src/HOL/HOL.ML
 author wenzelm Wed Aug 06 11:57:20 1997 +0200 (1997-08-06) changeset 3621 d3e248853428 parent 3615 e5322197cfea child 3646 a11338a5d2d4 permissions -rw-r--r--
1 (*  Title:      HOL/HOL.ML
2     ID:         \$Id\$
3     Author:     Tobias Nipkow
4     Copyright   1991  University of Cambridge
6 For HOL.thy
7 Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68
8 *)
10 open HOL;
13 (** Equality **)
14 section "=";
16 qed_goal "sym" HOL.thy "s=t ==> t=s"
17  (fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
19 (*calling "standard" reduces maxidx to 0*)
20 bind_thm ("ssubst", (sym RS subst));
22 qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
23  (fn prems =>
24         [rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
26 (*Useful with eresolve_tac for proving equalties from known equalities.
27         a = b
28         |   |
29         c = d   *)
30 qed_goal "box_equals" HOL.thy
31     "[| a=b;  a=c;  b=d |] ==> c=d"
32  (fn prems=>
33   [ (rtac trans 1),
34     (rtac trans 1),
35     (rtac sym 1),
36     (REPEAT (resolve_tac prems 1)) ]);
39 (** Congruence rules for meta-application **)
40 section "Congruence";
42 (*similar to AP_THM in Gordon's HOL*)
43 qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
44   (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
46 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
47 qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
48  (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
50 qed_goal "cong" HOL.thy
51    "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
52  (fn [prem1,prem2] =>
53    [rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
56 (** Equality of booleans -- iff **)
57 section "iff";
59 qed_goal "iffI" HOL.thy
60    "[| P ==> Q;  Q ==> P |] ==> P=Q"
61  (fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);
63 qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
64  (fn prems =>
65         [rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
67 val iffD1 = sym RS iffD2;
69 qed_goal "iffE" HOL.thy
70     "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
71  (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
74 (** True **)
75 section "True";
77 qed_goalw "TrueI" HOL.thy [True_def] "True"
78   (fn _ => [rtac refl 1]);
80 qed_goal "eqTrueI " HOL.thy "P ==> P=True"
81  (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
83 qed_goal "eqTrueE" HOL.thy "P=True ==> P"
84  (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
87 (** Universal quantifier **)
88 section "!";
90 qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
91  (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
93 qed_goalw "spec" HOL.thy [All_def] "! x::'a.P(x) ==> P(x)"
94  (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
96 qed_goal "allE" HOL.thy "[| !x.P(x);  P(x) ==> R |] ==> R"
97  (fn major::prems=>
98   [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
100 qed_goal "all_dupE" HOL.thy
101     "[| ! x.P(x);  [| P(x); ! x.P(x) |] ==> R |] ==> R"
102  (fn prems =>
103   [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);
106 (** False ** Depends upon spec; it is impossible to do propositional logic
107              before quantifiers! **)
108 section "False";
110 qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
111  (fn [major] => [rtac (major RS spec) 1]);
113 qed_goal "False_neq_True" HOL.thy "False=True ==> P"
114  (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
117 (** Negation **)
118 section "~";
120 qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
121  (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
123 qed_goalw "notE" HOL.thy [not_def] "[| ~P;  P |] ==> R"
124  (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
126 bind_thm ("classical2", notE RS notI);
128 qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
129  (fn _ => [REPEAT (ares_tac [notE] 1)]);
132 (** Implication **)
133 section "-->";
135 qed_goal "impE" HOL.thy "[| P-->Q;  P;  Q ==> R |] ==> R"
136  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
138 (* Reduces Q to P-->Q, allowing substitution in P. *)
139 qed_goal "rev_mp" HOL.thy "[| P;  P --> Q |] ==> Q"
140  (fn prems=>  [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
142 qed_goal "contrapos" HOL.thy "[| ~Q;  P==>Q |] ==> ~P"
143  (fn [major,minor]=>
144   [ (rtac (major RS notE RS notI) 1),
145     (etac minor 1) ]);
147 qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P"
148  (fn [major,minor]=>
149   [ (rtac (minor RS contrapos) 1), (etac major 1) ]);
151 (* ~(?t = ?s) ==> ~(?s = ?t) *)
152 bind_thm("not_sym", sym COMP rev_contrapos);
155 (** Existential quantifier **)
156 section "?";
158 qed_goalw "exI" HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)"
159  (fn prems => [rtac selectI 1, resolve_tac prems 1]);
161 qed_goalw "exE" HOL.thy [Ex_def]
162   "[| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q"
163   (fn prems => [REPEAT(resolve_tac prems 1)]);
166 (** Conjunction **)
167 section "&";
169 qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
170  (fn prems =>
171   [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
173 qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
174  (fn prems =>
175    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
177 qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
178  (fn prems =>
179    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
181 qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
182  (fn prems =>
183          [cut_facts_tac prems 1, resolve_tac prems 1,
184           etac conjunct1 1, etac conjunct2 1]);
187 (** Disjunction *)
188 section "|";
190 qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
191  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
193 qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
194  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
196 qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
197  (fn [a1,a2,a3] =>
198         [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
199          rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
202 (** CCONTR -- classical logic **)
203 section "classical logic";
205 qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
206  (fn [prem] =>
207    [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
208     rtac (impI RS prem RS eqTrueI) 1,
209     etac subst 1,  assume_tac 1]);
211 val ccontr = FalseE RS classical;
213 (*Double negation law*)
214 qed_goal "notnotD" HOL.thy "~~P ==> P"
215  (fn [major]=>
216   [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
218 qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
219         rtac classical 1,
220         dtac p2 1,
221         etac notE 1,
222         rtac p1 1]);
224 qed_goal "swap2" HOL.thy "[| P;  Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [
225         rtac notI 1,
226         dtac p2 1,
227         etac notE 1,
228         rtac p1 1]);
230 (** Unique existence **)
231 section "?!";
233 qed_goalw "ex1I" HOL.thy [Ex1_def]
234             "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
235  (fn prems =>
236   [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
238 (*Sometimes easier to use: the premises have no shared variables.  Safe!*)
239 qed_goal "ex_ex1I" HOL.thy
240     "[| ? x.P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)"
241  (fn [ex,eq] => [ (rtac (ex RS exE) 1),
242                   (REPEAT (ares_tac [ex1I,eq] 1)) ]);
244 qed_goalw "ex1E" HOL.thy [Ex1_def]
245     "[| ?! x.P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
246  (fn major::prems =>
247   [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
250 (** Select: Hilbert's Epsilon-operator **)
251 section "@";
253 (*Easier to apply than selectI: conclusion has only one occurrence of P*)
254 qed_goal "selectI2" HOL.thy
255     "[| P(a);  !!x. P(x) ==> Q(x) |] ==> Q(@x.P(x))"
256  (fn prems => [ resolve_tac prems 1,
257                 rtac selectI 1,
258                 resolve_tac prems 1 ]);
260 qed_goal "select_equality" HOL.thy
261     "[| P(a);  !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a"
262  (fn prems => [ rtac selectI2 1,
263                 REPEAT (ares_tac prems 1) ]);
265 (*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
266 qed_goal "selectI2EX" HOL.thy
267   "[| ? a.P a; !!x. P x ==> Q x |] ==> Q(Eps P)"
268 (fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
270 qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) =  (? x. P x)" (fn prems => [
271         rtac iffI 1,
272         etac exI 1,
273         etac exE 1,
274         etac selectI 1]);
277 (** Classical intro rules for disjunction and existential quantifiers *)
278 section "classical intro rules";
280 qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
281  (fn prems=>
282   [ (rtac classical 1),
283     (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
284     (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
286 qed_goal "excluded_middle" HOL.thy "~P | P"
287  (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
289 (*For disjunctive case analysis*)
290 fun excluded_middle_tac sP =
291     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
293 (*Classical implies (-->) elimination. *)
294 qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R"
295  (fn major::prems=>
296   [ rtac (excluded_middle RS disjE) 1,
297     REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
299 (*Classical <-> elimination. *)
300 qed_goal "iffCE" HOL.thy
301     "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
302  (fn major::prems =>
303   [ (rtac (major RS iffE) 1),
304     (REPEAT (DEPTH_SOLVE_1
305         (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
307 qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)"
308  (fn prems=>
309   [ (rtac ccontr 1),
310     (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);
313 (* case distinction *)
315 qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
316   (fn [p1,p2] => [cut_facts_tac [excluded_middle] 1, etac disjE 1,
317                   etac p2 1, etac p1 1]);
319 fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
322 (** Standard abbreviations **)
324 fun stac th = CHANGED o rtac (th RS ssubst);
325 fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
328 (** strip ! and --> from proved goal while preserving !-bound var names **)
330 local
332 (* Use XXX to avoid forall_intr failing because of duplicate variable name *)
333 val myspec = read_instantiate [("P","?XXX")] spec;
334 val _ \$ (_ \$ (vx as Var(_,vxT))) = concl_of myspec;
335 val cvx = cterm_of (#sign(rep_thm myspec)) vx;
336 val aspec = forall_intr cvx myspec;
338 in
340 fun RSspec th =
341   (case concl_of th of
342      _ \$ (Const("All",_) \$ Abs(a,_,_)) =>
343          let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
344          in th RS forall_elim ca aspec end
345   | _ => raise THM("RSspec",0,[th]));
347 fun RSmp th =
348   (case concl_of th of
349      _ \$ (Const("op -->",_)\$_\$_) => th RS mp
350   | _ => raise THM("RSmp",0,[th]));
352 fun normalize_thm funs =
353 let fun trans [] th = th
354       | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
355 in trans funs end;
357 fun qed_spec_mp name =
358   let val thm = normalize_thm [RSspec,RSmp] (result())
359   in bind_thm(name, thm) end;
361 fun qed_goal_spec_mp name thy s p =
362 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
364 fun qed_goalw_spec_mp name thy defs s p =
365 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
367 end;
370 (*Thus, assignments to the references claset and simpset are recorded
371   with theory "HOL".  These files cannot be loaded directly in ROOT.ML.*)
372 use "hologic.ML";