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# Library prosa.implementation.refinements.FP.refinements

Require Export prosa.implementation.refinements.refinements.
Require Export prosa.implementation.refinements.FP.fast_search_space.
Require Export prosa.implementation.refinements.arrival_bound.
Throughout this file, we work with Prosa's fixed-priority policy implementation.
#[local] Existing Instance NumericFPAscending.
We begin by defining a generic version of the functions we seek to define.
Section Definitions.
Consider a generic type T supporting basic arithmetic operations, neutral elements, and comparisons.
Context `{T : Type}.
Context `{!zero_of T, !one_of T, !sub_of T, !add_of T, !mul_of T,
!div_of T, !mod_of T, !eq_of T, !leq_of T, !lt_of T}.
Context `{!eq_of (seq T)}.
Context {eq_of2 : eq_of (@task_T T)}.
We define a generic version of higher-or-equal priority task, ...
Definition hep_task_T tsk_o tsk := (task_priority_T tsk ≤ @task_priority_T T tsk_o)%C.
... of total request-bound function of higher-or-equal priority task, ...
Definition total_hep_rbf_T (ts : seq task_T) (tsk : task_T) (Δ : T) : T :=
let hep_ts := filter (fun tsk' ⇒ hep_task_T tsk' tsk) ts in
let work_ts := map (fun tsk' ⇒ task_rbf_T tsk' Δ) hep_ts in
foldr +%C 0%C work_ts.
... an analogous version of hep_task in without symmetry, ...
Definition ohep_task_T tsk_o tsk :=
(hep_task_T tsk_o tsk) && (~~ (eq_of2 tsk_o tsk)%C).
... with a respective cumulative version.
Definition total_ohep_rbf_T (ts : seq task_T) (tsk : task_T) (Δ : T) : T :=
let hep_ts := filter (fun tsk' ⇒ ohep_task_T tsk' tsk) ts in
let work_ts := map (fun tsk' ⇒ task_rbf_T tsk' Δ) hep_ts in
foldr +%C 0%C work_ts.
Next, we define a generic version of check_point_FP, ...
Definition check_point_FP_T (ts : seq task_T) (tsk : task_T) (R : T) (P : T × T) : bool :=
(task_rbf_T tsk (P.1 + 1) + total_ohep_rbf_T ts tsk (P.1 + P.2)
≤ P.1 + P.2)%C && (P.2 ≤ R)%C.
... a generic version of blocking_bound_NP, ...
Definition blocking_bound_NP_T (ts : seq task_T) (tsk : task_T) :=
let lp_ts := filter (fun tsk_o ⇒ ~~ hep_task_T tsk_o tsk) ts in
let block_ts := map (fun tsk_o ⇒ task_cost_T tsk_o - 1)%C lp_ts in
foldr maxn_T 0%C block_ts.
... and of check_point_NP.
Definition check_point_NP_T (ts : seq task_T) (tsk : task_T) (R : T) (P : T × T) : bool :=
(blocking_bound_NP_T ts tsk
+ (task_rbf_T tsk (P.1 + 1) - (task_cost_T tsk - 1))
+ total_ohep_rbf_T ts tsk (P.1 + P.2) ≤ P.1 + P.2)%C
&& (P.2 + (task_cost_T tsk - 1) ≤ R)%C.
End Definitions.
We introduce some functions operating on binary numbers.
We provide a definition of iota, ...
Definition iota_N (a Δ : N) := iota_T a Δ.
... of search_space_emax_FP_h, ...
Definition search_space_emax_FP_h_N (tsk : task_T) (l r : N) : seq N :=
let h := get_horizon_of_task_T tsk in
let offsets := map (N.mul h) (iota_N l r) in
let emax_offsets := repeat_steps_with_offset_T tsk offsets in
map predn_T emax_offsets.
... and of search_space_emax_FP.
Definition search_space_emax_FP_N (tsk : task_T) (L : N) :=
let h := get_horizon_of_task_T tsk in
search_space_emax_FP_h_N tsk 0 (((L %/ h)+1))%C.

## Refinements

We now establish the desired refinements.
First, we prove a refinement for the search_space_emax_h function.
Local Instance refine_search_space_emax_h :
refines (Rtask ==> Rnat ==> Rnat ==> list_R Rnat)%rel
search_space_emax_FP_h
search_space_emax_FP_h_N.
Proof.
rewrite refinesE ⇒ tsk tsk' Rtsk l l' Rl r r' Rr.
apply refinesP; unfold search_space_emax_FP_h, search_space_emax_FP_h_N.
by refines_apply; refines_abstr.
Qed.
Next, we prove a refinement for the search_space_emax_FP function.
Global Instance refine_search_space_emax :
∀ tsk,
refines (Rnat ==> list_R Rnat)%rel
(search_space_emax_FP (taskT_to_task tsk))
(search_space_emax_FP_N tsk).
Proof.
intros.
rewrite refinesE ⇒ δ δ' Rδ.
unfold search_space_emax_FP, search_space_emax_FP_N.
apply refinesP; rewrite -addn1.
by refines_apply; rewrite refinesE; unfold Rtask, fun_hrel.
Qed.
Next, we prove a refinement for the hep_task function.
Global Instance refine_hep_task :
refines ( Rtask ==> Rtask ==> bool_R )%rel hep_task hep_task_T.
Proof.
apply refines_abstr2.
rewrite /hep_task /hep_task_T /NumericFPAscending
/numeric_fixed_priority.task_priority /TaskPriority.
move⇒ tsk1 tsk1' Rtsk1 tsk2 tsk2' Rtsk2.
by refines_apply.
Qed.
Next, we prove a refinement for the total_hep_rbf function.
Global Instance refine_total_hep_rbf :
refines ( list_R Rtask ==> Rtask ==> Rnat ==> Rnat )%rel total_hep_rbf total_hep_rbf_T.
Proof.
apply refines_abstr ⇒ ts ts' Rts.
apply refines_abstr ⇒ tsk tsk' Rtsk.
apply refines_abstr ⇒ Δ Δ' RΔ.
rewrite /total_hep_rbf /total_hep_request_bound_function_FP /total_hep_rbf_T.
eapply refine_foldr.
- by apply Rts.
- by apply refines_abstr ⇒ tsk1 tsk1' Rtsk1; tc.
- by apply refines_abstr ⇒ tsk1 tsk1' Rtsk1; tc.
Qed.
Next, we prove a special-case refinement for the total_hep_rbf function when applied to a refined task set and a refined task. This special case is required to guide the typeclass engine.
Global Instance refine_total_hep_rbf' :
∀ ts tsk,
refines (Rnat ==> Rnat)%rel
(total_hep_rbf (map taskT_to_task ts) (taskT_to_task tsk))
(total_hep_rbf_T ts tsk) | 0.
Proof.
move⇒ ts tsk; rewrite refinesE ⇒ δ δ' Rδ.
move: refine_task_set' ⇒ RTS.
rewrite refinesE in RTS.
specialize (RTS ts ts (unifyxx _)); simpl in RTS.
apply refinesP.
move: refine_total_hep_rbf ⇒ EQ.
refines_apply.
move: refine_task' ⇒ Rtsk.
rewrite refinesE; rewrite refinesE in Rtsk.
by apply Rtsk.
Qed.
Next, we provide equality comparisons between different pairs of objects manipulated in Poet's certificates.
Global Instance eq_listN : eq_of (seq N) := fun x y ⇒ x == y.
Global Instance eq_listNN : eq_of (seq (prod N N)) := fun x y ⇒ x == y.
Global Instance eq_NlistNN : eq_of (prod N (seq (prod N N))) := fun x y ⇒ x == y.
Global Instance eq_taskab : eq_of (@task_arrivals_bound_T N) := taskab_eqdef_T.
Global Instance eq_task : eq_of (@task_T N) := task_eqdef_T.
Next, we prove a refinement for the task_eqdef function.
Global Instance refine_task_eqdef :
refines (Rtask ==> Rtask ==> bool_R)%rel task_eqdef task_eqdef_T.
Proof.
apply refines_abstr2.
intros a a' Ra b b' Rb.
unfold task_eqdef, task_eqdef_T.
by rewrite refinesE; repeat apply andb_R; apply refinesP; refines_apply.
Qed.
Next, we prove a refinement for the ohep_task function.
Global Instance refine_ohep_task :
refines ( Rtask ==> Rtask ==> bool_R )%rel ohep_task ohep_task_T.
Proof.
apply refines_abstr2.
rewrite /ohep_task /ohep_task_T /NumericFPAscending
/numeric_fixed_priority.task_priority /TaskPriority.
move⇒ tsk1 tsk1' Rtsk1 tsk2 tsk2' Rtsk2.
by refines_apply.
Qed.
Next, we prove a refinement for the total_ohep_rbf function.
Global Instance refine_total_ohep_rbf :
refines ( list_R Rtask ==> Rtask ==> Rnat ==> Rnat )%rel total_ohep_rbf total_ohep_rbf_T.
Proof.
apply refines_abstr ⇒ ts ts' Rts.
apply refines_abstr ⇒ tsk tsk' Rtsk.
apply refines_abstr ⇒ Δ Δ' RΔ.
rewrite /total_hep_rbf /total_hep_request_bound_function_FP.
eapply refine_foldr.
- by apply Rts.
- by apply refines_abstr ⇒ tsk1 tsk1' Rtsk1; tc.
- by unfold ohep_task_T; apply refines_abstr ⇒ tsk1 tsk1' Rtsk1; tc.
Qed.
Next, we prove a refinement for the check_point_FP function.
Global Instance refine_check_point :
refines (list_R Rtask ==> Rtask ==> Rnat ==> prod_R Rnat Rnat ==> bool_R)%rel
check_point_FP check_point_FP_T.
Proof.
rewrite refinesE ⇒ ts ts' Rts tsk tsk' Rtsk l l' Rl p p' Rp.
by unfold check_point_FP, check_point_FP_T; apply andb_R; apply refinesP; tc.
Qed.
Next, we prove a special-case refinement for the check_point_FP function when applied to a refined task set and a refined task. This special case is required to guide the typeclass engine.
Global Instance refine_check_point' :
∀ ts tsk,
refines (Rnat ==> prod_R Rnat Rnat ==> bool_R)%rel
(check_point_FP (map taskT_to_task ts) (taskT_to_task tsk))
(check_point_FP_T ts tsk) | 0.
Proof.
intros ts tsk; rewrite refinesE ⇒ l l' Rl p p' Rp.
unfold check_point_FP, check_point_FP_T; apply andb_R; apply refinesP; tc.
move: refine_task' ⇒ RT.
move: refine_task_set' ⇒ RTS.
rewrite refinesE in RT; rewrite refinesE in RTS.
by refines_apply; rewrite refinesE; [apply RT | apply RTS | apply RT].
Qed.
Next, we prove a refinement for the blocking_bound_NP function.
Global Instance refine_blocking_bound :
refines ( list_R Rtask ==> Rtask ==> Rnat )%rel blocking_bound_NP blocking_bound_NP_T.
Proof.
rewrite refinesE ⇒ ts ts' Rts tsk tsk' Rtsk.
unfold blocking_bound, blocking_bound_NP_T.
apply refinesP; eapply refine_foldr_max.
- by rewrite refinesE; apply Rts.
- by apply refines_abstr ⇒ tsk1 tsk1' Rtsk1; tc.
- by apply refines_abstr ⇒ tsk1 tsk1' Rtsk1; tc.
Qed.
Next, we prove a special-case refinement for the blocking_bound_NP function when applied to a refined task set and a refined task. This special case is required to guide the typeclass engine.
Global Instance refine_blocking_bound' :
∀ ts tsk,
refines ( Rnat )%rel
(blocking_bound_NP (map taskT_to_task ts) (taskT_to_task tsk))
(blocking_bound_NP_T ts tsk) | 0.
Proof.
intros ts tsk; rewrite refinesE.
move: refine_task' ⇒ RT.
move: refine_task_set' ⇒ RTS.
rewrite refinesE in RT; rewrite refinesE in RTS.
by apply refinesP; refines_apply; (rewrite refinesE; (apply RT || apply RTS)).
Qed.
Next, we prove a refinement for the check_point_NP function.
Global Instance refine_check_point_NP :
refines (list_R Rtask ==> Rtask ==> Rnat ==> prod_R Rnat Rnat ==> bool_R)%rel
check_point_NP check_point_NP_T.
Proof.
rewrite refinesE ⇒ ts ts' Rts tsk tsk' Rtsk l l' Rl p p' Rp.
by unfold check_point_NP, check_point_NP_T; apply andb_R; apply refinesP; tc.
Qed.
Finally, we prove a special-case refinement for the check_point_NP function when applied to a refined task set and a refined task. This special case is required to guide the typeclass engine.
Global Instance refine_check_point_NP' :
∀ ts tsk,
refines (Rnat ==> prod_R Rnat Rnat ==> bool_R)%rel
(check_point_NP (map taskT_to_task ts) (taskT_to_task tsk))
(check_point_NP_T ts tsk) | 0.
Proof.
intros ts tsk; rewrite refinesE ⇒ l l' Rl p p' Rp.
move: refine_task' ⇒ RT.
move: refine_task_set' ⇒ RTS.
rewrite refinesE in RT; rewrite refinesE in RTS.
unfold check_point_NP, check_point_NP_T; apply andb_R; apply refinesP; refines_apply.
all: try by (rewrite refinesE; (apply RT || apply RTS)).
Qed.