, If g ? i f i is autonomous, then g ? f is autonomous; likewise, if g ? i 1 f i 1 ? i 2 f i 2 . . . is autonomous for some pairwise distinct i 1 , i 2
, Laws 1 and 2 are proved by induction on M , using algebraic reasoning and Lemma 2.2.10. Law 1 is needed to show law 2
,
,
, Since x is the eager normal-form of M and N , x ? y. Note that y can contain more names
, then the equation uses a purely-divergent term. We choose the encoding of ? for this
, ) I
, then the equation encodes an abstraction whose body refers to the normal forms of M , N , via the variable X M
, z). !z(x, q, p.p
, we separate the evaluation contexts and the values, as in Definition 2.2.3. In the body of the equation, this is achieved by: (i) rewriting C e [xV ] into (?z. C e [z])(xV ), for some fresh z, and similarly for C e and V (such a transformation is valid for , as per law 2.1, which is established in Lemma 2.2.22); and (ii) referring to the equation variable associated to the evaluation contexts
,
, pre-fixed point of E for S can be obtained from a pre-fixed point of E (for S) by removing the components corresponding to indices in J
, 2. the same as (1) with post-fixed points in place of pre-fixed points
, Consider two systems of equations E and E where E extends E with respect to tr . Furthermore, suppose E is guarded and has a divergence-free syntactic solution. If F is a pre-fixed point for tr of E, and G a post-fixed point
, -eager normal-form similarity, written ? ? . It is obtained by imposing that M ? ? N for all N , whenever M is divergent. Thus, with respect to the bisimilarity relation ? , we only have to change clause
, A relation R between ?-terms is an ?-eager normal-form simulation if
,
,
,
,
, C e [xV ] for some x, z, V and C e such that V Rz and C e [y]Ry for some y fresh in C e
, ?-eager normal form similarity is the largest ?-eager normal-form simulation
, For any M, N ? ?, we have M ? ? N iff I
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, =? Q for some Q, then there exists P and n such that
, ? Q for some Q, then there exists P and n such that
, Let E be a guarded system of equations, and K E its syntactic solution. Suppose K E has no divergence. If F is a post-fixed point for tr of E
, For simplicity, we only give the proof for a single equation E, rather than a system of equations. The generalisation to systems of equations is straightforward. Consider an equation E, a process abstraction F , and suppose F tr E[F ]. We fix a set of fresh names a, and write P for F a
, Sets of Operations and unique solution Theorem 1. O contains the identity function
, O is closed by composition (that is, f ? g ? O whenever f, g ? O)
, A 'symmetric variant' of clause 3 always holds: if f is autonomous, then so is f ? g: indeed, transitions of f (x) do not depend on x, hence transitions of f (g(x)) do not depend on either x or g. Clause 4 expresses congruence of the equivalence w.r.t. the set of contexts: here, the state functions in O. The autonomous transitions of a set of R-operations yield an LTS whose states are the operations themselves
, ?q (r q | q p) | ?y (z y | y x )) and z(r, z ). (r p | z x ) (for all non-continuation names x or x , z
, We show that this is an expansion up to expansion and contexts, using the previous laws (each of those processes has only one possible action)
, We have: 1. ?x (I
,
, By induction over the structure of M . We use Lemma 2.2.10, and usual laws of expansion, vol.01
, | x y) = ?x (p(z ). z z | x y) ? p(z ). z z = I
, | x y) = ?x (p(z). z x | x y) ? p(z). ?x (z x | x y) p(z). z y (by lemma 2.2.10) ?z. M : ?x (I, | x y) = ?x (p(z) . !z (z, q). I
, | x y) = ?x (?q (I
, ? ?q (?x (I
, | x y) | q(u). ?r (r(w). u(w , r ). (w w | r p) | ?x (I
, ?p (I[[x]] p | p q) = ?p (p(y). y x | p q) ?y (y x | q(z)
, ? q(z). ?y (z y | y x) q(z). z x (by Lemma 2.2.10) ?x. M : ?p (I
, | y(x, r). (w x | r p)) q(z). !z(w, p). ?x, r (I[[M ]] r | w x | r p) q(z). !z(w, p). I[[M {x/w}]] p (by induction hypothesis and using case 1) 2 : ?p (I
, | r s))) (by Lemma 2.2.10)
, | x y) ? !x(z, q). ?w, p (I, ? If V = ?x. M , then I V
, = !y(w, q). z(w , q ). (w w | q q) = y z
, Validity of ? v -reduction). For any M, N in ?, M ?? N implies I
, exploiting algebraic properties of replication; then the result follows by the compositionality of the encoding and the congruence of . We start by simplifying I
, = ?q (q(y). !y(x, s). I[[M ]] s | q(y). ?r ( r(w). I V, | r(w). y(w , p )
, ?x, p.p p | ?w
, | ?w (!w(z, s). I V
, ?x (I
, We now prove, by induction over M , that ?x (I
,
, p(z). ?x (z x | I V [[V ]] x ) p(z). I V [[V ]] z (by Lemma 2.2.12) ?w. M : ?x (I, = ?x (p(z). z x | I V
,
, is always replicated, so: I V, ? If M = M 1 M 2 . First recall that I V
, ? ?q (?x (I
, ). (w w | r p))) (by induction hypothesis, | q(u). ?r (I
, of the encoding and the congruence of , it follows that for all evaluation context C e I
, We have to show that I
, ?r (r(w). I V [[V ]] w | r(w). y(w , p ). (w w | p p))) ?y (y x | ?w (I V, p = ?p (q(y). y x | q(y)
, p (x(z, q). (z w | q p ) | ?w (!w(z, s). I V, ?w
, w ) (by Lemma 2.2.10) x(z, q). (q p | I V
,
, | q(y). I
, | q(y). ?s ( (I
, ?r (I
, ] r ) | r(w). u(w , r ). (w w | r p))) (by induction hypothesis) ?u (I V
, ). (w w | r p))))) (because I V
,
, N : assuming O, q , we have, by definition O, p.p
,
, ?r (O[[V ]] r | r(w). y(w , p ). (w w | p p))) (IH) ?(y, w) (y x | O V
, | z w | w w | q p | p p) x(z, q). ?w (O V
,
, = ?q (q(y). !y(x, p ). I
, | q(y). ?r (r(w). O V [[V ]] w | r(w). y(w , p ). (w w | p p))) (IH) ?(y, p , w) (!y(x, p ). I
, | y(x, p ). (x w | p p)) (IH) N : I
, y(w , p ). (w w | p p))) (IH) ?y (O V, r | r(w). y(w , p ). (w w | p p))) N V : I
, ?r (r(w). O V [[V ]] r | r(w). y(w , p )
, For any M ? ? and fresh p, process O
,
,
,
,
, Completeness: systems of equations We provide the full description of the systems of equations E R (Figure D.1) and E R (Figure D.2). There, y is assumed to be the ordering of fv(M, N )
,
, ) I
,
,
,
,
,
,
,