bedrock.lang.cpp.bi.split_cfrac_tests

(*
* Copyright (c) 2022 BlueRock Security, Inc.
*
* This software is distributed under the terms of the BedRock Open-Source License.
* See the LICENSE-BedRock file in the repository root for details.
*)

Require Import bedrock.prelude.base.
Require Import bedrock.lang.cpp.bi.split_cfrac.

#[local] Set Printing Coercions.

Split q q1 q2 succeeds if SplitCFrac q outputs q1, q2.

Class Split (q q1 q2 : cQp.t) : Prop := split : SplitCFrac q q1 q2.
#[global] Hint Mode Split + + + : typeclass_instances.
#[global] Instance split_cfrac q q'1 q'2 q1 q2 :
SplitCFrac q q'1 q'2 -> TCEq q1 q'1 -> TCEq q2 q'2 -> Split q q1 q2 | 10.
Proof. by rewrite !TCEq_eq=>? ->->. Qed.

Section split.

Splitting on cQp.add is immediate

  Lemma immediate p q : Split (p + q) p q.
  Proof. apply _. Abort.
  Lemma immediate p q : Split (cmra.op p q) p q.
  Proof. apply _. Abort.

Splitting on cQp.scale q preserves the factor q

  Lemma subterm q q1 q2 :
    Split (cQp.scale q (q1 + q2))
      (cQp.scale q q1)
      (cQp.scale q q2).
  Proof. apply _. Abort.

  Lemma subterm p q q1 q2 :
    Split (cQp.scale p (cQp.scale q (q1 + q2)))
      (cQp.scale p (cQp.scale q q1))
      (cQp.scale p (cQp.scale q q2)).
  Proof. apply _. Abort.

Otherwise, splitting works componentwise.

See ../../bi/split_andb_tests.v and ../../bi/split_frac_tests.v for more examples of the underlying splits.

  Lemma const q1 q2 : Split (cQp.const (q1 + q2)) (cQp.const q1) (cQp.const q2).
  Proof. apply _. Abort.

  Lemma mut q1 q2 : Split (cQp.mut (q1 + q2)) (cQp.mut q1) (cQp.mut q2).
  Proof. apply _. Abort.

  Lemma general c1 c2 q1 q2 :
    Split (cQp.mk (c1 && c2) (q1 + q2)) (cQp.mk c1 q1) (cQp.mk c2 q2).
  Proof. apply _. Abort.

  Lemma mk c q :
    let half := cQp.mk c (q / 2) in
    Split (cQp.mk c q) half half.
  Proof. cbn. apply _. Abort.

  Lemma var cq :
    let half := cQp.mk (cQp.is_const cq) (cQp.frac cq / 2) in
    Split cq half half.
  Proof. cbn. apply _. Abort.

  Lemma scale_var q cq :
    let half := cQp.scale q (cQp.mk (cQp.is_const cq) (cQp.frac cq / 2)) in
    Split (cQp.scale q cq) half half.
  Proof. cbn. apply _. Abort.

End split.

Combine q1 q2 q succeeds if CombineCFrac q1 q2 outputs q.

Class Combine (q1 q2 q : cQp.t) : Prop := combine : CombineCFrac q1 q2 q.
#[global] Hint Mode Combine + + + : typeclass_instances.
#[global] Instance combine_frac q1 q2 q' q :
CombineCFrac q1 q2 q' -> TCEq q q' -> Combine q1 q2 q | 10.
Proof. by rewrite TCEq_eq=>? ->. Qed.

Section combine.

Combining terms of the form cQp.scale q _ preserves that structure.

  Lemma scale q q1 q2 :
    Combine (cQp.scale q q1) (cQp.scale q q2) (cQp.scale q (q1 + q2)).
  Proof. apply _. Abort.
  Lemma scale p q q1 q2 :
    Combine
      (cQp.scale p (cQp.scale q q1))
      (cQp.scale p (cQp.scale q q2))
      (cQp.scale p (cQp.scale q (q1 + q2))).
  Proof. apply _. Abort.

  Lemma scale_mut q q1 q2 :
    Combine (cQp.scale q (cQp.mut q1)) (cQp.scale q (cQp.mut q2))
      (cQp.scale q (cQp.mut (q1 + q2))).
  Proof. apply _. Abort.

Otherwise, combining works componentwise and then folds cQp.add and eta reduces cQp.mk.

See ../../bi/split_andb_tests.v and ../../bi/split_frac_tests.v for more examples of how components combine.

  Lemma var q : Combine q q (q + q).
  Proof. apply _. Abort.
  Lemma var q1 q2 : Combine q1 q2 (q1 + q2).
  Proof. apply _. Abort.

  Lemma const q1 q2 : Combine (cQp.const q1) (cQp.const q2) (cQp.const (q1 + q2)).
  Proof. apply _. Abort.

  Lemma mut q1 q2 : Combine (cQp.mut q1) (cQp.mut q2) (cQp.mut (q1 + q2)).
  Proof. apply _. Abort.

  Lemma general c1 c2 q1 q2 :
    Combine (cQp.mk c1 q1) (cQp.mk c2 q2) (cQp.mk (c1 && c2) (q1 + q2)).
  Proof. apply _. Abort.

  Lemma mk c q :
    let half := cQp.mk c (q / 2) in
    Combine half half (cQp.mk c q) .
  Proof. cbn. apply _. Abort.

  Lemma var cq :
    let half := cQp.mk (cQp.is_const cq) (cQp.frac cq / 2) in
    Combine half half cq.
  Proof. cbn. apply _. Abort.

  Lemma scale_var q cq :
    let half := cQp.scale q (cQp.mk (cQp.is_const cq) (cQp.frac cq / 2)) in
    Combine half half (cQp.scale q cq).
  Proof. cbn. apply _. Abort.

End combine.