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Chapter 4 Augmented Cylindric Theory for Sets of Finite Sequences
CHAPTER 4
AUGMENTED CYLINDRIC THEORY FOR SETS OF FINITE SEQUENCES We shall now present a set E,, of equalities of L,, which can serve as an axiomatic basis for the set of Iw-valid equalities of Lpq.This requires that Epqbe sor4nd ( k i t h respect t o lo-validity) in the sense that each equality in Epq.and hence also each equality derivablefrom E,,. is /w-valid. I t also requires that E,, be complete (Mith respect t o lwvalidity) in the sense that each Iw-valid equality of L,, can be derived from E,,. This completeness of E,, will be proved in ch. 5. In the present ch. 4 we prepare for ch. 5. largely by deriving equalities. Since Lpqresults by augmenting the language for cylindric algebras, the set of equalities derivable from E,, will be called the augmented cylindric theory f o r sets of finite seqirences. Note that codol= 1, for example. is not lo-valid and hence not in this theory. yet derivable in the theory of o-dimensional cylindric algebras, which is concerned with sets of o-sequences. The Booletin ptrrt B P of E,, shall be a set of equational axioms for the theory of Boolean algebras such that -, -, 1, but no other function symbols are used. The remaining part of E,, shall consist of the following equalities l(a), ..., 3 ( c ) , where 0 s i < w and 0 G .j < w . Recall that 0 is the term -1, d j ( j , land ) d(i+l,jare the term q'd, and dicj+ 2 l and d , j ,2 ) j are the term ci+,(qid qi+'d).
+,
-
1. (a) cjo = 0. ( b ) x * C ~ X= X. (c) Ci(X * c,y) = c,x * cjy. ( d ) C ~ C ; X= C ; C ~ X .
2 . (a) q o = 0. ( b ) q(x + y) = qx + qy. ( c ) q(x * y) = qx qy. (d) qcjx = ci+tqx. (e) cj(-q'+'1 x) = -qi+'l ( f ) qpx = q l * c,,x. (g) pqx = x.
-
-
A6
- x.
CH.
41
A U Ci M E N T E D C Y L I N DRlC T H EORY
3. (a) cid = q21,for; < 2. (b) cid = d, fori 2 2. (c) dj,;* ci(dij x) = d i j * x, for i # j and -2
-
47
s i-,j s 2 .
We let L, be the sublanguage of L,, which results when one refrains from using p. We let E, be the set of equalities of L, which results from E,, when the equality 2(g) is removed and when 2(f) is replaced by the following equality: 2. (f), q x = coqx.
Let E be any set of equalities of L,, (of L,). An E-purh in L,, (in L,) f r o m T to T ' , or also u drrivution in L,, (in L,). htrscd on E. of T = T ' , shall be a finite sequence ( T ~ ,..., , 7 , ) of terms of I-,, (of L,) such that T , is ~ T , 7, is T' and, for each i < t , T ~ ' results , from T~ by the replacement of an occurrence of a term u by the occurrence of a term u', such that either u = u' or u' = u is obtained by substituting terms for variables throughout some equality in E. Reference to L,, or to L, will usually be omitted. We let E 1 T = T' if and only if there is a derivation, based on E, of T = 7 ' . As is well-known, E t- T = T' if and only if there is a conjunction @ of formulas Vxh,... Vx6@ = p ' ) where p = p' is in E such that 0 -+ T = T' is valid. One can verify that each T = T' in E,, and each 7 = 7' in E, is Iwvalid and that derivations preserve Iw-validity. Hence E,, and E, are sound with respect to Iw-validity. For the soundness of E,,, the proviso in the definition of lo-validity that U # @ is essential. For U = 69, one h a s k = '(W= { @},QY= 0,and hence P Q Y = @ f /y: From this soundness of E,, and from the completeness, to be proved in ch. 5 , it follows that k T = T' if and only if there is an E,,-path ( T ~.., ., from T to 7 ' . One may therefore think of E,, as a sound and complete set of one-premise rules with each premise T~ and conclusion T ~ +denoting , for each U # @ the same Iw-operation. Each premise T~ and conclusion T ~ + indicate, , or at least suggest, a different mode of determining the value of the Jw-operation for given arguments. For example, q(x * y) indicates that one first forms the intersection of two arguments and then applies Q , while q x qy indicates that one first applies Q to each argument and then forms the intersection. Yet, E,, is such that in every case the difference between the two modes is small. Hence any two modes which can be indicated in Lw of
-
48
AUGMENTED CYLINDRIC THEORY
[CH.
4
determining the value of a given Iw-operation can be transformed into one another by a sequence of small changes, each leaving the lw-operation itself unaltered. If one wishes t o distinguish in logic o r philosophy between the notion of being valid and a narrower notion of being rrnulytic. then the latter notion is perhaps best defined in terms of paths of this kind. Problems concerning proofs in first-order logic with equality can be formulated as problems concerning E,,-paths. This follows from the soundness and completeness of Epqand from Theorem 2.6(b). This reformulation may have advantages.' Although our main aim will be to prove the completeness of E,,, most of our time will be spent on proving a related result for E,. More precisely, we shall show in ch. 5 that E, is neurly complete (with respect t o Iw-vulidity) in the following sense: If 7 and T ' are terms of L, and if l= T = T ' , then there is some m such that E, I- q"'7= q V . For brevity, we henceforth let k T = T' if and only if E, T = 7 ' . Moreover. to avoid repetition and interruption we let, for example, k T = 7'= T" if and only if 1 T = T ' and 1 T' = 7". Also, w e shall sometimes make tacit use of 1 dij= dji, of well-known results of derivability from BP, such as 1 T T' = T' * T and k T * (7' 7") = (T 7 ' ) T", and of well-known relationships between derivability from BP, such as that 1(J -T = 0 and 1-(J T = 0 imply cr = 7. In subsequent lemmas we let 0 s i,m,n < o.
-
-
- -
-
LEMMA4.1: (a) ci - cjx = -cjx. (b) q"'+"l < q"'1. (c) k qm- x = q'"1 -q,nx. (d) F c i q V = q"'1. (e) ci(q"'l x) = q"'1 cjx. (f) ci(-q"l * x) = -q"'l * cix. (g) I f 1 7 < T ' , then 1qT 6 qT'. Proof: (a) By l(b) and BP, k x * -cix = 0 and hence
-
t-
-
-
k -cix
*
-ci-cix
= 0.
'As far as we know, a comparable reduction of first-order logic with equality was first explicitly achieved by Howard[l6]. Earlier, Bernays[l] obtained a reduction to a system in which there are no variable-binding operators, but in which there are nonequational axioms.
CH. 41
49
A U G M E N T E D CYLINDRIC THEORY
Conversely,
k --cix
*
ci - cix = ci - cjx cix = ci(-cix
CiX)
= Ci(0) = 0,
by BP, l(c), BP, and l(a), respectively. Hence, l-ci-cix=-cix BP. (b) By 2(c), and BP, respectively, t q m + " l . q"l = q 7 y q n l . 1) = qmq1.
by
(c) By BP, 2(b), BP, BP, 2(c), BP, and 2(a), respectively, -(ql
Also
- -qx)
*
-
+-(ql
*
-qx)
+
+
q - x = (-q(x -x) qx) * q - x = (-(qx+q-x)+qx)*q-x = ((-qx -q - x) * q - x) (qx * q - x) = qx q - x = q(x -x) = qo = 0.
-
-
+
+ q - x = -ql + qx + q - x =-ql+q(x+-x)=-ql+ql=
1
by BP, 2(b), BP and BP, respectively. Hence, by BP, k q - x = q l --qx.
Now assume as inductive hypothesis that
k q" - x = q"1 .- qmx,
which holds by BP when m = 0. Then
k q(q" - x) = q(q"1 * - q"x) =qm+ll
= q"+'l * q - q'nx
. (ql.- qm+lX)= q m + l l .-
qm+lx
by the inductive hypothesis, 2(c), the equality just derived and part (b), respectively. (d) If m = 0, then k c i q m l= q m l for each i < w, since k 1 cil = 1 by l(b) and hence k c J = 1 by BP for each i < w. Given m , assume now as inductive hypothesis that k ciq"l = q"l for each i < w. If i = 0, then kciqm+ll= qm+'l by 2(f),. If i = j +1. then l-cCi+lqm+ll = qcjqml= qq"1 by 2(d) and the inductive hypothesis respectively, so that again ciqm+ll= qm+'l.
-
50
A U G M F N T E D CYI.INDRIC
[CH. 4
THEORY
(e) By part (d), I(c) and BP, and part (d), respectively,
k c;(q”’l* x) = c,(c;q”’l- x) = c;q”‘l* cjx = q”’1 - cix.
( f ) By parts (d) and (a). respectively,
k c, - q”’1 = c, - c,q”’l= -c,q”’l. Using this equality and proceeding as in the proof of part (e), one obtains the desired equality. (g) Assume that k 7 7’ = 7. Then, by 2(c), F qr qr’ = q(r * 7‘) =q7.0
-
For each term p of L,, we now define inductively a term ( p ) # which, roughly speaking, is obtained from p by driving q inward as far as possible. We let (x6)*= x6, (1)* = 1 , (d)’ = d, (a+o’)*= ( a ) # + ( @ (’ )a#*,a ’ ) ”(a)#.(a’)#, = (-a)#= -(a)”, (c,u)#= C , ( U ) # , ( p a ) # = p(a)*,
and we let (qX,)#= qxij.
((I])#=
q l , (qd)”= qd,
-
( q ( o + a ‘ ) )= # ( q a ) # + (qa’)#.( q ( a a ’ ) )= # ( q a ) #* (qa’)#,
-
(q - a)#= q l * -(qa)#, (qc,a)#= ci+,(qo)x,( s p a ) # = q l (coo)#,
and we let
( q q a P = q(qu)# (qqa)# = (q(qa)#)#
if ( q a ) #= q(a)#, if (sa)#Z q(a)#.
Consider any r . For a unique r b 0 and p , p does not begin with p and 7 is p”p. By induction, either p does not occur in p o r else (qp)# contains fewer occurrences of p than p does. Thus there is a least k , 0 d k < w , such that (qkp)# contains no occurrences of p. We call P ” +(~qkp)#the normtrl transfirm of r , pr+k the p-prejix for r , and r k the length of the p-prefix. Note that if r is a term of L,, then the pprefix for 7 is of length 0 and the normal transform of 7 is also a term of L,. We call r normal if it is its own normal transform. Let 7 be normal. One proves easily that if qp is any subterm of r then qp is either q i l or q‘d or qixs for some i and 6. An occurrence of a
+
CH.
41
51
AUGMENTED CYLINDRIC THEORY
term u in T shall be quasi-atomic if and only if u is some qtl or q’d or qix,, 0 s i < o,and the occurrence is not an occurrence within qu. A quasi-atom of 7 shall be any u which has a quasi-atomic occurrence in T. It follows from the definition that if a is any term of Lpqand T’ is the normal transform of 7,then l=T = 7’.T h e next lemma gives a prooftheoretic version for Lq and E,. It is proved by using 2(b), 2(c), 4.1 (c) and 2(d). LEMMA4.2: If then I- T = 7’.
T
is a term of Lq and
T’ is
the normal transform of T,
T h e next lemma is proved by using induction, BP, and 4.l(e) o r (f), respectively. LEMMA4.3: Let u be a term of Lq and normal. Consider any occurrence in u of a term p which is not properly within a quasi-atomic occurrence. Let m < w and let u‘ and u“ result from u by replucement of this occurrence of p by q“l p and by -q”‘1 p respectively. Then q’”1 u = q”‘1 u’and 1-q’”l u = -q”’l a‘’.
-
--
-
.
-
Let p k u be normal, pk being the p-prefix. Evidently, (1) u is normal, and (2) u is a term of L,. Also, there is some n such that (3a) if q’l occurs in u then i S n , (4) if q‘x, occurs in u then i s n , ( 5 ) if q’d occurs in u then i < n - 1, and (6) if ci occurs in c then i < n. If all the above conditions hold, then u shall be weakly n-normal. If, in addition, (3) no 1 occurs in u, then u shall be n-normal. Evidently, if n s n’ and if u is (weak!y) n-normal then c is (weakly) n‘-normal. Let pku be normal, pk being the p-prefix. Given anv n < w , let u(,/) be the term resulting from u by the following changes: (3a) replacement by x -x of each quasi-atomic occurrence of any q’l such that i > n , (4) replacement by x -x of each quasi-atomic occurrence of any q‘x, such that i > n ,
-
-
52
[CH. 4
AUGMENTED CYLINDRIC THEORY
-
(5) replacement by x -x of each quasi-atomic occurrence of any q'd such that either n = 0 o r i 2 n - 1, and (6) deletion of each occurrence of any c, such that i 2 n. Given any n < w , let be the term resulting from u(ll) by ( 3b) replacement by x - x of each quasi-atomic occurrence of any q'l such that i d n. Evidently, each utn) is weakly n-normal and each u nis n-normal. Let T be any term and let p h u be its normal transform, pk being the p-prefix. Thus, k = 0 when T is a term of L,. Let n, be the least n such that u is weakly n-normal. Let m,, = max(0, n o - k ) . If m < m,,, then the rn-development of T shall not exist. N o w let m 2 m,). Then we form u h , ..., u h + , , ' from u by the steps (3a), (4), ( 5 ) , ( 6 ) ,and (3b) described. = U h + , , l , p + l = ... = u h + , , t . We let the m-development of Note that Uh+,ft(, T be the term
+
Let u be weakly (k+n)-normal. By induction one can show that 1u1 is uniform from k + n on. It follows that Iq"1 * p k a k + , l lis uniform from n on.
It follows from the definitions involved that if T is any term of L, and T ' is the rn-development of T , then I=T = 7 ' . T h e next lemma gives a proof-theoretic version for L, and E,.
LEMMA 4.4: Let T be a term of L, and, for some m , let T' be the m-development of 7 . Then T = 7 ' . Proc,$ Let T be a term of L,, let u be the normal transform of T , and let the m-development T ' of T be By 4.2,
q"l s-9'1 - u 0 + . . . + q " ' - ' l --q"1 * U , , ~ ' + q " ' l . cn. FT=U.
By BP and 4. I(b),
t
= qol .-qll.
...+q"'-ll
.-q'nl.
u+qml
U.
Since u yields a,, by changes of kind (3b) only, it follows from 4.3.4. I(b), and induction that
t q"'1
*
u = q"l * ufn.
N o w consider any m' < m. By induction on the total number of
CH.
41
53
AUGMENTED CYLINDRIC THEORY
changes of kinds (3a), (4), ( 5 ) , (6) and (3b), one can show that
t-
.
.
q"'l - q m ' + l l
.- q m ' + l l .u,n,.
= qm'l
For changes of kind (6) one uses 4.l(b), 4.3, and 2(e). For changes of kind (3a) or (4), one uses 4.l(b), 4.3, and BP. For changes of kind ( 5 ) , one shows that q'd = qi(d * cod) = q'd * q'cod= q'd * 9'9'1
using l(b), 2(c), and 3(a), respectively, and then argues as for changes ' of kind (3a). For changes of kind (3bL one uses 4.3 and 4.1(b). 0 T h e next lemma reduces the proof that E, is nearly complete to a special case. Recall that by E, being nearly complete is meant the T = T ' , then there is following: If T and T' are terms of L, and if some m such that qmT= q"'?'.
+
LEMMA 4.5: Assume that if k # 0, p is k-normal and does not conand k qkl co ... ck-lp = 0, then there is some m such that tain 1 qm(qkl co ... ck-,p) = q"0 = 0. Then E, is nearly complete. Proof: Assume the hypothesis of the lemma. Suppose that k q k l * p = 0 and that p is k-normal. If k = 0, then 1 q kl p = 0 by the completeness of BP for terms not containing q , p , d , c j , i < w . Now let k # 0. Then qkl co ... ck-,p = 0 and hence, for some m, k q m ( q k l co ... ck-,p) = 0. Inview of l(b), BP,and4.l(g),ourassumption therefore implies the following simpler hypothesis: If 0 s k < w , p is k-normal, and qkl p = 0, then there is some m such that qm(qkl p ) = q"0 = 0. Now let T and T' be any terms of L, such that l= T = 7'.Let u be the normal transform of (T * - T ' ) + ( T ' -7) and let u be weakly nnormal. Let
-
+,
-
-
-
-
-
-
901 -q'1
*
a,+
***
+ q"-'l
*
-q"l
+ q"1
* ul#-l
*
u,,
be the n-development u a o f u . Then t= u@= 0. Now consider any k s n. Then qkl .-qk++'l uk= 0. Since uk is k-normal, given any C I , (ukl is uniform from k on. Hence, qkl uk= 0. By the above simplification of the hypothesis of the lemma, there is some mk such
+
-
+ -
'Note that equalities 3(b) and 3(c)of E, have not been used in the proof.
54
[CH.
A U G M E N T E D CYLINDRIC 'THEORY
4
t-q"'(q"1 - a , , + . . . + q " - ' 1 * a , , ~ , + q " l ~ a , , ) = 0 .
Then, by BP and 4.1 (g),
k q"'u%= 0. Hence
k 0 = q"(((7 -7')+ (7'- -7)) = q y 7 - -7')+ q"'(7' *
.q" - 7')+ (q?' * q"' - 7) - (q"l7 . q"I1 .-q'fl7') + (qVf7' .qUI1 - -qff'7) .-qtri71) + (q~n7' - -q n w
*
-7)
= (q"7 (q7r~7
by 4.4, 2(b), 2(c), 4.l(c), and 4.l(g) and BP, respectively. By BP it follows that 1q"'7 = q'"7'. 0 Some further consequences of BP, 1 (a), ..., 2(d), 2(f), will be useful. Some of these are known from the theory of cylindric algebras.
LEMMA4.6:
(a) (b) (c) (d) (e) (f)
k cicix= cix.
lft- 7 + , t h e n k c j 7 t- Ci(X + y) = cix + ciy. Q
Q
ci+.
F q"kix = ci,,,,qmx.
t-
(8) ProoJ
... ~ , , - ~ q=~q'%for x i < m. q"dij = d(i+k)(j+k). cj-l(di(j-l)-d(j-,)j)= d,,fori < J - 1.
(a) By BP, l(c),and l(b), respectively,
k cicix = Ci(CiX * CiX) = cicix * cix = cix. (b) Assume that
k7
6 7'.
By l(b),
k 7'
C
ci7'. Then
k7
ci7'
CH.
41
55
A U G M E N T E D CYLINDRIC THEORY
and hence F T
- -C~T'
= 0 by BP. Then
by 4.1(a), 1(c), 4.1 (a), the equality just derived, and 1 (a), respectively. Hence, k ci7 C ~ T '= ci7 by BP. ( c ) By BP and 4.6(b), k cix C ci(x+ y) and likewise k c,yC ci(x+ y). Hence cix+ciy C c,(x+y) by BP. By l(b), k x S cix and 1 y ciy. Therefore
-
I-
Ci(X
+ y) C Ci(CiX + ciy) = cj - (- ci - ci(-cix
*
ci
-
cix -ciy)
- c1y) = - (-
cjx * -ciy) = cix
+ ciy
by BP and 4.6(b), BP, 4.l(a) and l(c), 4.l(a) and l(c), and BP, respectively. Hence, f ci(x+ y) = cix+ ciy by BP. (d) Let m > 0 and assume as inductive hypothesis that
by the inductive hypothesis, and 2(d), respectively. (e) Let m > i 2 0 and assume as inductive hypothesis that for any i' < m , t- Ci' Ci'+l ... cn,-2qm-1x = qm-lx. Then
... ~ ~ - 2 ~ ~ _ 1=qcici+l ~ - ~ Cm-2qm-' q ~ CoqX
c~c~+I
1.1
= qm-lcoqx = q"-'qx
by 4.6(d), the inductive hypothesis, and 2(f),, respectively. (f) If i = j , then dijis qi+'l and4i+lHj+l)i~qi+21,~~thatqdiiisc4i+~~j+~,. Now assume that i < j . Then
56
[CH.
A U G M E N T E D CYLINDRIC THEORY
4
by the definition of dij, 2(d), 2(c), and the definition of d(i+l)j+l), respectively. For j < i the proof is similar. It follows by induction that qkd.. = d (I+!,.)( i + k ) . II ( g ) Assume that i < j- I . Then
1 dij = c ~ _ _+_ ~~ ~ - ~ ~ ~ ... - ~- qj-'d ( q ' dqj-ld) = cj-lci+l ... cj-Z(q'd . ... . q'-'d qj-ld) *
*
*
... ~ ~ - ~ ( (* q...' d* q'-'d) * c ~ ... + Cj--2qj-1d) ~ - c,-,(c~+,... cj_,(q'd * ... - qj-'d) - c ~ ... + CjpPqj-ld) ~ - cj-l(ci+I ... ~j-,(q'd. ... . qj-'d). qj-'d) = c. c.1 + 1
= Cj-l(di(j-1)
*
d(j-l)j)
by the definition of dij, I(d), 4.6(e), l(c), 4.6(e), and the definitions of d;, and d( respectively. 0 We now turn to derivations from E, which involve one o r more of the equalities 3(a), 3(b), 3(c).
LEMMA4.7: (a) k ckdij= dij,fur k # i und k # j . (b) k ck(dij* x) = dij c,x,for k # i und k # j . (c) cidii= q t i i n . r . ( i . j ) t l 1. (d) k qk+'l d i j= d,;,for /< < mux (i,j ) . (e) k ci(d,,* x * y) = ci(dij x) * ci(di.; y)for i # jund-2 < i-j (f) k ck(dik* dk,J= qk+'l d,;,for k # i and k # j . dik dk,l= dik dij. (8) PrmJ (a) Assume that k # i and k # j . Case 1 : i =j . Then
-
-
-
-
-
C
2.
k cJij = c,,.qi+ll = qi+ll= dij by the definition of dii, 4.l(d), and the definition of dif, respectively. Cuse 2: i < j . Subcuse (a): k < i. Then ckdi,;= ckq'd,)(, ;-') = qkcoqq'-"-'d
.
l n . i - i ).
- qkqqi-k-1 do( - i = dij -
by 4.6(f), 2(d), 2(f),. and 4.6(f),respectively.
)
CH.
41
51
A U G M E N T E D CYLINDRIC THEORY
Subcuse (b): i
< k < j . Then
q d i i= cI,cil1... ~ ; - ~ ( q* ' ... d * qj-ld)
- cj+l ... c ~ - I c ~ c ~ c ~I +... c;-I(q'd * ... * qj-ld) - ~ i + ... l c ~ - I c ~ c ~ + ... I ~j-l(q'd*... * q'-'d) = dj.;
by the definition of dij, l(d), 4.6(a), and the definition of dtj, respectively. Subcase (c): i < j < k. First let j = i+ 1. Then
k qdii = ckqid= q'ck-,d = q'd = dij by the definition of di(j+lj, 2(d), 3(b) and k - i > 2, and the definition of di(i+l,, respectively. Now let j a i+ 1 and assume as inductive hypothesis that k ckdirjr = dip;,whenever i' < j ' < li and j ' - i' < j - i. Then ckdj = CkCj-l(di(j-1)* d(j-l)j)= Cj-lCk(di(j-l)* d(j-l)j) = Cj-lCk(Ckdi(j-i
j *
Ckd(j-1 j j ) = Cj-I(CkCkdj(j - 1 )
*
Ckd(j - 1
jj)
= Cj-l(Ckdi(j-1) * Ckd(j-ijj)= cj-i(dj(j-i) * d(j-ijj) = djj by 4.6(g), l(d), the inductive hypothesis, I (c), 4.6(a), the inductive hypothesis, and 4.6(g), respectively. Case 3: i > j . Since dtjis d,ii,the proof is the same as for case 2. (b) Let k # i and k # j . Then
C ck(dij
X)
= Ck(Ckdij * X) = ckdi,;
. C ~ =X dij
*
C ~ X
by 4.7(a), I(c) and BP, and 4.7(a), respectively. (c) Case 1 : i =j . Then c,d,,= c i q i + l l = q n r n s ( i . j ) + l l I 11 by the definition of dijand by 4. I(d) respectively. Case 2 : i < j . First let j = i+ 1. Then 2(d), 3(a), and rnax(i,j ) = i+ 1, respectively. by the definition of di(i+l), Now let j > i 1 and assume as inductive hypothesis that - q m o s ( i , j - l ) + l 1= qjl. k c.d. I I ( 3.- I )
+
58
A U G M E N T E D CYLINDRIC THEORY
Then
I- cdi,;= CiC,i-l(di(; - I )
*
-
4
.
4 - l ) ; ) = C,;-lCi(di(.;-t)d(j-l)j) 4 j-l)j) = cjPl(qjl d(j-l)j) = qjl * cj-,d( j - l ) j
= cj-,(cidi(j-l) - qjl q p i i f i . r ( j - l , j l + l l
.
[CH.
-
=.qj+ll
= qi,to.r(i.jl+l
by 4.6(g), 1 (d), 4.7(b), the inductive hypothesis, 4.1 (e), the above proof forj = i+ I , 4. I(b),and max(i,j) = j , respectively. Case 3 : i > j. First let i = j + 1. Then
t c.d. = c,j+,qjd= qjc,d = q j q z l 1
71
= qmns(i.j)+ll
by the definition of d( I ) , r 2(d), 3(a), and max(i,j) = j + 1 , respectively. Now let i > j + I and assume as inductive hypothesis that I- c.d.. - qfnms(i.j')+ll = q i + l l 7 11,wheneverj < j ' < i. Then 1cjdi.j= cjd,ji = ~j~;+l(d,j,j+l) * d(,j+l)i) = C.j+lCi(dj(j+l). d(j+I)j)
= C,i+l(di(j+l) *cid(jtI)O=c.itl(dj(i+l)* q"'1) = q i + l l c,.l + l dJ.( . .I + l ) - qi+ll qfnas(j.j+l)+ll = qi+ll = qmar(i,j)+ll
.
.
by the definition of d,ji,case 2 of 4.7(f) below and 4.7(d) below, l(d), 4.7(b) and BP, the inductive hypothesis, 4.l(e), the above proof for i=j+ 1,4.I(b),andmax(i,j) = i,respectively. Note that case 3 of 4.7(c) is not used for 4.7(d). Nor is 4.7(c) used for case 2 of 4.7(f), although it is used for some other cases. (d) Assume that k S max (i,j) and i S j . Then
.
-
dii= d j j cjdjj= dij
qpptar(i-j)+11
< d.. * qk+'l
by I(b), case 2 of 4.7(c), and 4. I(b), respectively. Hence, dj,;= q"+'l * di,;
by BP. For i > j , then use the definition of dij. (e) Assume that i # j and-2 i-j 2. Then
/-
cj(dj;. X) cj(di,* y) = Ci(dj,;* x * ci(djj. y)) = C ~ ( X* dij * Cj(dij * y)) = C ~ ( X* dj,;* y) = cj(djj * x y)
by I(c), BP, 3(c), and BP, respectively.
CH.
41
59
AUGMENTED CYLINDRIC THEORY
(f) Assume that X # i and X # j. Case 1 : i =j. Then
-
cA(d,, dh,)= c-d,,, = q~n~~~(I,h)+ll = qA+'l* q'+'l = q"I1
*
d,,
by i =j and BP, 4.7(c), 4. I(b), and i =j and the definition of d,,, respectively. Case 2 : i < k < j. First letj- k = 1. Then
[CH.
A U G M E N T E D C Y L I N D R I C THEORY
4
Then
k Cddik . dkj) = Ck(Ck+l(dk(k+l).d(k+l)i).Ck+l(dk(k+l)* &+Id) =ck,lch.(dk,/~+1).d(/~+l)i.d(kli)i)
= c k * l(Ckdk(k+l)* d(k+l)i* d(k+~);) = ~ kI(qk+21 + .dik, I ) j * d(k+l)j)= qk+21* CkLI(d(k+l)i* d(k.l , j ) - q k 1 2 1 .q k + 2 1 . d,. = q k + 2 1 di, 1.1
.
by case 2 above and 4.7(d), 4.7(e) and l(d), 4.7(b), 4.7(c), 4.l(e), the inductive hypothesis, and BP respectively. Case 4: i < j < k . Then the proof differs only in minor respects from that for case 3. For the cases where j < I, < i, k < j < i, or j < i < k , the proofs are the same as for the cases 2 , 3, and 4, respectively, since di, and d,iiare the same term.:' ( 8 ) If k # i and k # j , then dik dk,j= (dik . dkj) * Ck(djk * dk,j) = dik * dk,;* qk+'l di;
= dik * dk; * di,j by 1 (b), 4.7(f), and 4.7(d), respectively. If k
= i o r k = j, then
k dik - dk,;= dik * dk,;- dii
by BP. Since dik is dki,similarly
k dik * di,;= djk Hence
dij * dk,;.
-
dik * dkj = dik djj
by BP. 0
-
Next, we turn to equalities containing terms c,;(dji 7 ) where i # j. More briefly and more suggestively, we let S;T be ci(dii 7 ) . We shall employ 'DF' to appeal to this use or definition. In this use we shall assume tacitly, for the rest of ch. 4, that i # j ; similarly for other suband superscripts.
-
:'For cases I and 2, the proof does not use any equality 3 ( c ) of E,. For cases 3 and 4, it uses only those instances of 3 ( c )where either i =j + 1 or i = j - I .
CH. 41
61
A U G M E N T E D CYLINDRIC THEORY
LEMMA 4.8: (a) qk+'l * six = s:(qk++'lx) = s:x,for k G max (iJ. (b) I- qs:x = siz,!qx. (c) -t C;$X = s:x. (d) k c,s:x = s:ckx, f o r k # i and k # j. (e) t- cks:x 4 s:c,x, f o r k # i. s;si,'x = si)'six,forj # j ' , j # i', undj' # i. (f)
-
-
ttt-
s;s;,x = q'+'l $,x. s:s:'x = s$$'x, for i # j'. c.sjc.x t . l = qnr~~~(i,j)+ll cyix. = qntft.r(i.j)+ll c,;x. (j) C s!cjx 1 ' (k) dii syx = di,; s$x,for k # i und k # j . (8)
(h) (i)
t-
-
.
-
.
Prooj (a) Assume that ( i # j and)k qk+'l
- s;x
- cj(dji
= q"'l
*
G max
( i , j ) .Then
-
x) = C j ( q k + ' l
d.. 31 x) = c.(d.. 3 11
X)
= S!X
by DF, 4. I(e), 4.7(d), and DF, respectively. Also
k s:(q"+'l
*
X)
= c,;(djj * qk+'l* X) = c;(dj;
X)
= S:X
by DF, 4.7(d), and DF, respectively. (b) By DF, 2(d), 2(c), 4.6(f), and DF, respectively,
k q S $ = qCj(d,ii.
X) = C , j + l q ( d i i . X) = c,i+l(qdii * qx) = C.;+i(d(;+i)(i+i) * qx) = siI:SX.
(c) By DF, 4,6(a), and DF, respectively, cjsijx = c,;c,;(dji
* X)
= c,;(dji
*
X)
= S:X.
(d) Assume that k # i and k # j. Then
k CS,X:
= CkCj(djj
*
X)
= CjCk ( d ji
*
X)
= cj(d;;
*
C~X= ) S ~ C ~ X
by DF, l(d), 4.7(b), and DF, respectively. (e) Assume that k # i. If k # j , then
I-
c,s:x
*
s:clix
= c,s:x
by 4.8(d). Now assume that k =j.Then CS ,X :
= C,iC;(d,;i
= C,j(dji
* X) 6 C;Cj(dji * C ~ X )
.C ~ X )= S ~ C ~ X
62
A U C M E N l E D CYLINDRIC THEORY
[CH.
4
by D F and j = X , I(b) and BP and 4.6(b), 4.6(a), and;= k and D F , respectively. ( f ) Assume thatj # j ' , j # i';andj' # i. Then s/s:,'x = cj(d,i
*
C#,(d,,ic *
- cj,cj(dji - d,.;,
.
x)) = c;cjn(d;j d,,i, * X)
= Cjg(djrig
X)
- cj(dji - x)) = S$S$
by DF, 4.7(b), I(d), 4.7(b) and BP, and DF, respectively. (8) By DF, I(c). 4.7(c), 4.l(b),4.l(e),4.7(d), and D F , respectively, s~s,!,x = Cj(dji
-
qiiirr.r(
-
x)) = cjdji
Cj(djic
i .j) t 1
1 -c.;(dji.
X)
-
Cj(djit
-
-
X)
= qi+'l qj+'l * cj(dji,
*
X)
= q'+ll . c j ( q j + l l . d.i,. x) = q ; + l l . ~ , ~ ( d .x); ~=, q i + l l . s;,x.
( h ) Assume that i # j ' (and that i # j a n d j # j ' ) .Then
-
k S:S:'X
-
= c;(d;; cjr(d;tj x)) = c;c;t(d;i d,j,,; X) = cici,(d;i * d;ti * X) = c,;(d;i * ci,(dirj * x)) = S:S:'X
by DF, 4.7(b), 4.7(g), 4.7(b), and DF, respectively. (i) By DF, I(d), l(c), 4.7(c), and 4.1(e), respectively,
k C~S;C~X= cicj(di;
*
C ~ X )= cjci(dij * C ~ X )= cj(cidfj * C ~ X )
.
= c,j(qinns(i.i)+ll cix) = q i i i f i . r ( i . j ) + l l
.c,;cix
(j) By DF, l(c),and 4.7(c), respectively,
t-
S$X
.
J I . 1 . c.I . = ~ q ~ l r f ~ s ( i . j ) + l l . c,ix. = c.(d. J 0 cJ . ~ = ) c.d..
( k ) Assume that k # i and k # j. Then
k di;
-
= dij * Ck(dpi * X) = Ck(dij * dki X) = ck(dj,; * dkj * X) = dij Ck(dkj * X) = dfj * SFX
* S ~ X
by D F , 4.7(b), 4.7(g), 4.7(b), and DF, respectively. In 4.9(c) below we obtain a generalization of 4.7(e) for those cases where i < j and where x and y are of the form cku. The generalization consists in dropping the condition that -2 s i-j. T o prove 4.9(c), we use 4.9(b), which by induction implies that under suitable conditions s: can be replaced by combinations of s;,', each satisfying-2 s i' -j' c 2.
CH.
41
63
A U G M E N T E D C Y L I N D R I C THEORY
LEMMA 4.9: = q ~ ~ ~ x ( i 3 j )c.x +1 1) for-2 1 < i-j (a) 1 s2slcix 2. J 1 S $ ~ X = s ~ ~ ~ ~ j + ~ ~ ~ + , ~ 0 j , < , i ~ < ~ j ~- ~1.~ ~ - ~ x , f o r (b) (c) 1 sk(ctx c,,y) = sijcpx s:cty,for k < i < j. Proof: (a) Assume that -2 s i - j d 2. Then
-
-
-
~ s $ : c ~= x ci(di,i* ci(dii c~x))= ci(dii * cix) = cidii* cix - q l l t f t x ( i , j ) + l l . cix
by DF, 3(c), I(c), and 4.7(c), respectively. (b) Assume that 0 < i < j- 1. Then
1 S!
s!
s(+ly
t i 1 2+1
1-1
= s( si ,+I
I f ,
si,+~y = 1-1
sI
t+l
sjsi+ly i i-1
= s'
)+I
si+lsjy 1-1 i
by 4.8(f), 4.8(h), and 4.8(f), respectively. Hence
1s ~ ~ ~ s ~ + ' s ~ + l s j += l s s;;;si+'sj+ j ~ ~ c i - I ls;+~s~ci-lx x = sj~~sj+'sj+lci+lsj'~s~cj~lx = s!-l(qj+21 C. si+lsjci_lx) If, l + l i-1 i
.
= s ; ~ ~ c i + l s ~ ' ~= s ~s~;;s~?~s;Ci-,x ci~lx = S f - l S i + l C , s j - i + 1~ * Ci-lSijX (+I
i-1
= q'+21
7-1
.SJC.
ix- q x = S,Ci-,X I
1-1
by the equality just derived, 4.8(c), 4.9(a), 4.8(a), 4.8(c), 4.8(d), 4.9(a), 4.8(d), and 4.8(a), respectively. (c) Assume that k < i < j. We shall first show that
-
t-
-
s:(ci-,x Ciply)= s:ci-,x si'ci-,y.
If j - i = 1, then this holds by 4.7(e) and DF. Now let j- i > 1 and assume as inductive hypothesis that
t-
S;+&x
*
-
ciy) = s;+lcfx s;+Iciy.
64
AUGMENTED CYLINDRIC THLORY
Then
t-
s;(ci-lx * cj-ly) = s;cj-l(cj-lx
-
[CH.
4
Cj-'y)
= sj;;sj+ls;+I S ; + l S j ~ ~ C ; - I ( C I - l X* cj-ly) = si;;sj+ls;,
-
lsj+lsj~;(cj-lxci-ly)
= sj;:s;+lsij+$j+
Isj+:cj-Ix* sj+ls;:;cj..ly)
= s;;;sj+1sij, I(cjsj+lsj?;c;~lx * cjs~+lsj~;cj-ly) - I , IsI i + l ( s J I t I c 8. s; ~ + l s ~ ~ ; c j -sit l xl c j s ~ + l s j ~ ; c j - l y )
-
= sj;;sj+ls! I t 1 slI f 1 Sl+'Cj-,X r-1 - s,icj-lx* sijcj-Iy -
- sj;;sj+1s;+lsj+lsj+;cj-ly
by 4.6(a) and I(c), 4.9(b). l(c) and 4.6(a), 4.7(e) and D F (twice), 4.8(c), the inductive hypothesis, 4.7(e) and D F (twice) and 4.8(c), and 4.9(b), respectively. N o w let k < i - 1 and assume as inductive hypothesis that
k d ( C k , IX Then
cp, ly) = s;ck+lx* s:'cktIy.
-
s:(c/;x c,y) = s;ck(ckx* c,,.y) = s;(q"f21 * Ck(CkX * cky)) = s J skk , $.,+)I Ck(C/,.X CkY) = s;+Is;s;+'ck(c,x
-s;+1
-
sj(sk+lckx. I k
*
C,Y)
&+l p CkY)
= s;+lsij(cp,,s;+lcI;x *
Ck+ 'S;+'cky)
- s;+,(sijck, Is;+'ckx * sjck+,S;+'cky) - s;+ I s J s k + l CkX * s;, 's;S;+'cky -
-
= s;s;+ Is;+'ckx S;S;tIS;+'Cpy
- ,
= s$q"+"l c x) * s;(q"+'l
- c,y) = s;c,x
*
s;Cky
by 4.6(a) and l(c), 4.8(a), 4.9(a), 4.8(f), 4.7(e) and D F , 4.8(c), the inductive hypothesis, 4.7(e) and D F and 4.8(c), 4.8(f), 4.9(a), and 4.8(a), respectively. 0
LEMMA4.10: (a) IsijO = 0. (b) sijl = qttfo.r'Ci.j)+l 1. ( c ) 1s;(x y) = sijx sijy.
+
+
CH.
41
6.5
A U G M E N T E D C Y L I N D R I C THEORY
-
(d) 1s:(qx * qy) = skqx s:qy,for 0 < i < j. (e) t- s: - qx = q’+’I - s,’qx,for O < i < j. ( f ) If T s T ’ , then 1 S/T C S:T’. Proofi (a) By D F , BP and I(a), respectively,
-
1 s:O = c,j(d.ji
*
0) = cj0 = 0.
(b) By D F , 4.l(d) and I (c),and 4.7(c) and BP, respectively,
k s,’l = ci(d,jj * 1)= c.d.. . 1= qll~~~J’(i.j)+ll I I1
(c) By D F , BP, 4.6(c), and DF, respectively,
1 S:(X+Y)
+
= Ci(d,jj * (x+Y)) = C,j((d,jj . X) (d,ji .y)) = C;(dji * X) cj(dji y) = SX; s,’Y.
+
+
.
(d) Assume that 0 < i < j. Then
t-
$(qx
*
qy) = s;(c,,qx
*
c,qy) = s;c,qx
*
s:c,,qy
= s;qx
*
s;qy
by 2(f),, 4.9(c), and 2(f),, respectively. (e) Assume that 0 < i < j. Then
1s;-
qx * sjqx = si(q1
- - qx) - s;qx
= s;q - x * siqx
= s/(q-x* qx) = $0) = 0
by 4.8(a), 4.l(c), 4.1O(d), 2(c) and BP and 2(a), and 4.1O(a), respectively. Also,
k s; - qx + s;qx
= s/(-qx
+ qx) = $1 = qj+’l
by 4.10(c), BP, and 4.1O(b),respectively. Hence by BP, s; - qx = q’+’l - s,‘qx.
.
(f) Assume that
T
s 7’ and hence
7+7’
= T ‘ . Then by 4.10(c),
k $7 + S i 7 ’ = S i ( 7 + 7 ’ ) = S i 7 ’ .
Hence S:T
s Si7’. 0
We conclude with a few remarks on axiomatization and definability. Much of ch. 4 is a struggle for economy in the axioms involving d. In particular, 4.7(f) is derived rather than taken as an axiom scheme.
66
A U G M E N T E D CYLINDRIC THEORY
[CH.
4
Also, 3(c) is restricted to those i and j which are 1 or 2 integers apart, while similar equalities where i and j are further apart are derived. Thus, 3(c) consists of four 1-dimensional infinite arrays of axioms. We doubt that any of these four arrays is redundant or replaceable by a finite array. Only I(d) consists of an array of axioms which is infinite in 2 dimensions. We doubt that it can be replaced by a finite number of 1 -dimensional arrays. There are two schemes, namely 2(e) and 3(c),for which there is no finite upper bound to the length of instances of the scheme. This can be remedied by using d,,, d , . d2, ... as primitives in place of d. One then takes qd; = dj+las additional axioms, i < w, makes the obvious changes in 3(a), 3(b), 3(c), and in 2(e) either replaces q'+' 1 by pd; or, for i # 0. replaces q i t ' 1 by ci-ldj-l. Given any i , it follows from 2(d) and 2(g) that ctx = pcitlqx. Hence c, is definable in terms of p, q and ci +,. This redundancy of primitives of I L,, 1 appears to be unavoidable. There result certain redundancies of axioms. For example, cjc,x= cicix can be derived from c ~ + ~ c ~ + ~ x = c , + ,cj + /,.xtogether with certain other axioms. Finally, we note that the terms djkcould have been replaced by terms dj, that are defined using only d, -,q, and c,. Let q,,,T be c,(d 47) and, for i < k , let d:,,.and di., be the term qiq,l,k-i-'d.When k - i = 1 or k - i = - 1 then dj, is dikr and when k - i = 2 or k - i = - 2 then dj, = djk can be derived from E, without using 3(a), 3(b), or 3(c). Hence, replacement of 3(c) as an axiom scheme by the set of corresponding equalities which involve dl, instead does not affect the set of derivable equalities. Moreover, by the completeness of Epq,to be proved in ch. 5, if an equality involving terms dikis derivable, then the corresponding equality involving terms d,!, instead is also derivable. Whether derivations of the latter would have been easier, we do not know.
-
AUGMENTED CYLINDRIC THEORY FOR SETS OF FINITE SEQUENCES We shall now present a set E,, of equalities of L,, which can serve as an axiomatic basis for the set of Iw-valid equalities of Lpq.This requires that Epqbe sor4nd ( k i t h respect t o lo-validity) in the sense that each equality in Epq.and hence also each equality derivablefrom E,,. is /w-valid. I t also requires that E,, be complete (Mith respect t o lwvalidity) in the sense that each Iw-valid equality of L,, can be derived from E,,. This completeness of E,, will be proved in ch. 5. In the present ch. 4 we prepare for ch. 5. largely by deriving equalities. Since Lpqresults by augmenting the language for cylindric algebras, the set of equalities derivable from E,, will be called the augmented cylindric theory f o r sets of finite seqirences. Note that codol= 1, for example. is not lo-valid and hence not in this theory. yet derivable in the theory of o-dimensional cylindric algebras, which is concerned with sets of o-sequences. The Booletin ptrrt B P of E,, shall be a set of equational axioms for the theory of Boolean algebras such that -, -, 1, but no other function symbols are used. The remaining part of E,, shall consist of the following equalities l(a), ..., 3 ( c ) , where 0 s i < w and 0 G .j < w . Recall that 0 is the term -1, d j ( j , land ) d(i+l,jare the term q'd, and dicj+ 2 l and d , j ,2 ) j are the term ci+,(qid qi+'d).
+,
-
1. (a) cjo = 0. ( b ) x * C ~ X= X. (c) Ci(X * c,y) = c,x * cjy. ( d ) C ~ C ; X= C ; C ~ X .
2 . (a) q o = 0. ( b ) q(x + y) = qx + qy. ( c ) q(x * y) = qx qy. (d) qcjx = ci+tqx. (e) cj(-q'+'1 x) = -qi+'l ( f ) qpx = q l * c,,x. (g) pqx = x.
-
-
A6
- x.
CH.
41
A U Ci M E N T E D C Y L I N DRlC T H EORY
3. (a) cid = q21,for; < 2. (b) cid = d, fori 2 2. (c) dj,;* ci(dij x) = d i j * x, for i # j and -2
-
47
s i-,j s 2 .
We let L, be the sublanguage of L,, which results when one refrains from using p. We let E, be the set of equalities of L, which results from E,, when the equality 2(g) is removed and when 2(f) is replaced by the following equality: 2. (f), q x = coqx.
Let E be any set of equalities of L,, (of L,). An E-purh in L,, (in L,) f r o m T to T ' , or also u drrivution in L,, (in L,). htrscd on E. of T = T ' , shall be a finite sequence ( T ~ ,..., , 7 , ) of terms of I-,, (of L,) such that T , is ~ T , 7, is T' and, for each i < t , T ~ ' results , from T~ by the replacement of an occurrence of a term u by the occurrence of a term u', such that either u = u' or u' = u is obtained by substituting terms for variables throughout some equality in E. Reference to L,, or to L, will usually be omitted. We let E 1 T = T' if and only if there is a derivation, based on E, of T = 7 ' . As is well-known, E t- T = T' if and only if there is a conjunction @ of formulas Vxh,... Vx6@ = p ' ) where p = p' is in E such that 0 -+ T = T' is valid. One can verify that each T = T' in E,, and each 7 = 7' in E, is Iwvalid and that derivations preserve Iw-validity. Hence E,, and E, are sound with respect to Iw-validity. For the soundness of E,,, the proviso in the definition of lo-validity that U # @ is essential. For U = 69, one h a s k = '(W= { @},QY= 0,and hence P Q Y = @ f /y: From this soundness of E,, and from the completeness, to be proved in ch. 5 , it follows that k T = T' if and only if there is an E,,-path ( T ~.., ., from T to 7 ' . One may therefore think of E,, as a sound and complete set of one-premise rules with each premise T~ and conclusion T ~ +denoting , for each U # @ the same Iw-operation. Each premise T~ and conclusion T ~ + indicate, , or at least suggest, a different mode of determining the value of the Jw-operation for given arguments. For example, q(x * y) indicates that one first forms the intersection of two arguments and then applies Q , while q x qy indicates that one first applies Q to each argument and then forms the intersection. Yet, E,, is such that in every case the difference between the two modes is small. Hence any two modes which can be indicated in Lw of
-
48
AUGMENTED CYLINDRIC THEORY
[CH.
4
determining the value of a given Iw-operation can be transformed into one another by a sequence of small changes, each leaving the lw-operation itself unaltered. If one wishes t o distinguish in logic o r philosophy between the notion of being valid and a narrower notion of being rrnulytic. then the latter notion is perhaps best defined in terms of paths of this kind. Problems concerning proofs in first-order logic with equality can be formulated as problems concerning E,,-paths. This follows from the soundness and completeness of Epqand from Theorem 2.6(b). This reformulation may have advantages.' Although our main aim will be to prove the completeness of E,,, most of our time will be spent on proving a related result for E,. More precisely, we shall show in ch. 5 that E, is neurly complete (with respect t o Iw-vulidity) in the following sense: If 7 and T ' are terms of L, and if l= T = T ' , then there is some m such that E, I- q"'7= q V . For brevity, we henceforth let k T = T' if and only if E, T = 7 ' . Moreover. to avoid repetition and interruption we let, for example, k T = 7'= T" if and only if 1 T = T ' and 1 T' = 7". Also, w e shall sometimes make tacit use of 1 dij= dji, of well-known results of derivability from BP, such as 1 T T' = T' * T and k T * (7' 7") = (T 7 ' ) T", and of well-known relationships between derivability from BP, such as that 1(J -T = 0 and 1-(J T = 0 imply cr = 7. In subsequent lemmas we let 0 s i,m,n < o.
-
-
- -
-
LEMMA4.1: (a) ci - cjx = -cjx. (b) q"'+"l < q"'1. (c) k qm- x = q'"1 -q,nx. (d) F c i q V = q"'1. (e) ci(q"'l x) = q"'1 cjx. (f) ci(-q"l * x) = -q"'l * cix. (g) I f 1 7 < T ' , then 1qT 6 qT'. Proof: (a) By l(b) and BP, k x * -cix = 0 and hence
-
t-
-
-
k -cix
*
-ci-cix
= 0.
'As far as we know, a comparable reduction of first-order logic with equality was first explicitly achieved by Howard[l6]. Earlier, Bernays[l] obtained a reduction to a system in which there are no variable-binding operators, but in which there are nonequational axioms.
CH. 41
49
A U G M E N T E D CYLINDRIC THEORY
Conversely,
k --cix
*
ci - cix = ci - cjx cix = ci(-cix
CiX)
= Ci(0) = 0,
by BP, l(c), BP, and l(a), respectively. Hence, l-ci-cix=-cix BP. (b) By 2(c), and BP, respectively, t q m + " l . q"l = q 7 y q n l . 1) = qmq1.
by
(c) By BP, 2(b), BP, BP, 2(c), BP, and 2(a), respectively, -(ql
Also
- -qx)
*
-
+-(ql
*
-qx)
+
+
q - x = (-q(x -x) qx) * q - x = (-(qx+q-x)+qx)*q-x = ((-qx -q - x) * q - x) (qx * q - x) = qx q - x = q(x -x) = qo = 0.
-
-
+
+ q - x = -ql + qx + q - x =-ql+q(x+-x)=-ql+ql=
1
by BP, 2(b), BP and BP, respectively. Hence, by BP, k q - x = q l --qx.
Now assume as inductive hypothesis that
k q" - x = q"1 .- qmx,
which holds by BP when m = 0. Then
k q(q" - x) = q(q"1 * - q"x) =qm+ll
= q"+'l * q - q'nx
. (ql.- qm+lX)= q m + l l .-
qm+lx
by the inductive hypothesis, 2(c), the equality just derived and part (b), respectively. (d) If m = 0, then k c i q m l= q m l for each i < w, since k 1 cil = 1 by l(b) and hence k c J = 1 by BP for each i < w. Given m , assume now as inductive hypothesis that k ciq"l = q"l for each i < w. If i = 0, then kciqm+ll= qm+'l by 2(f),. If i = j +1. then l-cCi+lqm+ll = qcjqml= qq"1 by 2(d) and the inductive hypothesis respectively, so that again ciqm+ll= qm+'l.
-
50
A U G M F N T E D CYI.INDRIC
[CH. 4
THEORY
(e) By part (d), I(c) and BP, and part (d), respectively,
k c;(q”’l* x) = c,(c;q”’l- x) = c;q”‘l* cjx = q”’1 - cix.
( f ) By parts (d) and (a). respectively,
k c, - q”’1 = c, - c,q”’l= -c,q”’l. Using this equality and proceeding as in the proof of part (e), one obtains the desired equality. (g) Assume that k 7 7’ = 7. Then, by 2(c), F qr qr’ = q(r * 7‘) =q7.0
-
For each term p of L,, we now define inductively a term ( p ) # which, roughly speaking, is obtained from p by driving q inward as far as possible. We let (x6)*= x6, (1)* = 1 , (d)’ = d, (a+o’)*= ( a ) # + ( @ (’ )a#*,a ’ ) ”(a)#.(a’)#, = (-a)#= -(a)”, (c,u)#= C , ( U ) # , ( p a ) # = p(a)*,
and we let (qX,)#= qxij.
((I])#=
q l , (qd)”= qd,
-
( q ( o + a ‘ ) )= # ( q a ) # + (qa’)#.( q ( a a ’ ) )= # ( q a ) #* (qa’)#,
-
(q - a)#= q l * -(qa)#, (qc,a)#= ci+,(qo)x,( s p a ) # = q l (coo)#,
and we let
( q q a P = q(qu)# (qqa)# = (q(qa)#)#
if ( q a ) #= q(a)#, if (sa)#Z q(a)#.
Consider any r . For a unique r b 0 and p , p does not begin with p and 7 is p”p. By induction, either p does not occur in p o r else (qp)# contains fewer occurrences of p than p does. Thus there is a least k , 0 d k < w , such that (qkp)# contains no occurrences of p. We call P ” +(~qkp)#the normtrl transfirm of r , pr+k the p-prejix for r , and r k the length of the p-prefix. Note that if r is a term of L,, then the pprefix for 7 is of length 0 and the normal transform of 7 is also a term of L,. We call r normal if it is its own normal transform. Let 7 be normal. One proves easily that if qp is any subterm of r then qp is either q i l or q‘d or qixs for some i and 6. An occurrence of a
+
CH.
41
51
AUGMENTED CYLINDRIC THEORY
term u in T shall be quasi-atomic if and only if u is some qtl or q’d or qix,, 0 s i < o,and the occurrence is not an occurrence within qu. A quasi-atom of 7 shall be any u which has a quasi-atomic occurrence in T. It follows from the definition that if a is any term of Lpqand T’ is the normal transform of 7,then l=T = 7’.T h e next lemma gives a prooftheoretic version for Lq and E,. It is proved by using 2(b), 2(c), 4.1 (c) and 2(d). LEMMA4.2: If then I- T = 7’.
T
is a term of Lq and
T’ is
the normal transform of T,
T h e next lemma is proved by using induction, BP, and 4.l(e) o r (f), respectively. LEMMA4.3: Let u be a term of Lq and normal. Consider any occurrence in u of a term p which is not properly within a quasi-atomic occurrence. Let m < w and let u‘ and u“ result from u by replucement of this occurrence of p by q“l p and by -q”‘1 p respectively. Then q’”1 u = q”‘1 u’and 1-q’”l u = -q”’l a‘’.
-
--
-
.
-
Let p k u be normal, pk being the p-prefix. Evidently, (1) u is normal, and (2) u is a term of L,. Also, there is some n such that (3a) if q’l occurs in u then i S n , (4) if q‘x, occurs in u then i s n , ( 5 ) if q’d occurs in u then i < n - 1, and (6) if ci occurs in c then i < n. If all the above conditions hold, then u shall be weakly n-normal. If, in addition, (3) no 1 occurs in u, then u shall be n-normal. Evidently, if n s n’ and if u is (weak!y) n-normal then c is (weakly) n‘-normal. Let pku be normal, pk being the p-prefix. Given anv n < w , let u(,/) be the term resulting from u by the following changes: (3a) replacement by x -x of each quasi-atomic occurrence of any q’l such that i > n , (4) replacement by x -x of each quasi-atomic occurrence of any q‘x, such that i > n ,
-
-
52
[CH. 4
AUGMENTED CYLINDRIC THEORY
-
(5) replacement by x -x of each quasi-atomic occurrence of any q'd such that either n = 0 o r i 2 n - 1, and (6) deletion of each occurrence of any c, such that i 2 n. Given any n < w , let be the term resulting from u(ll) by ( 3b) replacement by x - x of each quasi-atomic occurrence of any q'l such that i d n. Evidently, each utn) is weakly n-normal and each u nis n-normal. Let T be any term and let p h u be its normal transform, pk being the p-prefix. Thus, k = 0 when T is a term of L,. Let n, be the least n such that u is weakly n-normal. Let m,, = max(0, n o - k ) . If m < m,,, then the rn-development of T shall not exist. N o w let m 2 m,). Then we form u h , ..., u h + , , ' from u by the steps (3a), (4), ( 5 ) , ( 6 ) ,and (3b) described. = U h + , , l , p + l = ... = u h + , , t . We let the m-development of Note that Uh+,ft(, T be the term
+
Let u be weakly (k+n)-normal. By induction one can show that 1u1 is uniform from k + n on. It follows that Iq"1 * p k a k + , l lis uniform from n on.
It follows from the definitions involved that if T is any term of L, and T ' is the rn-development of T , then I=T = 7 ' . T h e next lemma gives a proof-theoretic version for L, and E,.
LEMMA 4.4: Let T be a term of L, and, for some m , let T' be the m-development of 7 . Then T = 7 ' . Proc,$ Let T be a term of L,, let u be the normal transform of T , and let the m-development T ' of T be By 4.2,
q"l s-9'1 - u 0 + . . . + q " ' - ' l --q"1 * U , , ~ ' + q " ' l . cn. FT=U.
By BP and 4. I(b),
t
= qol .-qll.
...+q"'-ll
.-q'nl.
u+qml
U.
Since u yields a,, by changes of kind (3b) only, it follows from 4.3.4. I(b), and induction that
t q"'1
*
u = q"l * ufn.
N o w consider any m' < m. By induction on the total number of
CH.
41
53
AUGMENTED CYLINDRIC THEORY
changes of kinds (3a), (4), ( 5 ) , (6) and (3b), one can show that
t-
.
.
q"'l - q m ' + l l
.- q m ' + l l .u,n,.
= qm'l
For changes of kind (6) one uses 4.l(b), 4.3, and 2(e). For changes of kind (3a) or (4), one uses 4.l(b), 4.3, and BP. For changes of kind ( 5 ) , one shows that q'd = qi(d * cod) = q'd * q'cod= q'd * 9'9'1
using l(b), 2(c), and 3(a), respectively, and then argues as for changes ' of kind (3a). For changes of kind (3bL one uses 4.3 and 4.1(b). 0 T h e next lemma reduces the proof that E, is nearly complete to a special case. Recall that by E, being nearly complete is meant the T = T ' , then there is following: If T and T' are terms of L, and if some m such that qmT= q"'?'.
+
LEMMA 4.5: Assume that if k # 0, p is k-normal and does not conand k qkl co ... ck-lp = 0, then there is some m such that tain 1 qm(qkl co ... ck-,p) = q"0 = 0. Then E, is nearly complete. Proof: Assume the hypothesis of the lemma. Suppose that k q k l * p = 0 and that p is k-normal. If k = 0, then 1 q kl p = 0 by the completeness of BP for terms not containing q , p , d , c j , i < w . Now let k # 0. Then qkl co ... ck-,p = 0 and hence, for some m, k q m ( q k l co ... ck-,p) = 0. Inview of l(b), BP,and4.l(g),ourassumption therefore implies the following simpler hypothesis: If 0 s k < w , p is k-normal, and qkl p = 0, then there is some m such that qm(qkl p ) = q"0 = 0. Now let T and T' be any terms of L, such that l= T = 7'.Let u be the normal transform of (T * - T ' ) + ( T ' -7) and let u be weakly nnormal. Let
-
+,
-
-
-
-
-
-
901 -q'1
*
a,+
***
+ q"-'l
*
-q"l
+ q"1
* ul#-l
*
u,,
be the n-development u a o f u . Then t= u@= 0. Now consider any k s n. Then qkl .-qk++'l uk= 0. Since uk is k-normal, given any C I , (ukl is uniform from k on. Hence, qkl uk= 0. By the above simplification of the hypothesis of the lemma, there is some mk such
+
-
+ -
'Note that equalities 3(b) and 3(c)of E, have not been used in the proof.
54
[CH.
A U G M E N T E D CYLINDRIC 'THEORY
4
t-q"'(q"1 - a , , + . . . + q " - ' 1 * a , , ~ , + q " l ~ a , , ) = 0 .
Then, by BP and 4.1 (g),
k q"'u%= 0. Hence
k 0 = q"(((7 -7')+ (7'- -7)) = q y 7 - -7')+ q"'(7' *
.q" - 7')+ (q?' * q"' - 7) - (q"l7 . q"I1 .-q'fl7') + (qVf7' .qUI1 - -qff'7) .-qtri71) + (q~n7' - -q n w
*
-7)
= (q"7 (q7r~7
by 4.4, 2(b), 2(c), 4.l(c), and 4.l(g) and BP, respectively. By BP it follows that 1q"'7 = q'"7'. 0 Some further consequences of BP, 1 (a), ..., 2(d), 2(f), will be useful. Some of these are known from the theory of cylindric algebras.
LEMMA4.6:
(a) (b) (c) (d) (e) (f)
k cicix= cix.
lft- 7 + , t h e n k c j 7 t- Ci(X + y) = cix + ciy. Q
Q
ci+.
F q"kix = ci,,,,qmx.
t-
(8) ProoJ
... ~ , , - ~ q=~q'%for x i < m. q"dij = d(i+k)(j+k). cj-l(di(j-l)-d(j-,)j)= d,,fori < J - 1.
(a) By BP, l(c),and l(b), respectively,
k cicix = Ci(CiX * CiX) = cicix * cix = cix. (b) Assume that
k7
6 7'.
By l(b),
k 7'
C
ci7'. Then
k7
ci7'
CH.
41
55
A U G M E N T E D CYLINDRIC THEORY
and hence F T
- -C~T'
= 0 by BP. Then
by 4.1(a), 1(c), 4.1 (a), the equality just derived, and 1 (a), respectively. Hence, k ci7 C ~ T '= ci7 by BP. ( c ) By BP and 4.6(b), k cix C ci(x+ y) and likewise k c,yC ci(x+ y). Hence cix+ciy C c,(x+y) by BP. By l(b), k x S cix and 1 y ciy. Therefore
-
I-
Ci(X
+ y) C Ci(CiX + ciy) = cj - (- ci - ci(-cix
*
ci
-
cix -ciy)
- c1y) = - (-
cjx * -ciy) = cix
+ ciy
by BP and 4.6(b), BP, 4.l(a) and l(c), 4.l(a) and l(c), and BP, respectively. Hence, f ci(x+ y) = cix+ ciy by BP. (d) Let m > 0 and assume as inductive hypothesis that
by the inductive hypothesis, and 2(d), respectively. (e) Let m > i 2 0 and assume as inductive hypothesis that for any i' < m , t- Ci' Ci'+l ... cn,-2qm-1x = qm-lx. Then
... ~ ~ - 2 ~ ~ _ 1=qcici+l ~ - ~ Cm-2qm-' q ~ CoqX
c~c~+I
1.1
= qm-lcoqx = q"-'qx
by 4.6(d), the inductive hypothesis, and 2(f),, respectively. (f) If i = j , then dijis qi+'l and4i+lHj+l)i~qi+21,~~thatqdiiisc4i+~~j+~,. Now assume that i < j . Then
56
[CH.
A U G M E N T E D CYLINDRIC THEORY
4
by the definition of dij, 2(d), 2(c), and the definition of d(i+l)j+l), respectively. For j < i the proof is similar. It follows by induction that qkd.. = d (I+!,.)( i + k ) . II ( g ) Assume that i < j- I . Then
1 dij = c ~ _ _+_ ~~ ~ - ~ ~ ~ ... - ~- qj-'d ( q ' dqj-ld) = cj-lci+l ... cj-Z(q'd . ... . q'-'d qj-ld) *
*
*
... ~ ~ - ~ ( (* q...' d* q'-'d) * c ~ ... + Cj--2qj-1d) ~ - c,-,(c~+,... cj_,(q'd * ... - qj-'d) - c ~ ... + CjpPqj-ld) ~ - cj-l(ci+I ... ~j-,(q'd. ... . qj-'d). qj-'d) = c. c.1 + 1
= Cj-l(di(j-1)
*
d(j-l)j)
by the definition of dij, I(d), 4.6(e), l(c), 4.6(e), and the definitions of d;, and d( respectively. 0 We now turn to derivations from E, which involve one o r more of the equalities 3(a), 3(b), 3(c).
LEMMA4.7: (a) k ckdij= dij,fur k # i und k # j . (b) k ck(dij* x) = dij c,x,for k # i und k # j . (c) cidii= q t i i n . r . ( i . j ) t l 1. (d) k qk+'l d i j= d,;,for /< < mux (i,j ) . (e) k ci(d,,* x * y) = ci(dij x) * ci(di.; y)for i # jund-2 < i-j (f) k ck(dik* dk,J= qk+'l d,;,for k # i and k # j . dik dk,l= dik dij. (8) PrmJ (a) Assume that k # i and k # j . Case 1 : i =j . Then
-
-
-
-
-
C
2.
k cJij = c,,.qi+ll = qi+ll= dij by the definition of dii, 4.l(d), and the definition of dif, respectively. Cuse 2: i < j . Subcuse (a): k < i. Then ckdi,;= ckq'd,)(, ;-') = qkcoqq'-"-'d
.
l n . i - i ).
- qkqqi-k-1 do( - i = dij -
by 4.6(f), 2(d), 2(f),. and 4.6(f),respectively.
)
CH.
41
51
A U G M E N T E D CYLINDRIC THEORY
Subcuse (b): i
< k < j . Then
q d i i= cI,cil1... ~ ; - ~ ( q* ' ... d * qj-ld)
- cj+l ... c ~ - I c ~ c ~ c ~I +... c;-I(q'd * ... * qj-ld) - ~ i + ... l c ~ - I c ~ c ~ + ... I ~j-l(q'd*... * q'-'d) = dj.;
by the definition of dij, l(d), 4.6(a), and the definition of dtj, respectively. Subcase (c): i < j < k. First let j = i+ 1. Then
k qdii = ckqid= q'ck-,d = q'd = dij by the definition of di(j+lj, 2(d), 3(b) and k - i > 2, and the definition of di(i+l,, respectively. Now let j a i+ 1 and assume as inductive hypothesis that k ckdirjr = dip;,whenever i' < j ' < li and j ' - i' < j - i. Then ckdj = CkCj-l(di(j-1)* d(j-l)j)= Cj-lCk(di(j-l)* d(j-l)j) = Cj-lCk(Ckdi(j-i
j *
Ckd(j-1 j j ) = Cj-I(CkCkdj(j - 1 )
*
Ckd(j - 1
jj)
= Cj-l(Ckdi(j-1) * Ckd(j-ijj)= cj-i(dj(j-i) * d(j-ijj) = djj by 4.6(g), l(d), the inductive hypothesis, I (c), 4.6(a), the inductive hypothesis, and 4.6(g), respectively. Case 3: i > j . Since dtjis d,ii,the proof is the same as for case 2. (b) Let k # i and k # j . Then
C ck(dij
X)
= Ck(Ckdij * X) = ckdi,;
. C ~ =X dij
*
C ~ X
by 4.7(a), I(c) and BP, and 4.7(a), respectively. (c) Case 1 : i =j . Then c,d,,= c i q i + l l = q n r n s ( i . j ) + l l I 11 by the definition of dijand by 4. I(d) respectively. Case 2 : i < j . First let j = i+ 1. Then 2(d), 3(a), and rnax(i,j ) = i+ 1, respectively. by the definition of di(i+l), Now let j > i 1 and assume as inductive hypothesis that - q m o s ( i , j - l ) + l 1= qjl. k c.d. I I ( 3.- I )
+
58
A U G M E N T E D CYLINDRIC THEORY
Then
I- cdi,;= CiC,i-l(di(; - I )
*
-
4
.
4 - l ) ; ) = C,;-lCi(di(.;-t)d(j-l)j) 4 j-l)j) = cjPl(qjl d(j-l)j) = qjl * cj-,d( j - l ) j
= cj-,(cidi(j-l) - qjl q p i i f i . r ( j - l , j l + l l
.
[CH.
-
=.qj+ll
= qi,to.r(i.jl+l
by 4.6(g), 1 (d), 4.7(b), the inductive hypothesis, 4.1 (e), the above proof forj = i+ I , 4. I(b),and max(i,j) = j , respectively. Case 3 : i > j. First let i = j + 1. Then
t c.d. = c,j+,qjd= qjc,d = q j q z l 1
71
= qmns(i.j)+ll
by the definition of d( I ) , r 2(d), 3(a), and max(i,j) = j + 1 , respectively. Now let i > j + I and assume as inductive hypothesis that I- c.d.. - qfnms(i.j')+ll = q i + l l 7 11,wheneverj < j ' < i. Then 1cjdi.j= cjd,ji = ~j~;+l(d,j,j+l) * d(,j+l)i) = C.j+lCi(dj(j+l). d(j+I)j)
= C,i+l(di(j+l) *cid(jtI)O=c.itl(dj(i+l)* q"'1) = q i + l l c,.l + l dJ.( . .I + l ) - qi+ll qfnas(j.j+l)+ll = qi+ll = qmar(i,j)+ll
.
.
by the definition of d,ji,case 2 of 4.7(f) below and 4.7(d) below, l(d), 4.7(b) and BP, the inductive hypothesis, 4.l(e), the above proof for i=j+ 1,4.I(b),andmax(i,j) = i,respectively. Note that case 3 of 4.7(c) is not used for 4.7(d). Nor is 4.7(c) used for case 2 of 4.7(f), although it is used for some other cases. (d) Assume that k S max (i,j) and i S j . Then
.
-
dii= d j j cjdjj= dij
qpptar(i-j)+11
< d.. * qk+'l
by I(b), case 2 of 4.7(c), and 4. I(b), respectively. Hence, dj,;= q"+'l * di,;
by BP. For i > j , then use the definition of dij. (e) Assume that i # j and-2 i-j 2. Then
/-
cj(dj;. X) cj(di,* y) = Ci(dj,;* x * ci(djj. y)) = C ~ ( X* dij * Cj(dij * y)) = C ~ ( X* dj,;* y) = cj(djj * x y)
by I(c), BP, 3(c), and BP, respectively.
CH.
41
59
AUGMENTED CYLINDRIC THEORY
(f) Assume that X # i and X # j. Case 1 : i =j. Then
-
cA(d,, dh,)= c-d,,, = q~n~~~(I,h)+ll = qA+'l* q'+'l = q"I1
*
d,,
by i =j and BP, 4.7(c), 4. I(b), and i =j and the definition of d,,, respectively. Case 2 : i < k < j. First letj- k = 1. Then
[CH.
A U G M E N T E D C Y L I N D R I C THEORY
4
Then
k Cddik . dkj) = Ck(Ck+l(dk(k+l).d(k+l)i).Ck+l(dk(k+l)* &+Id) =ck,lch.(dk,/~+1).d(/~+l)i.d(kli)i)
= c k * l(Ckdk(k+l)* d(k+l)i* d(k+~);) = ~ kI(qk+21 + .dik, I ) j * d(k+l)j)= qk+21* CkLI(d(k+l)i* d(k.l , j ) - q k 1 2 1 .q k + 2 1 . d,. = q k + 2 1 di, 1.1
.
by case 2 above and 4.7(d), 4.7(e) and l(d), 4.7(b), 4.7(c), 4.l(e), the inductive hypothesis, and BP respectively. Case 4: i < j < k . Then the proof differs only in minor respects from that for case 3. For the cases where j < I, < i, k < j < i, or j < i < k , the proofs are the same as for the cases 2 , 3, and 4, respectively, since di, and d,iiare the same term.:' ( 8 ) If k # i and k # j , then dik dk,j= (dik . dkj) * Ck(djk * dk,j) = dik * dk,;* qk+'l di;
= dik * dk; * di,j by 1 (b), 4.7(f), and 4.7(d), respectively. If k
= i o r k = j, then
k dik - dk,;= dik * dk,;- dii
by BP. Since dik is dki,similarly
k dik * di,;= djk Hence
dij * dk,;.
-
dik * dkj = dik djj
by BP. 0
-
Next, we turn to equalities containing terms c,;(dji 7 ) where i # j. More briefly and more suggestively, we let S;T be ci(dii 7 ) . We shall employ 'DF' to appeal to this use or definition. In this use we shall assume tacitly, for the rest of ch. 4, that i # j ; similarly for other suband superscripts.
-
:'For cases I and 2, the proof does not use any equality 3 ( c ) of E,. For cases 3 and 4, it uses only those instances of 3 ( c )where either i =j + 1 or i = j - I .
CH. 41
61
A U G M E N T E D CYLINDRIC THEORY
LEMMA 4.8: (a) qk+'l * six = s:(qk++'lx) = s:x,for k G max (iJ. (b) I- qs:x = siz,!qx. (c) -t C;$X = s:x. (d) k c,s:x = s:ckx, f o r k # i and k # j. (e) t- cks:x 4 s:c,x, f o r k # i. s;si,'x = si)'six,forj # j ' , j # i', undj' # i. (f)
-
-
ttt-
s;s;,x = q'+'l $,x. s:s:'x = s$$'x, for i # j'. c.sjc.x t . l = qnr~~~(i,j)+ll cyix. = qntft.r(i.j)+ll c,;x. (j) C s!cjx 1 ' (k) dii syx = di,; s$x,for k # i und k # j . (8)
(h) (i)
t-
-
.
-
.
Prooj (a) Assume that ( i # j and)k qk+'l
- s;x
- cj(dji
= q"'l
*
G max
( i , j ) .Then
-
x) = C j ( q k + ' l
d.. 31 x) = c.(d.. 3 11
X)
= S!X
by DF, 4. I(e), 4.7(d), and DF, respectively. Also
k s:(q"+'l
*
X)
= c,;(djj * qk+'l* X) = c;(dj;
X)
= S:X
by DF, 4.7(d), and DF, respectively. (b) By DF, 2(d), 2(c), 4.6(f), and DF, respectively,
k q S $ = qCj(d,ii.
X) = C , j + l q ( d i i . X) = c,i+l(qdii * qx) = C.;+i(d(;+i)(i+i) * qx) = siI:SX.
(c) By DF, 4,6(a), and DF, respectively, cjsijx = c,;c,;(dji
* X)
= c,;(dji
*
X)
= S:X.
(d) Assume that k # i and k # j. Then
k CS,X:
= CkCj(djj
*
X)
= CjCk ( d ji
*
X)
= cj(d;;
*
C~X= ) S ~ C ~ X
by DF, l(d), 4.7(b), and DF, respectively. (e) Assume that k # i. If k # j , then
I-
c,s:x
*
s:clix
= c,s:x
by 4.8(d). Now assume that k =j.Then CS ,X :
= C,iC;(d,;i
= C,j(dji
* X) 6 C;Cj(dji * C ~ X )
.C ~ X )= S ~ C ~ X
62
A U C M E N l E D CYLINDRIC THEORY
[CH.
4
by D F and j = X , I(b) and BP and 4.6(b), 4.6(a), and;= k and D F , respectively. ( f ) Assume thatj # j ' , j # i';andj' # i. Then s/s:,'x = cj(d,i
*
C#,(d,,ic *
- cj,cj(dji - d,.;,
.
x)) = c;cjn(d;j d,,i, * X)
= Cjg(djrig
X)
- cj(dji - x)) = S$S$
by DF, 4.7(b), I(d), 4.7(b) and BP, and DF, respectively. (8) By DF, I(c). 4.7(c), 4.l(b),4.l(e),4.7(d), and D F , respectively, s~s,!,x = Cj(dji
-
qiiirr.r(
-
x)) = cjdji
Cj(djic
i .j) t 1
1 -c.;(dji.
X)
-
Cj(djit
-
-
X)
= qi+'l qj+'l * cj(dji,
*
X)
= q'+ll . c j ( q j + l l . d.i,. x) = q ; + l l . ~ , ~ ( d .x); ~=, q i + l l . s;,x.
( h ) Assume that i # j ' (and that i # j a n d j # j ' ) .Then
-
k S:S:'X
-
= c;(d;; cjr(d;tj x)) = c;c;t(d;i d,j,,; X) = cici,(d;i * d;ti * X) = c,;(d;i * ci,(dirj * x)) = S:S:'X
by DF, 4.7(b), 4.7(g), 4.7(b), and DF, respectively. (i) By DF, I(d), l(c), 4.7(c), and 4.1(e), respectively,
k C~S;C~X= cicj(di;
*
C ~ X )= cjci(dij * C ~ X )= cj(cidfj * C ~ X )
.
= c,j(qinns(i.i)+ll cix) = q i i i f i . r ( i . j ) + l l
.c,;cix
(j) By DF, l(c),and 4.7(c), respectively,
t-
S$X
.
J I . 1 . c.I . = ~ q ~ l r f ~ s ( i . j ) + l l . c,ix. = c.(d. J 0 cJ . ~ = ) c.d..
( k ) Assume that k # i and k # j. Then
k di;
-
= dij * Ck(dpi * X) = Ck(dij * dki X) = ck(dj,; * dkj * X) = dij Ck(dkj * X) = dfj * SFX
* S ~ X
by D F , 4.7(b), 4.7(g), 4.7(b), and DF, respectively. In 4.9(c) below we obtain a generalization of 4.7(e) for those cases where i < j and where x and y are of the form cku. The generalization consists in dropping the condition that -2 s i-j. T o prove 4.9(c), we use 4.9(b), which by induction implies that under suitable conditions s: can be replaced by combinations of s;,', each satisfying-2 s i' -j' c 2.
CH.
41
63
A U G M E N T E D C Y L I N D R I C THEORY
LEMMA 4.9: = q ~ ~ ~ x ( i 3 j )c.x +1 1) for-2 1 < i-j (a) 1 s2slcix 2. J 1 S $ ~ X = s ~ ~ ~ ~ j + ~ ~ ~ + , ~ 0 j , < , i ~ < ~ j ~- ~1.~ ~ - ~ x , f o r (b) (c) 1 sk(ctx c,,y) = sijcpx s:cty,for k < i < j. Proof: (a) Assume that -2 s i - j d 2. Then
-
-
-
~ s $ : c ~= x ci(di,i* ci(dii c~x))= ci(dii * cix) = cidii* cix - q l l t f t x ( i , j ) + l l . cix
by DF, 3(c), I(c), and 4.7(c), respectively. (b) Assume that 0 < i < j- 1. Then
1 S!
s!
s(+ly
t i 1 2+1
1-1
= s( si ,+I
I f ,
si,+~y = 1-1
sI
t+l
sjsi+ly i i-1
= s'
)+I
si+lsjy 1-1 i
by 4.8(f), 4.8(h), and 4.8(f), respectively. Hence
1s ~ ~ ~ s ~ + ' s ~ + l s j += l s s;;;si+'sj+ j ~ ~ c i - I ls;+~s~ci-lx x = sj~~sj+'sj+lci+lsj'~s~cj~lx = s!-l(qj+21 C. si+lsjci_lx) If, l + l i-1 i
.
= s ; ~ ~ c i + l s ~ ' ~= s ~s~;;s~?~s;Ci-,x ci~lx = S f - l S i + l C , s j - i + 1~ * Ci-lSijX (+I
i-1
= q'+21
7-1
.SJC.
ix- q x = S,Ci-,X I
1-1
by the equality just derived, 4.8(c), 4.9(a), 4.8(a), 4.8(c), 4.8(d), 4.9(a), 4.8(d), and 4.8(a), respectively. (c) Assume that k < i < j. We shall first show that
-
t-
-
s:(ci-,x Ciply)= s:ci-,x si'ci-,y.
If j - i = 1, then this holds by 4.7(e) and DF. Now let j- i > 1 and assume as inductive hypothesis that
t-
S;+&x
*
-
ciy) = s;+lcfx s;+Iciy.
64
AUGMENTED CYLINDRIC THLORY
Then
t-
s;(ci-lx * cj-ly) = s;cj-l(cj-lx
-
[CH.
4
Cj-'y)
= sj;;sj+ls;+I S ; + l S j ~ ~ C ; - I ( C I - l X* cj-ly) = si;;sj+ls;,
-
lsj+lsj~;(cj-lxci-ly)
= sj;:s;+lsij+$j+
Isj+:cj-Ix* sj+ls;:;cj..ly)
= s;;;sj+1sij, I(cjsj+lsj?;c;~lx * cjs~+lsj~;cj-ly) - I , IsI i + l ( s J I t I c 8. s; ~ + l s ~ ~ ; c j -sit l xl c j s ~ + l s j ~ ; c j - l y )
-
= sj;;sj+ls! I t 1 slI f 1 Sl+'Cj-,X r-1 - s,icj-lx* sijcj-Iy -
- sj;;sj+1s;+lsj+lsj+;cj-ly
by 4.6(a) and I(c), 4.9(b). l(c) and 4.6(a), 4.7(e) and D F (twice), 4.8(c), the inductive hypothesis, 4.7(e) and D F (twice) and 4.8(c), and 4.9(b), respectively. N o w let k < i - 1 and assume as inductive hypothesis that
k d ( C k , IX Then
cp, ly) = s;ck+lx* s:'cktIy.
-
s:(c/;x c,y) = s;ck(ckx* c,,.y) = s;(q"f21 * Ck(CkX * cky)) = s J skk , $.,+)I Ck(C/,.X CkY) = s;+Is;s;+'ck(c,x
-s;+1
-
sj(sk+lckx. I k
*
C,Y)
&+l p CkY)
= s;+lsij(cp,,s;+lcI;x *
Ck+ 'S;+'cky)
- s;+,(sijck, Is;+'ckx * sjck+,S;+'cky) - s;+ I s J s k + l CkX * s;, 's;S;+'cky -
-
= s;s;+ Is;+'ckx S;S;tIS;+'Cpy
- ,
= s$q"+"l c x) * s;(q"+'l
- c,y) = s;c,x
*
s;Cky
by 4.6(a) and l(c), 4.8(a), 4.9(a), 4.8(f), 4.7(e) and D F , 4.8(c), the inductive hypothesis, 4.7(e) and D F and 4.8(c), 4.8(f), 4.9(a), and 4.8(a), respectively. 0
LEMMA4.10: (a) IsijO = 0. (b) sijl = qttfo.r'Ci.j)+l 1. ( c ) 1s;(x y) = sijx sijy.
+
+
CH.
41
6.5
A U G M E N T E D C Y L I N D R I C THEORY
-
(d) 1s:(qx * qy) = skqx s:qy,for 0 < i < j. (e) t- s: - qx = q’+’I - s,’qx,for O < i < j. ( f ) If T s T ’ , then 1 S/T C S:T’. Proofi (a) By D F , BP and I(a), respectively,
-
1 s:O = c,j(d.ji
*
0) = cj0 = 0.
(b) By D F , 4.l(d) and I (c),and 4.7(c) and BP, respectively,
k s,’l = ci(d,jj * 1)= c.d.. . 1= qll~~~J’(i.j)+ll I I1
(c) By D F , BP, 4.6(c), and DF, respectively,
1 S:(X+Y)
+
= Ci(d,jj * (x+Y)) = C,j((d,jj . X) (d,ji .y)) = C;(dji * X) cj(dji y) = SX; s,’Y.
+
+
.
(d) Assume that 0 < i < j. Then
t-
$(qx
*
qy) = s;(c,,qx
*
c,qy) = s;c,qx
*
s:c,,qy
= s;qx
*
s;qy
by 2(f),, 4.9(c), and 2(f),, respectively. (e) Assume that 0 < i < j. Then
1s;-
qx * sjqx = si(q1
- - qx) - s;qx
= s;q - x * siqx
= s/(q-x* qx) = $0) = 0
by 4.8(a), 4.l(c), 4.1O(d), 2(c) and BP and 2(a), and 4.1O(a), respectively. Also,
k s; - qx + s;qx
= s/(-qx
+ qx) = $1 = qj+’l
by 4.10(c), BP, and 4.1O(b),respectively. Hence by BP, s; - qx = q’+’l - s,‘qx.
.
(f) Assume that
T
s 7’ and hence
7+7’
= T ‘ . Then by 4.10(c),
k $7 + S i 7 ’ = S i ( 7 + 7 ’ ) = S i 7 ’ .
Hence S:T
s Si7’. 0
We conclude with a few remarks on axiomatization and definability. Much of ch. 4 is a struggle for economy in the axioms involving d. In particular, 4.7(f) is derived rather than taken as an axiom scheme.
66
A U G M E N T E D CYLINDRIC THEORY
[CH.
4
Also, 3(c) is restricted to those i and j which are 1 or 2 integers apart, while similar equalities where i and j are further apart are derived. Thus, 3(c) consists of four 1-dimensional infinite arrays of axioms. We doubt that any of these four arrays is redundant or replaceable by a finite array. Only I(d) consists of an array of axioms which is infinite in 2 dimensions. We doubt that it can be replaced by a finite number of 1 -dimensional arrays. There are two schemes, namely 2(e) and 3(c),for which there is no finite upper bound to the length of instances of the scheme. This can be remedied by using d,,, d , . d2, ... as primitives in place of d. One then takes qd; = dj+las additional axioms, i < w, makes the obvious changes in 3(a), 3(b), 3(c), and in 2(e) either replaces q'+' 1 by pd; or, for i # 0. replaces q i t ' 1 by ci-ldj-l. Given any i , it follows from 2(d) and 2(g) that ctx = pcitlqx. Hence c, is definable in terms of p, q and ci +,. This redundancy of primitives of I L,, 1 appears to be unavoidable. There result certain redundancies of axioms. For example, cjc,x= cicix can be derived from c ~ + ~ c ~ + ~ x = c , + ,cj + /,.xtogether with certain other axioms. Finally, we note that the terms djkcould have been replaced by terms dj, that are defined using only d, -,q, and c,. Let q,,,T be c,(d 47) and, for i < k , let d:,,.and di., be the term qiq,l,k-i-'d.When k - i = 1 or k - i = - 1 then dj, is dikr and when k - i = 2 or k - i = - 2 then dj, = djk can be derived from E, without using 3(a), 3(b), or 3(c). Hence, replacement of 3(c) as an axiom scheme by the set of corresponding equalities which involve dl, instead does not affect the set of derivable equalities. Moreover, by the completeness of Epq,to be proved in ch. 5, if an equality involving terms dikis derivable, then the corresponding equality involving terms d,!, instead is also derivable. Whether derivations of the latter would have been easier, we do not know.
-