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fix definition of InnerProductSpace #17

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Feb 2, 2017
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fix definition of InnerProductSpace #17

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3e697ee
fix definition of InnerProductSpace
langston-barrett Nov 22, 2016
054cce9
better names for vectorspace axioms
langston-barrett Dec 5, 2016
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17 changes: 10 additions & 7 deletions interfaces/vectorspace.v
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Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
Require Import
Require Import
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.orders.

(** Scalar multiplication function class *)
Expand All @@ -14,6 +14,7 @@ Notation "(· x )" := (λ y, y · x) (only parsing) : mc_scope.
Class Inproduct K V := inprod : V → V → K.
Instance: Params (@inprod) 3.

Notation "(⟨⟩)" := (inprod) (only parsing) : mc_scope.
Notation "⟨ u , v ⟩" := (inprod u v) (at level 51) : mc_scope.
Notation "⟨ u , ⟩" := (λ v, ⟨u,v⟩) (at level 50, only parsing) : mc_scope.
Notation "⟨ , v ⟩" := (λ u, ⟨u,v⟩) (at level 50, only parsing) : mc_scope.
Expand Down Expand Up @@ -83,12 +84,14 @@ Class InnerProductSpace (K V : Type)
{Ve Vop Vunit Vnegate} (* vector operations *)
{sm : ScalarMult K V} {inp: Inproduct K V} {Kle: Le K}
:=
{ in_vectorspace :>> @VectorSpace K V Ke Kplus Kmult Kzero Kone Knegate
Krecip Ve Vop Vunit Vnegate sm
; in_srorder :>> SemiRingOrder Kle
; in_comm :> Commutative inprod
; in_linear_l : ∀ a u v, ⟨a·u,v⟩ = a*⟨u,v⟩
; in_nonneg :> ∀ v, PropHolds (0 ≤ ⟨v,v⟩) (* TODO Le to strong? *)
{ in_vectorspace :>> @VectorSpace K V Ke Kplus Kmult Kzero Kone Knegate
Krecip Ve Vop Vunit Vnegate sm
; in_srorder :>> SemiRingOrder Kle
; in_comm :> Commutative inprod
; in_linear_l : ∀ a u v, ⟨a·u,v⟩ = a*⟨u,v⟩
; in_nonneg :> ∀ v, PropHolds (0 ≤ ⟨v,v⟩) (* TODO Le to strong? *)
; in_mon_unit_zero :> ∀ v, 0 = ⟨v,v⟩ <-> v = mon_unit
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Slightly offtopic: is there any good example of a module over a ring with something but "+" as the group operation? If not, why do we use a generic AbGroup instead of Add V and AbGroup (Aop := (+))?

; inprod_proper :> Proper ((=) ==> (=) ==> (=)) (⟨⟩)
}.

(* TODO complex conjugate?
Expand Down