Appendix D The Normalization Algorithm

Appendix D The Normalization Algorithm

APPENDIX D THE NORMALIZATION ALGORITHM In this appendix, we extend the relation > of Appendix C to a global reducibility relation L which determines...

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APPENDIX D

THE NORMALIZATION ALGORITHM

In this appendix, we extend the relation > of Appendix C to a global reducibility relation L which determines the choice and order of application of Rules (R2)-(R17) in cases of syntactic ambiguity. The relation L is defined on three subclasses of Der(A(X)) whose elements represent the arrows of (a) non-Cartesian, non-symmetric, (b) nonCartesian, symmetric, and (c) Cartesian categories, respectively. The adopted priorities may be summarized as follows: (a)

(R8) 5 (R14) 5 (R9) 5 (R2), (R15) I(R14) I(R9) I(R2), (R17) I(R14) I(R9) I(R2).

(b)

(R8) I(R14) I(R9) I(R2) I(R4), (R15) I(R14) I(R9) I(R2) I(R4).

(c) (R5) 5 (R13) 5 (R10) I(R12) 5 (R14) I(R3) I( R l l ) 5 (R2) 5 (R4), (R15) I(R12) I(R14) I(R3) 5 (R11) I(R2) I(R4), where I is transitive, and (Ri)<(Rj) indicates that (Ri) takes precedence over (Rj) whenever there exists a choice in their order of application. An argument by cases shows that the selected priorities are compatible. Specifically, the relation 2 is the smallest monotone partial ordering on Der(A(X)) containing > and satisfying the following further conditions:

(R2-R2) f

I

244

T H E N O R M A L I Z A T I O N AI.GOR1THM

245

(R2-R3)

(D.2.1)

(D.2.2) h TaA+ @ (D.2.3)

(D.2.4)

(R2-R4)

(D.3.1)

(D.3.2)

(D.3.3)

(D.3.4)

246

APPENDIX D

(R2-R5

(R2-R8)

(D.5.1)

(D.5.2) (R2-R9)

(D.6.1)

(D.6.2) (R2-R 1 0)

T H E NORMAL.IZATION AI-GORITHM

247

(R2-Rll)

(D.8.1)

(R2-R 12)

(R2-R 13)

(D. 10)

248

APPENDIX D

(D.12.1)

(D. 12.2) (R2-R 16)

(D.13.1)

(D.13.2)

(R2-R 17)

(D.14.1)

provided that in (D.14.2),I is the only atomic subformula of a.

T H E NORMALIZATION ALGORITHM

249

(D.15.2) where n L 3, A = ( Y I . * an, ai = a ( 1 5 i 5 n ) , and (a)and ( 7 ) denote n - 1 instances of (R3), with ai active before aj in ( 7 ) if i < j.

(D.16.1)

(D.16.2)

(D.16.3) (R3-R5)

(R3-R10)

250

APPENDIX D

(R3-Rll)

(D. 19.1)

(D. 19.2)

(D. 19.3)

(R3-R 12)

25 1

T H E N O R M A L I Z A T I O N ALGORIT HM

(R3-R14)

(D.22.1)

(D.22.2)

(D.22.3)

(R3-Rl5)

(D.23.1)

(D.23.2) (R4-R4) For any permutation r of the integers 1 , . successions (cr) and ( 7 ) of instances of (R4),

r +f @

A+@

. . , n , and

any

(D.24.1)

provided that T = a1 * * an,A = PI * . . p., pi = aa(i)for 1 5 i 5 n, and ( 7 ) is the unique string of interchanges which first moves a,,(~) to PI, then a,(~)to 0 2 , etc. If r is the identity permutation, the right-hand side denotes f.

252

APPENDIX D

If I is the only atomic subformula of a,or if a = T, then

and

where g and k are the unique cut-free derivations obtained by applying the cut elimination algorithm of Appendix C to the derivations

where m is the derivation of -+ (Y described in 8.6.2, 9.6.2, and 10.6.4, and where ((T)involves no instances of (Rl) and (R4), and contains at most instances of (R8) or (R10) whose premisses have empty antecedents, and at most instances of (R14) whose left premisses have empty antecedents.

(R4-RS)

(D.25) (R4-R8)

(D.26.1)

(D.26.2)

T H E NORMALIZATION ALGORITHM

253

(R4-R9)

(D.27.1)

( D. 27.2)

(D.27.3)

(D.27.4)

(R4-RlO)

(R4-Rll)

(D.29.1)

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APPENDIX D

(D.29.2)

(D.29.3)

(D.29.4)

(R4-R12)

(R4-R13)

(D.31)

T H E NORMALIZATION ALGORITHM

255

(D.32.4)

(D.32.5) (R4R15) (D.33.1)

(D.33.2)

(D.33.3)

256

APPENDlX D

(D.33.4) (R5-R5)

(D.34)

(R5-Rl0)

(D.35.2)

(D.35.3)

(D.35.4)

(D.35.5)

T H E NORMALIZATION ALGORITHM

257

(D.35.9)

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APPENDIX D

(R5-R 1 1 )

(D.36)

(R5-RI2)

(R5-Rl3)

(D.38.1)

(D.38.2) (R5-RI4)

(D.39)

T H E NORMALIZATION ALGORITHM

259

(D.40.1)

n

(D.40.2)

(R8-R 14)

(R8-R 16)

(R9-R9)

(D.43)

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APPENDIX D

(R9-R 14)

(R9-R 15)

(D.45.1)

(R9-R 16)

T H E NORMALIZATION ALGORIT HM

26 I

(R9-R 17)

(D.47) (R 10-R 10) c

262

APPENDIX D

(D.51.2)

(R 10-R 14)

(D.52)

(R1I-R11)

(D.53) (R11-R12)

T H E NORMALIZATION ALGORITHM

263

(R11-RI3)

(D.55) (Rll-R14)

(R11-Rl5)

264

APPENDIX D

(D.57.2)

THE NORMALIZATION ALGORITHM

265

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APPENDIX D

(R14-R 15)

(D.65.3)

(D.65.4)

(D.65.5)

T H E NORMALIZATION ALGORITHM

-2

261

(D.65.7)

(R14-R 16)

(R14-R17)

(D.67) (R15-R 16)

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APPENDIX D

(R16-R 16)

(R 16-R 17)

(D.70) (R2-R2-R 10)

(R2-R2-R 12)

I'HE N O K M A I I 7 A T I O N A I . ( i O R I T H M

(R2-R3-R 10)

(R2-R3-R 12)

269

270

(R2-R4-R 10)

(R2-R4-R 12)

APPENDIX D

T H E NORMALIZATION ALGORITH M

27 I

(D.76.9)

(D.76.10)

(D.76.I I )

(D.76.12)

(D.76.13)

(D.76.14)

(D.76.15)

(D.76.16)

THE NOKMAI 1 / 4 1 1 0 N

(R3-R3-R 10)

(R3-R3-R 12)

A I CnOKIIHM

273

214

APPENDIX D

(R4-R4-R 12)

n

r S y A a v PA-+ @

(D.80.

(D.81.1)

(D.8 1.2)

T H E NORMALIZATION ALGORITHM

rrAajph+ rra 3 P A A + r a 4 rra+pA+

Tct+PA-f2 If a

= p, and

ra+$A+

if I is the only atomic subformula of a,or if a

275

(D.81.3)

= T, then (D.81.4)

t

(D.82.2)

-

P

(D.82.4)

(R8) If I is the only atomic subformula of a and /? and of the terms of r and A, then r m n +a &(,) r+a A+P, - + a n p (D.83) TA+auP rA+anP

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APPENDIX D

with r, s, and (a)involving at most instances of (R8), resp. (R14), whose premisses, resp. left premisses, have empty antecedents, as described in Lemma 8.6.2.

(R14) If I is the only atomic subformula of a and p and of the terms of A, and A, then

r,

(D.84) with r, s,(a), and (7) involving at most instances of (R8), resp. (R14), whose premisses, resp. left premisses, have empty antecedents, as described in Lemma 8.6.2.

(R15) If I is the only subformula of a,or if a =T, then

(D.85) where g is the unique cut-free derivation obtained by applying the cut elimination algorithm of Appendix C to the derivation

where h is the derivation of + a described in 8.6.2,9.6.2,and 10.6.4, and where (a)involves no instances of (Rl) and (R4), and contains at most instances of (R8) or (R10) whose premisses have empty antecedents, and at most instances of (R14) whose left premisses have empty antecedents. This completes the description of the global defining conditions of 2 . All additional local requirements ensuring the uniqueness of representation of unique arrows are listed in the main body of the text. The effectiveness of 2 requires that the semantic condition in (D.12.3), (D.12.4), (D.24.2), (D.81.4). and (D.85) is given a syntactic

T H E NORMALIZATION ALGORITHM

277

characterization. Corollaries 4.6.3, 5.6.6, 6.6.6, and Lemmas 9.6.3 and 10.6.2 achieve this purpose. We define a derivation f to be normal if it is cut-free and if f 2 g implies that f = g . We say that f reduces to g if there exists a finite sequence ( f l , . . . f n > of derivations of A(X) such that f = f l , fn = g, and f l 2 f z 2 . . . 2 fn, and say that f reduces immediately to g , written as f % g , if f = g or if f 2 g by virtue of precisely one of the defining conditions of 2 . An argument by cases shows that if f % g and f % h, then there exists a derivation p such that g 8 p and h ~ p and , an induction on n + m extends this result to show that if f % f l % . . . % fn and f % g l % . * * + gm, there exists a derivation q with the property that fn 2 q and gm 2 q. Hence any derivation fEDer(A(X)) reduces to at most one normal derivation g . Since the relation > is contained in 2 , every fEDer(A(X)) which represents an arrow of one of the categories constructed in Chapters 2-12, reduces to a cut-free derivation g , and an induction on the number of violations of Conditions (D.l-85) in g shows that f reduces to a normal, and hence unique normal, derivation h. The relation 2 thus defines an algorithm for normalizing any f E Der(A(X)) belonging to one of the subclasses Der(xA(X)) of Der(A(X)) mentioned at the beginning of this appendix. Let = be the equivalence relation on Der(A(X)) generated by 2 . Then an induction on the length of the proof that f - g shows that for all f, g E Der(A(X)) representing an arrow of one of the categories constructed in Chapters 2-12, there exists a derivation h such that f 2 h and g 2 h. This property is called the Church-Rosser property of 2 . The proofs of the normalization theorems for the various subsystems xA(X) of A(X) in the main body of the text consist of the appropriate verifications that the relation z Xis contained in where = x is the equivalence relation on Der(xA(X)) induced by the interpretation S : Der(xA(X))+ ArFx(X), and where Z x is the reducibility relation on Der(xA(X)) generated by the local refinement of the restriction of 2 to Der(xA(X)). The proofs of the Church-Rosser theorems for the various subsystems xA(X) of A(X) in the body of the text, on the other hand, consist of the appropriate verifications that distinct normal derivations f and g of a sequent r + @represent distinct arrows of Fx(X), i.e., that f = * g iff f - x g , where z Xis the equivalence relation on Der(xA(X)) generated by zx.