u) let coord : (Fin (m + n) → ℝ) → EReal := helperForTheorem_6_29_4_coordinateGraphFunction BF have hcoordProper : IsProperEReal coord := by simpa [coord, BF, helperForTheorem_6_29_4_coordinateGraphFunction, bifunctionGraphFunction] using hgraphProper have hcoordConv : ConvexFunction coord := by simpa [coord] using helperForTheorem_6_29_4_coordinateGraphFunction_convex BF have hcoordConvE : IsERealConvex coord := by simpa [ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hcoordConv have hcoordPC : ProperConvexFunctionOn (Set.univ : Set (Fin (m + n) → ℝ)) coord := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex coord hcoordProper hcoordConvE have hclosureEq : convexFunctionClosure coord = coord := by funext z let u : Fin m → ℝ := fun i => z (Fin.castAdd n i) let x : Fin n → ℝ := fun j => z (Fin.natAdd m j) have hz : Fin.append u x = z := by ext i cases i using Fin.addCases with | left i => simp [u] | right i => simp [x] have hu : u ∈ euclideanRelativeInterior_fin m (bifunctionEffectiveDomain BF.1) := by rw [hdom, helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] exact (interior_subset_intrinsicInterior (s := (Set.univ : Set (Fin m → ℝ)))) (by simp) have hsection := (theorem_29_4_convex_bifunction_closure_section_and_domain BF).1 u hu have hsliceEq : convexFunctionClosure (F u) = F u := convexFunctionClosure_eq_of_closedConvexFunction (hsliceClosed u) (hcof u).2.1.1 rw [hsliceEq] at hsection have hxEq := congrFun hsection x simpa [coord, helperForTheorem_6_29_4_define_section29_bifunctionClosure, helperForTheorem_6_29_4_coordinateGraphFunction, BF, hz] using hxEq have hclosureClosed : ClosedConvexFunction (convexFunctionClosure coord) := (convexFunctionClosure_closed_properConvexFunctionOn_and_agrees_on_ri (f := coord) hcoordPC).1.1 have hcoordClosed : ClosedConvexFunction coord := by simpa [hclosureEq] using hclosureClosed refine ⟨hF, ?_⟩ simpa [coord, BF, helperForTheorem_6_29_4_coordinateGraphFunction, bifunctionGraphFunction] using hcoordClosed
set_option synthInstance.maxHeartbeats 80000 in /-- The adjoint of a co-finite composition is the supremal composition of the adjoints. -/ lemma cofinite_compose_adjoint_identity {m n p : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) (G : (Fin n → ℝ) → (Fin p → ℝ) → EReal) (hG_joint : ConvexBifunction G) (hG : ∀ x : Fin n → ℝ, CoFiniteERealFunction (G x)) : bifunctionAdjointInner (bifunctionComposeInfGeneric G F) = bifunctionComposeSup (bifunctionAdjointInner F) (bifunctionAdjointInner G) := by have hFproper0 := properConvexBifunction_of_joint_cofinite F hF_joint hF have hGproper0 := properConvexBifunction_of_joint_cofinite G hG_joint hG obtain ⟨Fpkg, hFpkg⟩ := fiberwiseProperConvex_of_properConvexBifunction F hFproper0 obtain ⟨Gpkg, hGpkg⟩ := fiberwiseProperConvex_of_properConvexBifunction G hGproper0 subst F subst G have hFproper : ProperConvexBifunction Fpkg.toFun := hFproper0 have hGproper : ProperConvexBifunction Gpkg.toFun := hGproper0 have hdomG : bifunctionDom Gpkg.toFun = (Set.univ : Set (Fin n → ℝ)) := (helperForCoFiniteBifunction_dom_and_adjointDom_eq_univ Gpkg.toFun hG_joint hG).1 let D := effectiveDomain (Set.univ : Set (Fin (m + n) → ℝ)) (bifunctionGraphFunction Fpkg.toFun) have hDconv : Convex ℝ D := by exact effectiveDomain_convex (S := (Set.univ : Set (Fin (m + n) → ℝ))) (f := bifunctionGraphFunction Fpkg.toFun) (by simpa [ConvexFunction] using hF_joint) have hsetEq : bifunctionDomBot (bifunctionInverse Fpkg.toFun) = projLamLinearMap '' D := by rw [helperForCorollary_38_5_1_bifunctionDomBot_inverse_eq_exists_ne_top] ext x constructor · rintro ⟨u, hu⟩ refine ⟨Fin.append u x, ?_, ?_⟩ · simpa [D, effectiveDomain_eq, bifunctionGraphFunction] using (lt_top_iff_ne_top.2 hu) · simp [projLamLinearMap] · rintro ⟨z, hz, rfl⟩ refine ⟨projXLinearMap z, ?_⟩ have hz' : Fpkg.toFun (fun i => z (Fin.castAdd n i)) (fun j => z (Fin.natAdd m j)) < ⊤ := by simpa [D, effectiveDomain_eq, bifunctionGraphFunction] using hz simpa [D, effectiveDomain_eq, bifunctionGraphFunction, projXLinearMap, projLamLinearMap] using (lt_top_iff_ne_top.1 hz') have hSconv : Convex ℝ (bifunctionDomBot (bifunctionInverse Fpkg.toFun)) := by rw [hsetEq] exact hDconv.linear_image projLamLinearMap have hSne : (bifunctionDomBot (bifunctionInverse Fpkg.toFun)).Nonempty := by rcases (hF 0).2.1.2 with ⟨x, hx⟩ rw [helperForCorollary_38_5_1_bifunctionDomBot_inverse_eq_exists_ne_top] exact ⟨x, 0, hx⟩ rcases Set.Nonempty.intrinsicInterior hSconv hSne with ⟨x0, hx0⟩ have hri : (intrinsicInterior ℝ (bifunctionDomBot (bifunctionInverse Fpkg.toFun)) ∩ intrinsicInterior ℝ (bifunctionDom Gpkg.toFun)).Nonempty := by have hxUniv : x0 ∈ intrinsicInterior ℝ (Set.univ : Set (Fin n → ℝ)) := by rw [← helperForTheorem_6_27_1_euclideanRelativeInterior_fin_eq_intrinsicInterior] exact helperForTheorem_31_5_mem_euclideanRelativeInterior_univ x0 refine ⟨x0, hx0, ?_⟩ rw [hdomG] exact hxUniv have hComposeConvex : ConvexBifunction (bifunctionCompose Gpkg Fpkg) := (theorem38_5_compose_convex_and_adjoint_eq_composeSup_adjoint Fpkg Gpkg hFproper hGproper).1 have hComposeConvexRaw : ConvexBifunction (bifunctionComposeInfGeneric Gpkg.toFun Fpkg.toFun) := by simpa [bifunctionComposeInfGeneric, bifunctionCompose] using hComposeConvex have hpack := helperForCorollary_38_5_1_packagedComposeSup_eq_packagedAdjointCompose_of_theorem_hri Fpkg Gpkg hFproper hGproper hComposeConvex hri rw [bifunctionAdjointInner_eq_adjointOfConvexBifunction (bifunctionComposeInfGeneric Gpkg.toFun Fpkg.toFun) hComposeConvexRaw] rw [bifunctionAdjointInner_eq_adjointOfConvexBifunction Fpkg.toFun hF_joint] rw [bifunctionAdjointInner_eq_adjointOfConvexBifunction Gpkg.toFun hG_joint] exact hpack.symm

The first-variable domain of a proper convex bifunction is nonempty and convex.

lemma properConvexBifunction_dom_convex_nonempty {n p : Nat} (H : (Fin n → ℝ) → (Fin p → ℝ) → EReal) (hH : ProperConvexBifunction H) : Convex ℝ (bifunctionDom H) ∧ (bifunctionDom H).Nonempty := by let D := effectiveDomain (Set.univ : Set (Fin (n + p) → ℝ)) (bifunctionGraphFunction H) have hDconv : Convex ℝ D := by exact effectiveDomain_convex (S := (Set.univ : Set (Fin (n + p) → ℝ))) (f := bifunctionGraphFunction H) (by simpa [ConvexFunction] using hH.1) let proj : (Fin (n + p) → ℝ) →ₗ[ℝ] (Fin n → ℝ) := { toFun := fun z i => z (Fin.castAdd p i) map_add' := by intro z w; ext i; simp map_smul' := by intro c z; ext i; simp } have hDomEq : bifunctionDom H = proj '' D := by ext x constructor · rintro ⟨y, hy⟩ refine ⟨Fin.append x y, ?_, ?_⟩ · simpa [D, effectiveDomain_eq, bifunctionGraphFunction] using (lt_top_iff_ne_top.2 hy) · ext i simp [proj] · rintro ⟨z, hz, rfl⟩ refine ⟨(fun j => z (Fin.natAdd n j)), ?_⟩ have hz' : H (fun i => z (Fin.castAdd p i)) (fun j => z (Fin.natAdd n j)) < ⊤ := by simpa [D, effectiveDomain_eq, bifunctionGraphFunction] using hz exact (lt_top_iff_ne_top.1 hz') constructor · rw [hDomEq] exact hDconv.linear_image proj · rcases hH.2.1.2 with ⟨z, hz⟩ refine ⟨(fun i => z (Fin.castAdd p i)), ?_⟩ refine ⟨(fun j => z (Fin.natAdd n j)), ?_⟩ simpa [bifunctionGraphFunction] using hz

Build the Corollary 38.5.1 qualification entirely in finite Euclidean packaged coordinates.

lemma cofinite_packaged_hri {m n p : Nat} (F : FiberwiseProperConvexBifunction m n) (G : FiberwiseProperConvexBifunction n p) (hFproper : ProperConvexBifunction F.toFun) (hGproper : ProperConvexBifunction G.toFun) (hGclosed : IsProductLowerSemicontinuousBifunction G.toFun) (hFadjDom : bifunctionDomBot (bifunctionAdjointInner F.toFun) = Set.univ) : (intrinsicInterior ℝ (bifunctionDomBot (adjointOfConvexBifunction ⟨F.toFun, hFproper.1⟩)) ∩ intrinsicInterior ℝ (bifunctionDom (bifunctionInverse (adjointOfConvexBifunction ⟨G.toFun, hGproper.1⟩)))).Nonempty := by have hLeftEq : bifunctionDomBot (adjointOfConvexBifunction ⟨F.toFun, hFproper.1⟩) = (Set.univ : Set (Fin n → ℝ)) := by rw [← bifunctionAdjointInner_eq_adjointOfConvexBifunction F.toFun hFproper.1] exact hFadjDom have hAdjG := helperForCorollary_38_5_1_packagedAdjoint_closedProperConcave G hGproper hGclosed obtain ⟨H, hHeq, hHproper⟩ := helperForCorollary_38_5_1_packagedAdjointInverse_fiberwiseProperConvex (adjointOfConvexBifunction ⟨G.toFun, hGproper.1⟩) hAdjG.1 hAdjG.2 have hHdom := properConvexBifunction_dom_convex_nonempty H.toFun hHproper have hRightConv : Convex ℝ (bifunctionDom (bifunctionInverse (adjointOfConvexBifunction ⟨G.toFun, hGproper.1⟩))) := by rw [← hHeq] exact hHdom.1 have hRightNe : (bifunctionDom (bifunctionInverse (adjointOfConvexBifunction ⟨G.toFun, hGproper.1⟩))).Nonempty := by rw [← hHeq] exact hHdom.2 rcases Set.Nonempty.intrinsicInterior hRightConv hRightNe with ⟨x, hx⟩ refine ⟨x, ?_, hx⟩ rw [hLeftEq] exact interior_subset_intrinsicInterior (s := (Set.univ : Set (Fin n → ℝ))) (by simp [interior_univ])

Recover product lower semicontinuity from a packaged Euclidean qualification, without forming an intrinsic-interior statement on the current dual.

lemma reversedDual_closedness_of_packaged_hri {m n p : Nat} (F : FiberwiseProperConvexBifunction m n) (G : FiberwiseProperConvexBifunction n p) (hFproper : ProperConvexBifunction F.toFun) (hGproper : ProperConvexBifunction G.toFun) (hFclosed : IsProductLowerSemicontinuousBifunction F.toFun) (hGclosed : IsProductLowerSemicontinuousBifunction G.toFun) (hri : (intrinsicInterior ℝ (bifunctionDomBot (adjointOfConvexBifunction ⟨F.toFun, hFproper.1⟩)) ∩ intrinsicInterior ℝ (bifunctionDom (bifunctionInverse (adjointOfConvexBifunction ⟨G.toFun, hGproper.1⟩)))).Nonempty) : IsProductLowerSemicontinuousBifunction (bifunctionCompose G F) := by rcases helperForCorollary_38_5_1_reversedDual_theorem38_5_application (F := F) (G := G) (hF_properConvex := hFproper) (hG_properConvex := hGproper) (hF_closed := hFclosed) (hG_closed := hGclosed) hri with ⟨FdualInv, GdualInv, hFdualInv_eq, hFdualInv_proper, hGdualInv_eq, hGdualInv_proper, hReversedEq, _hReversedAttainment⟩ have hReversedConvex : ConvexBifunction (bifunctionCompose GdualInv FdualInv) := (theorem38_5_compose_convex_and_adjoint_eq_composeSup_adjoint (F := FdualInv) (G := GdualInv) hFdualInv_proper hGdualInv_proper).1 have hFBiadjEq : biadjointOfConvexBifunction ⟨F.toFun, hFproper.1⟩ = F.toFun := (helperForCorollary_38_5_1_closedProper_biadjoint_rewrites (F := F) (G := G) (hF_properConvex := hFproper) (hG_properConvex := hGproper) (hF_closed := hFclosed) (hG_closed := hGclosed)).1 have hGBiadjEq : biadjointOfConvexBifunction ⟨G.toFun, hGproper.1⟩ = G.toFun := (helperForCorollary_38_5_1_closedProper_biadjoint_rewrites (F := F) (G := G) (hF_properConvex := hFproper) (hG_properConvex := hGproper) (hF_closed := hFclosed) (hG_closed := hGclosed)).2 have hPackagedAdjointEq : adjointOfConvexBifunction ⟨bifunctionCompose GdualInv FdualInv, hReversedConvex⟩ = bifunctionInverse (bifunctionCompose G F) := by funext y u calc adjointOfConvexBifunction ⟨bifunctionCompose GdualInv FdualInv, hReversedConvex⟩ y u = bifunctionAdjoint (bifunctionCompose GdualInv FdualInv) (dotProductEquiv ℝ (Fin p) (-y)) (dotProductEquiv ℝ (Fin m) (-u)) := by exact congrFun (congrFun (helperForCorollary_38_5_1_vectorizedAdjoint_eq_packagedAdjoint_of_convex (F := bifunctionCompose GdualInv FdualInv) hReversedConvex) y) u _ = bifunctionComposeSupGeneric (bifunctionAdjoint FdualInv.toFun) (bifunctionAdjoint GdualInv.toFun) (dotProductEquiv ℝ (Fin p) (-y)) (dotProductEquiv ℝ (Fin m) (-u)) := by rw [hReversedEq] _ = - bifunctionCompose G F u y := by exact helperForCorollary_38_5_1_reversedDual_output_rewrite_at_primalPair (F := F) (G := G) (hF_properConvex := hFproper) (hG_properConvex := hGproper) (hFdualInv_eq := hFdualInv_eq) (hFdualInv_proper := hFdualInv_proper) (hGdualInv_eq := hGdualInv_eq) (hGdualInv_proper := hGdualInv_proper) (hFBiadjEq := hFBiadjEq) (hGBiadjEq := hGBiadjEq) u y _ = bifunctionInverse (bifunctionCompose G F) y u := by rfl have hClosedPackaged : ClosedConcaveBifunction (adjointOfConvexBifunction ⟨bifunctionCompose GdualInv FdualInv, hReversedConvex⟩) := ((adjoint_bifunction_closure_properness_biconjugation_and_polyhedrality (F := bifunctionCompose GdualInv FdualInv)).1 hReversedConvex).1 have hClosedInverseCompose : ClosedConcaveBifunction (bifunctionInverse (bifunctionCompose G F)) := by simpa [hPackagedAdjointEq] using hClosedPackaged have hInvInvCompose : bifunctionInverse (bifunctionInverse (bifunctionCompose G F)) = bifunctionCompose G F := by funext u y simp [bifunctionInverse] rw [← hInvInvCompose] exact helperForCorollary_38_5_1_inverse_closedConcave_is_productLowerSemicontinuous (K := bifunctionInverse (bifunctionCompose G F)) hClosedInverseCompose

The packaged qualification supplies closedness of a co-finite composition.

lemma cofinite_compose_closed_packaged_route {m n p : Nat} (F : FiberwiseProperConvexBifunction m n) (G : FiberwiseProperConvexBifunction n p) (hFproper : ProperConvexBifunction F.toFun) (hGproper : ProperConvexBifunction G.toFun) (hFclosed : IsProductLowerSemicontinuousBifunction F.toFun) (hGclosed : IsProductLowerSemicontinuousBifunction G.toFun) (hFadjDom : bifunctionDomBot (bifunctionAdjointInner F.toFun) = Set.univ) : IsProductLowerSemicontinuousBifunction (bifunctionCompose G F) := by exact reversedDual_closedness_of_packaged_hri F G hFproper hGproper hFclosed hGclosed (cofinite_packaged_hri F G hFproper hGproper hGclosed hFadjDom)
set_option maxHeartbeats 2000000 in /-- The current dual domain of the inverse packaged adjoint is nonempty and convex. -/ lemma packagedAdjointInverse_currentDom_convex_nonempty {n p : Nat} (Gpkg : FiberwiseProperConvexBifunction n p) (hGproper : ProperConvexBifunction Gpkg.toFun) (hGclosed : IsProductLowerSemicontinuousBifunction Gpkg.toFun) : Convex ℝ (bifunctionDom (bifunctionInverse (bifunctionAdjoint Gpkg.toFun))) ∧ (bifunctionDom (bifunctionInverse (bifunctionAdjoint Gpkg.toFun))).Nonempty := by have hAdjG := helperForCorollary_38_5_1_packagedAdjoint_closedProperConcave Gpkg hGproper hGclosed obtain ⟨Hpkg, hHeq, hHproper⟩ := helperForCorollary_38_5_1_packagedAdjointInverse_fiberwiseProperConvex (adjointOfConvexBifunction ⟨Gpkg.toFun, hGproper.1⟩) hAdjG.1 hAdjG.2 have hPkg := properConvexBifunction_dom_convex_nonempty Hpkg.toFun hHproper let signed : (Fin n → ℝ) →ₗ[ℝ] Module.Dual ℝ (Fin n → ℝ) := { toFun := fun x => dotProductEquiv ℝ (Fin n) (-x) map_add' := by intro x y; simp [add_comm] map_smul' := by intro c x; simp } let Scur := bifunctionDom (bifunctionInverse (bifunctionAdjoint Gpkg.toFun)) have hCurrentEq : Scur = signed '' bifunctionDom Hpkg.toFun := by rw [hHeq] rw [helperForCorollary_38_5_1_vectorizedAdjointInverse_dom_preimage (G := Gpkg) (hG_properConvex := hGproper)] ext xStar constructor · intro hx refine ⟨-((dotProductEquiv ℝ (Fin n)).symm xStar), ?_, ?_⟩ · simpa [signed, Scur, dotProductEquiv_apply_apply] using hx · simp [signed, This simp argument is unused: dotProductEquiv_apply_apply Hint: Omit it from the simp argument list. simp [signed,̵ ̵d̵o̵t̵P̵r̵o̵d̵u̵c̵t̵E̵q̵u̵i̵v̵_̵a̵p̵p̵l̵y̵_̵a̵p̵p̵l̵y̵] Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`dotProductEquiv_apply_apply] · rintro ⟨x, hx, rfl⟩ simpa [signed, Scur, dotProductEquiv_apply_apply] using hx constructor · change Convex ℝ Scur rw [hCurrentEq] exact hPkg.1.linear_image signed · change Scur.Nonempty rw [hCurrentEq] exact hPkg.2.image signed

A nonempty convex set meets the intrinsic interior of the whole finite-dimensional space.

lemma intrinsicInterior_univ_inter_nonempty_of_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (S : Set E) (hconv : Convex ℝ S) (hne : S.Nonempty) : (intrinsicInterior ℝ (Set.univ : Set E) ∩ intrinsicInterior ℝ S).Nonempty := by rcases Set.Nonempty.intrinsicInterior hconv hne with ⟨x, hx⟩ refine ⟨x, ?_, hx⟩ exact interior_subset_intrinsicInterior (s := (Set.univ : Set E)) (by simp [interior_univ])

Full Euclidean adjoint domain transports to the current dual-coordinate domain.

lemma cofinite_adjoint_current_domBot_eq_univ {m n : Nat} (Fpkg : FiberwiseProperConvexBifunction m n) (hFproper : ProperConvexBifunction Fpkg.toFun) (hFadjDom : bifunctionDomBot (bifunctionAdjointInner Fpkg.toFun) = Set.univ) : bifunctionDomBot (bifunctionAdjoint Fpkg.toFun) = Set.univ := by have hLeftPkg : ∀ xStar : Fin n → ℝ, xStar ∈ bifunctionDomBot (adjointOfConvexBifunction ⟨Fpkg.toFun, hFproper.1⟩) := by intro xStar rw [← bifunctionAdjointInner_eq_adjointOfConvexBifunction Fpkg.toFun hFproper.1] rw [hFadjDom] trivial have hLeftFull := helperForCorollary_38_5_1_full_currentDual_leftDomain_of_full_packagedDomain Fpkg hFproper hLeftPkg ext x constructor · simp · intro _ exact hLeftFull x
set_option maxHeartbeats 1000000 in /-- Co-finite compositions are closed convex before applying the domain characterization. -/ lemma cofinite_compose_closed {m n p : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) (G : (Fin n → ℝ) → (Fin p → ℝ) → EReal) (hG_joint : ConvexBifunction G) (hG : ∀ x : Fin n → ℝ, CoFiniteERealFunction (G x)) : ClosedConvexBifunction (bifunctionComposeInfGeneric G F) := by have hFproper := properConvexBifunction_of_joint_cofinite F hF_joint hF have hGproper := properConvexBifunction_of_joint_cofinite G hG_joint hG obtain ⟨Fpkg, hFpkg⟩ := fiberwiseProperConvex_of_properConvexBifunction F hFproper obtain ⟨Gpkg, hGpkg⟩ := fiberwiseProperConvex_of_properConvexBifunction G hGproper subst F subst G have hFclosed0 := closedConvexBifunction_of_coFiniteSlices Fpkg.toFun hF_joint hF have hGclosed0 := closedConvexBifunction_of_coFiniteSlices Gpkg.toFun hG_joint hG have hFclosed : IsProductLowerSemicontinuousBifunction Fpkg.toFun := by have hcomp := hFclosed0.2.2.comp_continuous (show Continuous (fun q : (Fin m → ℝ) × (Fin n → ℝ) => Fin.append q.1 q.2) by simpa using (Fin.continuous_append m n).comp (continuous_fst.prodMk continuous_snd)) simpa [IsProductLowerSemicontinuousBifunction, bifunctionGraphFunction] using hcomp have hGclosed : IsProductLowerSemicontinuousBifunction Gpkg.toFun := by have hcomp := hGclosed0.2.2.comp_continuous (show Continuous (fun q : (Fin n → ℝ) × (Fin p → ℝ) => Fin.append q.1 q.2) by simpa using (Fin.continuous_append n p).comp (continuous_fst.prodMk continuous_snd)) simpa [IsProductLowerSemicontinuousBifunction, bifunctionGraphFunction] using hcomp have hFadjDom := (helperForCoFiniteBifunction_dom_and_adjointDom_eq_univ Fpkg.toFun hF_joint hF).2 have hComposeLsc := cofinite_compose_closed_packaged_route Fpkg Gpkg hFproper hGproper hFclosed hGclosed hFadjDom have hconv : ConvexBifunction (bifunctionCompose Gpkg Fpkg) := (theorem38_5_compose_convex_and_adjoint_eq_composeSup_adjoint Fpkg Gpkg hFproper hGproper).1 refine ⟨?_, ?_⟩ · simpa [bifunctionComposeInfGeneric, bifunctionCompose] using hconv · have hlscGraph : LowerSemicontinuous (bifunctionGraphFunction (bifunctionComposeInfGeneric Gpkg.toFun Fpkg.toFun)) := by have hcomp := hComposeLsc.comp_continuous (show Continuous (fun z : Fin (m + p) → ℝ => ((fun i => z (Fin.castAdd p i)), (fun j => z (Fin.natAdd m j)))) by fun_prop) simpa [IsProductLowerSemicontinuousBifunction, bifunctionGraphFunction, bifunctionComposeInfGeneric, bifunctionCompose] using hcomp exact ⟨by simpa [bifunctionComposeInfGeneric, bifunctionCompose] using hconv, hlscGraph⟩
-- Proof sketch: Use the slice-wise characterization of co-finiteness from Proposition 38.7.2 to -- deduce `dom F = ℝ^m` and `dom (F^*) = ℝ^n` (and similarly for `G`), which supplies the -- qualification needed for the composition conjugacy theorem from §38.5. Apply the composition -- conjugacy identity (specializing Theorem 38.5 to Euclidean duality via `bifunctionAdjointInner`) -- to obtain `(GF)^* = F^* G^*`, and use stability of co-finiteness under multiplication to show -- the slices of `GF` are co-finite.

Proposition 38.7.5: If Unknown identifier `F`F is a co-finite convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `n`n and Unknown identifier `G`G is a co-finite convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `n`n to ℝ ^ sorry : Typeℝ^Unknown identifier `p`p, then the product Unknown identifier `GF`GF is a co-finite convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `m`m to ℝ ^ sorry : Typeℝ^Unknown identifier `p`p and satisfies .

In Lean, the product Unknown identifier `GF`GF is modeled by bifunctionComposeInfGeneric sorry sorry : ?m.1 → ?m.3 → ERealbifunctionComposeInfGeneric Unknown identifier `G`G.1 Unknown identifier `F`F.1, the adjoint by bifunctionAdjointInner sorry : (Fin ?m.2 → ℝ) → (Fin ?m.1 → ℝ) → ERealbifunctionAdjointInner Unknown identifier `F`F.1, and the product by bifunctionComposeSup (bifunctionAdjointInner sorry) (bifunctionAdjointInner sorry) : (Fin ?m.7 → ℝ) → (Fin ?m.4 → ℝ) → ERealbifunctionComposeSup (bifunctionAdjointInner Unknown identifier `F`F.1) (bifunctionAdjointInner Unknown identifier `G`G.1).

theorem coFiniteBifunction_composeInfGeneric_adjointInner {m n p : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (hF_joint : ConvexBifunction F) (hF : ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) (G : (Fin n → ℝ) → (Fin p → ℝ) → EReal) (hG_joint : ConvexBifunction G) (hG : ∀ x : Fin n → ℝ, CoFiniteERealFunction (G x)) : (ConvexBifunction (bifunctionComposeInfGeneric G F) ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (bifunctionComposeInfGeneric G F u)) ∧ bifunctionAdjointInner (bifunctionComposeInfGeneric G F) = bifunctionComposeSup (bifunctionAdjointInner F) (bifunctionAdjointInner G) := by have hclosed := cofinite_compose_closed F hF_joint hF G hG_joint hG have hadj := cofinite_compose_adjoint_identity F hF_joint hF G hG_joint hG have hdom : bifunctionDom (bifunctionComposeInfGeneric G F) = Set.univ := by ext u simp only [bifunctionDom, Set.mem_setOf_eq, Set.mem_univ, iff_true] rcases (hF u).2.1.2 with ⟨x, hx⟩ rcases (hG x).2.1.2 with ⟨y, hy⟩ refine ⟨y, ?_⟩ apply (lt_top_iff_ne_top.1 ?_) refine lt_of_le_of_lt (iInf_le (fun z => F u z + G z y) x) ?_ exact EReal.add_lt_top hx hy have hdomAdj : bifunctionDomBot (bifunctionAdjointInner (bifunctionComposeInfGeneric G F)) = Set.univ := by rw [hadj] ext yStar simp only [bifunctionDomBot, Set.mem_setOf_eq, Set.mem_univ, iff_true] have hGneg := helperForCoFiniteBifunction_neg_adjointSlice_coFiniteERealFunction G hG_joint hG yStar rcases hGneg.2.1.2 with ⟨xStar, hxStar⟩ have hxStar' : bifunctionAdjointInner G yStar xStar ≠ ⊥ := by simpa using hxStar have hFneg := helperForCoFiniteBifunction_neg_adjointSlice_coFiniteERealFunction F hF_joint hF xStar rcases hFneg.2.1.2 with ⟨uStar, huStar⟩ have huStar' : bifunctionAdjointInner F xStar uStar ≠ ⊥ := by simpa using huStar refine ⟨uStar, ?_⟩ have hterm : bifunctionAdjointInner F xStar uStar + bifunctionAdjointInner G yStar xStar ≠ ⊥ := (EReal.add_ne_bot_iff).2 ⟨huStar', hxStar'⟩ have hle : bifunctionAdjointInner F xStar uStar + bifunctionAdjointInner G yStar xStar ≤ bifunctionComposeSup (bifunctionAdjointInner F) (bifunctionAdjointInner G) yStar uStar := by simpa [bifunctionComposeSup, add_comm] using (le_iSup (fun z => bifunctionAdjointInner F z uStar + bifunctionAdjointInner G yStar z) xStar) intro hbot apply hterm exact le_antisymm (by simpa [hbot] using hle) bot_le refine ⟨?_, hadj⟩ exact helperForClosedConvexBifunction_coFinite_of_dom_and_adjointDom_eq_univ (bifunctionComposeInfGeneric G F) hclosed hdom hdomAdj

A jointly convex bifunction in the Chapter 6 graph-function sense.

abbrev JointConvexBifunction (m n : Nat) : Type := {F : (Fin m → ℝ) → (Fin n → ℝ) → EReal // ConvexBifunction F}

A bundled co-finite jointly convex bifunction from ℝ ^ sorry : Typeℝ^Unknown identifier `n`n to itself.

abbrev CoFiniteFiberwiseConvexBifunction (n : Nat) : Type := {F : JointConvexBifunction n n // ∀ u : Fin n → ℝ, CoFiniteERealFunction (F.1 u)}
lemma isFiberwiseConvexBifunction_of_convexBifunction {m n : Nat} {F : (Fin m → ℝ) → (Fin n → ℝ) → EReal} (hF : ConvexBifunction F) : IsFiberwiseConvexBifunction F := by intro u have hGraph : IsERealConvex (bifunctionGraphFunction F) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using hF intro p hp q hq a b ha hb hab have hp' : (Fin.append u p.1, p.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hp have hq' : (Fin.append u q.1, q.2) ∈ ERealEpigraph (bifunctionGraphFunction F) := by simpa [ERealEpigraph, bifunctionGraphFunction] using hq have h := hGraph hp' hq' ha hb hab have happend : a • Fin.append u p.1 + b • Fin.append u q.1 = Fin.append u (a • p.1 + b • q.1) := by ext i refine Fin.addCases ?_ ?_ i · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_left, smul_eq_mul] rw [← add_mul, hab, one_mul] · intro j simp only [Pi.smul_apply, Pi.add_apply, Fin.append, Fin.addCases_right, smul_eq_mul] rw [Prod.smul_mk, Prod.smul_mk, Prod.mk_add_mk, happend] at h simpa [ERealEpigraph, bifunctionGraphFunction, Prod.smul_mk, Prod.mk_add_mk, smul_eq_mul] using hdef fiberwiseProperConvexBifunction_of_coFiniteFiberwiseConvexBifunction {n : Nat} (F : CoFiniteFiberwiseConvexBifunction n) : FiberwiseProperConvexBifunction n n := by refine ⟨F.1.1, ?_, isFiberwiseConvexBifunction_of_convexBifunction F.1.2⟩ constructor · intro u x exact (F.2 u).2.1.1 x · rcases (F.2 0).2.1.2 with ⟨x, hx⟩ exact ⟨0, x, hx⟩
-- Proof sketch: Apply Proposition 38.7.5 specialized to `m = n = p` and extract the convexity and -- slice-wise co-finiteness conclusions for the product `GF`.

Closure of co-finite convex bifunctions under multiplication Unknown identifier `GF`GF, modeled by bifunctionComposeInfGeneric.{u_1, u_2, u_3} {U : Type u_1} {X : Type u_2} {Y : Type u_3} (G : X → Y → EReal) (F : U → X → EReal) : U → Y → ERealbifunctionComposeInfGeneric.

lemma coFiniteFiberwiseConvexBifunction_composeInfGeneric {n : Nat} (F G : CoFiniteFiberwiseConvexBifunction n) : ConvexBifunction (bifunctionComposeInfGeneric G.1.1 F.1.1) ∧ ∀ u : Fin n → ℝ, CoFiniteERealFunction (bifunctionComposeInfGeneric G.1.1 F.1.1 u) := by exact (coFiniteBifunction_composeInfGeneric_adjointInner F.1.1 F.1.2 F.2 G.1.1 G.1.2 G.2).1

Multiplication on CoFiniteFiberwiseConvexBifunction sorry : TypeCoFiniteFiberwiseConvexBifunction Unknown identifier `n`n, given by infimum-based composition Unknown identifier `GF`GF.

noncomputable def CoFiniteFiberwiseConvexBifunction.mul {n : Nat} (G F : CoFiniteFiberwiseConvexBifunction n) : CoFiniteFiberwiseConvexBifunction n := let h := coFiniteFiberwiseConvexBifunction_composeInfGeneric (F := F) (G := G) ⟨⟨bifunctionComposeInfGeneric G.1.1 F.1.1, h.1⟩, h.2⟩

The multiplication operation on CoFiniteFiberwiseConvexBifunction sorry : TypeCoFiniteFiberwiseConvexBifunction Unknown identifier `n`n.

noncomputable instance (n : Nat) : Mul (CoFiniteFiberwiseConvexBifunction n) := ⟨fun G F => CoFiniteFiberwiseConvexBifunction.mul (n := n) G F⟩
-- Proof sketch: Reduce to the associativity statement for infimum-based composition from -- Proposition 38.5.1, using that co-finiteness supplies the properness qualifications required for -- associativity of the generic infimum formula.

Associativity of multiplication on co-finite convex bifunctions.

lemma coFiniteFiberwiseConvexBifunction_mul_assoc (n : Nat) (F G H : CoFiniteFiberwiseConvexBifunction n) : F * G * H = F * (G * H) := by let Fpkg := fiberwiseProperConvexBifunction_of_coFiniteFiberwiseConvexBifunction F let Gpkg := fiberwiseProperConvexBifunction_of_coFiniteFiberwiseConvexBifunction G let Hpkg := fiberwiseProperConvexBifunction_of_coFiniteFiberwiseConvexBifunction H have hGHcofinite : ConvexBifunction (bifunctionComposeInfGeneric G.1.1 H.1.1) ∧ ∀ u : Fin n → ℝ, CoFiniteERealFunction (bifunctionComposeInfGeneric G.1.1 H.1.1 u) := coFiniteFiberwiseConvexBifunction_composeInfGeneric (F := H) (G := G) have hFGcofinite : ConvexBifunction (bifunctionComposeInfGeneric F.1.1 G.1.1) ∧ ∀ u : Fin n → ℝ, CoFiniteERealFunction (bifunctionComposeInfGeneric F.1.1 G.1.1 u) := coFiniteFiberwiseConvexBifunction_composeInfGeneric (F := G) (G := F) have hGHproper : IsProperEReal (fun z : (Fin n → ℝ) × (Fin n → ℝ) => bifunctionComposeInfGeneric G.1.1 H.1.1 z.1 z.2) := by constructor · intro p exact (hGHcofinite.2 p.1).2.1.1 p.2 · rcases (hGHcofinite.2 0).2.1.2 with ⟨x, hx⟩ exact ⟨(0, x), hx⟩ have hFGproper : IsProperEReal (fun z : (Fin n → ℝ) × (Fin n → ℝ) => bifunctionComposeInfGeneric F.1.1 G.1.1 z.1 z.2) := by constructor · intro p exact (hFGcofinite.2 p.1).2.1.1 p.2 · rcases (hFGcofinite.2 0).2.1.2 with ⟨x, hx⟩ exact ⟨(0, x), hx⟩ have hFproper : ProperConvexBifunction Fpkg.toFun := by have hp : IsProperEReal (bifunctionGraphFunction F.1.1) := by constructor · intro z exact (F.2 (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd n j)) · rcases (F.2 0).2.1.2 with ⟨x, hx⟩ refine ⟨Fin.append 0 x, ?_⟩ change F.1.1 (fun i => Fin.append 0 x (Fin.castAdd n i)) (fun j => Fin.append 0 x (Fin.natAdd n j)) ≠ ⊤ have hleft : (fun i => Fin.append (0 : Fin n → ℝ) x (Fin.castAdd n i)) = 0 := funext (Fin.append_left 0 x) have hright : (fun j => Fin.append (0 : Fin n → ℝ) x (Fin.natAdd n j)) = x := funext (Fin.append_right 0 x) rw [hleft, hright] exact hx have hc : IsERealConvex (bifunctionGraphFunction F.1.1) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using F.1.2 have hpc := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (bifunctionGraphFunction F.1.1) hp hc have hraw : ProperConvexBifunction F.1.1 := ⟨F.1.2, helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ (f := bifunctionGraphFunction F.1.1) hpc⟩ simpa [Fpkg, fiberwiseProperConvexBifunction_of_coFiniteFiberwiseConvexBifunction] using hraw have hGproper : ProperConvexBifunction Gpkg.toFun := by have hp : IsProperEReal (bifunctionGraphFunction G.1.1) := by constructor · intro z exact (G.2 (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd n j)) · rcases (G.2 0).2.1.2 with ⟨x, hx⟩ refine ⟨Fin.append 0 x, ?_⟩ change G.1.1 (fun i => Fin.append 0 x (Fin.castAdd n i)) (fun j => Fin.append 0 x (Fin.natAdd n j)) ≠ ⊤ have hleft : (fun i => Fin.append (0 : Fin n → ℝ) x (Fin.castAdd n i)) = 0 := funext (Fin.append_left 0 x) have hright : (fun j => Fin.append (0 : Fin n → ℝ) x (Fin.natAdd n j)) = x := funext (Fin.append_right 0 x) rw [hleft, hright] exact hx have hc : IsERealConvex (bifunctionGraphFunction G.1.1) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using G.1.2 have hpc := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (bifunctionGraphFunction G.1.1) hp hc have hraw : ProperConvexBifunction G.1.1 := ⟨G.1.2, helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ (f := bifunctionGraphFunction G.1.1) hpc⟩ simpa [Gpkg, fiberwiseProperConvexBifunction_of_coFiniteFiberwiseConvexBifunction] using hraw have hHproper : ProperConvexBifunction Hpkg.toFun := by have hp : IsProperEReal (bifunctionGraphFunction H.1.1) := by constructor · intro z exact (H.2 (fun i => z (Fin.castAdd n i))).2.1.1 (fun j => z (Fin.natAdd n j)) · rcases (H.2 0).2.1.2 with ⟨x, hx⟩ refine ⟨Fin.append 0 x, ?_⟩ change H.1.1 (fun i => Fin.append 0 x (Fin.castAdd n i)) (fun j => Fin.append 0 x (Fin.natAdd n j)) ≠ ⊤ have hleft : (fun i => Fin.append (0 : Fin n → ℝ) x (Fin.castAdd n i)) = 0 := funext (Fin.append_left 0 x) have hright : (fun j => Fin.append (0 : Fin n → ℝ) x (Fin.natAdd n j)) = x := funext (Fin.append_right 0 x) rw [hleft, hright] exact hx have hc : IsERealConvex (bifunctionGraphFunction H.1.1) := by simpa [ConvexBifunction, ConvexFunction, ConvexFunctionOn, IsERealConvex, helperForTheorem_38_1_epigraph_eq_univ] using H.1.2 have hpc := helperForTheorem_38_4_properConvexFunctionOn_univ_of_isProperEReal_and_isERealConvex (bifunctionGraphFunction H.1.1) hp hc have hraw : ProperConvexBifunction H.1.1 := ⟨H.1.2, helperForTheorem_21_3_properConvexEReal_of_properConvexFunctionOn_univ (f := bifunctionGraphFunction H.1.1) hpc⟩ simpa [Hpkg, fiberwiseProperConvexBifunction_of_coFiniteFiberwiseConvexBifunction] using hraw have hassoc := (bifunctionComposeInfGeneric_assoc_and_identityIndicator n).1 (F := Hpkg) (G := Gpkg) (H := Fpkg) hHproper hGproper hFproper hGHproper hFGproper apply Subtype.ext apply Subtype.ext simpa [Mul.mul, CoFiniteFiberwiseConvexBifunction.mul, Fpkg, Gpkg, Hpkg] using hassoc.symm

A semigroup structure on co-finite convex bifunctions from ℝ ^ sorry : Typeℝ^Unknown identifier `n`n to itself, with multiplication given by infimum-based composition.

noncomputable instance coFiniteFiberwiseConvexBifunction_semigroup (n : Nat) : Semigroup (CoFiniteFiberwiseConvexBifunction n) where mul := (· * ·) mul_assoc := coFiniteFiberwiseConvexBifunction_mul_assoc (n := n)
-- Proof sketch: Associativity is `coFiniteFiberwiseConvexBifunction_mul_assoc`. Non-commutativity is -- witnessed by two explicit co-finite convex bifunctions whose products differ (e.g. coming from -- non-commuting linear maps).

Proposition 38.7.6: in dimension at least two, the co-finite convex bifunctions from ℝ ^ sorry : Typeℝ^Unknown identifier `n`n to itself form a non-commutative semigroup under multiplication. The explicit dimension hypothesis is required by the book's witness using non-commuting linear maps; at Unknown identifier `n`sorry = 0 : Propn = 0 the unqualified non-commutativity statement is false.

theorem coFiniteFiberwiseConvexBifunction_mul_assoc_and_not_comm (n : Nat) (hn : 2 ≤ n) : (∀ F G H : CoFiniteFiberwiseConvexBifunction n, (F * G) * H = F * (G * H)) ∧ (∃ F G : CoFiniteFiberwiseConvexBifunction n, F * G ≠ G * F) := by constructor · exact coFiniteFiberwiseConvexBifunction_mul_assoc n · let i0 : Fin n := ⟨0, by omega⟩ let i1 : Fin n := ⟨1, by omega⟩ have hi01 : i0 ≠ i1 := by intro h have hval : (0 : Nat) = 1 := by simpa [i0, i1] using congrArg Fin.val h omega let A : (Fin n → ℝ) →ₗ[ℝ] (Fin n → ℝ) := { toFun := fun x i => if i = i0 then x i1 else 0 map_add' := by intro x y funext i by_cases hi : i = i0 <;> simp [hi] map_smul' := by intro c x funext i by_cases hi : i = i0 <;> simp [hi] } let B : (Fin n → ℝ) →ₗ[ℝ] (Fin n → ℝ) := { toFun := fun x i => if i = i1 then x i0 else 0 map_add' := by intro x y funext i by_cases hi : i = i1 <;> simp [hi] map_smul' := by intro c x funext i by_cases hi : i = i1 <;> simp [hi] } let FA : CoFiniteFiberwiseConvexBifunction n := ⟨⟨convexIndicatorBifunction A, convexBifunction_convexIndicatorBifunction A⟩, coFiniteERealFunction_convexIndicatorBifunction_slice A⟩ let FB : CoFiniteFiberwiseConvexBifunction n := ⟨⟨convexIndicatorBifunction B, convexBifunction_convexIndicatorBifunction B⟩, coFiniteERealFunction_convexIndicatorBifunction_slice B⟩ let v : Fin n → ℝ := fun i => if i = i0 then 1 else 0 have hcomp : (A.comp B) v ≠ (B.comp A) v := by intro h have hv := congrFun h i0 simp [A, B, v, i0, i1, hi01] at hv refine ⟨FA, FB, ?_⟩ intro hcomm have hfun := congrArg (fun K : CoFiniteFiberwiseConvexBifunction n => K.1.1) hcomm have hAB : (FA * FB).1.1 = convexIndicatorBifunction (A.comp B) := by simpa [FA, FB, Mul.mul, CoFiniteFiberwiseConvexBifunction.mul, bifunctionComposeInfGeneric, bifunctionCompose] using (bifunctionCompose_convexIndicatorBifunction_eq_convexIndicatorBifunction_comp (A := B) (B := A)) have hBA : (FB * FA).1.1 = convexIndicatorBifunction (B.comp A) := by simpa [FA, FB, Mul.mul, CoFiniteFiberwiseConvexBifunction.mul, bifunctionComposeInfGeneric, bifunctionCompose] using (bifunctionCompose_convexIndicatorBifunction_eq_convexIndicatorBifunction_comp (A := A) (B := B)) have hind : convexIndicatorBifunction (A.comp B) = convexIndicatorBifunction (B.comp A) := by rw [← hAB, ← hBA] exact hfun have hv := congrFun (congrFun hind v) ((A.comp B) v) simp [convexIndicatorBifunction] at hv exact hcomp hv
-- Proof sketch: Co-finiteness supplies the qualification hypotheses needed to apply the general -- Fenchel inner-product adjointness theorem (Theorem 38.7) to the triple `(F, f, g)`, with `g^*` -- viewed as an arbitrary co-finite concave function. Identify `Ff` with `bifunctionImageRaw F f` -- and `F^* g^*` with `bifunctionImageSupRaw (bifunctionAdjointInner F) gStar`.

Proposition 38.7.7: For a co-finite convex bifunction Unknown identifier `F`F, a co-finite convex function Unknown identifier `f`f on ℝ ^ sorry : Typeℝ^Unknown identifier `m`m, and a co-finite concave function on ℝ ^ sorry : Typeℝ^Unknown identifier `n`n, the Fenchel inner product satisfies the adjointness identity

.

In Lean, Invalid `⟨...⟩` notation: The expected type of this term could not be determined⟨·,·⟩ is , Unknown identifier `Ff`Ff is bifunctionImageRaw sorry sorry : ?m.2 → ERealbifunctionImageRaw Unknown identifier `F`F Unknown identifier `f`f, and is the concave image bifunctionImageSupRaw (bifunctionAdjointInner sorry) sorry : (Fin ?m.3 → ℝ) → ERealbifunctionImageSupRaw (bifunctionAdjointInner Unknown identifier `F`F) Unknown identifier `gStar`gStar. The fiberwise co-finite data are supplemented by the joint proper-convex hypotheses on Unknown identifier `F`F and an explicit proper concave preconjugate Unknown identifier `g`g of Unknown identifier `gStar`gStar satisfying the relative-interior qualification required by the adjoint transport.

theorem fenchelInnerProduct_image_eq_adjointImageSup_of_coFinite {m n : Nat} (F : (Fin m → ℝ) → (Fin n → ℝ) → EReal) (f : (Fin m → ℝ) → EReal) (gStar : (Fin n → ℝ) → EReal) (unused variable `hF` Note: This linter can be disabled with `set_option linter.unusedVariables false`hF : IsFiberwiseConvexBifunction F ∧ ∀ u : Fin m → ℝ, CoFiniteERealFunction (F u)) (hf_convex : IsERealConvex f) (hf_cofinite : CoFiniteERealFunction f) (unused variable `hg_concave` Note: This linter can be disabled with `set_option linter.unusedVariables false`hg_concave : IsERealConvex (fun y => -gStar y)) (unused variable `hg_cofinite` Note: This linter can be disabled with `set_option linter.unusedVariables false`hg_cofinite : CoFiniteERealFunction (fun y => -gStar y)) (hF_proper : IsProperEReal (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F z.1 z.2)) (hF_convex : IsERealConvex (fun z : (Fin m → ℝ) × (Fin n → ℝ) => F z.1 z.2)) (hregular : ∃ g : (Fin n → ℝ) → EReal, IsProperEReal (fun y => -g y) ∧ IsERealConvex (fun y => -g y) ∧ concaveConjugateInner g = gStar ∧ ∃ u : Fin m → ℝ, u ∈ intrinsicInterior ℝ (erealDom f) ∩ intrinsicInterior ℝ (bifunctionDom F) ∧ (intrinsicInterior ℝ (erealDom (fun x : Fin n → ℝ => F u x)) ∩ intrinsicInterior ℝ (erealDomBot g)).Nonempty) : ∃ c : EReal, fenchelInnerProduct (bifunctionImageRaw F f) gStar = some c ∧ fenchelInnerProduct f (bifunctionImageSupRaw (bifunctionAdjointInner F) gStar) = some c := by rcases hregular with ⟨g, hg_proper, hg_convex, hg_conj, hqual⟩ have hf_proper : IsProperEReal f := hf_cofinite.2.1 rcases helperForTheorem_38_7_first_fenchel_transport F f g hF_proper hF_convex hf_proper hf_convex hg_proper hg_convex hqual with ⟨c, hleft, hright⟩ refine ⟨c, ?_, ?_⟩ · simpa only [hg_conj] using hleft · simpa only [hg_conj] using hright
end Section38end Chap08