Library RelationAlgebra.kat_tac

kat_tac: decision tactics for KA, KAT, and KAT with Hoare hypotheses

corollary of kat_untyping and kat_completeness:

one can decide typed KAT by reasoning about untyped languages

Section j.
Notation Sigma := positive.
Variable Pred: nat.
Variables src tgt: Sigma positive.

Corollary kat_untype_weq n m (e f: gregex Pred src tgt n m): gerase e == gerase f e == f.
   intro H. apply kat_complete_weq.
   rewrite 2untype_glang.
   now apply traces.restrict_weq.
  apply gerase_weq.

End j.

we can thus refine the reification lemma provided in kat_reification
Corollary kat_weq_dec `{L: laws} f' fs fp n m (e f: @kat_expr X f' fs n m):
  e_level e + e_level f << BKA
  (let v := vars (e_pls e f) in
    eqb_kat (gerase (to_gregex v n m e)) (gerase (to_gregex v n m f)) = Some true)
  eval fp n m e == eval fp n m f.
  intros Hl H. apply to_gregex_weq. assumption.
  apply kat_untype_weq, eqb_kat_correct, H.

kat tactic, for Kleene algebra with tests

the kat tactic solves KAT (in)equalities; it proceeds as follows:
0. possibly convert inequalities into equalities, using leq_iff_cup; 1. look for the kat structure and laws, by calling typeclass resolution using (catch_kat_weq); 2. call the ML reification tactic
  • > the equivalence will be checked in OCaml, an a counter-example will be displayed if it does not hold
  • > this will otherwise produce a sequence of let-in's for the various reification ingredients
3. introduce those ingredients and apply the above reification lemma 4. close the two generated subgoals by reflexivity (the first one corresponds to checking that reified terms did no include forbidden constructions - it should never fail unless there is a bug in reification; the second one corresponds to the execution of the Coa KAT decision algorithm - it should never fail since the equation was already checked in OCaml)

Lemma catch_kat_weq {X} {L: laws X} n m (x y: X n m):
  (let L:=L in x == y) x == y.
Proof. trivial. Qed.

parametrised tactic to do the work share by ka, kat, and hkat b indicates whether we are in KAT (true) or KA (false)
Ltac pre_dec b :=
  let L := fresh "L" in intro L;
  let tenv := fresh "tenv" in
  let env := fresh "env" in
  let penv := fresh "penv" in
  let lhs := fresh "lhs" in
  let rhs := fresh "rhs" in
  ra_kat_reify b; intros tenv env penv ? ? lhs rhs;
  apply (@kat_weq_dec _ L tenv env penv _ _ lhs rhs); [
    (reflexivity || fail 1 "Bug in KAT reification, please report") |
    (close_by_reflection (Some true) ||
        fail 1 "Not a KAT theorem, but no counter-example can be displayed. Please report.") ].

kat tactic
Ltac kat :=
  intros; rewrite ?leq_iff_cup;
    (apply catch_kat_weq || fail "could not find a KAT structure");
    pre_dec true.

ka tactic, for Kleene algebra

we use the fact that any Kleene algebras with bottom faithfully embed into KAT, using trivial tests
Module bka_to_kat.
Definition ops (X: monoid.ops) := mk_ops X (fun _bool_lattice_ops) (fun _ofbool).
Lemma laws `{monoid.laws} `{BKA<<l}: kat.laws (ops X).
  apply lower_laws.
  intro. apply lower_lattice_laws.
  intro. constructor; try discriminate.
  apply ofbool_leq.
  apply op_leq_weq_1.
  intros _ ? ?. apply orb_pls.
  intros _. reflexivity.
  intros ?. reflexivity.
  intros ? ? ?. apply andb_dot.
End bka_to_kat.

the tactic is really similar to kat, except that we catch the KAT laws using the lemma below, exploiting the above embedding
Lemma catch_ka_weq {X l} {L: monoid.laws l X} {Hl: BKA<<l} n m (x y: X n m):
  (let L:=@bka_to_kat.laws l X L Hl in @weq (@kar (bka_to_kat.ops X) n m) x y) x == y.
Proof. trivial. Qed.

Ltac ka :=
  intros; rewrite ?leq_iff_cup;
    (apply catch_ka_weq || fail "could not find a KA structure");
    pre_dec false.

hkat tactic, for KAT with Hoare hypotheses

Hypotheses of the shape x == 0, called "Hoare hypotheses", can be eliminated in KAT.
In other words, the Horn theory of KAT restricted to the clauses wher all hypotheses have the above shape, reduces to the equational theory of KAT, and is thus decidable.
Moreover, some other kinds of hypotheses can be transformed into Hoare ones, and hypotheses of the shape [c];p == [c] can also be eliminated.
All in all, a non-trivial class of hypotheses can be handled in practice.
We perform such an elimination of hypotheses to obtain the hkat tactic below.
For more details, see Chris Hardin and Dexter Kozen. "On the elimination of hypotheses in Kleene algebra with tests". TR2002-1879, Computer Science Department, Cornell University, October 2002
converting various kinds of hypotheses to Hoare ones
Lemma ab_to_hoare `{L: laws} {n} (b c: tst n): b == c [b\cap !c \cup !b\cap c] <== 0.
Proof. intro H. rewrite H. kat. Qed.

Lemma ab'_to_hoare `{L: laws} {n} (b c: tst n): b <== c [b\cap !c] <== 0.
Proof. intro H. rewrite H. kat. Qed.

Lemma bpqc_to_hoare `{L: laws} {n m} (b: tst n) (c: tst m) p q:
  [b]×p <== q×[c] [b]×p×[!c] <== 0.
Proof. intro H. rewrite H. kat. Qed.

Lemma pbcq_to_hoare `{L: laws} {n m} (b: tst n) (c: tst m) p q:
  p×[b] <== [c]×q [!c]×p×[b] <== 0.
Proof. rewrite <-dotA. dual @bpqc_to_hoare. Qed.

Lemma qpc_to_hoare `{L: laws} {n m} (c: tst m) (p q: X n m):
  q <== p×[c] q×[!c] <== 0.
Proof. intro H. rewrite H. kat. Qed.

Lemma qcp_to_hoare `{L: laws} {n m} (c: tst m) (p q: X m n):
  q <== [c]×p [!c]×q <== 0.
Proof. dual @qpc_to_hoare. Qed.

Lemma cp_c `{L: laws} {n} (c: tst n) (p: X n n):
  [c]×p == [c] p == [!c]×p+[c].
Proof. intro H. rewrite <-H. kat. Qed.

Lemma pc_c `{L: laws} {n} (c: tst n) (p: X n n):
  p×[c] == [c] p == p×[!c]+[c].
Proof. dual @cp_c. Qed.

merging Hoare hypotheses
Lemma join_leq `{lattice.laws} `{CUP<<l} (x y z: X): x<==z y<==z x\cup y<==z.
Proof. rewrite cup_spec. tauto. Qed.

eliminating Hoare hypotheses ; u and v are intended to be the universal expressions of the appropriate type
Lemma elim_hoare_hypotheses_weq `{L: laws} {n m p q} (u: X n p) (v: X q m) (z: X p q) (x y: X n m):
  z <== 0 x+u×z×v == y+u×z×v x==y.
Proof. rewrite leq_xb_iff. intro Hz. now rewrite Hz, dotx0, dot0x, 2cupxb. Qed.

Lemma elim_hoare_hypotheses_leq `{L: laws} {n m p q} (u: X n p) (v: X q m) (z: X p q) (x y: X n m):
  z <== 0 x <== y+u×z×v x<==y.
Proof. intro Hz. now rewrite Hz, dotx0, dot0x, cupxb. Qed.

tactic used to aggregate Hoare hypotheses: convert hypotheses into Hoare ones using the above lemas, then merge them into a single one
Ltac aggregate_hoare_hypotheses :=
    match goal with
      | H: _ == _ |- _
        apply ab_to_hoare in H ||
        (rewrite (cp_c _ _ H); clear H) ||
        (rewrite (pc_c _ _ H); clear H) ||
        apply weq_spec in H as [? ?]
    match goal with
      | H: _ <== _ |- _
        apply ab'_to_hoare in H ||
        apply bpqc_to_hoare in H ||
        apply pbcq_to_hoare in H ||
        apply qcp_to_hoare in H ||
        apply qpc_to_hoare in H
      | H: _ <== 0, H': _ <== 0 |- _
        apply (join_leq _ _ _ H') in H; clear H'

final hkat tactic: aggregate Hoare hypotheses, possibly transform inequalities into equalities, get the alphabet to construct the universal expression, eliminate Hoare hypotheses using the above lemma, and finally call the kat tactic
Ltac hkat :=
  intros; aggregate_hoare_hypotheses; rewrite ?leq_iff_cup;
  (apply catch_kat_weq || fail "could not find a KAT structure");
  let L := fresh "L" in intro L;
  let u := fresh "u" in
  ((ra_get_kat_alphabet; intro u;
    eapply (elim_hoare_hypotheses_weq (u^*) (u^*)); [eassumption|])
  || fail "typed hkat is not supported yet");
  subst u; revert L; pre_dec true.