1 (* Title: HOL/prod |
|
2 ID: $Id$ |
|
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 Copyright 1991 University of Cambridge |
|
5 |
|
6 For prod.thy. Ordered Pairs, the Cartesian product type, the unit type |
|
7 *) |
|
8 |
|
9 open Prod; |
|
10 |
|
11 (*This counts as a non-emptiness result for admitting 'a * 'b as a type*) |
|
12 goalw Prod.thy [Prod_def] "Pair_Rep(a,b) : Prod"; |
|
13 by (EVERY1 [rtac CollectI, rtac exI, rtac exI, rtac refl]); |
|
14 qed "ProdI"; |
|
15 |
|
16 val [major] = goalw Prod.thy [Pair_Rep_def] |
|
17 "Pair_Rep(a, b) = Pair_Rep(a',b') ==> a=a' & b=b'"; |
|
18 by (EVERY1 [rtac (major RS fun_cong RS fun_cong RS subst), |
|
19 rtac conjI, rtac refl, rtac refl]); |
|
20 qed "Pair_Rep_inject"; |
|
21 |
|
22 goal Prod.thy "inj_onto(Abs_Prod,Prod)"; |
|
23 by (rtac inj_onto_inverseI 1); |
|
24 by (etac Abs_Prod_inverse 1); |
|
25 qed "inj_onto_Abs_Prod"; |
|
26 |
|
27 val prems = goalw Prod.thy [Pair_def] |
|
28 "[| <a, b> = <a',b'>; [| a=a'; b=b' |] ==> R |] ==> R"; |
|
29 by (rtac (inj_onto_Abs_Prod RS inj_ontoD RS Pair_Rep_inject RS conjE) 1); |
|
30 by (REPEAT (ares_tac (prems@[ProdI]) 1)); |
|
31 qed "Pair_inject"; |
|
32 |
|
33 goal Prod.thy "(<a,b> = <a',b'>) = (a=a' & b=b')"; |
|
34 by (fast_tac (set_cs addIs [Pair_inject]) 1); |
|
35 qed "Pair_eq"; |
|
36 |
|
37 goalw Prod.thy [fst_def] "fst(<a,b>) = a"; |
|
38 by (fast_tac (set_cs addIs [select_equality] addSEs [Pair_inject]) 1); |
|
39 qed "fst_conv"; |
|
40 |
|
41 goalw Prod.thy [snd_def] "snd(<a,b>) = b"; |
|
42 by (fast_tac (set_cs addIs [select_equality] addSEs [Pair_inject]) 1); |
|
43 qed "snd_conv"; |
|
44 |
|
45 goalw Prod.thy [Pair_def] "? x y. p = <x,y>"; |
|
46 by (rtac (rewrite_rule [Prod_def] Rep_Prod RS CollectE) 1); |
|
47 by (EVERY1[etac exE, etac exE, rtac exI, rtac exI, |
|
48 rtac (Rep_Prod_inverse RS sym RS trans), etac arg_cong]); |
|
49 qed "PairE_lemma"; |
|
50 |
|
51 val [prem] = goal Prod.thy "[| !!x y. p = <x,y> ==> Q |] ==> Q"; |
|
52 by (rtac (PairE_lemma RS exE) 1); |
|
53 by (REPEAT (eresolve_tac [prem,exE] 1)); |
|
54 qed "PairE"; |
|
55 |
|
56 goalw Prod.thy [split_def] "split(c, <a,b>) = c(a,b)"; |
|
57 by (sstac [fst_conv, snd_conv] 1); |
|
58 by (rtac refl 1); |
|
59 qed "split"; |
|
60 |
|
61 val prod_ss = set_ss addsimps [fst_conv, snd_conv, split, Pair_eq]; |
|
62 |
|
63 goal Prod.thy "(s=t) = (fst(s)=fst(t) & snd(s)=snd(t))"; |
|
64 by (res_inst_tac[("p","s")] PairE 1); |
|
65 by (res_inst_tac[("p","t")] PairE 1); |
|
66 by (asm_simp_tac prod_ss 1); |
|
67 qed "Pair_fst_snd_eq"; |
|
68 |
|
69 (*Prevents simplification of c: much faster*) |
|
70 qed_goal "split_weak_cong" Prod.thy |
|
71 "p=q ==> split(c,p) = split(c,q)" |
|
72 (fn [prem] => [rtac (prem RS arg_cong) 1]); |
|
73 |
|
74 (* Do not add as rewrite rule: invalidates some proofs in IMP *) |
|
75 goal Prod.thy "p = <fst(p),snd(p)>"; |
|
76 by (res_inst_tac [("p","p")] PairE 1); |
|
77 by (asm_simp_tac prod_ss 1); |
|
78 qed "surjective_pairing"; |
|
79 |
|
80 goal Prod.thy "p = split(%x y.<x,y>, p)"; |
|
81 by (res_inst_tac [("p","p")] PairE 1); |
|
82 by (asm_simp_tac prod_ss 1); |
|
83 qed "surjective_pairing2"; |
|
84 |
|
85 (*For use with split_tac and the simplifier*) |
|
86 goal Prod.thy "R(split(c,p)) = (! x y. p = <x,y> --> R(c(x,y)))"; |
|
87 by (stac surjective_pairing 1); |
|
88 by (stac split 1); |
|
89 by (fast_tac (HOL_cs addSEs [Pair_inject]) 1); |
|
90 qed "expand_split"; |
|
91 |
|
92 (** split used as a logical connective or set former **) |
|
93 |
|
94 (*These rules are for use with fast_tac. |
|
95 Could instead call simp_tac/asm_full_simp_tac using split as rewrite.*) |
|
96 |
|
97 goal Prod.thy "!!a b c. c(a,b) ==> split(c, <a,b>)"; |
|
98 by (asm_simp_tac prod_ss 1); |
|
99 qed "splitI"; |
|
100 |
|
101 val prems = goalw Prod.thy [split_def] |
|
102 "[| split(c,p); !!x y. [| p = <x,y>; c(x,y) |] ==> Q |] ==> Q"; |
|
103 by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1)); |
|
104 qed "splitE"; |
|
105 |
|
106 goal Prod.thy "!!R a b. split(R,<a,b>) ==> R(a,b)"; |
|
107 by (etac (split RS iffD1) 1); |
|
108 qed "splitD"; |
|
109 |
|
110 goal Prod.thy "!!a b c. z: c(a,b) ==> z: split(c, <a,b>)"; |
|
111 by (asm_simp_tac prod_ss 1); |
|
112 qed "mem_splitI"; |
|
113 |
|
114 val prems = goalw Prod.thy [split_def] |
|
115 "[| z: split(c,p); !!x y. [| p = <x,y>; z: c(x,y) |] ==> Q |] ==> Q"; |
|
116 by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1)); |
|
117 qed "mem_splitE"; |
|
118 |
|
119 (*** prod_fun -- action of the product functor upon functions ***) |
|
120 |
|
121 goalw Prod.thy [prod_fun_def] "prod_fun(f,g,<a,b>) = <f(a),g(b)>"; |
|
122 by (rtac split 1); |
|
123 qed "prod_fun"; |
|
124 |
|
125 goal Prod.thy |
|
126 "prod_fun(f1 o f2, g1 o g2) = (prod_fun(f1,g1) o prod_fun(f2,g2))"; |
|
127 by (rtac ext 1); |
|
128 by (res_inst_tac [("p","x")] PairE 1); |
|
129 by (asm_simp_tac (prod_ss addsimps [prod_fun,o_def]) 1); |
|
130 qed "prod_fun_compose"; |
|
131 |
|
132 goal Prod.thy "prod_fun(%x.x, %y.y) = (%z.z)"; |
|
133 by (rtac ext 1); |
|
134 by (res_inst_tac [("p","z")] PairE 1); |
|
135 by (asm_simp_tac (prod_ss addsimps [prod_fun]) 1); |
|
136 qed "prod_fun_ident"; |
|
137 |
|
138 val prems = goal Prod.thy "<a,b>:r ==> <f(a),g(b)> : prod_fun(f,g)``r"; |
|
139 by (rtac image_eqI 1); |
|
140 by (rtac (prod_fun RS sym) 1); |
|
141 by (resolve_tac prems 1); |
|
142 qed "prod_fun_imageI"; |
|
143 |
|
144 val major::prems = goal Prod.thy |
|
145 "[| c: prod_fun(f,g)``r; !!x y. [| c=<f(x),g(y)>; <x,y>:r |] ==> P \ |
|
146 \ |] ==> P"; |
|
147 by (rtac (major RS imageE) 1); |
|
148 by (res_inst_tac [("p","x")] PairE 1); |
|
149 by (resolve_tac prems 1); |
|
150 by (fast_tac HOL_cs 2); |
|
151 by (fast_tac (HOL_cs addIs [prod_fun]) 1); |
|
152 qed "prod_fun_imageE"; |
|
153 |
|
154 (*** Disjoint union of a family of sets - Sigma ***) |
|
155 |
|
156 qed_goalw "SigmaI" Prod.thy [Sigma_def] |
|
157 "[| a:A; b:B(a) |] ==> <a,b> : Sigma(A,B)" |
|
158 (fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]); |
|
159 |
|
160 (*The general elimination rule*) |
|
161 qed_goalw "SigmaE" Prod.thy [Sigma_def] |
|
162 "[| c: Sigma(A,B); \ |
|
163 \ !!x y.[| x:A; y:B(x); c=<x,y> |] ==> P \ |
|
164 \ |] ==> P" |
|
165 (fn major::prems=> |
|
166 [ (cut_facts_tac [major] 1), |
|
167 (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ]); |
|
168 |
|
169 (** Elimination of <a,b>:A*B -- introduces no eigenvariables **) |
|
170 qed_goal "SigmaD1" Prod.thy "<a,b> : Sigma(A,B) ==> a : A" |
|
171 (fn [major]=> |
|
172 [ (rtac (major RS SigmaE) 1), |
|
173 (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]); |
|
174 |
|
175 qed_goal "SigmaD2" Prod.thy "<a,b> : Sigma(A,B) ==> b : B(a)" |
|
176 (fn [major]=> |
|
177 [ (rtac (major RS SigmaE) 1), |
|
178 (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]); |
|
179 |
|
180 qed_goal "SigmaE2" Prod.thy |
|
181 "[| <a,b> : Sigma(A,B); \ |
|
182 \ [| a:A; b:B(a) |] ==> P \ |
|
183 \ |] ==> P" |
|
184 (fn [major,minor]=> |
|
185 [ (rtac minor 1), |
|
186 (rtac (major RS SigmaD1) 1), |
|
187 (rtac (major RS SigmaD2) 1) ]); |
|
188 |
|
189 (*** Domain of a relation ***) |
|
190 |
|
191 val prems = goalw Prod.thy [image_def] "<a,b> : r ==> a : fst``r"; |
|
192 by (rtac CollectI 1); |
|
193 by (rtac bexI 1); |
|
194 by (rtac (fst_conv RS sym) 1); |
|
195 by (resolve_tac prems 1); |
|
196 qed "fst_imageI"; |
|
197 |
|
198 val major::prems = goal Prod.thy |
|
199 "[| a : fst``r; !!y.[| <a,y> : r |] ==> P |] ==> P"; |
|
200 by (rtac (major RS imageE) 1); |
|
201 by (resolve_tac prems 1); |
|
202 by (etac ssubst 1); |
|
203 by (rtac (surjective_pairing RS subst) 1); |
|
204 by (assume_tac 1); |
|
205 qed "fst_imageE"; |
|
206 |
|
207 (*** Range of a relation ***) |
|
208 |
|
209 val prems = goalw Prod.thy [image_def] "<a,b> : r ==> b : snd``r"; |
|
210 by (rtac CollectI 1); |
|
211 by (rtac bexI 1); |
|
212 by (rtac (snd_conv RS sym) 1); |
|
213 by (resolve_tac prems 1); |
|
214 qed "snd_imageI"; |
|
215 |
|
216 val major::prems = goal Prod.thy |
|
217 "[| a : snd``r; !!y.[| <y,a> : r |] ==> P |] ==> P"; |
|
218 by (rtac (major RS imageE) 1); |
|
219 by (resolve_tac prems 1); |
|
220 by (etac ssubst 1); |
|
221 by (rtac (surjective_pairing RS subst) 1); |
|
222 by (assume_tac 1); |
|
223 qed "snd_imageE"; |
|
224 |
|
225 (** Exhaustion rule for unit -- a degenerate form of induction **) |
|
226 |
|
227 goalw Prod.thy [Unity_def] |
|
228 "u = Unity"; |
|
229 by (stac (rewrite_rule [Unit_def] Rep_Unit RS CollectD RS sym) 1); |
|
230 by (rtac (Rep_Unit_inverse RS sym) 1); |
|
231 qed "unit_eq"; |
|
232 |
|
233 val prod_cs = set_cs addSIs [SigmaI, mem_splitI] |
|
234 addIs [fst_imageI, snd_imageI, prod_fun_imageI] |
|
235 addSEs [SigmaE2, SigmaE, mem_splitE, |
|
236 fst_imageE, snd_imageE, prod_fun_imageE, |
|
237 Pair_inject]; |
|