src/HOL/HOL.ML
 author paulson Fri Jul 17 10:50:01 1998 +0200 (1998-07-17) changeset 5154 40fd46f3d3a1 parent 5139 013ea0f023e3 child 5185 d1067e2c3f9f permissions -rw-r--r--
tidying
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 qed_goal "rev_iffD2" HOL.thy "!!P. [| Q; P=Q |] ==> P"
68  (fn _ => [etac iffD2 1, assume_tac 1]);
70 bind_thm ("iffD1", sym RS iffD2);
71 bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));
73 qed_goal "iffE" HOL.thy
74     "[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
75  (fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
78 (** True **)
79 section "True";
81 qed_goalw "TrueI" HOL.thy [True_def] "True"
82   (fn _ => [rtac refl 1]);
84 qed_goal "eqTrueI" HOL.thy "P ==> P=True"
85  (fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
87 qed_goal "eqTrueE" HOL.thy "P=True ==> P"
88  (fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
91 (** Universal quantifier **)
92 section "!";
94 qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
95  (fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
97 qed_goalw "spec" HOL.thy [All_def] "! x::'a. P(x) ==> P(x)"
98  (fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
100 qed_goal "allE" HOL.thy "[| !x. P(x);  P(x) ==> R |] ==> R"
101  (fn major::prems=>
102   [ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
104 qed_goal "all_dupE" HOL.thy
105     "[| ! x. P(x);  [| P(x); ! x. P(x) |] ==> R |] ==> R"
106  (fn prems =>
107   [ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);
110 (** False ** Depends upon spec; it is impossible to do propositional logic
111              before quantifiers! **)
112 section "False";
114 qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
115  (fn [major] => [rtac (major RS spec) 1]);
117 qed_goal "False_neq_True" HOL.thy "False=True ==> P"
118  (fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
121 (** Negation **)
122 section "~";
124 qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
125  (fn prems=> [rtac impI 1, eresolve_tac prems 1]);
127 qed_goalw "notE" HOL.thy [not_def] "[| ~P;  P |] ==> R"
128  (fn prems => [rtac (prems MRS mp RS FalseE) 1]);
130 bind_thm ("classical2", notE RS notI);
132 qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
133  (fn _ => [REPEAT (ares_tac [notE] 1)]);
136 (** Implication **)
137 section "-->";
139 qed_goal "impE" HOL.thy "[| P-->Q;  P;  Q ==> R |] ==> R"
140  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
142 (* Reduces Q to P-->Q, allowing substitution in P. *)
143 qed_goal "rev_mp" HOL.thy "[| P;  P --> Q |] ==> Q"
144  (fn prems=>  [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
146 qed_goal "contrapos" HOL.thy "[| ~Q;  P==>Q |] ==> ~P"
147  (fn [major,minor]=>
148   [ (rtac (major RS notE RS notI) 1),
149     (etac minor 1) ]);
151 qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P"
152  (fn [major,minor]=>
153   [ (rtac (minor RS contrapos) 1), (etac major 1) ]);
155 (* ~(?t = ?s) ==> ~(?s = ?t) *)
156 bind_thm("not_sym", sym COMP rev_contrapos);
159 (** Existential quantifier **)
160 section "?";
162 qed_goalw "exI" HOL.thy [Ex_def] "P x ==> ? x::'a. P x"
163  (fn prems => [rtac selectI 1, resolve_tac prems 1]);
165 qed_goalw "exE" HOL.thy [Ex_def]
166   "[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"
167   (fn prems => [REPEAT(resolve_tac prems 1)]);
170 (** Conjunction **)
171 section "&";
173 qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
174  (fn prems =>
175   [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
177 qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
178  (fn prems =>
179    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
181 qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
182  (fn prems =>
183    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
185 qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
186  (fn prems =>
187          [cut_facts_tac prems 1, resolve_tac prems 1,
188           etac conjunct1 1, etac conjunct2 1]);
191 (** Disjunction *)
192 section "|";
194 qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
195  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
197 qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
198  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
200 qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
201  (fn [a1,a2,a3] =>
202         [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
203          rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
206 (** CCONTR -- classical logic **)
207 section "classical logic";
209 qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
210  (fn [prem] =>
211    [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
212     rtac (impI RS prem RS eqTrueI) 1,
213     etac subst 1,  assume_tac 1]);
215 val ccontr = FalseE RS classical;
217 (*Double negation law*)
218 qed_goal "notnotD" HOL.thy "~~P ==> P"
219  (fn [major]=>
220   [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
222 qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
223         rtac classical 1,
224         dtac p2 1,
225         etac notE 1,
226         rtac p1 1]);
228 qed_goal "swap2" HOL.thy "[| P;  Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [
229         rtac notI 1,
230         dtac p2 1,
231         etac notE 1,
232         rtac p1 1]);
234 (** Unique existence **)
235 section "?!";
237 qed_goalw "ex1I" HOL.thy [Ex1_def]
238             "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
239  (fn prems =>
240   [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
242 (*Sometimes easier to use: the premises have no shared variables.  Safe!*)
243 qed_goal "ex_ex1I" HOL.thy
244     "[| ? x. P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)"
245  (fn [ex,eq] => [ (rtac (ex RS exE) 1),
246                   (REPEAT (ares_tac [ex1I,eq] 1)) ]);
248 qed_goalw "ex1E" HOL.thy [Ex1_def]
249     "[| ?! x. P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
250  (fn major::prems =>
251   [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
254 (** Select: Hilbert's Epsilon-operator **)
255 section "@";
257 (*Easier to apply than selectI: conclusion has only one occurrence of P*)
258 qed_goal "selectI2" HOL.thy
259     "[| P a;  !!x. P x ==> Q x |] ==> Q (@x. P x)"
260  (fn prems => [ resolve_tac prems 1,
261                 rtac selectI 1,
262                 resolve_tac prems 1 ]);
264 (*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
265 qed_goal "selectI2EX" HOL.thy
266   "[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
267 (fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
269 qed_goal "select_equality" HOL.thy
270     "[| P a;  !!x. P x ==> x=a |] ==> (@x. P x) = a"
271  (fn prems => [ rtac selectI2 1,
272                 REPEAT (ares_tac prems 1) ]);
274 qed_goalw "select1_equality" HOL.thy [Ex1_def]
275   "!!P. [| ?!x. P x; P a |] ==> (@x. P x) = a" (K [
276 	  rtac select_equality 1, atac 1,
277           etac exE 1, etac conjE 1,
278           rtac allE 1, atac 1,
279           etac impE 1, atac 1, etac ssubst 1,
280           etac allE 1, etac impE 1, atac 1, etac ssubst 1,
281           rtac refl 1]);
283 qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) =  (? x. P x)" (K [
284         rtac iffI 1,
285         etac exI 1,
286         etac exE 1,
287         etac selectI 1]);
290 (** Classical intro rules for disjunction and existential quantifiers *)
291 section "classical intro rules";
293 qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
294  (fn prems=>
295   [ (rtac classical 1),
296     (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
297     (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
299 qed_goal "excluded_middle" HOL.thy "~P | P"
300  (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
302 (*For disjunctive case analysis*)
303 fun excluded_middle_tac sP =
304     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
306 (*Classical implies (-->) elimination. *)
307 qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R"
308  (fn major::prems=>
309   [ rtac (excluded_middle RS disjE) 1,
310     REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
312 (*This version of --> elimination works on Q before P.  It works best for
313   those cases in which P holds "almost everywhere".  Can't install as
314   default: would break old proofs.*)
315 qed_goal "impCE'" thy
316     "[| P-->Q;  Q ==> R;  ~P ==> R |] ==> R"
317  (fn major::prems=>
318   [ (resolve_tac [excluded_middle RS disjE] 1),
319     (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
321 (*Classical <-> elimination. *)
322 qed_goal "iffCE" HOL.thy
323     "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
324  (fn major::prems =>
325   [ (rtac (major RS iffE) 1),
326     (REPEAT (DEPTH_SOLVE_1
327         (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
329 qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)"
330  (fn prems=>
331   [ (rtac ccontr 1),
332     (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);
335 (* case distinction *)
337 qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
338   (fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
339                   etac p2 1, etac p1 1]);
341 fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
344 (** Standard abbreviations **)
346 (*Fails unless the substitution has an effect*)
347 fun stac th = CHANGED_GOAL (rtac (th RS ssubst));
349 fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
352 (** strip ! and --> from proved goal while preserving !-bound var names **)
354 local
356 (* Use XXX to avoid forall_intr failing because of duplicate variable name *)
357 val myspec = read_instantiate [("P","?XXX")] spec;
358 val _ \$ (_ \$ (vx as Var(_,vxT))) = concl_of myspec;
359 val cvx = cterm_of (#sign(rep_thm myspec)) vx;
360 val aspec = forall_intr cvx myspec;
362 in
364 fun RSspec th =
365   (case concl_of th of
366      _ \$ (Const("All",_) \$ Abs(a,_,_)) =>
367          let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
368          in th RS forall_elim ca aspec end
369   | _ => raise THM("RSspec",0,[th]));
371 fun RSmp th =
372   (case concl_of th of
373      _ \$ (Const("op -->",_)\$_\$_) => th RS mp
374   | _ => raise THM("RSmp",0,[th]));
376 fun normalize_thm funs =
377 let fun trans [] th = th
378       | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
379 in trans funs end;
381 fun qed_spec_mp name =
382   let val thm = normalize_thm [RSspec,RSmp] (result())
383   in bind_thm(name, thm) end;
385 fun qed_goal_spec_mp name thy s p =
386 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
388 fun qed_goalw_spec_mp name thy defs s p =
389 	bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
391 end;