author | wenzelm |
Sat, 15 Apr 2000 15:00:57 +0200 | |
changeset 8717 | 20c42415c07d |
parent 55 | 331d93292ee0 |
permissions | -rw-r--r-- |
0 | 1 |
(* Title: ZF/qpair.ML |
2 |
ID: $Id$ |
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 |
Copyright 1993 University of Cambridge |
|
5 |
||
6 |
For qpair.thy. |
|
7 |
||
8 |
Quine-inspired ordered pairs and disjoint sums, for non-well-founded data |
|
9 |
structures in ZF. Does not precisely follow Quine's construction. Thanks |
|
10 |
to Thomas Forster for suggesting this approach! |
|
11 |
||
12 |
W. V. Quine, On Ordered Pairs and Relations, in Selected Logic Papers, |
|
13 |
1966. |
|
14 |
||
15 |
Many proofs are borrowed from pair.ML and sum.ML |
|
16 |
||
17 |
Do we EVER have rank(a) < rank(<a;b>) ? Perhaps if the latter rank |
|
18 |
is not a limit ordinal? |
|
19 |
*) |
|
20 |
||
21 |
||
22 |
open QPair; |
|
23 |
||
24 |
(**** Quine ordered pairing ****) |
|
25 |
||
26 |
(** Lemmas for showing that <a;b> uniquely determines a and b **) |
|
27 |
||
28 |
val QPair_iff = prove_goalw QPair.thy [QPair_def] |
|
29 |
"<a;b> = <c;d> <-> a=c & b=d" |
|
30 |
(fn _=> [rtac sum_equal_iff 1]); |
|
31 |
||
32 |
val QPair_inject = standard (QPair_iff RS iffD1 RS conjE); |
|
33 |
||
34 |
val QPair_inject1 = prove_goal QPair.thy "<a;b> = <c;d> ==> a=c" |
|
35 |
(fn [major]=> |
|
36 |
[ (rtac (major RS QPair_inject) 1), (assume_tac 1) ]); |
|
37 |
||
38 |
val QPair_inject2 = prove_goal QPair.thy "<a;b> = <c;d> ==> b=d" |
|
39 |
(fn [major]=> |
|
40 |
[ (rtac (major RS QPair_inject) 1), (assume_tac 1) ]); |
|
41 |
||
42 |
||
43 |
(*** QSigma: Disjoint union of a family of sets |
|
44 |
Generalizes Cartesian product ***) |
|
45 |
||
46 |
val QSigmaI = prove_goalw QPair.thy [QSigma_def] |
|
47 |
"[| a:A; b:B(a) |] ==> <a;b> : QSigma(A,B)" |
|
48 |
(fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]); |
|
49 |
||
50 |
(*The general elimination rule*) |
|
51 |
val QSigmaE = prove_goalw QPair.thy [QSigma_def] |
|
52 |
"[| c: QSigma(A,B); \ |
|
53 |
\ !!x y.[| x:A; y:B(x); c=<x;y> |] ==> P \ |
|
54 |
\ |] ==> P" |
|
55 |
(fn major::prems=> |
|
56 |
[ (cut_facts_tac [major] 1), |
|
57 |
(REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ]); |
|
58 |
||
59 |
(** Elimination rules for <a;b>:A*B -- introducing no eigenvariables **) |
|
60 |
||
61 |
val QSigmaE2 = |
|
62 |
rule_by_tactic (REPEAT_FIRST (etac QPair_inject ORELSE' bound_hyp_subst_tac) |
|
63 |
THEN prune_params_tac) |
|
64 |
(read_instantiate [("c","<a;b>")] QSigmaE); |
|
65 |
||
66 |
val QSigmaD1 = prove_goal QPair.thy "<a;b> : QSigma(A,B) ==> a : A" |
|
67 |
(fn [major]=> |
|
68 |
[ (rtac (major RS QSigmaE2) 1), (assume_tac 1) ]); |
|
69 |
||
70 |
val QSigmaD2 = prove_goal QPair.thy "<a;b> : QSigma(A,B) ==> b : B(a)" |
|
71 |
(fn [major]=> |
|
72 |
[ (rtac (major RS QSigmaE2) 1), (assume_tac 1) ]); |
|
73 |
||
74 |
val QSigma_cong = prove_goalw QPair.thy [QSigma_def] |
|
75 |
"[| A=A'; !!x. x:A' ==> B(x)=B'(x) |] ==> \ |
|
76 |
\ QSigma(A,B) = QSigma(A',B')" |
|
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
|
77 |
(fn prems=> [ (simp_tac (ZF_ss addsimps prems) 1) ]); |
0 | 78 |
|
79 |
val QSigma_empty1 = prove_goal QPair.thy "QSigma(0,B) = 0" |
|
80 |
(fn _ => [ (fast_tac (ZF_cs addIs [equalityI] addSEs [QSigmaE]) 1) ]); |
|
81 |
||
82 |
val QSigma_empty2 = prove_goal QPair.thy "A <*> 0 = 0" |
|
83 |
(fn _ => [ (fast_tac (ZF_cs addIs [equalityI] addSEs [QSigmaE]) 1) ]); |
|
84 |
||
85 |
||
86 |
(*** Eliminator - qsplit ***) |
|
87 |
||
88 |
val qsplit = prove_goalw QPair.thy [qsplit_def] |
|
89 |
"qsplit(%x y.c(x,y), <a;b>) = c(a,b)" |
|
90 |
(fn _ => [ (fast_tac (ZF_cs addIs [the_equality] addEs [QPair_inject]) 1) ]); |
|
91 |
||
92 |
val qsplit_type = prove_goal QPair.thy |
|
93 |
"[| p:QSigma(A,B); \ |
|
94 |
\ !!x y.[| x:A; y:B(x) |] ==> c(x,y):C(<x;y>) \ |
|
95 |
\ |] ==> qsplit(%x y.c(x,y), p) : C(p)" |
|
96 |
(fn major::prems=> |
|
97 |
[ (rtac (major RS QSigmaE) 1), |
|
98 |
(etac ssubst 1), |
|
99 |
(REPEAT (ares_tac (prems @ [qsplit RS ssubst]) 1)) ]); |
|
100 |
||
101 |
||
102 |
val qpair_cs = ZF_cs addSIs [QSigmaI] addSEs [QSigmaE2, QSigmaE, QPair_inject]; |
|
103 |
||
104 |
(*** qconverse ***) |
|
105 |
||
106 |
val qconverseI = prove_goalw QPair.thy [qconverse_def] |
|
107 |
"!!a b r. <a;b>:r ==> <b;a>:qconverse(r)" |
|
108 |
(fn _ => [ (fast_tac qpair_cs 1) ]); |
|
109 |
||
110 |
val qconverseD = prove_goalw QPair.thy [qconverse_def] |
|
111 |
"!!a b r. <a;b> : qconverse(r) ==> <b;a> : r" |
|
112 |
(fn _ => [ (fast_tac qpair_cs 1) ]); |
|
113 |
||
114 |
val qconverseE = prove_goalw QPair.thy [qconverse_def] |
|
115 |
"[| yx : qconverse(r); \ |
|
116 |
\ !!x y. [| yx=<y;x>; <x;y>:r |] ==> P \ |
|
117 |
\ |] ==> P" |
|
118 |
(fn [major,minor]=> |
|
119 |
[ (rtac (major RS ReplaceE) 1), |
|
120 |
(REPEAT (eresolve_tac [exE, conjE, minor] 1)), |
|
121 |
(hyp_subst_tac 1), |
|
122 |
(assume_tac 1) ]); |
|
123 |
||
124 |
val qconverse_cs = qpair_cs addSIs [qconverseI] |
|
125 |
addSEs [qconverseD,qconverseE]; |
|
126 |
||
127 |
val qconverse_of_qconverse = prove_goal QPair.thy |
|
128 |
"!!A B r. r<=QSigma(A,B) ==> qconverse(qconverse(r)) = r" |
|
129 |
(fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]); |
|
130 |
||
131 |
val qconverse_type = prove_goal QPair.thy |
|
132 |
"!!A B r. r <= A <*> B ==> qconverse(r) <= B <*> A" |
|
133 |
(fn _ => [ (fast_tac qconverse_cs 1) ]); |
|
134 |
||
135 |
val qconverse_of_prod = prove_goal QPair.thy "qconverse(A <*> B) = B <*> A" |
|
136 |
(fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]); |
|
137 |
||
138 |
val qconverse_empty = prove_goal QPair.thy "qconverse(0) = 0" |
|
139 |
(fn _ => [ (fast_tac (qconverse_cs addSIs [equalityI]) 1) ]); |
|
140 |
||
141 |
||
142 |
(*** qsplit for predicates: result type o ***) |
|
143 |
||
144 |
goalw QPair.thy [qfsplit_def] "!!R a b. R(a,b) ==> qfsplit(R, <a;b>)"; |
|
145 |
by (REPEAT (ares_tac [refl,exI,conjI] 1)); |
|
146 |
val qfsplitI = result(); |
|
147 |
||
148 |
val major::prems = goalw QPair.thy [qfsplit_def] |
|
149 |
"[| qfsplit(R,z); !!x y. [| z = <x;y>; R(x,y) |] ==> P |] ==> P"; |
|
150 |
by (cut_facts_tac [major] 1); |
|
151 |
by (REPEAT (eresolve_tac (prems@[asm_rl,exE,conjE]) 1)); |
|
152 |
val qfsplitE = result(); |
|
153 |
||
154 |
goal QPair.thy "!!R a b. qfsplit(R,<a;b>) ==> R(a,b)"; |
|
155 |
by (REPEAT (eresolve_tac [asm_rl,qfsplitE,QPair_inject,ssubst] 1)); |
|
156 |
val qfsplitD = result(); |
|
157 |
||
158 |
||
159 |
(**** The Quine-inspired notion of disjoint sum ****) |
|
160 |
||
161 |
val qsum_defs = [qsum_def,QInl_def,QInr_def,qcase_def]; |
|
162 |
||
163 |
(** Introduction rules for the injections **) |
|
164 |
||
165 |
goalw QPair.thy qsum_defs "!!a A B. a : A ==> QInl(a) : A <+> B"; |
|
166 |
by (REPEAT (ares_tac [UnI1,QSigmaI,singletonI] 1)); |
|
167 |
val QInlI = result(); |
|
168 |
||
169 |
goalw QPair.thy qsum_defs "!!b A B. b : B ==> QInr(b) : A <+> B"; |
|
170 |
by (REPEAT (ares_tac [UnI2,QSigmaI,singletonI] 1)); |
|
171 |
val QInrI = result(); |
|
172 |
||
173 |
(** Elimination rules **) |
|
174 |
||
175 |
val major::prems = goalw QPair.thy qsum_defs |
|
176 |
"[| u: A <+> B; \ |
|
177 |
\ !!x. [| x:A; u=QInl(x) |] ==> P; \ |
|
178 |
\ !!y. [| y:B; u=QInr(y) |] ==> P \ |
|
179 |
\ |] ==> P"; |
|
180 |
by (rtac (major RS UnE) 1); |
|
181 |
by (REPEAT (rtac refl 1 |
|
182 |
ORELSE eresolve_tac (prems@[QSigmaE,singletonE,ssubst]) 1)); |
|
183 |
val qsumE = result(); |
|
184 |
||
185 |
(** Injection and freeness equivalences, for rewriting **) |
|
186 |
||
55 | 187 |
goalw QPair.thy qsum_defs "QInl(a)=QInl(b) <-> a=b"; |
188 |
by (simp_tac (ZF_ss addsimps [QPair_iff]) 1); |
|
0 | 189 |
val QInl_iff = result(); |
190 |
||
55 | 191 |
goalw QPair.thy qsum_defs "QInr(a)=QInr(b) <-> a=b"; |
192 |
by (simp_tac (ZF_ss addsimps [QPair_iff]) 1); |
|
0 | 193 |
val QInr_iff = result(); |
194 |
||
55 | 195 |
goalw QPair.thy qsum_defs "QInl(a)=QInr(b) <-> False"; |
196 |
by (simp_tac (ZF_ss addsimps [QPair_iff, one_not_0 RS not_sym]) 1); |
|
0 | 197 |
val QInl_QInr_iff = result(); |
198 |
||
55 | 199 |
goalw QPair.thy qsum_defs "QInr(b)=QInl(a) <-> False"; |
200 |
by (simp_tac (ZF_ss addsimps [QPair_iff, one_not_0]) 1); |
|
0 | 201 |
val QInr_QInl_iff = result(); |
202 |
||
55 | 203 |
(*Injection and freeness rules*) |
204 |
||
205 |
val QInl_inject = standard (QInl_iff RS iffD1); |
|
206 |
val QInr_inject = standard (QInr_iff RS iffD1); |
|
207 |
val QInl_neq_QInr = standard (QInl_QInr_iff RS iffD1 RS FalseE); |
|
208 |
val QInr_neq_QInl = standard (QInr_QInl_iff RS iffD1 RS FalseE); |
|
209 |
||
0 | 210 |
val qsum_cs = |
211 |
ZF_cs addIs [QInlI,QInrI] addSEs [qsumE,QInl_neq_QInr,QInr_neq_QInl] |
|
212 |
addSDs [QInl_inject,QInr_inject]; |
|
213 |
||
55 | 214 |
goal QPair.thy "!!A B. QInl(a): A<+>B ==> a: A"; |
215 |
by (fast_tac qsum_cs 1); |
|
216 |
val QInlD = result(); |
|
217 |
||
218 |
goal QPair.thy "!!A B. QInr(b): A<+>B ==> b: B"; |
|
219 |
by (fast_tac qsum_cs 1); |
|
220 |
val QInrD = result(); |
|
221 |
||
0 | 222 |
(** <+> is itself injective... who cares?? **) |
223 |
||
224 |
goal QPair.thy |
|
225 |
"u: A <+> B <-> (EX x. x:A & u=QInl(x)) | (EX y. y:B & u=QInr(y))"; |
|
226 |
by (fast_tac qsum_cs 1); |
|
227 |
val qsum_iff = result(); |
|
228 |
||
229 |
goal QPair.thy "A <+> B <= C <+> D <-> A<=C & B<=D"; |
|
230 |
by (fast_tac qsum_cs 1); |
|
231 |
val qsum_subset_iff = result(); |
|
232 |
||
233 |
goal QPair.thy "A <+> B = C <+> D <-> A=C & B=D"; |
|
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
|
234 |
by (simp_tac (ZF_ss addsimps [extension,qsum_subset_iff]) 1); |
0 | 235 |
by (fast_tac ZF_cs 1); |
236 |
val qsum_equal_iff = result(); |
|
237 |
||
238 |
(*** Eliminator -- qcase ***) |
|
239 |
||
240 |
goalw QPair.thy qsum_defs "qcase(c, d, QInl(a)) = c(a)"; |
|
241 |
by (rtac (qsplit RS trans) 1); |
|
242 |
by (rtac cond_0 1); |
|
243 |
val qcase_QInl = result(); |
|
244 |
||
245 |
goalw QPair.thy qsum_defs "qcase(c, d, QInr(b)) = d(b)"; |
|
246 |
by (rtac (qsplit RS trans) 1); |
|
247 |
by (rtac cond_1 1); |
|
248 |
val qcase_QInr = result(); |
|
249 |
||
250 |
val major::prems = goal QPair.thy |
|
251 |
"[| u: A <+> B; \ |
|
252 |
\ !!x. x: A ==> c(x): C(QInl(x)); \ |
|
253 |
\ !!y. y: B ==> d(y): C(QInr(y)) \ |
|
254 |
\ |] ==> qcase(c,d,u) : C(u)"; |
|
255 |
by (rtac (major RS qsumE) 1); |
|
256 |
by (ALLGOALS (etac ssubst)); |
|
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
|
257 |
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps |
0 | 258 |
(prems@[qcase_QInl,qcase_QInr])))); |
259 |
val qcase_type = result(); |
|
260 |
||
261 |
(** Rules for the Part primitive **) |
|
262 |
||
263 |
goal QPair.thy "Part(A <+> B,QInl) = {QInl(x). x: A}"; |
|
264 |
by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1); |
|
265 |
val Part_QInl = result(); |
|
266 |
||
267 |
goal QPair.thy "Part(A <+> B,QInr) = {QInr(y). y: B}"; |
|
268 |
by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1); |
|
269 |
val Part_QInr = result(); |
|
270 |
||
271 |
goal QPair.thy "Part(A <+> B, %x.QInr(h(x))) = {QInr(y). y: Part(B,h)}"; |
|
272 |
by (fast_tac (qsum_cs addIs [PartI,equalityI] addSEs [PartE]) 1); |
|
273 |
val Part_QInr2 = result(); |
|
274 |
||
275 |
goal QPair.thy "!!A B C. C <= A <+> B ==> Part(C,QInl) Un Part(C,QInr) = C"; |
|
276 |
by (rtac equalityI 1); |
|
277 |
by (rtac Un_least 1); |
|
278 |
by (rtac Part_subset 1); |
|
279 |
by (rtac Part_subset 1); |
|
280 |
by (fast_tac (ZF_cs addIs [PartI] addSEs [qsumE]) 1); |
|
281 |
val Part_qsum_equality = result(); |