Library ATBR.Examples


Examples about uses of the ATBR library


Require Import ATBR.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Setting


Section Setting.

We assume a typed Kleene algebra (as described in Classes)
  Context `{KA: KleeneAlgebra}.

This means we can write expressions like the following one, where
  • a # is the Kleene star of a,
  • a + b is the sum of a and b,
  • a × b is their product (or concatenation)
  • 1 is the neutral element for ×
  • 0 is the neutral element for +
  • == is the equality associated to this algebraic structure
  • <== is the preorder associated to + (x <== y iff x + y == y)

  Check A (a b c: X A A), 1+a#×b+(c×0+a×b×c)# == 1.

In the previous expression, A is a "type", it makes sense in situations like the following, where
  • R can be thought of as a relation from a set A to a set B,
  • S as a relation from set B to set A,
  • T as a relation from A to C
(As shown below, these `types' can also be seen as matrix dimensions.)

  Check A B C (R: X A B) (S: X B A) (T: X A C), ((R×S)# + 1) × T == 0.

End Setting.

Decision tactics


Section Tactics.

The main tactic of this library is kleene_reflexivity from file DecideKleeneAlgebra.v. This is a reflexive tactic that through automata constructions in order to solve (in)equations that are valid in any Kleene algebras:

  Section DKA.
  Context `{KA: KleeneAlgebra}.

  Variables A B: T.
  Variables a b c: X A A.
  Variable d: X A B.
  Variable e: X B A.

  Lemma star_distr: (a+b)# == a#×(b×a#)#.
    kleene_reflexivity.
  Qed.

  Goal (d×e)#×d == d×(e×d)#.
    kleene_reflexivity.
  Qed.

  Goal a#×(b+a#*(1+c))# == (a+b+c)#.
    kleene_reflexivity.
  Qed.

  Goal a×b×c×a×b×c×a# <== a#×(b×c+a)#.
    kleene_reflexivity.
  Qed.

Note that kleene_reflexivity cannot use hypotheses (Horn theory of KA is undecidable)

  Goal a×b <== ca×(b×a)# <== c#×a.
    intro H.
    try kleene_reflexivity.
    rewrite <- H.
    kleene_reflexivity.
  Qed.
  End DKA.

We also implemented reflexive decision tactics for the intermediate structures (lighter, faster). They work as soon as we provide enough structure (e.g. an idempotent semi-ring IdemSemiRing, or even a Monoid or a SemiLattice); they can of course be used to solve simple goals in richer settings, like Kleene Agebras.

  Section ISR.
  Context `{ISR: IdemSemiRing}.

  Variables A B: T.
  Variables a b c: X A A.
  Variable d: X A B.

  Goal (a+b)*(a+b) == a×a+a×b+b×a+b×b.
    semiring_reflexivity.
  Qed.

  Goal 0+b×a <== (a+b)*(1+a).
    semiring_reflexivity.
  Qed.

  Goal a*(b×1)*(c×d) == a×1×b×c×d.
    semiring_reflexivity.
  Qed.

  Goal a+(b+1)+(a+c) == 1+c+b+a+0.
    semiring_reflexivity.
  Qed.

  Goal a <== 1+c+b+a+0.
    semiring_reflexivity.
  Qed.

On these simpler structures, we also have `normalisation' tactics:
normalize expressions by expanding them
    semiring_normalize.
just remove zeros and ones
    Restart. semiring_clean.
remove zeros and ones and normalize parentheses
    Restart. semiring_cleanassoc.
    semiring_reflexivity.
  Qed.

Rewriting tactics

When rewriting terms, handling associativity and commutativity explicitly can be tedious. We implemented ad-hoc tactics for rewriting *closed* equations modulo A and/or AC; we plan to investigate this problem in a more systematic way.

  Goal c×d <== 0 → a×b×c×d <== 0.
    intro H.
parentheses do not match
    try rewrite H.
    monoid_rewrite H.
    semiring_reflexivity.
  Qed.

  Goal d <== c×da×b×d <== a×b×c×d.
    intro H.
parentheses do not match
    try rewrite <- H.
    monoid_rewrite <- H.
    semiring_reflexivity.
  Qed.

  Goal a+b+c <== cb+a+c+c <== c.
    intro H.
    ac_rewrite H.
    semiring_reflexivity.
  Qed.

  End ISR.
End Tactics.

Examples about matrices

Our development makes an heavy use of matrices, so that we had to develop a bit of matrix theory, that could be reused in other contexts. We give some examples about how to work with matrices in our setting.

Section Matrices.

  Require Import ATBR_Matrices.

Assume an underlying idempotent semi-ring
  Context `{ISR: IdemSemiRing}.

Notations are overloaded, thanks to the typeclass mechanism. We introduce specific notations to avoid confusion betwen matrices MX and their underlying elements X.

  Variable A: T.
type of matrices over (X A A)
  Notation MX := (@X (mx_Graph A)).
type of elements
  Notation X := (@X G).

Constant-to-a 2x2 matrix, with elements of type X A A
  Definition constant (a: X A A): MX 2 2 := box 2 2 (fun i ja).

To retrieve the elements of a matrix, we use "!" (a notation for the "get" operation)
  Goal a, !(constant a) O O == a.
  Proof.
    intros. reflexivity.
  Qed.

Dummy lemma, notice overloading of notations for ×
  Lemma square_constant a: constant a × constant a == constant (a×a).
  Proof.
since the dimensions are known (and finite), the matricial product can be computed
    simpl.
the mx_intros simple tactic introduces indices to prove a matricial equality; it is useful when considering vectors: only one dimension is introduced
    mx_intros i j Hi Hj.
    simpl.
    semiring_reflexivity.
  Qed.

Our tactics automatically work for matrices (matrices are just another idempotent semiring)
  Goal n m p (M: MX n m) (N: MX m p) (P: MX n p),
    M×1×N + P == P+M×N.
  Proof.
    intros.
    semiring_reflexivity.
  Qed.

Block matrices manipulation
  Lemma square_triangular_blocks n m (M: MX n n) (N: MX n m) (P: MX m m):
    mx_blocks M N 0 P × mx_blocks M N 0 P == mx_blocks (M×M) (M×N+N×P) 0 (P×P).
  Proof.
    intros.
    rewrite mx_blocks_dot.
    apply mx_blocks_compat; semiring_reflexivity.
  Qed.

(We will clean-up and document this library for matrices at some point, so that we do not give further details for now.)

End Matrices.

Using concrete structures

To work with a concrete given struture, you need to show that it satisfies the corresponding axioms. Examples are given in files Model*.v
For example, it is shown in Model_Relations.v that (heterogeneous) binary relations form a Kleene algebra with converse. This file can easily be adapted to use other definitions.

Section Concrete.

  Require Import Model_Relations.
  Import Load.

Any theorem we proved in an abstract structure now applies to binary relations
  Variable A: Set.
  Variables R S: rel A A.
  Check (star_distr R S).

tactics work out of the box when using our notations
  Goal R×S==RR*(S+R#) == R#×R.
  Proof.
    intro H.
    rewrite dot_distr_right, H.
    kleene_reflexivity.
  Qed.

  Canonical Structure rel_Monoid_Ops.
  Goal R×S==Rrel_comp R (S+R#) == rel_comp (R#) R.
  Proof.
    intro H.
    rewrite dot_distr_right, H.
    fold_relAlg.
    kleene_reflexivity.
  Qed.

End Concrete.

Similarly, homogeneous relations (from the standard library) are declared in Model_StdRelations, so that one can use our tactics to reason about these.

Section Concrete´.

  Require Import Relations.
  Require Import Model_StdRelations.
  Import Load.

  Variable A: Set.
  Variables R S: relation A.

  Lemma example: same_relation _
    (clos_refl_trans _ (union _ R S))
    (comp (clos_refl_trans _ R) (clos_refl_trans _ (comp S (clos_refl_trans _ R)))).
  Proof.
    intros.
    fold_relAlg A.
    kleene_reflexivity.
  Qed.
  Print Assumptions example.

End Concrete´.