maintain generic context naming in structure Name_Space (NB: empty = default_naming, init = local_naming);
more explicit Context.generic for Name_Space.declare/define and derivatives (NB: naming changed after Proof_Context.init_global);
prefer Context.pretty in low-level operations of structure Sorts/Type (defer full Syntax.init_pretty until error output);
simplified signatures;
2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
5 The very core of Isabelle's Meta Logic: certified types and terms,
6 derivations, theorems, framework rules (including lifting and
18 sorts: sort Ord_List.T}
19 val theory_of_ctyp: ctyp -> theory
20 val typ_of: ctyp -> typ
21 val ctyp_of: theory -> typ -> ctyp
25 exception CTERM of string * cterm list
26 val rep_cterm: cterm ->
31 sorts: sort Ord_List.T}
32 val crep_cterm: cterm ->
33 {thy_ref: theory_ref, t: term, T: ctyp, maxidx: int, sorts: sort Ord_List.T}
34 val theory_of_cterm: cterm -> theory
35 val term_of: cterm -> term
36 val cterm_of: theory -> term -> cterm
37 val ctyp_of_term: cterm -> ctyp
41 type conv = cterm -> thm
46 shyps: sort Ord_List.T,
47 hyps: term Ord_List.T,
48 tpairs: (term * term) list,
54 shyps: sort Ord_List.T,
55 hyps: cterm Ord_List.T,
56 tpairs: (cterm * cterm) list,
58 exception THM of string * int * thm list
59 val theory_of_thm: thm -> theory
60 val prop_of: thm -> term
61 val concl_of: thm -> term
62 val prems_of: thm -> term list
63 val nprems_of: thm -> int
64 val cprop_of: thm -> cterm
65 val cprem_of: thm -> int -> cterm
71 val dest_ctyp: ctyp -> ctyp list
72 val dest_comb: cterm -> cterm * cterm
73 val dest_fun: cterm -> cterm
74 val dest_arg: cterm -> cterm
75 val dest_fun2: cterm -> cterm
76 val dest_arg1: cterm -> cterm
77 val dest_abs: string option -> cterm -> cterm * cterm
78 val apply: cterm -> cterm -> cterm
79 val lambda_name: string * cterm -> cterm -> cterm
80 val lambda: cterm -> cterm -> cterm
81 val adjust_maxidx_cterm: int -> cterm -> cterm
82 val incr_indexes_cterm: int -> cterm -> cterm
83 val match: cterm * cterm -> (ctyp * ctyp) list * (cterm * cterm) list
84 val first_order_match: cterm * cterm -> (ctyp * ctyp) list * (cterm * cterm) list
85 val fold_terms: (term -> 'a -> 'a) -> thm -> 'a -> 'a
86 val terms_of_tpairs: (term * term) list -> term list
87 val full_prop_of: thm -> term
88 val maxidx_of: thm -> int
89 val maxidx_thm: thm -> int -> int
90 val hyps_of: thm -> term list
91 val tpairs_of: thm -> (term * term) list
92 val no_prems: thm -> bool
93 val major_prem_of: thm -> term
94 val transfer: theory -> thm -> thm
95 val weaken: cterm -> thm -> thm
96 val weaken_sorts: sort list -> cterm -> cterm
97 val extra_shyps: thm -> sort list
98 val proof_bodies_of: thm list -> proof_body list
99 val proof_body_of: thm -> proof_body
100 val proof_of: thm -> proof
101 val join_proofs: thm list -> unit
102 val status_of: thm -> {oracle: bool, unfinished: bool, failed: bool}
103 val future: thm future -> cterm -> thm
104 val derivation_name: thm -> string
105 val name_derivation: string -> thm -> thm
106 val axiom: theory -> string -> thm
107 val axioms_of: theory -> (string * thm) list
108 val get_tags: thm -> Properties.T
109 val map_tags: (Properties.T -> Properties.T) -> thm -> thm
110 val norm_proof: thm -> thm
111 val adjust_maxidx_thm: int -> thm -> thm
113 val assume: cterm -> thm
114 val implies_intr: cterm -> thm -> thm
115 val implies_elim: thm -> thm -> thm
116 val forall_intr: cterm -> thm -> thm
117 val forall_elim: cterm -> thm -> thm
118 val reflexive: cterm -> thm
119 val symmetric: thm -> thm
120 val transitive: thm -> thm -> thm
121 val beta_conversion: bool -> conv
122 val eta_conversion: conv
123 val eta_long_conversion: conv
124 val abstract_rule: string -> cterm -> thm -> thm
125 val combination: thm -> thm -> thm
126 val equal_intr: thm -> thm -> thm
127 val equal_elim: thm -> thm -> thm
128 val flexflex_rule: thm -> thm Seq.seq
129 val generalize: string list * string list -> int -> thm -> thm
130 val instantiate: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm
131 val instantiate_cterm: (ctyp * ctyp) list * (cterm * cterm) list -> cterm -> cterm
132 val trivial: cterm -> thm
133 val of_class: ctyp * class -> thm
134 val strip_shyps: thm -> thm
135 val unconstrainT: thm -> thm
136 val varifyT_global': (string * sort) list -> thm -> ((string * sort) * indexname) list * thm
137 val varifyT_global: thm -> thm
138 val legacy_freezeT: thm -> thm
139 val lift_rule: cterm -> thm -> thm
140 val incr_indexes: int -> thm -> thm
141 val assumption: int -> thm -> thm Seq.seq
142 val eq_assumption: int -> thm -> thm
143 val rotate_rule: int -> int -> thm -> thm
144 val permute_prems: int -> int -> thm -> thm
145 val rename_params_rule: string list * int -> thm -> thm
146 val rename_boundvars: term -> term -> thm -> thm
147 val compose_no_flatten: bool -> thm * int -> int -> thm -> thm Seq.seq
148 val bicompose: bool -> bool * thm * int -> int -> thm -> thm Seq.seq
149 val biresolution: bool -> (bool * thm) list -> int -> thm -> thm Seq.seq
150 val extern_oracles: Proof.context -> xstring list
151 val add_oracle: binding * ('a -> cterm) -> theory -> (string * ('a -> thm)) * theory
157 (*** Certified terms and types ***)
159 (** certified types **)
161 abstype ctyp = Ctyp of
162 {thy_ref: theory_ref,
165 sorts: sort Ord_List.T}
168 fun rep_ctyp (Ctyp args) = args;
169 fun theory_of_ctyp (Ctyp {thy_ref, ...}) = Theory.deref thy_ref;
170 fun typ_of (Ctyp {T, ...}) = T;
172 fun ctyp_of thy raw_T =
174 val T = Sign.certify_typ thy raw_T;
175 val maxidx = Term.maxidx_of_typ T;
176 val sorts = Sorts.insert_typ T [];
177 in Ctyp {thy_ref = Theory.check_thy thy, T = T, maxidx = maxidx, sorts = sorts} end;
179 fun dest_ctyp (Ctyp {thy_ref, T = Type (_, Ts), maxidx, sorts}) =
180 map (fn T => Ctyp {thy_ref = thy_ref, T = T, maxidx = maxidx, sorts = sorts}) Ts
181 | dest_ctyp cT = raise TYPE ("dest_ctyp", [typ_of cT], []);
185 (** certified terms **)
187 (*certified terms with checked typ, maxidx, and sorts*)
188 abstype cterm = Cterm of
189 {thy_ref: theory_ref,
193 sorts: sort Ord_List.T}
196 exception CTERM of string * cterm list;
198 fun rep_cterm (Cterm args) = args;
200 fun crep_cterm (Cterm {thy_ref, t, T, maxidx, sorts}) =
201 {thy_ref = thy_ref, t = t, maxidx = maxidx, sorts = sorts,
202 T = Ctyp {thy_ref = thy_ref, T = T, maxidx = maxidx, sorts = sorts}};
204 fun theory_of_cterm (Cterm {thy_ref, ...}) = Theory.deref thy_ref;
205 fun term_of (Cterm {t, ...}) = t;
207 fun ctyp_of_term (Cterm {thy_ref, T, maxidx, sorts, ...}) =
208 Ctyp {thy_ref = thy_ref, T = T, maxidx = maxidx, sorts = sorts};
210 fun cterm_of thy tm =
212 val (t, T, maxidx) = Sign.certify_term thy tm;
213 val sorts = Sorts.insert_term t [];
214 in Cterm {thy_ref = Theory.check_thy thy, t = t, T = T, maxidx = maxidx, sorts = sorts} end;
216 fun merge_thys0 (Cterm {thy_ref = r1, ...}) (Cterm {thy_ref = r2, ...}) =
217 Theory.merge_refs (r1, r2);
222 fun dest_comb (Cterm {t = c $ a, T, thy_ref, maxidx, sorts}) =
223 let val A = Term.argument_type_of c 0 in
224 (Cterm {t = c, T = A --> T, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts},
225 Cterm {t = a, T = A, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts})
227 | dest_comb ct = raise CTERM ("dest_comb", [ct]);
229 fun dest_fun (Cterm {t = c $ _, T, thy_ref, maxidx, sorts}) =
230 let val A = Term.argument_type_of c 0
231 in Cterm {t = c, T = A --> T, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
232 | dest_fun ct = raise CTERM ("dest_fun", [ct]);
234 fun dest_arg (Cterm {t = c $ a, T = _, thy_ref, maxidx, sorts}) =
235 let val A = Term.argument_type_of c 0
236 in Cterm {t = a, T = A, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
237 | dest_arg ct = raise CTERM ("dest_arg", [ct]);
240 fun dest_fun2 (Cterm {t = c $ _ $ _, T, thy_ref, maxidx, sorts}) =
242 val A = Term.argument_type_of c 0;
243 val B = Term.argument_type_of c 1;
244 in Cterm {t = c, T = A --> B --> T, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
245 | dest_fun2 ct = raise CTERM ("dest_fun2", [ct]);
247 fun dest_arg1 (Cterm {t = c $ a $ _, T = _, thy_ref, maxidx, sorts}) =
248 let val A = Term.argument_type_of c 0
249 in Cterm {t = a, T = A, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
250 | dest_arg1 ct = raise CTERM ("dest_arg1", [ct]);
252 fun dest_abs a (Cterm {t = Abs (x, T, t), T = Type ("fun", [_, U]), thy_ref, maxidx, sorts}) =
253 let val (y', t') = Term.dest_abs (the_default x a, T, t) in
254 (Cterm {t = Free (y', T), T = T, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts},
255 Cterm {t = t', T = U, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts})
257 | dest_abs _ ct = raise CTERM ("dest_abs", [ct]);
263 (cf as Cterm {t = f, T = Type ("fun", [dty, rty]), maxidx = maxidx1, sorts = sorts1, ...})
264 (cx as Cterm {t = x, T, maxidx = maxidx2, sorts = sorts2, ...}) =
266 Cterm {thy_ref = merge_thys0 cf cx,
269 maxidx = Int.max (maxidx1, maxidx2),
270 sorts = Sorts.union sorts1 sorts2}
271 else raise CTERM ("apply: types don't agree", [cf, cx])
272 | apply cf cx = raise CTERM ("apply: first arg is not a function", [cf, cx]);
275 (x, ct1 as Cterm {t = t1, T = T1, maxidx = maxidx1, sorts = sorts1, ...})
276 (ct2 as Cterm {t = t2, T = T2, maxidx = maxidx2, sorts = sorts2, ...}) =
277 let val t = Term.lambda_name (x, t1) t2 in
278 Cterm {thy_ref = merge_thys0 ct1 ct2,
279 t = t, T = T1 --> T2,
280 maxidx = Int.max (maxidx1, maxidx2),
281 sorts = Sorts.union sorts1 sorts2}
284 fun lambda t u = lambda_name ("", t) u;
289 fun adjust_maxidx_cterm i (ct as Cterm {thy_ref, t, T, maxidx, sorts}) =
290 if maxidx = i then ct
291 else if maxidx < i then
292 Cterm {maxidx = i, thy_ref = thy_ref, t = t, T = T, sorts = sorts}
294 Cterm {maxidx = Int.max (maxidx_of_term t, i), thy_ref = thy_ref, t = t, T = T, sorts = sorts};
296 fun incr_indexes_cterm i (ct as Cterm {thy_ref, t, T, maxidx, sorts}) =
297 if i < 0 then raise CTERM ("negative increment", [ct])
298 else if i = 0 then ct
299 else Cterm {thy_ref = thy_ref, t = Logic.incr_indexes ([], i) t,
300 T = Logic.incr_tvar i T, maxidx = maxidx + i, sorts = sorts};
308 (ct1 as Cterm {t = t1, sorts = sorts1, ...},
309 ct2 as Cterm {t = t2, sorts = sorts2, maxidx = maxidx2, ...}) =
311 val thy = Theory.deref (merge_thys0 ct1 ct2);
312 val (Tinsts, tinsts) = match thy (t1, t2) (Vartab.empty, Vartab.empty);
313 val sorts = Sorts.union sorts1 sorts2;
314 fun mk_cTinst ((a, i), (S, T)) =
315 (Ctyp {T = TVar ((a, i), S), thy_ref = Theory.check_thy thy, maxidx = i, sorts = sorts},
316 Ctyp {T = T, thy_ref = Theory.check_thy thy, maxidx = maxidx2, sorts = sorts});
317 fun mk_ctinst ((x, i), (T, t)) =
318 let val T = Envir.subst_type Tinsts T in
319 (Cterm {t = Var ((x, i), T), T = T, thy_ref = Theory.check_thy thy,
320 maxidx = i, sorts = sorts},
321 Cterm {t = t, T = T, thy_ref = Theory.check_thy thy, maxidx = maxidx2, sorts = sorts})
323 in (Vartab.fold (cons o mk_cTinst) Tinsts [], Vartab.fold (cons o mk_ctinst) tinsts []) end;
327 val match = gen_match Pattern.match;
328 val first_order_match = gen_match Pattern.first_order_match;
334 (*** Derivations and Theorems ***)
337 deriv * (*derivation*)
338 {thy_ref: theory_ref, (*dynamic reference to theory*)
339 tags: Properties.T, (*additional annotations/comments*)
340 maxidx: int, (*maximum index of any Var or TVar*)
341 shyps: sort Ord_List.T, (*sort hypotheses*)
342 hyps: term Ord_List.T, (*hypotheses*)
343 tpairs: (term * term) list, (*flex-flex pairs*)
344 prop: term} (*conclusion*)
346 {promises: (serial * thm future) Ord_List.T,
347 body: Proofterm.proof_body}
350 type conv = cterm -> thm;
352 (*errors involving theorems*)
353 exception THM of string * int * thm list;
355 fun rep_thm (Thm (_, args)) = args;
357 fun crep_thm (Thm (_, {thy_ref, tags, maxidx, shyps, hyps, tpairs, prop})) =
358 let fun cterm max t = Cterm {thy_ref = thy_ref, t = t, T = propT, maxidx = max, sorts = shyps} in
359 {thy_ref = thy_ref, tags = tags, maxidx = maxidx, shyps = shyps,
360 hyps = map (cterm ~1) hyps,
361 tpairs = map (pairself (cterm maxidx)) tpairs,
362 prop = cterm maxidx prop}
365 fun fold_terms f (Thm (_, {tpairs, prop, hyps, ...})) =
366 fold (fn (t, u) => f t #> f u) tpairs #> f prop #> fold f hyps;
368 fun terms_of_tpairs tpairs = fold_rev (fn (t, u) => cons t o cons u) tpairs [];
370 fun eq_tpairs ((t, u), (t', u')) = t aconv t' andalso u aconv u';
371 fun union_tpairs ts us = Library.merge eq_tpairs (ts, us);
372 val maxidx_tpairs = fold (fn (t, u) => Term.maxidx_term t #> Term.maxidx_term u);
374 fun attach_tpairs tpairs prop =
375 Logic.list_implies (map Logic.mk_equals tpairs, prop);
377 fun full_prop_of (Thm (_, {tpairs, prop, ...})) = attach_tpairs tpairs prop;
379 val union_hyps = Ord_List.union Term_Ord.fast_term_ord;
380 val insert_hyps = Ord_List.insert Term_Ord.fast_term_ord;
381 val remove_hyps = Ord_List.remove Term_Ord.fast_term_ord;
384 (* merge theories of cterms/thms -- trivial absorption only *)
386 fun merge_thys1 (Cterm {thy_ref = r1, ...}) (Thm (_, {thy_ref = r2, ...})) =
387 Theory.merge_refs (r1, r2);
389 fun merge_thys2 (Thm (_, {thy_ref = r1, ...})) (Thm (_, {thy_ref = r2, ...})) =
390 Theory.merge_refs (r1, r2);
393 (* basic components *)
395 val theory_of_thm = Theory.deref o #thy_ref o rep_thm;
396 val maxidx_of = #maxidx o rep_thm;
397 fun maxidx_thm th i = Int.max (maxidx_of th, i);
398 val hyps_of = #hyps o rep_thm;
399 val prop_of = #prop o rep_thm;
400 val tpairs_of = #tpairs o rep_thm;
402 val concl_of = Logic.strip_imp_concl o prop_of;
403 val prems_of = Logic.strip_imp_prems o prop_of;
404 val nprems_of = Logic.count_prems o prop_of;
405 fun no_prems th = nprems_of th = 0;
407 fun major_prem_of th =
409 prem :: _ => Logic.strip_assums_concl prem
410 | [] => raise THM ("major_prem_of: rule with no premises", 0, [th]));
412 (*the statement of any thm is a cterm*)
413 fun cprop_of (Thm (_, {thy_ref, maxidx, shyps, prop, ...})) =
414 Cterm {thy_ref = thy_ref, maxidx = maxidx, T = propT, t = prop, sorts = shyps};
416 fun cprem_of (th as Thm (_, {thy_ref, maxidx, shyps, prop, ...})) i =
417 Cterm {thy_ref = thy_ref, maxidx = maxidx, T = propT, sorts = shyps,
418 t = Logic.nth_prem (i, prop) handle TERM _ => raise THM ("cprem_of", i, [th])};
420 (*explicit transfer to a super theory*)
421 fun transfer thy' thm =
423 val Thm (der, {thy_ref, tags, maxidx, shyps, hyps, tpairs, prop}) = thm;
424 val thy = Theory.deref thy_ref;
425 val _ = Theory.subthy (thy, thy') orelse raise THM ("transfer: not a super theory", 0, [thm]);
426 val is_eq = Theory.eq_thy (thy, thy');
427 val _ = Theory.check_thy thy;
432 {thy_ref = Theory.check_thy thy',
441 (*explicit weakening: maps |- B to A |- B*)
442 fun weaken raw_ct th =
444 val ct as Cterm {t = A, T, sorts, maxidx = maxidxA, ...} = adjust_maxidx_cterm ~1 raw_ct;
445 val Thm (der, {tags, maxidx, shyps, hyps, tpairs, prop, ...}) = th;
448 raise THM ("weaken: assumptions must have type prop", 0, [])
449 else if maxidxA <> ~1 then
450 raise THM ("weaken: assumptions may not contain schematic variables", maxidxA, [])
453 {thy_ref = merge_thys1 ct th,
456 shyps = Sorts.union sorts shyps,
457 hyps = insert_hyps A hyps,
462 fun weaken_sorts raw_sorts ct =
464 val Cterm {thy_ref, t, T, maxidx, sorts} = ct;
465 val thy = Theory.deref thy_ref;
466 val more_sorts = Sorts.make (map (Sign.certify_sort thy) raw_sorts);
467 val sorts' = Sorts.union sorts more_sorts;
468 in Cterm {thy_ref = Theory.check_thy thy, t = t, T = T, maxidx = maxidx, sorts = sorts'} end;
470 (*dangling sort constraints of a thm*)
471 fun extra_shyps (th as Thm (_, {shyps, ...})) =
472 Sorts.subtract (fold_terms Sorts.insert_term th []) shyps;
476 (** derivations and promised proofs **)
478 fun make_deriv promises oracles thms proof =
479 Deriv {promises = promises, body = PBody {oracles = oracles, thms = thms, proof = proof}};
481 val empty_deriv = make_deriv [] [] [] Proofterm.MinProof;
484 (* inference rules *)
486 fun promise_ord ((i, _), (j, _)) = int_ord (j, i);
489 (Deriv {promises = ps1, body = PBody {oracles = oras1, thms = thms1, proof = prf1}})
490 (Deriv {promises = ps2, body = PBody {oracles = oras2, thms = thms2, proof = prf2}}) =
492 val ps = Ord_List.union promise_ord ps1 ps2;
493 val oras = Proofterm.unions_oracles [oras1, oras2];
494 val thms = Proofterm.unions_thms [thms1, thms2];
496 (case ! Proofterm.proofs of
500 | i => error ("Illegal level of detail for proof objects: " ^ string_of_int i));
501 in make_deriv ps oras thms prf end;
503 fun deriv_rule1 f = deriv_rule2 (K f) empty_deriv;
504 fun deriv_rule0 prf = deriv_rule1 I (make_deriv [] [] [] prf);
506 fun deriv_rule_unconditional f (Deriv {promises, body = PBody {oracles, thms, proof}}) =
507 make_deriv promises oracles thms (f proof);
510 (* fulfilled proofs *)
512 fun raw_body_of (Thm (Deriv {body, ...}, _)) = body;
513 fun raw_promises_of (Thm (Deriv {promises, ...}, _)) = promises;
515 fun join_promises [] = ()
516 | join_promises promises = join_promises_of (Future.joins (map snd promises))
517 and join_promises_of thms = join_promises (maps raw_promises_of thms);
519 fun fulfill_body (Thm (Deriv {promises, body}, {thy_ref, ...})) =
520 Proofterm.fulfill_norm_proof (Theory.deref thy_ref) (fulfill_promises promises) body
521 and fulfill_promises promises =
522 map fst promises ~~ map fulfill_body (Future.joins (map snd promises));
524 fun proof_bodies_of thms =
526 val _ = join_promises_of thms;
527 val bodies = map fulfill_body thms;
528 val _ = Proofterm.join_bodies bodies;
531 val proof_body_of = singleton proof_bodies_of;
532 val proof_of = Proofterm.proof_of o proof_body_of;
534 val join_proofs = ignore o proof_bodies_of;
537 (* derivation status *)
539 fun status_of (Thm (Deriv {promises, body}, _)) =
541 val ps = map (Future.peek o snd) promises;
543 map_filter (fn SOME (Exn.Res th) => SOME (raw_body_of th) | _ => NONE) ps;
544 val {oracle, unfinished, failed} = Proofterm.status_of bodies;
547 unfinished = unfinished orelse exists is_none ps,
548 failed = failed orelse exists (fn SOME (Exn.Exn _) => true | _ => false) ps}
554 fun future_result i orig_thy orig_shyps orig_prop thm =
556 fun err msg = raise THM ("future_result: " ^ msg, 0, [thm]);
557 val Thm (Deriv {promises, ...}, {thy_ref, shyps, hyps, tpairs, prop, ...}) = thm;
559 val _ = Theory.eq_thy (Theory.deref thy_ref, orig_thy) orelse err "bad theory";
560 val _ = Theory.check_thy orig_thy;
561 val _ = prop aconv orig_prop orelse err "bad prop";
562 val _ = null tpairs orelse err "bad tpairs";
563 val _ = null hyps orelse err "bad hyps";
564 val _ = Sorts.subset (shyps, orig_shyps) orelse err "bad shyps";
565 val _ = forall (fn (j, _) => i <> j) promises orelse err "bad dependencies";
566 val _ = join_promises promises;
569 fun future future_thm ct =
571 val Cterm {thy_ref = thy_ref, t = prop, T, maxidx, sorts} = ct;
572 val thy = Context.reject_draft (Theory.deref thy_ref);
573 val _ = T <> propT andalso raise CTERM ("future: prop expected", [ct]);
576 val future = future_thm |> Future.map (future_result i thy sorts prop);
578 Thm (make_deriv [(i, future)] [] [] (Proofterm.promise_proof thy i prop),
589 (* closed derivations with official name *)
591 (*non-deterministic, depends on unknown promises*)
592 fun derivation_name (Thm (Deriv {body, ...}, {shyps, hyps, prop, ...})) =
593 Proofterm.get_name shyps hyps prop (Proofterm.proof_of body);
595 fun name_derivation name (thm as Thm (der, args)) =
597 val Deriv {promises, body} = der;
598 val {thy_ref, shyps, hyps, prop, tpairs, ...} = args;
599 val _ = null tpairs orelse raise THM ("put_name: unsolved flex-flex constraints", 0, [thm]);
601 val ps = map (apsnd (Future.map fulfill_body)) promises;
602 val thy = Theory.deref thy_ref;
603 val (pthm, proof) = Proofterm.thm_proof thy name shyps hyps prop ps body;
604 val der' = make_deriv [] [] [pthm] proof;
605 val _ = Theory.check_thy thy;
606 in Thm (der', args) end;
612 fun axiom theory name =
615 Symtab.lookup (Theory.axiom_table thy) name
616 |> Option.map (fn prop =>
618 val der = deriv_rule0 (Proofterm.axm_proof name prop);
619 val maxidx = maxidx_of_term prop;
620 val shyps = Sorts.insert_term prop [];
622 Thm (der, {thy_ref = Theory.check_thy thy, tags = [],
623 maxidx = maxidx, shyps = shyps, hyps = [], tpairs = [], prop = prop})
626 (case get_first get_ax (Theory.nodes_of theory) of
628 | NONE => raise THEORY ("No axiom " ^ quote name, [theory]))
631 (*return additional axioms of this theory node*)
633 map (fn s => (s, axiom thy s)) (Symtab.keys (Theory.axiom_table thy));
638 val get_tags = #tags o rep_thm;
640 fun map_tags f (Thm (der, {thy_ref, tags, maxidx, shyps, hyps, tpairs, prop})) =
641 Thm (der, {thy_ref = thy_ref, tags = f tags, maxidx = maxidx,
642 shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop});
645 (* technical adjustments *)
647 fun norm_proof (Thm (der, args as {thy_ref, ...})) =
649 val thy = Theory.deref thy_ref;
650 val der' = deriv_rule1 (Proofterm.rew_proof thy) der;
651 val _ = Theory.check_thy thy;
652 in Thm (der', args) end;
654 fun adjust_maxidx_thm i (th as Thm (der, {thy_ref, tags, maxidx, shyps, hyps, tpairs, prop})) =
655 if maxidx = i then th
656 else if maxidx < i then
657 Thm (der, {maxidx = i, thy_ref = thy_ref, tags = tags, shyps = shyps,
658 hyps = hyps, tpairs = tpairs, prop = prop})
660 Thm (der, {maxidx = Int.max (maxidx_tpairs tpairs (maxidx_of_term prop), i), thy_ref = thy_ref,
661 tags = tags, shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop});
667 (** primitive rules **)
669 (*The assumption rule A |- A*)
671 let val Cterm {thy_ref, t = prop, T, maxidx, sorts} = adjust_maxidx_cterm ~1 raw_ct in
673 raise THM ("assume: prop", 0, [])
674 else if maxidx <> ~1 then
675 raise THM ("assume: variables", maxidx, [])
676 else Thm (deriv_rule0 (Proofterm.Hyp prop),
686 (*Implication introduction
694 (ct as Cterm {t = A, T, maxidx = maxidxA, sorts, ...})
695 (th as Thm (der, {maxidx, hyps, shyps, tpairs, prop, ...})) =
697 raise THM ("implies_intr: assumptions must have type prop", 0, [th])
699 Thm (deriv_rule1 (Proofterm.implies_intr_proof A) der,
700 {thy_ref = merge_thys1 ct th,
702 maxidx = Int.max (maxidxA, maxidx),
703 shyps = Sorts.union sorts shyps,
704 hyps = remove_hyps A hyps,
706 prop = Logic.mk_implies (A, prop)});
709 (*Implication elimination
714 fun implies_elim thAB thA =
716 val Thm (derA, {maxidx = maxA, hyps = hypsA, shyps = shypsA, tpairs = tpairsA,
717 prop = propA, ...}) = thA
718 and Thm (der, {maxidx, hyps, shyps, tpairs, prop, ...}) = thAB;
719 fun err () = raise THM ("implies_elim: major premise", 0, [thAB, thA]);
722 Const ("==>", _) $ A $ B =>
723 if A aconv propA then
724 Thm (deriv_rule2 (curry Proofterm.%%) der derA,
725 {thy_ref = merge_thys2 thAB thA,
727 maxidx = Int.max (maxA, maxidx),
728 shyps = Sorts.union shypsA shyps,
729 hyps = union_hyps hypsA hyps,
730 tpairs = union_tpairs tpairsA tpairs,
736 (*Forall introduction. The Free or Var x must not be free in the hypotheses.
744 (ct as Cterm {t = x, T, sorts, ...})
745 (th as Thm (der, {maxidx, shyps, hyps, tpairs, prop, ...})) =
748 Thm (deriv_rule1 (Proofterm.forall_intr_proof x a) der,
749 {thy_ref = merge_thys1 ct th,
752 shyps = Sorts.union sorts shyps,
755 prop = Logic.all_const T $ Abs (a, T, abstract_over (x, prop))});
756 fun check_occs a x ts =
757 if exists (fn t => Logic.occs (x, t)) ts then
758 raise THM ("forall_intr: variable " ^ quote a ^ " free in assumptions", 0, [th])
762 Free (a, _) => (check_occs a x hyps; check_occs a x (terms_of_tpairs tpairs); result a)
763 | Var ((a, _), _) => (check_occs a x (terms_of_tpairs tpairs); result a)
764 | _ => raise THM ("forall_intr: not a variable", 0, [th])
773 (ct as Cterm {t, T, maxidx = maxt, sorts, ...})
774 (th as Thm (der, {maxidx, shyps, hyps, tpairs, prop, ...})) =
776 Const ("all", Type ("fun", [Type ("fun", [qary, _]), _])) $ A =>
778 raise THM ("forall_elim: type mismatch", 0, [th])
780 Thm (deriv_rule1 (Proofterm.% o rpair (SOME t)) der,
781 {thy_ref = merge_thys1 ct th,
783 maxidx = Int.max (maxidx, maxt),
784 shyps = Sorts.union sorts shyps,
787 prop = Term.betapply (A, t)})
788 | _ => raise THM ("forall_elim: not quantified", 0, [th]));
796 fun reflexive (Cterm {thy_ref, t, T = _, maxidx, sorts}) =
797 Thm (deriv_rule0 Proofterm.reflexive,
804 prop = Logic.mk_equals (t, t)});
811 fun symmetric (th as Thm (der, {thy_ref, maxidx, shyps, hyps, tpairs, prop, ...})) =
813 (eq as Const ("==", _)) $ t $ u =>
814 Thm (deriv_rule1 Proofterm.symmetric der,
822 | _ => raise THM ("symmetric", 0, [th]));
829 fun transitive th1 th2 =
831 val Thm (der1, {maxidx = max1, hyps = hyps1, shyps = shyps1, tpairs = tpairs1,
832 prop = prop1, ...}) = th1
833 and Thm (der2, {maxidx = max2, hyps = hyps2, shyps = shyps2, tpairs = tpairs2,
834 prop = prop2, ...}) = th2;
835 fun err msg = raise THM ("transitive: " ^ msg, 0, [th1, th2]);
837 case (prop1, prop2) of
838 ((eq as Const ("==", Type (_, [T, _]))) $ t1 $ u, Const ("==", _) $ u' $ t2) =>
839 if not (u aconv u') then err "middle term"
841 Thm (deriv_rule2 (Proofterm.transitive u T) der1 der2,
842 {thy_ref = merge_thys2 th1 th2,
844 maxidx = Int.max (max1, max2),
845 shyps = Sorts.union shyps1 shyps2,
846 hyps = union_hyps hyps1 hyps2,
847 tpairs = union_tpairs tpairs1 tpairs2,
848 prop = eq $ t1 $ t2})
849 | _ => err "premises"
854 fully beta-reduces the term if full = true
856 fun beta_conversion full (Cterm {thy_ref, t, T = _, maxidx, sorts}) =
858 if full then Envir.beta_norm t
860 (case t of Abs (_, _, bodt) $ u => subst_bound (u, bodt)
861 | _ => raise THM ("beta_conversion: not a redex", 0, []));
863 Thm (deriv_rule0 Proofterm.reflexive,
870 prop = Logic.mk_equals (t, t')})
873 fun eta_conversion (Cterm {thy_ref, t, T = _, maxidx, sorts}) =
874 Thm (deriv_rule0 Proofterm.reflexive,
881 prop = Logic.mk_equals (t, Envir.eta_contract t)});
883 fun eta_long_conversion (Cterm {thy_ref, t, T = _, maxidx, sorts}) =
884 Thm (deriv_rule0 Proofterm.reflexive,
891 prop = Logic.mk_equals (t, Pattern.eta_long [] t)});
893 (*The abstraction rule. The Free or Var x must not be free in the hypotheses.
894 The bound variable will be named "a" (since x will be something like x320)
900 (Cterm {t = x, T, sorts, ...})
901 (th as Thm (der, {thy_ref, maxidx, hyps, shyps, tpairs, prop, ...})) =
903 val (t, u) = Logic.dest_equals prop
904 handle TERM _ => raise THM ("abstract_rule: premise not an equality", 0, [th]);
906 Thm (deriv_rule1 (Proofterm.abstract_rule x a) der,
910 shyps = Sorts.union sorts shyps,
913 prop = Logic.mk_equals
914 (Abs (a, T, abstract_over (x, t)), Abs (a, T, abstract_over (x, u)))});
915 fun check_occs a x ts =
916 if exists (fn t => Logic.occs (x, t)) ts then
917 raise THM ("abstract_rule: variable " ^ quote a ^ " free in assumptions", 0, [th])
921 Free (a, _) => (check_occs a x hyps; check_occs a x (terms_of_tpairs tpairs); result)
922 | Var ((a, _), _) => (check_occs a x (terms_of_tpairs tpairs); result)
923 | _ => raise THM ("abstract_rule: not a variable", 0, [th])
926 (*The combination rule
931 fun combination th1 th2 =
933 val Thm (der1, {maxidx = max1, shyps = shyps1, hyps = hyps1, tpairs = tpairs1,
934 prop = prop1, ...}) = th1
935 and Thm (der2, {maxidx = max2, shyps = shyps2, hyps = hyps2, tpairs = tpairs2,
936 prop = prop2, ...}) = th2;
939 Type ("fun", [T1, _]) =>
941 raise THM ("combination: types", 0, [th1, th2])
943 | _ => raise THM ("combination: not function type", 0, [th1, th2]));
945 case (prop1, prop2) of
946 (Const ("==", Type ("fun", [fT, _])) $ f $ g,
947 Const ("==", Type ("fun", [tT, _])) $ t $ u) =>
949 Thm (deriv_rule2 (Proofterm.combination f g t u fT) der1 der2,
950 {thy_ref = merge_thys2 th1 th2,
952 maxidx = Int.max (max1, max2),
953 shyps = Sorts.union shyps1 shyps2,
954 hyps = union_hyps hyps1 hyps2,
955 tpairs = union_tpairs tpairs1 tpairs2,
956 prop = Logic.mk_equals (f $ t, g $ u)}))
957 | _ => raise THM ("combination: premises", 0, [th1, th2])
960 (*Equality introduction
965 fun equal_intr th1 th2 =
967 val Thm (der1, {maxidx = max1, shyps = shyps1, hyps = hyps1, tpairs = tpairs1,
968 prop = prop1, ...}) = th1
969 and Thm (der2, {maxidx = max2, shyps = shyps2, hyps = hyps2, tpairs = tpairs2,
970 prop = prop2, ...}) = th2;
971 fun err msg = raise THM ("equal_intr: " ^ msg, 0, [th1, th2]);
973 case (prop1, prop2) of
974 (Const("==>", _) $ A $ B, Const("==>", _) $ B' $ A') =>
975 if A aconv A' andalso B aconv B' then
976 Thm (deriv_rule2 (Proofterm.equal_intr A B) der1 der2,
977 {thy_ref = merge_thys2 th1 th2,
979 maxidx = Int.max (max1, max2),
980 shyps = Sorts.union shyps1 shyps2,
981 hyps = union_hyps hyps1 hyps2,
982 tpairs = union_tpairs tpairs1 tpairs2,
983 prop = Logic.mk_equals (A, B)})
985 | _ => err "premises"
988 (*The equal propositions rule
993 fun equal_elim th1 th2 =
995 val Thm (der1, {maxidx = max1, shyps = shyps1, hyps = hyps1,
996 tpairs = tpairs1, prop = prop1, ...}) = th1
997 and Thm (der2, {maxidx = max2, shyps = shyps2, hyps = hyps2,
998 tpairs = tpairs2, prop = prop2, ...}) = th2;
999 fun err msg = raise THM ("equal_elim: " ^ msg, 0, [th1, th2]);
1002 Const ("==", _) $ A $ B =>
1003 if prop2 aconv A then
1004 Thm (deriv_rule2 (Proofterm.equal_elim A B) der1 der2,
1005 {thy_ref = merge_thys2 th1 th2,
1007 maxidx = Int.max (max1, max2),
1008 shyps = Sorts.union shyps1 shyps2,
1009 hyps = union_hyps hyps1 hyps2,
1010 tpairs = union_tpairs tpairs1 tpairs2,
1012 else err "not equal"
1013 | _ => err"major premise"
1018 (**** Derived rules ****)
1020 (*Smash unifies the list of term pairs leaving no flex-flex pairs.
1021 Instantiates the theorem and deletes trivial tpairs. Resulting
1022 sequence may contain multiple elements if the tpairs are not all
1024 fun flexflex_rule (th as Thm (der, {thy_ref, maxidx, shyps, hyps, tpairs, prop, ...})) =
1025 let val thy = Theory.deref thy_ref in
1026 Unify.smash_unifiers thy tpairs (Envir.empty maxidx)
1027 |> Seq.map (fn env =>
1028 if Envir.is_empty env then th
1031 val tpairs' = tpairs |> map (pairself (Envir.norm_term env))
1032 (*remove trivial tpairs, of the form t==t*)
1033 |> filter_out (op aconv);
1034 val der' = deriv_rule1 (Proofterm.norm_proof' env) der;
1035 val prop' = Envir.norm_term env prop;
1036 val maxidx = maxidx_tpairs tpairs' (maxidx_of_term prop');
1037 val shyps = Envir.insert_sorts env shyps;
1039 Thm (der', {thy_ref = Theory.check_thy thy, tags = [], maxidx = maxidx,
1040 shyps = shyps, hyps = hyps, tpairs = tpairs', prop = prop'})
1045 (*Generalization of fixed variables
1047 --------------------
1048 A[?'a/'a, ?x/x, ...]
1051 fun generalize ([], []) _ th = th
1052 | generalize (tfrees, frees) idx th =
1054 val Thm (der, {thy_ref, maxidx, shyps, hyps, tpairs, prop, ...}) = th;
1055 val _ = idx <= maxidx andalso raise THM ("generalize: bad index", idx, [th]);
1058 if null tfrees then K false
1059 else Term.exists_subtype (fn TFree (a, _) => member (op =) tfrees a | _ => false);
1060 fun bad_term (Free (x, T)) = bad_type T orelse member (op =) frees x
1061 | bad_term (Var (_, T)) = bad_type T
1062 | bad_term (Const (_, T)) = bad_type T
1063 | bad_term (Abs (_, T, t)) = bad_type T orelse bad_term t
1064 | bad_term (t $ u) = bad_term t orelse bad_term u
1065 | bad_term (Bound _) = false;
1066 val _ = exists bad_term hyps andalso
1067 raise THM ("generalize: variable free in assumptions", 0, [th]);
1069 val gen = Term_Subst.generalize (tfrees, frees) idx;
1070 val prop' = gen prop;
1071 val tpairs' = map (pairself gen) tpairs;
1072 val maxidx' = maxidx_tpairs tpairs' (maxidx_of_term prop');
1074 Thm (deriv_rule1 (Proofterm.generalize (tfrees, frees) idx) der,
1085 (*Instantiation of schematic variables
1087 --------------------
1088 A[t1/v1, ..., tn/vn]
1093 fun pretty_typing thy t T = Pretty.block
1094 [Syntax.pretty_term_global thy t, Pretty.str " ::", Pretty.brk 1, Syntax.pretty_typ_global thy T];
1096 fun add_inst (ct, cu) (thy_ref, sorts) =
1098 val Cterm {t = t, T = T, ...} = ct;
1099 val Cterm {t = u, T = U, sorts = sorts_u, maxidx = maxidx_u, ...} = cu;
1100 val thy_ref' = Theory.merge_refs (thy_ref, merge_thys0 ct cu);
1101 val sorts' = Sorts.union sorts_u sorts;
1104 if T = U then ((v, (u, maxidx_u)), (thy_ref', sorts'))
1105 else raise TYPE (Pretty.string_of (Pretty.block
1106 [Pretty.str "instantiate: type conflict",
1107 Pretty.fbrk, pretty_typing (Theory.deref thy_ref') t T,
1108 Pretty.fbrk, pretty_typing (Theory.deref thy_ref') u U]), [T, U], [t, u])
1109 | _ => raise TYPE (Pretty.string_of (Pretty.block
1110 [Pretty.str "instantiate: not a variable",
1111 Pretty.fbrk, Syntax.pretty_term_global (Theory.deref thy_ref') t]), [], [t]))
1114 fun add_instT (cT, cU) (thy_ref, sorts) =
1116 val Ctyp {T, thy_ref = thy_ref1, ...} = cT
1117 and Ctyp {T = U, thy_ref = thy_ref2, sorts = sorts_U, maxidx = maxidx_U, ...} = cU;
1118 val thy' = Theory.deref (Theory.merge_refs (thy_ref, Theory.merge_refs (thy_ref1, thy_ref2)));
1119 val sorts' = Sorts.union sorts_U sorts;
1121 (case T of TVar (v as (_, S)) =>
1122 if Sign.of_sort thy' (U, S) then ((v, (U, maxidx_U)), (Theory.check_thy thy', sorts'))
1123 else raise TYPE ("Type not of sort " ^ Syntax.string_of_sort_global thy' S, [U], [])
1124 | _ => raise TYPE (Pretty.string_of (Pretty.block
1125 [Pretty.str "instantiate: not a type variable",
1126 Pretty.fbrk, Syntax.pretty_typ_global thy' T]), [T], []))
1131 (*Left-to-right replacements: ctpairs = [..., (vi, ti), ...].
1132 Instantiates distinct Vars by terms of same type.
1133 Does NOT normalize the resulting theorem!*)
1134 fun instantiate ([], []) th = th
1135 | instantiate (instT, inst) th =
1137 val Thm (der, {thy_ref, hyps, shyps, tpairs, prop, ...}) = th;
1138 val (inst', (instT', (thy_ref', shyps'))) =
1139 (thy_ref, shyps) |> fold_map add_inst inst ||> fold_map add_instT instT;
1140 val subst = Term_Subst.instantiate_maxidx (instT', inst');
1141 val (prop', maxidx1) = subst prop ~1;
1142 val (tpairs', maxidx') =
1143 fold_map (fn (t, u) => fn i => subst t i ||>> subst u) tpairs maxidx1;
1146 (fn d => Proofterm.instantiate (map (apsnd #1) instT', map (apsnd #1) inst') d) der,
1147 {thy_ref = thy_ref',
1155 handle TYPE (msg, _, _) => raise THM (msg, 0, [th]);
1157 fun instantiate_cterm ([], []) ct = ct
1158 | instantiate_cterm (instT, inst) ct =
1160 val Cterm {thy_ref, t, T, sorts, ...} = ct;
1161 val (inst', (instT', (thy_ref', sorts'))) =
1162 (thy_ref, sorts) |> fold_map add_inst inst ||> fold_map add_instT instT;
1163 val subst = Term_Subst.instantiate_maxidx (instT', inst');
1164 val substT = Term_Subst.instantiateT_maxidx instT';
1165 val (t', maxidx1) = subst t ~1;
1166 val (T', maxidx') = substT T maxidx1;
1167 in Cterm {thy_ref = thy_ref', t = t', T = T', sorts = sorts', maxidx = maxidx'} end
1168 handle TYPE (msg, _, _) => raise CTERM (msg, [ct]);
1173 (*The trivial implication A ==> A, justified by assume and forall rules.
1174 A can contain Vars, not so for assume!*)
1175 fun trivial (Cterm {thy_ref, t =A, T, maxidx, sorts}) =
1177 raise THM ("trivial: the term must have type prop", 0, [])
1179 Thm (deriv_rule0 (Proofterm.AbsP ("H", NONE, Proofterm.PBound 0)),
1186 prop = Logic.mk_implies (A, A)});
1188 (*Axiom-scheme reflecting signature contents
1193 fun of_class (cT, raw_c) =
1195 val Ctyp {thy_ref, T, ...} = cT;
1196 val thy = Theory.deref thy_ref;
1197 val c = Sign.certify_class thy raw_c;
1198 val Cterm {t = prop, maxidx, sorts, ...} = cterm_of thy (Logic.mk_of_class (T, c));
1200 if Sign.of_sort thy (T, [c]) then
1201 Thm (deriv_rule0 (Proofterm.OfClass (T, c)),
1202 {thy_ref = Theory.check_thy thy,
1209 else raise THM ("of_class: type not of class " ^ Syntax.string_of_sort_global thy [c], 0, [])
1212 (*Remove extra sorts that are witnessed by type signature information*)
1213 fun strip_shyps (thm as Thm (_, {shyps = [], ...})) = thm
1214 | strip_shyps (thm as Thm (der, {thy_ref, tags, maxidx, shyps, hyps, tpairs, prop})) =
1216 val thy = Theory.deref thy_ref;
1217 val algebra = Sign.classes_of thy;
1219 val present = (fold_terms o fold_types o fold_atyps_sorts) (insert (eq_fst op =)) thm [];
1220 val extra = fold (Sorts.remove_sort o #2) present shyps;
1221 val witnessed = Sign.witness_sorts thy present extra;
1222 val extra' = fold (Sorts.remove_sort o #2) witnessed extra
1223 |> Sorts.minimal_sorts algebra;
1224 val shyps' = fold (Sorts.insert_sort o #2) present extra';
1226 Thm (deriv_rule_unconditional
1227 (Proofterm.strip_shyps_proof algebra present witnessed extra') der,
1228 {thy_ref = Theory.check_thy thy, tags = tags, maxidx = maxidx,
1229 shyps = shyps', hyps = hyps, tpairs = tpairs, prop = prop})
1232 (*Internalize sort constraints of type variables*)
1233 fun unconstrainT (thm as Thm (der, args)) =
1235 val Deriv {promises, body} = der;
1236 val {thy_ref, shyps, hyps, tpairs, prop, ...} = args;
1238 fun err msg = raise THM ("unconstrainT: " ^ msg, 0, [thm]);
1239 val _ = null hyps orelse err "illegal hyps";
1240 val _ = null tpairs orelse err "unsolved flex-flex constraints";
1241 val tfrees = rev (Term.add_tfree_names prop []);
1242 val _ = null tfrees orelse err ("illegal free type variables " ^ commas_quote tfrees);
1244 val ps = map (apsnd (Future.map fulfill_body)) promises;
1245 val thy = Theory.deref thy_ref;
1246 val (pthm as (_, (_, prop', _)), proof) =
1247 Proofterm.unconstrain_thm_proof thy shyps prop ps body;
1248 val der' = make_deriv [] [] [pthm] proof;
1249 val _ = Theory.check_thy thy;
1254 maxidx = maxidx_of_term prop',
1255 shyps = [[]], (*potentially redundant*)
1261 (* Replace all TFrees not fixed or in the hyps by new TVars *)
1262 fun varifyT_global' fixed (Thm (der, {thy_ref, maxidx, shyps, hyps, tpairs, prop, ...})) =
1264 val tfrees = fold Term.add_tfrees hyps fixed;
1265 val prop1 = attach_tpairs tpairs prop;
1266 val (al, prop2) = Type.varify_global tfrees prop1;
1267 val (ts, prop3) = Logic.strip_prems (length tpairs, [], prop2);
1269 (al, Thm (deriv_rule1 (Proofterm.varify_proof prop tfrees) der,
1272 maxidx = Int.max (0, maxidx),
1275 tpairs = rev (map Logic.dest_equals ts),
1279 val varifyT_global = #2 o varifyT_global' [];
1281 (* Replace all TVars by TFrees that are often new *)
1282 fun legacy_freezeT (Thm (der, {thy_ref, shyps, hyps, tpairs, prop, ...})) =
1284 val prop1 = attach_tpairs tpairs prop;
1285 val prop2 = Type.legacy_freeze prop1;
1286 val (ts, prop3) = Logic.strip_prems (length tpairs, [], prop2);
1288 Thm (deriv_rule1 (Proofterm.legacy_freezeT prop1) der,
1291 maxidx = maxidx_of_term prop2,
1294 tpairs = rev (map Logic.dest_equals ts),
1299 (*** Inference rules for tactics ***)
1301 (*Destruct proof state into constraints, other goals, goal(i), rest *)
1302 fun dest_state (state as Thm (_, {prop,tpairs,...}), i) =
1303 (case Logic.strip_prems(i, [], prop) of
1304 (B::rBs, C) => (tpairs, rev rBs, B, C)
1305 | _ => raise THM("dest_state", i, [state]))
1306 handle TERM _ => raise THM("dest_state", i, [state]);
1308 (*Prepare orule for resolution by lifting it over the parameters and
1309 assumptions of goal.*)
1310 fun lift_rule goal orule =
1312 val Cterm {t = gprop, T, maxidx = gmax, sorts, ...} = goal;
1314 val lift_abs = Logic.lift_abs inc gprop;
1315 val lift_all = Logic.lift_all inc gprop;
1316 val Thm (der, {maxidx, shyps, hyps, tpairs, prop, ...}) = orule;
1317 val (As, B) = Logic.strip_horn prop;
1319 if T <> propT then raise THM ("lift_rule: the term must have type prop", 0, [])
1321 Thm (deriv_rule1 (Proofterm.lift_proof gprop inc prop) der,
1322 {thy_ref = merge_thys1 goal orule,
1324 maxidx = maxidx + inc,
1325 shyps = Sorts.union shyps sorts, (*sic!*)
1327 tpairs = map (pairself lift_abs) tpairs,
1328 prop = Logic.list_implies (map lift_all As, lift_all B)})
1331 fun incr_indexes i (thm as Thm (der, {thy_ref, maxidx, shyps, hyps, tpairs, prop, ...})) =
1332 if i < 0 then raise THM ("negative increment", 0, [thm])
1333 else if i = 0 then thm
1335 Thm (deriv_rule1 (Proofterm.incr_indexes i) der,
1338 maxidx = maxidx + i,
1341 tpairs = map (pairself (Logic.incr_indexes ([], i))) tpairs,
1342 prop = Logic.incr_indexes ([], i) prop});
1344 (*Solve subgoal Bi of proof state B1...Bn/C by assumption. *)
1345 fun assumption i state =
1347 val Thm (der, {thy_ref, maxidx, shyps, hyps, ...}) = state;
1348 val thy = Theory.deref thy_ref;
1349 val (tpairs, Bs, Bi, C) = dest_state (state, i);
1350 fun newth n (env, tpairs) =
1352 ((if Envir.is_empty env then I else (Proofterm.norm_proof' env)) o
1353 Proofterm.assumption_proof Bs Bi n) der,
1355 maxidx = Envir.maxidx_of env,
1356 shyps = Envir.insert_sorts env shyps,
1359 if Envir.is_empty env then tpairs
1360 else map (pairself (Envir.norm_term env)) tpairs,
1362 if Envir.is_empty env then (*avoid wasted normalizations*)
1363 Logic.list_implies (Bs, C)
1364 else (*normalize the new rule fully*)
1365 Envir.norm_term env (Logic.list_implies (Bs, C)),
1366 thy_ref = Theory.check_thy thy});
1368 val (close, asms, concl) = Logic.assum_problems (~1, Bi);
1369 val concl' = close concl;
1370 fun addprfs [] _ = Seq.empty
1371 | addprfs (asm :: rest) n = Seq.make (fn () => Seq.pull
1373 (if Term.could_unify (asm, concl) then
1374 (Unify.unifiers (thy, Envir.empty maxidx, (close asm, concl') :: tpairs))
1376 (addprfs rest (n + 1))))
1377 in addprfs asms 1 end;
1379 (*Solve subgoal Bi of proof state B1...Bn/C by assumption.
1380 Checks if Bi's conclusion is alpha-convertible to one of its assumptions*)
1381 fun eq_assumption i state =
1383 val Thm (der, {thy_ref, maxidx, shyps, hyps, ...}) = state;
1384 val (tpairs, Bs, Bi, C) = dest_state (state, i);
1385 val (_, asms, concl) = Logic.assum_problems (~1, Bi);
1387 (case find_index (fn asm => Pattern.aeconv (asm, concl)) asms of
1388 ~1 => raise THM ("eq_assumption", 0, [state])
1390 Thm (deriv_rule1 (Proofterm.assumption_proof Bs Bi (n + 1)) der,
1397 prop = Logic.list_implies (Bs, C)}))
1401 (*For rotate_tac: fast rotation of assumptions of subgoal i*)
1402 fun rotate_rule k i state =
1404 val Thm (der, {thy_ref, maxidx, shyps, hyps, ...}) = state;
1405 val (tpairs, Bs, Bi, C) = dest_state (state, i);
1406 val params = Term.strip_all_vars Bi;
1407 val rest = Term.strip_all_body Bi;
1408 val asms = Logic.strip_imp_prems rest
1409 val concl = Logic.strip_imp_concl rest;
1410 val n = length asms;
1411 val m = if k < 0 then n + k else k;
1413 if 0 = m orelse m = n then Bi
1414 else if 0 < m andalso m < n then
1415 let val (ps, qs) = chop m asms
1416 in Logic.list_all (params, Logic.list_implies (qs @ ps, concl)) end
1417 else raise THM ("rotate_rule", k, [state]);
1419 Thm (deriv_rule1 (Proofterm.rotate_proof Bs Bi m) der,
1426 prop = Logic.list_implies (Bs @ [Bi'], C)})
1430 (*Rotates a rule's premises to the left by k, leaving the first j premises
1431 unchanged. Does nothing if k=0 or if k equals n-j, where n is the
1432 number of premises. Useful with etac and underlies defer_tac*)
1433 fun permute_prems j k rl =
1435 val Thm (der, {thy_ref, maxidx, shyps, hyps, tpairs, prop, ...}) = rl;
1436 val prems = Logic.strip_imp_prems prop
1437 and concl = Logic.strip_imp_concl prop;
1438 val moved_prems = List.drop (prems, j)
1439 and fixed_prems = List.take (prems, j)
1440 handle General.Subscript => raise THM ("permute_prems: j", j, [rl]);
1441 val n_j = length moved_prems;
1442 val m = if k < 0 then n_j + k else k;
1444 if 0 = m orelse m = n_j then prop
1445 else if 0 < m andalso m < n_j then
1446 let val (ps, qs) = chop m moved_prems
1447 in Logic.list_implies (fixed_prems @ qs @ ps, concl) end
1448 else raise THM ("permute_prems: k", k, [rl]);
1450 Thm (deriv_rule1 (Proofterm.permute_prems_proof prems j m) der,
1461 (** User renaming of parameters in a subgoal **)
1463 (*Calls error rather than raising an exception because it is intended
1464 for top-level use -- exception handling would not make sense here.
1465 The names in cs, if distinct, are used for the innermost parameters;
1466 preceding parameters may be renamed to make all params distinct.*)
1467 fun rename_params_rule (cs, i) state =
1469 val Thm (der, {thy_ref, tags, maxidx, shyps, hyps, ...}) = state;
1470 val (tpairs, Bs, Bi, C) = dest_state (state, i);
1471 val iparams = map #1 (Logic.strip_params Bi);
1472 val short = length iparams - length cs;
1474 if short < 0 then error "More names than abstractions!"
1475 else Name.variant_list cs (take short iparams) @ cs;
1476 val freenames = Term.fold_aterms (fn Free (x, _) => insert (op =) x | _ => I) Bi [];
1477 val newBi = Logic.list_rename_params newnames Bi;
1479 (case duplicates (op =) cs of
1480 a :: _ => (warning ("Can't rename. Bound variables not distinct: " ^ a); state)
1482 (case inter (op =) cs freenames of
1483 a :: _ => (warning ("Can't rename. Bound/Free variable clash: " ^ a); state)
1492 prop = Logic.list_implies (Bs @ [newBi], C)})))
1496 (*** Preservation of bound variable names ***)
1498 fun rename_boundvars pat obj (thm as Thm (der, {thy_ref, tags, maxidx, shyps, hyps, tpairs, prop})) =
1499 (case Term.rename_abs pat obj prop of
1501 | SOME prop' => Thm (der,
1511 (* strip_apply f B A strips off all assumptions/parameters from A
1512 introduced by lifting over B, and applies f to remaining part of A*)
1514 let fun strip (Const ("==>", _) $ _ $ B1)
1515 (Const ("==>", _) $ A2 $ B2) = Logic.mk_implies (A2, strip B1 B2)
1516 | strip ((c as Const ("all", _)) $ Abs (_, _, t1))
1517 ( Const ("all", _) $ Abs (a, T, t2)) = c $ Abs (a, T, strip t1 t2)
1521 fun strip_lifted (Const ("==>", _) $ _ $ B1)
1522 (Const ("==>", _) $ _ $ B2) = strip_lifted B1 B2
1523 | strip_lifted (Const ("all", _) $ Abs (_, _, t1))
1524 (Const ("all", _) $ Abs (_, _, t2)) = strip_lifted t1 t2
1525 | strip_lifted _ A = A;
1527 (*Use the alist to rename all bound variables and some unknowns in a term
1528 dpairs = current disagreement pairs; tpairs = permanent ones (flexflex);
1529 Preserves unknowns in tpairs and on lhs of dpairs. *)
1530 fun rename_bvs [] _ _ _ _ = K I
1531 | rename_bvs al dpairs tpairs B As =
1533 val add_var = fold_aterms (fn Var ((x, _), _) => insert (op =) x | _ => I);
1535 |> fold (add_var o fst) dpairs
1536 |> fold (add_var o fst) tpairs
1537 |> fold (add_var o snd) tpairs;
1538 val vids' = fold (add_var o strip_lifted B) As [];
1539 (*unknowns appearing elsewhere be preserved!*)
1540 val al' = distinct ((op =) o pairself fst)
1541 (filter_out (fn (x, y) =>
1542 not (member (op =) vids' x) orelse
1543 member (op =) vids x orelse member (op =) vids y) al);
1544 val unchanged = filter_out (AList.defined (op =) al') vids';
1545 fun del_clashing clash xs _ [] qs =
1546 if clash then del_clashing false xs xs qs [] else qs
1547 | del_clashing clash xs ys ((p as (x, y)) :: ps) qs =
1548 if member (op =) ys y
1549 then del_clashing true (x :: xs) (x :: ys) ps qs
1550 else del_clashing clash xs (y :: ys) ps (p :: qs);
1551 val al'' = del_clashing false unchanged unchanged al' [];
1552 fun rename (t as Var ((x, i), T)) =
1553 (case AList.lookup (op =) al'' x of
1554 SOME y => Var ((y, i), T)
1556 | rename (Abs (x, T, t)) =
1557 Abs (the_default x (AList.lookup (op =) al x), T, rename t)
1558 | rename (f $ t) = rename f $ rename t
1560 fun strip_ren f Ai = f rename B Ai
1563 (*Function to rename bounds/unknowns in the argument, lifted over B*)
1564 fun rename_bvars dpairs =
1565 rename_bvs (List.foldr Term.match_bvars [] dpairs) dpairs;
1568 (*** RESOLUTION ***)
1570 (** Lifting optimizations **)
1572 (*strip off pairs of assumptions/parameters in parallel -- they are
1573 identical because of lifting*)
1574 fun strip_assums2 (Const("==>", _) $ _ $ B1,
1575 Const("==>", _) $ _ $ B2) = strip_assums2 (B1,B2)
1576 | strip_assums2 (Const("all",_)$Abs(a,T,t1),
1577 Const("all",_)$Abs(_,_,t2)) =
1578 let val (B1,B2) = strip_assums2 (t1,t2)
1579 in (Abs(a,T,B1), Abs(a,T,B2)) end
1580 | strip_assums2 BB = BB;
1583 (*Faster normalization: skip assumptions that were lifted over*)
1584 fun norm_term_skip env 0 t = Envir.norm_term env t
1585 | norm_term_skip env n (Const ("all", _) $ Abs (a, T, t)) =
1587 val T' = Envir.subst_type (Envir.type_env env) T
1588 (*Must instantiate types of parameters because they are flattened;
1589 this could be a NEW parameter*)
1590 in Logic.all_const T' $ Abs (a, T', norm_term_skip env n t) end
1591 | norm_term_skip env n (Const ("==>", _) $ A $ B) =
1592 Logic.mk_implies (A, norm_term_skip env (n - 1) B)
1593 | norm_term_skip _ _ _ = error "norm_term_skip: too few assumptions??";
1596 (*Composition of object rule r=(A1...Am/B) with proof state s=(B1...Bn/C)
1597 Unifies B with Bi, replacing subgoal i (1 <= i <= n)
1598 If match then forbid instantiations in proof state
1599 If lifted then shorten the dpair using strip_assums2.
1600 If eres_flg then simultaneously proves A1 by assumption.
1601 nsubgoal is the number of new subgoals (written m above).
1602 Curried so that resolution calls dest_state only once.
1604 local exception COMPOSE
1606 fun bicompose_aux flatten match (state, (stpairs, Bs, Bi, C), lifted)
1607 (eres_flg, orule, nsubgoal) =
1608 let val Thm (sder, {maxidx=smax, shyps=sshyps, hyps=shyps, ...}) = state
1609 and Thm (rder, {maxidx=rmax, shyps=rshyps, hyps=rhyps,
1610 tpairs=rtpairs, prop=rprop,...}) = orule
1611 (*How many hyps to skip over during normalization*)
1612 and nlift = Logic.count_prems (strip_all_body Bi) + (if eres_flg then ~1 else 0)
1613 val thy = Theory.deref (merge_thys2 state orule);
1614 (** Add new theorem with prop = '[| Bs; As |] ==> C' to thq **)
1615 fun addth A (As, oldAs, rder', n) ((env, tpairs), thq) =
1616 let val normt = Envir.norm_term env;
1617 (*perform minimal copying here by examining env*)
1618 val (ntpairs, normp) =
1619 if Envir.is_empty env then (tpairs, (Bs @ As, C))
1621 let val ntps = map (pairself normt) tpairs
1622 in if Envir.above env smax then
1623 (*no assignments in state; normalize the rule only*)
1625 then (ntps, (Bs @ map (norm_term_skip env nlift) As, C))
1626 else (ntps, (Bs @ map normt As, C))
1627 else if match then raise COMPOSE
1628 else (*normalize the new rule fully*)
1629 (ntps, (map normt (Bs @ As), normt C))
1633 ((if Envir.is_empty env then I
1634 else if Envir.above env smax then
1635 (fn f => fn der => f (Proofterm.norm_proof' env der))
1637 curry op oo (Proofterm.norm_proof' env))
1638 (Proofterm.bicompose_proof flatten Bs oldAs As A n (nlift+1))) rder' sder,
1640 maxidx = Envir.maxidx_of env,
1641 shyps = Envir.insert_sorts env (Sorts.union rshyps sshyps),
1642 hyps = union_hyps rhyps shyps,
1644 prop = Logic.list_implies normp,
1645 thy_ref = Theory.check_thy thy})
1646 in Seq.cons th thq end handle COMPOSE => thq;
1647 val (rAs,B) = Logic.strip_prems(nsubgoal, [], rprop)
1648 handle TERM _ => raise THM("bicompose: rule", 0, [orule,state]);
1649 (*Modify assumptions, deleting n-th if n>0 for e-resolution*)
1650 fun newAs(As0, n, dpairs, tpairs) =
1651 let val (As1, rder') =
1652 if not lifted then (As0, rder)
1654 let val rename = rename_bvars dpairs tpairs B As0
1655 in (map (rename strip_apply) As0,
1656 deriv_rule1 (Proofterm.map_proof_terms (rename K) I) rder)
1658 in (map (if flatten then (Logic.flatten_params n) else I) As1, As1, rder', n)
1660 raise THM("bicompose: 1st premise", 0, [orule])
1662 val env = Envir.empty(Int.max(rmax,smax));
1663 val BBi = if lifted then strip_assums2(B,Bi) else (B,Bi);
1664 val dpairs = BBi :: (rtpairs@stpairs);
1666 (*elim-resolution: try each assumption in turn*)
1667 fun eres [] = raise THM ("bicompose: no premises", 0, [orule, state])
1671 val (close, asms, concl) = Logic.assum_problems (nlift + 1, A1);
1672 val concl' = close concl;
1673 fun tryasms [] _ = Seq.empty
1674 | tryasms (asm :: rest) n =
1675 if Term.could_unify (asm, concl) then
1676 let val asm' = close asm in
1677 (case Seq.pull (Unify.unifiers (thy, env, (asm', concl') :: dpairs)) of
1678 NONE => tryasms rest (n + 1)
1679 | cell as SOME ((_, tpairs), _) =>
1680 Seq.it_right (addth A (newAs (As, n, [BBi, (concl', asm')], tpairs)))
1681 (Seq.make (fn () => cell),
1682 Seq.make (fn () => Seq.pull (tryasms rest (n + 1)))))
1684 else tryasms rest (n + 1);
1685 in tryasms asms 1 end;
1687 (*ordinary resolution*)
1689 (case Seq.pull (Unify.unifiers (thy, env, dpairs)) of
1691 | cell as SOME ((_, tpairs), _) =>
1692 Seq.it_right (addth NONE (newAs (rev rAs, 0, [BBi], tpairs)))
1693 (Seq.make (fn () => cell), Seq.empty));
1695 if eres_flg then eres (rev rAs) else res ()
1699 fun compose_no_flatten match (orule, nsubgoal) i state =
1700 bicompose_aux false match (state, dest_state (state, i), false) (false, orule, nsubgoal);
1702 fun bicompose match arg i state =
1703 bicompose_aux true match (state, dest_state (state,i), false) arg;
1705 (*Quick test whether rule is resolvable with the subgoal with hyps Hs
1706 and conclusion B. If eres_flg then checks 1st premise of rule also*)
1707 fun could_bires (Hs, B, eres_flg, rule) =
1708 let fun could_reshyp (A1::_) = exists (fn H => Term.could_unify (A1, H)) Hs
1709 | could_reshyp [] = false; (*no premise -- illegal*)
1710 in Term.could_unify(concl_of rule, B) andalso
1711 (not eres_flg orelse could_reshyp (prems_of rule))
1714 (*Bi-resolution of a state with a list of (flag,rule) pairs.
1715 Puts the rule above: rule/state. Renames vars in the rules. *)
1716 fun biresolution match brules i state =
1717 let val (stpairs, Bs, Bi, C) = dest_state(state,i);
1718 val lift = lift_rule (cprem_of state i);
1719 val B = Logic.strip_assums_concl Bi;
1720 val Hs = Logic.strip_assums_hyp Bi;
1721 val compose = bicompose_aux true match (state, (stpairs, Bs, Bi, C), true);
1722 fun res [] = Seq.empty
1723 | res ((eres_flg, rule)::brules) =
1724 if !Pattern.trace_unify_fail orelse
1725 could_bires (Hs, B, eres_flg, rule)
1726 then Seq.make (*delay processing remainder till needed*)
1727 (fn()=> SOME(compose (eres_flg, lift rule, nprems_of rule),
1730 in Seq.flat (res brules) end;
1738 fun invoke_oracle thy_ref1 name oracle arg =
1739 let val Cterm {thy_ref = thy_ref2, t = prop, T, maxidx, sorts} = oracle arg in
1741 raise THM ("Oracle's result must have type prop: " ^ name, 0, [])
1743 let val (ora, prf) = Proofterm.oracle_proof name prop in
1744 Thm (make_deriv [] [ora] [] prf,
1745 {thy_ref = Theory.merge_refs (thy_ref1, thy_ref2),
1760 (* authentic derivation names *)
1762 structure Oracles = Theory_Data
1764 type T = unit Name_Space.table;
1765 val empty : T = Name_Space.empty_table "oracle";
1767 fun merge data : T = Name_Space.merge_tables data;
1770 fun extern_oracles ctxt =
1771 map #1 (Name_Space.extern_table ctxt (Oracles.get (Proof_Context.theory_of ctxt)));
1773 fun add_oracle (b, oracle) thy =
1775 val (name, tab') = Name_Space.define (Context.Theory thy) true (b, ()) (Oracles.get thy);
1776 val thy' = Oracles.put tab' thy;
1777 in ((name, invoke_oracle (Theory.check_thy thy') name oracle), thy') end;
1781 structure Basic_Thm: BASIC_THM = Thm;