src/HOL/HOL.ML
 author paulson Fri Jun 28 15:27:53 1996 +0200 (1996-06-28) changeset 1840 149b2e69633e parent 1725 8b7414384396 child 1918 d898eb4beb96 permissions -rw-r--r--
Added rev_notE by analogy with rev_mp
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 qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
127  (fn _ => [REPEAT (ares_tac [notE] 1)]);
130 (** Implication **)
131 section "-->";
133 qed_goal "impE" HOL.thy "[| P-->Q;  P;  Q ==> R |] ==> R"
134  (fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
136 (* Reduces Q to P-->Q, allowing substitution in P. *)
137 qed_goal "rev_mp" HOL.thy "[| P;  P --> Q |] ==> Q"
138  (fn prems=>  [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
140 qed_goal "contrapos" HOL.thy "[| ~Q;  P==>Q |] ==> ~P"
141  (fn [major,minor]=>
142   [ (rtac (major RS notE RS notI) 1),
143     (etac minor 1) ]);
145 qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P"
146  (fn [major,minor]=>
147   [ (rtac (minor RS contrapos) 1), (etac major 1) ]);
149 (* ~(?t = ?s) ==> ~(?s = ?t) *)
150 bind_thm("not_sym", sym COMP rev_contrapos);
153 (** Existential quantifier **)
154 section "?";
156 qed_goalw "exI" HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)"
157  (fn prems => [rtac selectI 1, resolve_tac prems 1]);
159 qed_goalw "exE" HOL.thy [Ex_def]
160   "[| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q"
161   (fn prems => [REPEAT(resolve_tac prems 1)]);
164 (** Conjunction **)
165 section "&";
167 qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
168  (fn prems =>
169   [REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
171 qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
172  (fn prems =>
173    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
175 qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
176  (fn prems =>
177    [resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
179 qed_goal "conjE" HOL.thy "[| P&Q;  [| P; Q |] ==> R |] ==> R"
180  (fn prems =>
181          [cut_facts_tac prems 1, resolve_tac prems 1,
182           etac conjunct1 1, etac conjunct2 1]);
185 (** Disjunction *)
186 section "|";
188 qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
189  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
191 qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
192  (fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
194 qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
195  (fn [a1,a2,a3] =>
196         [rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
197          rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
200 (** CCONTR -- classical logic **)
201 section "classical logic";
203 qed_goalw "classical" HOL.thy [not_def]  "(~P ==> P) ==> P"
204  (fn [prem] =>
205    [rtac (True_or_False RS (disjE RS eqTrueE)) 1,  assume_tac 1,
206     rtac (impI RS prem RS eqTrueI) 1,
207     etac subst 1,  assume_tac 1]);
209 val ccontr = FalseE RS classical;
211 (*Double negation law*)
212 qed_goal "notnotD" HOL.thy "~~P ==> P"
213  (fn [major]=>
214   [ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
216 qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
217 	rtac classical 1,
218 	dtac p2 1,
219 	etac notE 1,
220 	rtac p1 1]);
222 qed_goal "swap2" HOL.thy "[| P;  Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [
223 	rtac notI 1,
224 	dtac p2 1,
225 	etac notE 1,
226 	rtac p1 1]);
228 (** Unique existence **)
229 section "?!";
231 qed_goalw "ex1I" HOL.thy [Ex1_def]
232 	    "[| P(a);  !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
233  (fn prems =>
234   [REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
236 qed_goalw "ex1E" HOL.thy [Ex1_def]
237     "[| ?! x.P(x);  !!x. [| P(x);  ! y. P(y) --> y=x |] ==> R |] ==> R"
238  (fn major::prems =>
239   [rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
242 (** Select: Hilbert's Epsilon-operator **)
243 section "@";
245 (*Easier to apply than selectI: conclusion has only one occurrence of P*)
246 qed_goal "selectI2" HOL.thy
247     "[| P(a);  !!x. P(x) ==> Q(x) |] ==> Q(@x.P(x))"
248  (fn prems => [ resolve_tac prems 1,
249                 rtac selectI 1,
250                 resolve_tac prems 1 ]);
252 qed_goal "select_equality" HOL.thy
253     "[| P(a);  !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a"
254  (fn prems => [ rtac selectI2 1,
255                 REPEAT (ares_tac prems 1) ]);
257 qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) =  (? x. P x)" (fn prems => [
258         rtac iffI 1,
259         etac exI 1,
260         etac exE 1,
261         etac selectI 1]);
264 (** Classical intro rules for disjunction and existential quantifiers *)
265 section "classical intro rules";
267 qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
268  (fn prems=>
269   [ (rtac classical 1),
270     (REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
271     (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
273 qed_goal "excluded_middle" HOL.thy "~P | P"
274  (fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
276 (*For disjunctive case analysis*)
277 fun excluded_middle_tac sP =
278     res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
280 (*Classical implies (-->) elimination. *)
281 qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R"
282  (fn major::prems=>
283   [ rtac (excluded_middle RS disjE) 1,
284     REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
286 (*Classical <-> elimination. *)
287 qed_goal "iffCE" HOL.thy
288     "[| P=Q;  [| P; Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R"
289  (fn major::prems =>
290   [ (rtac (major RS iffE) 1),
291     (REPEAT (DEPTH_SOLVE_1
292         (eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
294 qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)"
295  (fn prems=>
296   [ (rtac ccontr 1),
297     (REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1))  ]);
300 (* case distinction *)
302 qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
303   (fn [p1,p2] => [cut_facts_tac [excluded_middle] 1, etac disjE 1,
304                   etac p2 1, etac p1 1]);
306 fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
309 (** Standard abbreviations **)
311 fun stac th = CHANGED o rtac (th RS ssubst);
312 fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
314 (** strip proved goal while preserving !-bound var names **)
316 local
318 (* Use XXX to avoid forall_intr failing because of duplicate variable name *)
319 val myspec = read_instantiate [("P","?XXX")] spec;
320 val _ \$ (_ \$ (vx as Var(_,vxT))) = concl_of myspec;
321 val cvx = cterm_of (#sign(rep_thm myspec)) vx;
322 val aspec = forall_intr cvx myspec;
324 in
326 fun RSspec th =
327   (case concl_of th of
328      _ \$ (Const("All",_) \$ Abs(a,_,_)) =>
329          let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
330          in th RS forall_elim ca aspec end
331   | _ => raise THM("RSspec",0,[th]));
333 fun RSmp th =
334   (case concl_of th of
335      _ \$ (Const("op -->",_)\$_\$_) => th RS mp
336   | _ => raise THM("RSmp",0,[th]));
338 fun normalize_thm funs =
339 let fun trans [] th = th
340       | trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
341 in trans funs end;
343 fun qed_spec_mp name =
344   let val thm = normalize_thm [RSspec,RSmp] (result())
345   in bind_thm(name, thm) end;
347 end;
351 (*** Load simpdata.ML to be able to initialize HOL's simpset ***)
354 (** Applying HypsubstFun to generate hyp_subst_tac **)
355 section "Classical Reasoner";
357 structure Hypsubst_Data =
358   struct
359   structure Simplifier = Simplifier
360   (*Take apart an equality judgement; otherwise raise Match!*)
361   fun dest_eq (Const("Trueprop",_) \$ (Const("op =",_)  \$ t \$ u)) = (t,u);
362   val eq_reflection = eq_reflection
363   val imp_intr = impI
364   val rev_mp = rev_mp
365   val subst = subst
366   val sym = sym
367   end;
369 structure Hypsubst = HypsubstFun(Hypsubst_Data);
370 open Hypsubst;
372 (*** Applying ClassicalFun to create a classical prover ***)
373 structure Classical_Data =
374   struct
375   val sizef     = size_of_thm
376   val mp        = mp
377   val not_elim  = notE
378   val classical = classical
379   val hyp_subst_tacs=[hyp_subst_tac]
380   end;
382 structure Classical = ClassicalFun(Classical_Data);
383 open Classical;
385 (*Propositional rules*)
386 val prop_cs = empty_cs addSIs [refl,TrueI,conjI,disjCI,impI,notI,iffI]
389 (*Quantifier rules*)
390 val HOL_cs = prop_cs addSIs [allI] addIs [exI,ex1I]
393 exception CS_DATA of claset;
395 let fun merge [] = CS_DATA empty_cs
396       | merge cs = let val cs = map (fn CS_DATA x => x) cs;
397                    in CS_DATA (foldl merge_cs (hd cs, tl cs)) end;
399     fun put (CS_DATA cs) = claset := cs;
401     fun get () = CS_DATA (!claset);
402 in add_thydata "HOL"
403      ("claset", ThyMethods {merge = merge, put = put, get = get})
404 end;
406 claset := HOL_cs;
408 section "Simplifier";
410 use     "simpdata.ML";
411 simpset := HOL_ss;
414 (** Install simpsets and datatypes in theory structure **)
415 exception SS_DATA of simpset;
417 let fun merge [] = SS_DATA empty_ss
418       | merge ss = let val ss = map (fn SS_DATA x => x) ss;
419                    in SS_DATA (foldl merge_ss (hd ss, tl ss)) end;
421     fun put (SS_DATA ss) = simpset := ss;
423     fun get () = SS_DATA (!simpset);
424 in add_thydata "HOL"
425      ("simpset", ThyMethods {merge = merge, put = put, get = get})
426 end;
428 type dtype_info = {case_const:term, case_rewrites:thm list,
429                    constructors:term list, nchotomy:thm, case_cong:thm};
431 exception DT_DATA of (string * dtype_info) list;
432 val datatypes = ref [] : (string * dtype_info) list ref;
434 let fun merge [] = DT_DATA []
435       | merge ds =
436           let val ds = map (fn DT_DATA x => x) ds;
437           in DT_DATA (foldl (gen_union eq_fst) (hd ds, tl ds)) end;
439     fun put (DT_DATA ds) = datatypes := ds;
441     fun get () = DT_DATA (!datatypes);
442 in add_thydata "HOL"
443      ("datatypes", ThyMethods {merge = merge, put = put, get = get})
444 end;