By Irène Guessarian (auth.)

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**Example text**

The d i s t i n c t i o n is e s p e c i a l l y these two types of s u b s t i t u t i o n i m p o r t a n t in the semantics /AI, AN, AH, BO, ES/. ductions between It also Occurs of n o n - d e t e r m i n i s m in the study of tree trans- /AD,DN,ES,RO,T/. 23: Let f be in F, t a tree in M(F,V) in M ( F , { w I .... ,Wr(f)}~. titutingfor t(t'/f) each o c c u r r e n c e w h e r e ~" is the r ( f ) - t u p l e m In other words, t(t'/f) denotes and t' a tree the tree obtained by subs m of f in t the tree t' (~m(t'/f) of trees such that tim = f ( ~ ) .

T' can be ~ in T by Moreover, we can easily verify that ~ c o i n c i d e s w i t h the order d e f i n e d on M~(F,V) and iff T is thus g r e a t e r than any one of the t's in E, and c l e a r l y is the least such tree. {DTCDT, to since for any m in D T Es ~ ~ previously M (F,V) <=> EcE' => E' M (F ,V) T(m)~ => T' (m)=T(m)} => T ~ T' 37 We thus have c o n s t r u c t e d a m o r p h i s m from of trees (finite or infinite) M~(F,V) on FuV. We leave it to the reader to check that it is an i s o m o r p h i s m its i n v e r s e ) .

The proof Lemma (~/~) . t2 S lemma and its c o r o l l a r y Let t , t 2, t 3 be in M(Fu~,V) such that: 1 S ........ > t 3 then, there exists t 4 in M(Fu~,V) such > t 4 ~* ''> t 3• (i) suppose first tI > t2 . Then, for some tree t, + and some o c c u r r e n c e s m i and mj in t: t 1 = t(Gi(t')/m i) t2=t(~/m i) 41 tl t2 t4 = t3 / t j ~ Gi case (a) V tj (~/~) tI t2 t3 t4 = • V ...... 25 I 42 and t 2 = t(Gj(~")/mj) (a) suppose and t 3 = t 2 ( t j ( ~ " / ~ ) / m j ) = t ( ~ / m i ) ( t first mj is not a left factor t4=t(Gi(~')/m i) (b) suppose (tj(~"/v)/mj), clearly 3 then let of mi; tI S > t4 (~"/~)/mj) ~ > t3 .