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Set Automatic Coercions Import.
Require Import util c_util stability.
Require geometry abstract abstract_cont_trans_over.
Require Import Morphisms.
Set Implicit Arguments.
Open Scope CR_scope.
Section contents.
Variables
(chs: concrete.System)
(component: morpher (@cs_eq (concrete.Point chs) ==> @cs_eq CRasCSetoid)%signature)
(Region: Set).
Context
{Region_eq_dec: EquivDec.EqDec Region eq}
{regions: ExhaustiveList Region}
(NoDup_regions: NoDup regions)
(bounds: Region -> geometry.OpenQRange)
(select_interval: forall l p, concrete.invariant (l, p) ->
DN (sig (fun i => geometry.in_orange (bounds i) (component p)))).
Definition in_region (p: concrete.Point chs) (r: Region): Prop :=
geometry.in_orange (bounds r) (component p).
Instance in_region_mor: Proper (@cs_eq _ ==> eq ==> iff) in_region.
Proof with auto.
unfold in_region. repeat intro.
apply geometry.in_orange_mor.
subst. split...
rewrite H...
Qed.
Definition space: abstract.Space chs :=
abstract.Build_Space chs _ _ NoDup_regions
in_region_mor select_interval.
Definition hints (l: concrete.Location chs) (r r': abstract.Region space) (E: r <> r'):
(forall x, strongly_increasing (component ∘ concrete.flow chs l x)) ->
option (abstract_cont_trans_over.strong_redundant space l E).
Proof with auto.
intros.
unfold abstract_cont_trans_over.strong_redundant, containers.In.
simpl abstract.in_region.
unfold in_region.
destruct (bounds r) as [[ci_lo ci_hi] ci_le].
destruct (bounds r') as [[ci'_lo ci'_hi] ci'_le].
unfold geometry.in_orange, geometry.orange_left, geometry.orange_right.
simpl @fst. simpl @snd.
destruct ci_lo; [ | exact None].
destruct ci'_hi; [ | exact None].
destruct (Qeq_dec q q0) as [A|B]; [ | exact None].
apply Some.
intros p [H0 H4] t [H1 H5].
apply (strongly_increasing_inv_mild (X p)).
unfold Basics.compose.
rewrite flow.flow_zero.
apply CRle_trans with ('q)...
rewrite A...
Defined.
End contents.