author | paulson |
Fri, 27 Nov 1998 10:40:29 +0100 | |
changeset 5978 | fa2c2dd74f8c |
parent 5809 | bacf85370ce0 |
child 6171 | cd237a10cbf8 |
permissions | -rw-r--r-- |
2935 | 1 |
(* Title: HOL/Univ |
923 | 2 |
ID: $Id$ |
1465 | 3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
923 | 4 |
Copyright 1991 University of Cambridge |
5 |
*) |
|
6 |
||
7 |
(** apfst -- can be used in similar type definitions **) |
|
8 |
||
5069 | 9 |
Goalw [apfst_def] "apfst f (a,b) = (f(a),b)"; |
923 | 10 |
by (rtac split 1); |
976
14b55f7fbf15
renamed theorem "apfst" to "apfst_conv" to avoid conflict with function
clasohm
parents:
972
diff
changeset
|
11 |
qed "apfst_conv"; |
923 | 12 |
|
5316 | 13 |
val [major,minor] = Goal |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
14 |
"[| q = apfst f p; !!x y. [| p = (x,y); q = (f(x),y) |] ==> R \ |
923 | 15 |
\ |] ==> R"; |
16 |
by (rtac PairE 1); |
|
17 |
by (rtac minor 1); |
|
18 |
by (assume_tac 1); |
|
19 |
by (rtac (major RS trans) 1); |
|
20 |
by (etac ssubst 1); |
|
976
14b55f7fbf15
renamed theorem "apfst" to "apfst_conv" to avoid conflict with function
clasohm
parents:
972
diff
changeset
|
21 |
by (rtac apfst_conv 1); |
14b55f7fbf15
renamed theorem "apfst" to "apfst_conv" to avoid conflict with function
clasohm
parents:
972
diff
changeset
|
22 |
qed "apfst_convE"; |
923 | 23 |
|
24 |
(** Push -- an injection, analogous to Cons on lists **) |
|
25 |
||
5316 | 26 |
Goalw [Push_def] "Push i f = Push j g ==> i=j"; |
27 |
by (etac (fun_cong RS box_equals RS Suc_inject) 1); |
|
923 | 28 |
by (rtac nat_case_0 1); |
29 |
by (rtac nat_case_0 1); |
|
30 |
qed "Push_inject1"; |
|
31 |
||
5316 | 32 |
Goalw [Push_def] "Push i f = Push j g ==> f=g"; |
33 |
by (rtac (ext RS box_equals) 1); |
|
34 |
by (etac fun_cong 1); |
|
923 | 35 |
by (rtac (nat_case_Suc RS ext) 1); |
36 |
by (rtac (nat_case_Suc RS ext) 1); |
|
37 |
qed "Push_inject2"; |
|
38 |
||
5316 | 39 |
val [major,minor] = Goal |
923 | 40 |
"[| Push i f =Push j g; [| i=j; f=g |] ==> P \ |
41 |
\ |] ==> P"; |
|
42 |
by (rtac ((major RS Push_inject2) RS ((major RS Push_inject1) RS minor)) 1); |
|
43 |
qed "Push_inject"; |
|
44 |
||
5316 | 45 |
Goalw [Push_def] "Push k f =(%z.0) ==> P"; |
46 |
by (etac (fun_cong RS box_equals RS Suc_neq_Zero) 1); |
|
923 | 47 |
by (rtac nat_case_0 1); |
48 |
by (rtac refl 1); |
|
49 |
qed "Push_neq_K0"; |
|
50 |
||
51 |
(*** Isomorphisms ***) |
|
52 |
||
5069 | 53 |
Goal "inj(Rep_Node)"; |
1465 | 54 |
by (rtac inj_inverseI 1); (*cannot combine by RS: multiple unifiers*) |
923 | 55 |
by (rtac Rep_Node_inverse 1); |
56 |
qed "inj_Rep_Node"; |
|
57 |
||
5069 | 58 |
Goal "inj_on Abs_Node Node"; |
4830 | 59 |
by (rtac inj_on_inverseI 1); |
923 | 60 |
by (etac Abs_Node_inverse 1); |
4830 | 61 |
qed "inj_on_Abs_Node"; |
923 | 62 |
|
4830 | 63 |
val Abs_Node_inject = inj_on_Abs_Node RS inj_onD; |
923 | 64 |
|
65 |
||
66 |
(*** Introduction rules for Node ***) |
|
67 |
||
5069 | 68 |
Goalw [Node_def] "(%k. 0,a) : Node"; |
2891 | 69 |
by (Blast_tac 1); |
923 | 70 |
qed "Node_K0_I"; |
71 |
||
5069 | 72 |
Goalw [Node_def,Push_def] |
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|
73 |
"p: Node ==> apfst (Push i) p : Node"; |
4089 | 74 |
by (blast_tac (claset() addSIs [apfst_conv, nat_case_Suc RS trans]) 1); |
923 | 75 |
qed "Node_Push_I"; |
76 |
||
77 |
||
78 |
(*** Distinctness of constructors ***) |
|
79 |
||
80 |
(** Scons vs Atom **) |
|
81 |
||
5191
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berghofe
parents:
5148
diff
changeset
|
82 |
Goalw [Atom_def,Scons_def,Push_Node_def] "Scons M N ~= Atom(a)"; |
923 | 83 |
by (rtac notI 1); |
84 |
by (etac (equalityD2 RS subsetD RS UnE) 1); |
|
85 |
by (rtac singletonI 1); |
|
976
14b55f7fbf15
renamed theorem "apfst" to "apfst_conv" to avoid conflict with function
clasohm
parents:
972
diff
changeset
|
86 |
by (REPEAT (eresolve_tac [imageE, Abs_Node_inject RS apfst_convE, |
1465 | 87 |
Pair_inject, sym RS Push_neq_K0] 1 |
923 | 88 |
ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1)); |
89 |
qed "Scons_not_Atom"; |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
90 |
bind_thm ("Atom_not_Scons", Scons_not_Atom RS not_sym); |
923 | 91 |
|
92 |
||
93 |
(*** Injectiveness ***) |
|
94 |
||
95 |
(** Atomic nodes **) |
|
96 |
||
5069 | 97 |
Goalw [Atom_def, inj_def] "inj(Atom)"; |
4089 | 98 |
by (blast_tac (claset() addSIs [Node_K0_I] addSDs [Abs_Node_inject]) 1); |
923 | 99 |
qed "inj_Atom"; |
100 |
val Atom_inject = inj_Atom RS injD; |
|
101 |
||
5069 | 102 |
Goal "(Atom(a)=Atom(b)) = (a=b)"; |
4089 | 103 |
by (blast_tac (claset() addSDs [Atom_inject]) 1); |
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
104 |
qed "Atom_Atom_eq"; |
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
105 |
AddIffs [Atom_Atom_eq]; |
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
106 |
|
5069 | 107 |
Goalw [Leaf_def,o_def] "inj(Leaf)"; |
923 | 108 |
by (rtac injI 1); |
109 |
by (etac (Atom_inject RS Inl_inject) 1); |
|
110 |
qed "inj_Leaf"; |
|
111 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
112 |
bind_thm ("Leaf_inject", inj_Leaf RS injD); |
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
113 |
AddSDs [Leaf_inject]; |
923 | 114 |
|
5069 | 115 |
Goalw [Numb_def,o_def] "inj(Numb)"; |
923 | 116 |
by (rtac injI 1); |
117 |
by (etac (Atom_inject RS Inr_inject) 1); |
|
118 |
qed "inj_Numb"; |
|
119 |
||
120 |
val Numb_inject = inj_Numb RS injD; |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
121 |
AddSDs [Numb_inject]; |
923 | 122 |
|
123 |
(** Injectiveness of Push_Node **) |
|
124 |
||
5316 | 125 |
val [major,minor] = Goalw [Push_Node_def] |
923 | 126 |
"[| Push_Node i m =Push_Node j n; [| i=j; m=n |] ==> P \ |
127 |
\ |] ==> P"; |
|
976
14b55f7fbf15
renamed theorem "apfst" to "apfst_conv" to avoid conflict with function
clasohm
parents:
972
diff
changeset
|
128 |
by (rtac (major RS Abs_Node_inject RS apfst_convE) 1); |
923 | 129 |
by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1)); |
976
14b55f7fbf15
renamed theorem "apfst" to "apfst_conv" to avoid conflict with function
clasohm
parents:
972
diff
changeset
|
130 |
by (etac (sym RS apfst_convE) 1); |
923 | 131 |
by (rtac minor 1); |
132 |
by (etac Pair_inject 1); |
|
133 |
by (etac (Push_inject1 RS sym) 1); |
|
134 |
by (rtac (inj_Rep_Node RS injD) 1); |
|
135 |
by (etac trans 1); |
|
4089 | 136 |
by (safe_tac (claset() addSEs [Push_inject,sym])); |
923 | 137 |
qed "Push_Node_inject"; |
138 |
||
139 |
||
140 |
(** Injectiveness of Scons **) |
|
141 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
142 |
Goalw [Scons_def] "Scons M N <= Scons M' N' ==> M<=M'"; |
4089 | 143 |
by (blast_tac (claset() addSDs [Push_Node_inject]) 1); |
923 | 144 |
qed "Scons_inject_lemma1"; |
145 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
146 |
Goalw [Scons_def] "Scons M N <= Scons M' N' ==> N<=N'"; |
4089 | 147 |
by (blast_tac (claset() addSDs [Push_Node_inject]) 1); |
923 | 148 |
qed "Scons_inject_lemma2"; |
149 |
||
5316 | 150 |
Goal "Scons M N = Scons M' N' ==> M=M'"; |
151 |
by (etac equalityE 1); |
|
923 | 152 |
by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1)); |
153 |
qed "Scons_inject1"; |
|
154 |
||
5316 | 155 |
Goal "Scons M N = Scons M' N' ==> N=N'"; |
156 |
by (etac equalityE 1); |
|
923 | 157 |
by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1)); |
158 |
qed "Scons_inject2"; |
|
159 |
||
5316 | 160 |
val [major,minor] = Goal |
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
161 |
"[| Scons M N = Scons M' N'; [| M=M'; N=N' |] ==> P \ |
923 | 162 |
\ |] ==> P"; |
163 |
by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1); |
|
164 |
qed "Scons_inject"; |
|
165 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
166 |
Goal "(Scons M N = Scons M' N') = (M=M' & N=N')"; |
4089 | 167 |
by (blast_tac (claset() addSEs [Scons_inject]) 1); |
923 | 168 |
qed "Scons_Scons_eq"; |
169 |
||
170 |
(*** Distinctness involving Leaf and Numb ***) |
|
171 |
||
172 |
(** Scons vs Leaf **) |
|
173 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
174 |
Goalw [Leaf_def,o_def] "Scons M N ~= Leaf(a)"; |
923 | 175 |
by (rtac Scons_not_Atom 1); |
176 |
qed "Scons_not_Leaf"; |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
177 |
bind_thm ("Leaf_not_Scons", Scons_not_Leaf RS not_sym); |
923 | 178 |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
179 |
AddIffs [Scons_not_Leaf, Leaf_not_Scons]; |
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
180 |
|
923 | 181 |
|
182 |
(** Scons vs Numb **) |
|
183 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
184 |
Goalw [Numb_def,o_def] "Scons M N ~= Numb(k)"; |
923 | 185 |
by (rtac Scons_not_Atom 1); |
186 |
qed "Scons_not_Numb"; |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
187 |
bind_thm ("Numb_not_Scons", Scons_not_Numb RS not_sym); |
923 | 188 |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
189 |
AddIffs [Scons_not_Numb, Numb_not_Scons]; |
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
190 |
|
923 | 191 |
|
192 |
(** Leaf vs Numb **) |
|
193 |
||
5069 | 194 |
Goalw [Leaf_def,Numb_def] "Leaf(a) ~= Numb(k)"; |
4089 | 195 |
by (simp_tac (simpset() addsimps [Inl_not_Inr]) 1); |
923 | 196 |
qed "Leaf_not_Numb"; |
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
197 |
bind_thm ("Numb_not_Leaf", Leaf_not_Numb RS not_sym); |
923 | 198 |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
199 |
AddIffs [Leaf_not_Numb, Numb_not_Leaf]; |
923 | 200 |
|
201 |
||
202 |
(*** ndepth -- the depth of a node ***) |
|
203 |
||
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
204 |
Addsimps [apfst_conv]; |
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
205 |
AddIffs [Scons_not_Atom, Atom_not_Scons, Scons_Scons_eq]; |
923 | 206 |
|
207 |
||
5069 | 208 |
Goalw [ndepth_def] "ndepth (Abs_Node((%k.0, x))) = 0"; |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
209 |
by (EVERY1[stac (Node_K0_I RS Abs_Node_inverse), stac split]); |
923 | 210 |
by (rtac Least_equality 1); |
211 |
by (rtac refl 1); |
|
212 |
by (etac less_zeroE 1); |
|
213 |
qed "ndepth_K0"; |
|
214 |
||
5069 | 215 |
Goal "k < Suc(LEAST x. f(x)=0) --> 0 < nat_case (Suc i) f k"; |
923 | 216 |
by (nat_ind_tac "k" 1); |
1264 | 217 |
by (ALLGOALS Simp_tac); |
923 | 218 |
by (rtac impI 1); |
4356 | 219 |
by (dtac not_less_Least 1); |
220 |
by (Asm_full_simp_tac 1); |
|
221 |
val lemma = result(); |
|
923 | 222 |
|
5069 | 223 |
Goalw [ndepth_def,Push_Node_def] |
923 | 224 |
"ndepth (Push_Node i n) = Suc(ndepth(n))"; |
225 |
by (stac (Rep_Node RS Node_Push_I RS Abs_Node_inverse) 1); |
|
226 |
by (cut_facts_tac [rewrite_rule [Node_def] Rep_Node] 1); |
|
4153 | 227 |
by Safe_tac; |
1465 | 228 |
by (etac ssubst 1); (*instantiates type variables!*) |
1264 | 229 |
by (Simp_tac 1); |
923 | 230 |
by (rtac Least_equality 1); |
231 |
by (rewtac Push_def); |
|
232 |
by (rtac (nat_case_Suc RS trans) 1); |
|
233 |
by (etac LeastI 1); |
|
4356 | 234 |
by (asm_simp_tac (simpset() addsimps [lemma]) 1); |
923 | 235 |
qed "ndepth_Push_Node"; |
236 |
||
237 |
||
238 |
(*** ntrunc applied to the various node sets ***) |
|
239 |
||
5069 | 240 |
Goalw [ntrunc_def] "ntrunc 0 M = {}"; |
2891 | 241 |
by (Blast_tac 1); |
923 | 242 |
qed "ntrunc_0"; |
243 |
||
5069 | 244 |
Goalw [Atom_def,ntrunc_def] "ntrunc (Suc k) (Atom a) = Atom(a)"; |
4089 | 245 |
by (fast_tac (claset() addss (simpset() addsimps [ndepth_K0])) 1); |
923 | 246 |
qed "ntrunc_Atom"; |
247 |
||
5069 | 248 |
Goalw [Leaf_def,o_def] "ntrunc (Suc k) (Leaf a) = Leaf(a)"; |
923 | 249 |
by (rtac ntrunc_Atom 1); |
250 |
qed "ntrunc_Leaf"; |
|
251 |
||
5069 | 252 |
Goalw [Numb_def,o_def] "ntrunc (Suc k) (Numb i) = Numb(i)"; |
923 | 253 |
by (rtac ntrunc_Atom 1); |
254 |
qed "ntrunc_Numb"; |
|
255 |
||
5069 | 256 |
Goalw [Scons_def,ntrunc_def] |
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
257 |
"ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"; |
4089 | 258 |
by (safe_tac (claset() addSIs [imageI])); |
923 | 259 |
by (REPEAT (stac ndepth_Push_Node 3 THEN etac Suc_mono 3)); |
260 |
by (REPEAT (rtac Suc_less_SucD 1 THEN |
|
1465 | 261 |
rtac (ndepth_Push_Node RS subst) 1 THEN |
262 |
assume_tac 1)); |
|
923 | 263 |
qed "ntrunc_Scons"; |
264 |
||
4521 | 265 |
Addsimps [ntrunc_0, ntrunc_Atom, ntrunc_Leaf, ntrunc_Numb, ntrunc_Scons]; |
266 |
||
267 |
||
923 | 268 |
(** Injection nodes **) |
269 |
||
5069 | 270 |
Goalw [In0_def] "ntrunc (Suc 0) (In0 M) = {}"; |
4521 | 271 |
by (Simp_tac 1); |
923 | 272 |
by (rewtac Scons_def); |
2891 | 273 |
by (Blast_tac 1); |
923 | 274 |
qed "ntrunc_one_In0"; |
275 |
||
5069 | 276 |
Goalw [In0_def] |
923 | 277 |
"ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"; |
4521 | 278 |
by (Simp_tac 1); |
923 | 279 |
qed "ntrunc_In0"; |
280 |
||
5069 | 281 |
Goalw [In1_def] "ntrunc (Suc 0) (In1 M) = {}"; |
4521 | 282 |
by (Simp_tac 1); |
923 | 283 |
by (rewtac Scons_def); |
2891 | 284 |
by (Blast_tac 1); |
923 | 285 |
qed "ntrunc_one_In1"; |
286 |
||
5069 | 287 |
Goalw [In1_def] |
923 | 288 |
"ntrunc (Suc (Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"; |
4521 | 289 |
by (Simp_tac 1); |
923 | 290 |
qed "ntrunc_In1"; |
291 |
||
4521 | 292 |
Addsimps [ntrunc_one_In0, ntrunc_In0, ntrunc_one_In1, ntrunc_In1]; |
293 |
||
923 | 294 |
|
295 |
(*** Cartesian Product ***) |
|
296 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
297 |
Goalw [uprod_def] "[| M:A; N:B |] ==> Scons M N : A<*>B"; |
923 | 298 |
by (REPEAT (ares_tac [singletonI,UN_I] 1)); |
299 |
qed "uprodI"; |
|
300 |
||
301 |
(*The general elimination rule*) |
|
5316 | 302 |
val major::prems = Goalw [uprod_def] |
923 | 303 |
"[| c : A<*>B; \ |
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
304 |
\ !!x y. [| x:A; y:B; c = Scons x y |] ==> P \ |
923 | 305 |
\ |] ==> P"; |
306 |
by (cut_facts_tac [major] 1); |
|
307 |
by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1 |
|
308 |
ORELSE resolve_tac prems 1)); |
|
309 |
qed "uprodE"; |
|
310 |
||
311 |
(*Elimination of a pair -- introduces no eigenvariables*) |
|
5316 | 312 |
val prems = Goal |
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
313 |
"[| Scons M N : A<*>B; [| M:A; N:B |] ==> P \ |
923 | 314 |
\ |] ==> P"; |
315 |
by (rtac uprodE 1); |
|
316 |
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1)); |
|
317 |
qed "uprodE2"; |
|
318 |
||
319 |
||
320 |
(*** Disjoint Sum ***) |
|
321 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
322 |
Goalw [usum_def] "M:A ==> In0(M) : A<+>B"; |
2891 | 323 |
by (Blast_tac 1); |
923 | 324 |
qed "usum_In0I"; |
325 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
326 |
Goalw [usum_def] "N:B ==> In1(N) : A<+>B"; |
2891 | 327 |
by (Blast_tac 1); |
923 | 328 |
qed "usum_In1I"; |
329 |
||
5316 | 330 |
val major::prems = Goalw [usum_def] |
923 | 331 |
"[| u : A<+>B; \ |
332 |
\ !!x. [| x:A; u=In0(x) |] ==> P; \ |
|
333 |
\ !!y. [| y:B; u=In1(y) |] ==> P \ |
|
334 |
\ |] ==> P"; |
|
335 |
by (rtac (major RS UnE) 1); |
|
336 |
by (REPEAT (rtac refl 1 |
|
337 |
ORELSE eresolve_tac (prems@[imageE,ssubst]) 1)); |
|
338 |
qed "usumE"; |
|
339 |
||
340 |
||
341 |
(** Injection **) |
|
342 |
||
5069 | 343 |
Goalw [In0_def,In1_def] "In0(M) ~= In1(N)"; |
923 | 344 |
by (rtac notI 1); |
345 |
by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1); |
|
346 |
qed "In0_not_In1"; |
|
347 |
||
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
348 |
bind_thm ("In1_not_In0", In0_not_In1 RS not_sym); |
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
349 |
|
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1786
diff
changeset
|
350 |
AddIffs [In0_not_In1, In1_not_In0]; |
923 | 351 |
|
5316 | 352 |
Goalw [In0_def] "In0(M) = In0(N) ==> M=N"; |
353 |
by (etac (Scons_inject2) 1); |
|
923 | 354 |
qed "In0_inject"; |
355 |
||
5316 | 356 |
Goalw [In1_def] "In1(M) = In1(N) ==> M=N"; |
357 |
by (etac (Scons_inject2) 1); |
|
923 | 358 |
qed "In1_inject"; |
359 |
||
5069 | 360 |
Goal "(In0 M = In0 N) = (M=N)"; |
4089 | 361 |
by (blast_tac (claset() addSDs [In0_inject]) 1); |
3421 | 362 |
qed "In0_eq"; |
363 |
||
5069 | 364 |
Goal "(In1 M = In1 N) = (M=N)"; |
4089 | 365 |
by (blast_tac (claset() addSDs [In1_inject]) 1); |
3421 | 366 |
qed "In1_eq"; |
367 |
||
368 |
AddIffs [In0_eq, In1_eq]; |
|
369 |
||
5069 | 370 |
Goalw [inj_def] "inj In0"; |
3421 | 371 |
by (Blast_tac 1); |
372 |
qed "inj_In0"; |
|
373 |
||
5069 | 374 |
Goalw [inj_def] "inj In1"; |
3421 | 375 |
by (Blast_tac 1); |
376 |
qed "inj_In1"; |
|
377 |
||
923 | 378 |
|
379 |
(*** proving equality of sets and functions using ntrunc ***) |
|
380 |
||
5069 | 381 |
Goalw [ntrunc_def] "ntrunc k M <= M"; |
2891 | 382 |
by (Blast_tac 1); |
923 | 383 |
qed "ntrunc_subsetI"; |
384 |
||
5316 | 385 |
val [major] = Goalw [ntrunc_def] "(!!k. ntrunc k M <= N) ==> M<=N"; |
4089 | 386 |
by (blast_tac (claset() addIs [less_add_Suc1, less_add_Suc2, |
4521 | 387 |
major RS subsetD]) 1); |
923 | 388 |
qed "ntrunc_subsetD"; |
389 |
||
390 |
(*A generalized form of the take-lemma*) |
|
5316 | 391 |
val [major] = Goal "(!!k. ntrunc k M = ntrunc k N) ==> M=N"; |
923 | 392 |
by (rtac equalityI 1); |
393 |
by (ALLGOALS (rtac ntrunc_subsetD)); |
|
394 |
by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans)))); |
|
395 |
by (rtac (major RS equalityD1) 1); |
|
396 |
by (rtac (major RS equalityD2) 1); |
|
397 |
qed "ntrunc_equality"; |
|
398 |
||
5316 | 399 |
val [major] = Goalw [o_def] |
923 | 400 |
"[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"; |
401 |
by (rtac (ntrunc_equality RS ext) 1); |
|
402 |
by (rtac (major RS fun_cong) 1); |
|
403 |
qed "ntrunc_o_equality"; |
|
404 |
||
405 |
(*** Monotonicity ***) |
|
406 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
407 |
Goalw [uprod_def] "[| A<=A'; B<=B' |] ==> A<*>B <= A'<*>B'"; |
2891 | 408 |
by (Blast_tac 1); |
923 | 409 |
qed "uprod_mono"; |
410 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
411 |
Goalw [usum_def] "[| A<=A'; B<=B' |] ==> A<+>B <= A'<+>B'"; |
2891 | 412 |
by (Blast_tac 1); |
923 | 413 |
qed "usum_mono"; |
414 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
415 |
Goalw [Scons_def] "[| M<=M'; N<=N' |] ==> Scons M N <= Scons M' N'"; |
2891 | 416 |
by (Blast_tac 1); |
923 | 417 |
qed "Scons_mono"; |
418 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
419 |
Goalw [In0_def] "M<=N ==> In0(M) <= In0(N)"; |
923 | 420 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1)); |
421 |
qed "In0_mono"; |
|
422 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
423 |
Goalw [In1_def] "M<=N ==> In1(M) <= In1(N)"; |
923 | 424 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1)); |
425 |
qed "In1_mono"; |
|
426 |
||
427 |
||
428 |
(*** Split and Case ***) |
|
429 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
430 |
Goalw [Split_def] "Split c (Scons M N) = c M N"; |
4535 | 431 |
by (Blast_tac 1); |
923 | 432 |
qed "Split"; |
433 |
||
5069 | 434 |
Goalw [Case_def] "Case c d (In0 M) = c(M)"; |
4535 | 435 |
by (Blast_tac 1); |
923 | 436 |
qed "Case_In0"; |
437 |
||
5069 | 438 |
Goalw [Case_def] "Case c d (In1 N) = d(N)"; |
4535 | 439 |
by (Blast_tac 1); |
923 | 440 |
qed "Case_In1"; |
441 |
||
4521 | 442 |
Addsimps [Split, Case_In0, Case_In1]; |
443 |
||
444 |
||
923 | 445 |
(**** UN x. B(x) rules ****) |
446 |
||
5069 | 447 |
Goalw [ntrunc_def] "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"; |
2891 | 448 |
by (Blast_tac 1); |
923 | 449 |
qed "ntrunc_UN1"; |
450 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
451 |
Goalw [Scons_def] "Scons (UN x. f x) M = (UN x. Scons (f x) M)"; |
2891 | 452 |
by (Blast_tac 1); |
923 | 453 |
qed "Scons_UN1_x"; |
454 |
||
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
455 |
Goalw [Scons_def] "Scons M (UN x. f x) = (UN x. Scons M (f x))"; |
2891 | 456 |
by (Blast_tac 1); |
923 | 457 |
qed "Scons_UN1_y"; |
458 |
||
5069 | 459 |
Goalw [In0_def] "In0(UN x. f(x)) = (UN x. In0(f(x)))"; |
1465 | 460 |
by (rtac Scons_UN1_y 1); |
923 | 461 |
qed "In0_UN1"; |
462 |
||
5069 | 463 |
Goalw [In1_def] "In1(UN x. f(x)) = (UN x. In1(f(x)))"; |
1465 | 464 |
by (rtac Scons_UN1_y 1); |
923 | 465 |
qed "In1_UN1"; |
466 |
||
467 |
||
468 |
(*** Equality for Cartesian Product ***) |
|
469 |
||
5069 | 470 |
Goalw [dprod_def] |
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
471 |
"[| (M,M'):r; (N,N'):s |] ==> (Scons M N, Scons M' N') : r<**>s"; |
2891 | 472 |
by (Blast_tac 1); |
923 | 473 |
qed "dprodI"; |
474 |
||
475 |
(*The general elimination rule*) |
|
5316 | 476 |
val major::prems = Goalw [dprod_def] |
923 | 477 |
"[| c : r<**>s; \ |
5191
8ceaa19f7717
Renamed '$' to 'Scons' because of clashes with constants of the same
berghofe
parents:
5148
diff
changeset
|
478 |
\ !!x y x' y'. [| (x,x') : r; (y,y') : s; c = (Scons x y, Scons x' y') |] ==> P \ |
923 | 479 |
\ |] ==> P"; |
480 |
by (cut_facts_tac [major] 1); |
|
481 |
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE])); |
|
482 |
by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1)); |
|
483 |
qed "dprodE"; |
|
484 |
||
485 |
||
486 |
(*** Equality for Disjoint Sum ***) |
|
487 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
488 |
Goalw [dsum_def] "(M,M'):r ==> (In0(M), In0(M')) : r<++>s"; |
2891 | 489 |
by (Blast_tac 1); |
923 | 490 |
qed "dsum_In0I"; |
491 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
492 |
Goalw [dsum_def] "(N,N'):s ==> (In1(N), In1(N')) : r<++>s"; |
2891 | 493 |
by (Blast_tac 1); |
923 | 494 |
qed "dsum_In1I"; |
495 |
||
5316 | 496 |
val major::prems = Goalw [dsum_def] |
923 | 497 |
"[| w : r<++>s; \ |
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
498 |
\ !!x x'. [| (x,x') : r; w = (In0(x), In0(x')) |] ==> P; \ |
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
923
diff
changeset
|
499 |
\ !!y y'. [| (y,y') : s; w = (In1(y), In1(y')) |] ==> P \ |
923 | 500 |
\ |] ==> P"; |
501 |
by (cut_facts_tac [major] 1); |
|
502 |
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE])); |
|
503 |
by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1)); |
|
504 |
qed "dsumE"; |
|
505 |
||
5978
fa2c2dd74f8c
moved diag (diagonal relation) from Univ to Relation
paulson
parents:
5809
diff
changeset
|
506 |
AddSIs [uprodI, dprodI]; |
fa2c2dd74f8c
moved diag (diagonal relation) from Univ to Relation
paulson
parents:
5809
diff
changeset
|
507 |
AddIs [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I]; |
fa2c2dd74f8c
moved diag (diagonal relation) from Univ to Relation
paulson
parents:
5809
diff
changeset
|
508 |
AddSEs [uprodE, dprodE, usumE, dsumE]; |
923 | 509 |
|
510 |
||
511 |
(*** Monotonicity ***) |
|
512 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
513 |
Goal "[| r<=r'; s<=s' |] ==> r<**>s <= r'<**>s'"; |
2891 | 514 |
by (Blast_tac 1); |
923 | 515 |
qed "dprod_mono"; |
516 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
517 |
Goal "[| r<=r'; s<=s' |] ==> r<++>s <= r'<++>s'"; |
2891 | 518 |
by (Blast_tac 1); |
923 | 519 |
qed "dsum_mono"; |
520 |
||
521 |
||
522 |
(*** Bounding theorems ***) |
|
523 |
||
5069 | 524 |
Goal "((A Times B) <**> (C Times D)) <= (A<*>C) Times (B<*>D)"; |
2891 | 525 |
by (Blast_tac 1); |
923 | 526 |
qed "dprod_Sigma"; |
527 |
||
528 |
val dprod_subset_Sigma = [dprod_mono, dprod_Sigma] MRS subset_trans |>standard; |
|
529 |
||
530 |
(*Dependent version*) |
|
5278 | 531 |
Goal "(Sigma A B <**> Sigma C D) <= Sigma (A<*>C) (Split(%x y. B(x)<*>D(y)))"; |
4153 | 532 |
by Safe_tac; |
923 | 533 |
by (stac Split 1); |
2891 | 534 |
by (Blast_tac 1); |
923 | 535 |
qed "dprod_subset_Sigma2"; |
536 |
||
5069 | 537 |
Goal "(A Times B <++> C Times D) <= (A<+>C) Times (B<+>D)"; |
2891 | 538 |
by (Blast_tac 1); |
923 | 539 |
qed "dsum_Sigma"; |
540 |
||
541 |
val dsum_subset_Sigma = [dsum_mono, dsum_Sigma] MRS subset_trans |> standard; |
|
542 |
||
543 |
||
544 |
(*** Domain ***) |
|
545 |
||
5788 | 546 |
Goal "Domain (r<**>s) = (Domain r) <*> (Domain s)"; |
4521 | 547 |
by Auto_tac; |
5788 | 548 |
qed "Domain_dprod"; |
923 | 549 |
|
5788 | 550 |
Goal "Domain (r<++>s) = (Domain r) <+> (Domain s)"; |
4521 | 551 |
by Auto_tac; |
5788 | 552 |
qed "Domain_dsum"; |
923 | 553 |
|
5978
fa2c2dd74f8c
moved diag (diagonal relation) from Univ to Relation
paulson
parents:
5809
diff
changeset
|
554 |
Addsimps [Domain_dprod, Domain_dsum]; |