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ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | import OAI.Analysis.DiamondDistortion.Vanishing
namespace OAI
universe u v
noncomputable section
open scoped BigOperators Classical Topology
open Set
namespace DiamondDistortion
open scoped ENNReal
namespace InfimalCost
variable {E F : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E]
[AddCommGroup F] [Module ℝ F]
/-- Quotient construction of the exact infimum over all linear representations.
No infimum or supremum is assumed to be attained. -/
structure Presentation (E F : Type*) [SeminormedAddCommGroup E] [NormedSpace ℝ E]
[AddCommGroup F] [Module ℝ F] where
output : E →ₗ[ℝ] F
sectionMap : F →ₗ[ℝ] E
right_inverse : ∀ v,output (sectionMap v)=v
namespace Presentation
variable (A : Presentation E F)
def seminorm : Seminorm ℝ F :=
(normSeminorm ℝ (E ⧸ LinearMap.ker A.output)).comp ((LinearMap.ker A.output).mkQ.comp A.sectionMap)
lemma representation_eq {d : E} {v : F} (hd : A.output d=v) :
(LinearMap.ker A.output).mkQ d=(LinearMap.ker A.output).mkQ (A.sectionMap v) := by
change (Submodule.Quotient.mk d : E ⧸ LinearMap.ker A.output)=Submodule.Quotient.mk (A.sectionMap v)
rw [Submodule.Quotient.eq]
change A.output (d-A.sectionMap v)=0
rw [map_sub,hd,A.right_inverse,sub_self]
lemma le_representation {d : E} {v : F} (hd : A.output d=v) : A.seminorm v ≤ ‖d‖ := by
change ‖(LinearMap.ker A.output).mkQ (A.sectionMap v)‖ ≤ ‖d‖
rw [← A.representation_eq hd]
exact Submodule.Quotient.norm_mk_le _ d
lemma near_representation (v : F) {ε : ℝ} (hε : 0 < ε) :
∃ d : E,A.output d=v ∧ ‖d‖ < A.seminorm v+ε := by
obtain ⟨d,hd,hb⟩ := Submodule.Quotient.norm_mk_lt
((LinearMap.ker A.output).mkQ (A.sectionMap v)) hε
refine ⟨d,?_,hb⟩
have hh : d-A.sectionMap v ∈ LinearMap.ker A.output := (Submodule.Quotient.eq _).mp hd
change A.output (d-A.sectionMap v)=0 at hh
rw [map_sub,A.right_inverse,sub_eq_zero] at hh
exact hh
lemma eq_sInf (v : F) : A.seminorm v=sInf {r : ℝ | ∃ d : E,A.output d=v ∧ ‖d‖=r} := by
let costs := {r : ℝ | ∃ d : E,A.output d=v ∧ ‖d‖=r}
have hn : costs.Nonempty := ⟨‖A.sectionMap v‖,A.sectionMap v,A.right_inverse v,rfl⟩
have hb : BddBelow costs := ⟨0,by rintro r ⟨d,hd,rfl⟩; exact norm_nonneg d⟩
apply le_antisymm
· exact le_csInf hn (by rintro r ⟨d,hd,rfl⟩; exact A.le_representation hd)
· apply le_of_forall_pos_le_add
intro ε hε
obtain ⟨d,hd,hh⟩ := A.near_representation v hε
exact (csInf_le hb ⟨d,hd,rfl⟩).trans hh.le
lemma bound (v : F) {C b : ℝ} (hC : 0 ≤ C) (hb : ∀ d : E,A.output d=v → b ≤ C*‖d‖) :
b ≤ C*A.seminorm v := by
by_cases hC0 : C=0
· simpa only [hC0,zero_mul] using hb (A.sectionMap v) (A.right_inverse v)
have hCpos : 0 < C := lt_of_le_of_ne hC (Ne.symm hC0)
apply le_of_forall_pos_le_add
intro ε hε
obtain ⟨d,hd,hh⟩ := A.near_representation v (div_pos hε hCpos)
have he := (hb d hd).trans (mul_le_mul_of_nonneg_left hh.le hC)
calc
_ ≤ C*(A.seminorm v+ε/C) := he
_ = C*A.seminorm v+ε := by field_simp
end Presentation
end InfimalCost
end DiamondDistortion
end
end OAI
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