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src/ZF/WF.ML

author | clasohm |

Thu, 16 Sep 1993 12:20:38 +0200 | |

changeset 0 | a5a9c433f639 |

child 6 | 8ce8c4d13d4d |

permissions | -rw-r--r-- |

Initial revision

(* Title: ZF/wf.ML ID: $Id$ Author: Tobias Nipkow and Lawrence C Paulson Copyright 1992 University of Cambridge For wf.thy. Well-founded Recursion Derived first for transitive relations, and finally for arbitrary WF relations via wf_trancl and trans_trancl. It is difficult to derive this general case directly, using r^+ instead of r. In is_recfun, the two occurrences of the relation must have the same form. Inserting r^+ in the_recfun or wftrec yields a recursion rule with r^+ -`` {a} instead of r-``{a}. This recursion rule is stronger in principle, but harder to use, especially to prove wfrec_eclose_eq in epsilon.ML. Expanding out the definition of wftrec in wfrec would yield a mess. *) open WF; val [H_cong] = mk_typed_congs WF.thy[("H","[i,i]=>i")]; val wf_ss = ZF_ss addcongs [H_cong]; (*** Well-founded relations ***) (*Are these two theorems at all useful??*) (*If every subset of field(r) possesses an r-minimal element then wf(r). Seems impossible to prove this for domain(r) or range(r) instead... Consider in particular finite wf relations!*) val [prem1,prem2] = goalw WF.thy [wf_def] "[| field(r)<=A; \ \ !!Z u. [| Z<=A; u:Z; ALL x:Z. EX y:Z. <y,x>:r |] ==> False |] \ \ ==> wf(r)"; by (rtac (equals0I RS disjCI RS allI) 1); by (rtac prem2 1); by (res_inst_tac [ ("B1", "Z") ] (prem1 RS (Int_lower1 RS subset_trans)) 1); by (fast_tac ZF_cs 1); by (fast_tac ZF_cs 1); val wfI = result(); (*If r allows well-founded induction then wf(r)*) val [prem1,prem2] = goal WF.thy "[| field(r)<=A; \ \ !!B. ALL x:A. (ALL y. <y,x>: r --> y:B) --> x:B ==> A<=B |] \ \ ==> wf(r)"; by (rtac (prem1 RS wfI) 1); by (res_inst_tac [ ("B", "A-Z") ] (prem2 RS subsetCE) 1); by (fast_tac ZF_cs 3); by (fast_tac ZF_cs 2); by (fast_tac ZF_cs 1); val wfI2 = result(); (** Well-founded Induction **) (*Consider the least z in domain(r) Un {a} such that P(z) does not hold...*) val major::prems = goalw WF.thy [wf_def] "[| wf(r); \ \ !!x.[| ALL y. <y,x>: r --> P(y) |] ==> P(x) \ \ |] ==> P(a)"; by (res_inst_tac [ ("x", "{z:domain(r) Un {a}. ~P(z)}") ] (major RS allE) 1); by (etac disjE 1); by (rtac classical 1); by (etac equals0D 1); by (etac (singletonI RS UnI2 RS CollectI) 1); by (etac bexE 1); by (etac CollectE 1); by (etac swap 1); by (resolve_tac prems 1); by (fast_tac ZF_cs 1); val wf_induct = result(); (*Perform induction on i, then prove the wf(r) subgoal using prems. *) fun wf_ind_tac a prems i = EVERY [res_inst_tac [("a",a)] wf_induct i, rename_last_tac a ["1"] (i+1), ares_tac prems i]; (*The form of this rule is designed to match wfI2*) val wfr::amem::prems = goal WF.thy "[| wf(r); a:A; field(r)<=A; \ \ !!x.[| x: A; ALL y. <y,x>: r --> P(y) |] ==> P(x) \ \ |] ==> P(a)"; by (rtac (amem RS rev_mp) 1); by (wf_ind_tac "a" [wfr] 1); by (rtac impI 1); by (eresolve_tac prems 1); by (fast_tac (ZF_cs addIs (prems RL [subsetD])) 1); val wf_induct2 = result(); val prems = goal WF.thy "[| wf(r); <a,x>:r; <x,a>:r |] ==> False"; by (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> False" 1); by (wf_ind_tac "a" prems 2); by (fast_tac ZF_cs 2); by (fast_tac (FOL_cs addIs prems) 1); val wf_anti_sym = result(); (*transitive closure of a WF relation is WF!*) val [prem] = goal WF.thy "wf(r) ==> wf(r^+)"; by (rtac (trancl_type RS field_rel_subset RS wfI2) 1); by (rtac subsetI 1); (*must retain the universal formula for later use!*) by (rtac (bspec RS mp) 1 THEN assume_tac 1 THEN assume_tac 1); by (eres_inst_tac [("a","x")] (prem RS wf_induct2) 1); by (rtac subset_refl 1); by (rtac (impI RS allI) 1); by (etac tranclE 1); by (etac (bspec RS mp) 1); by (etac fieldI1 1); by (fast_tac ZF_cs 1); by (fast_tac ZF_cs 1); val wf_trancl = result(); (** r-``{a} is the set of everything under a in r **) val underI = standard (vimage_singleton_iff RS iffD2); val underD = standard (vimage_singleton_iff RS iffD1); (** is_recfun **) val [major] = goalw WF.thy [is_recfun_def] "is_recfun(r,a,H,f) ==> f: r-``{a} -> range(f)"; by (rtac (major RS ssubst) 1); by (rtac (lamI RS rangeI RS lam_type) 1); by (assume_tac 1); val is_recfun_type = result(); val [isrec,rel] = goalw WF.thy [is_recfun_def] "[| is_recfun(r,a,H,f); <x,a>:r |] ==> f`x = H(x, restrict(f,r-``{x}))"; by (res_inst_tac [("P", "%x.?t(x) = ?u::i")] (isrec RS ssubst) 1); by (rtac (rel RS underI RS beta) 1); val apply_recfun = result(); (*eresolve_tac transD solves <a,b>:r using transitivity AT MOST ONCE spec RS mp instantiates induction hypotheses*) fun indhyp_tac hyps = ares_tac (TrueI::hyps) ORELSE' (cut_facts_tac hyps THEN' DEPTH_SOLVE_1 o (ares_tac [TrueI, ballI] ORELSE' eresolve_tac [underD, transD, spec RS mp])); (*** NOTE! some simplifications need a different auto_tac!! ***) val wf_super_ss = wf_ss setauto indhyp_tac; val prems = goalw WF.thy [is_recfun_def] "[| wf(r); trans(r); is_recfun(r,a,H,f); is_recfun(r,b,H,g) |] ==> \ \ <x,a>:r --> <x,b>:r --> f`x=g`x"; by (cut_facts_tac prems 1); by (wf_ind_tac "x" prems 1); by (REPEAT (rtac impI 1 ORELSE etac ssubst 1)); by (rewtac restrict_def); by (ASM_SIMP_TAC (wf_super_ss addrews [vimage_singleton_iff]) 1); val is_recfun_equal_lemma = result(); val is_recfun_equal = standard (is_recfun_equal_lemma RS mp RS mp); val prems as [wfr,transr,recf,recg,_] = goal WF.thy "[| wf(r); trans(r); \ \ is_recfun(r,a,H,f); is_recfun(r,b,H,g); <b,a>:r |] ==> \ \ restrict(f, r-``{b}) = g"; by (cut_facts_tac prems 1); by (rtac (consI1 RS restrict_type RS fun_extension) 1); by (etac is_recfun_type 1); by (ALLGOALS (ASM_SIMP_TAC (wf_super_ss addrews [ [wfr,transr,recf,recg] MRS is_recfun_equal ]))); val is_recfun_cut = result(); (*** Main Existence Lemma ***) val prems = goal WF.thy "[| wf(r); trans(r); is_recfun(r,a,H,f); is_recfun(r,a,H,g) |] ==> f=g"; by (cut_facts_tac prems 1); by (rtac fun_extension 1); by (REPEAT (ares_tac [is_recfun_equal] 1 ORELSE eresolve_tac [is_recfun_type,underD] 1)); val is_recfun_functional = result(); (*If some f satisfies is_recfun(r,a,H,-) then so does the_recfun(r,a,H) *) val prems = goalw WF.thy [the_recfun_def] "[| is_recfun(r,a,H,f); wf(r); trans(r) |] \ \ ==> is_recfun(r, a, H, the_recfun(r,a,H))"; by (rtac (ex1I RS theI) 1); by (REPEAT (ares_tac (prems@[is_recfun_functional]) 1)); val is_the_recfun = result(); val prems = goal WF.thy "[| wf(r); trans(r) |] ==> is_recfun(r, a, H, the_recfun(r,a,H))"; by (cut_facts_tac prems 1); by (wf_ind_tac "a" prems 1); by (res_inst_tac [("f", "lam y: r-``{a1}. wftrec(r,y,H)")] is_the_recfun 1); by (REPEAT (assume_tac 2)); by (rewrite_goals_tac [is_recfun_def, wftrec_def]); (*Applying the substitution: must keep the quantified assumption!!*) by (REPEAT (dtac underD 1 ORELSE resolve_tac [refl, lam_cong, H_cong] 1)); by (fold_tac [is_recfun_def]); by (rtac (consI1 RS restrict_type RSN (2,fun_extension)) 1); by (rtac is_recfun_type 1); by (ALLGOALS (ASM_SIMP_TAC (wf_super_ss addrews [underI RS beta, apply_recfun, is_recfun_cut]))); val unfold_the_recfun = result(); (*** Unfolding wftrec ***) val prems = goal WF.thy "[| wf(r); trans(r); <b,a>:r |] ==> \ \ restrict(the_recfun(r,a,H), r-``{b}) = the_recfun(r,b,H)"; by (REPEAT (ares_tac (prems @ [is_recfun_cut, unfold_the_recfun]) 1)); val the_recfun_cut = result(); (*NOT SUITABLE FOR REWRITING since it is recursive!*) val prems = goalw WF.thy [wftrec_def] "[| wf(r); trans(r) |] ==> \ \ wftrec(r,a,H) = H(a, lam x: r-``{a}. wftrec(r,x,H))"; by (rtac (rewrite_rule [is_recfun_def] unfold_the_recfun RS ssubst) 1); by (ALLGOALS (ASM_SIMP_TAC (wf_ss addrews (prems@[vimage_singleton_iff RS iff_sym, the_recfun_cut])))); val wftrec = result(); (** Removal of the premise trans(r) **) (*NOT SUITABLE FOR REWRITING since it is recursive!*) val [wfr] = goalw WF.thy [wfrec_def] "wf(r) ==> wfrec(r,a,H) = H(a, lam x:r-``{a}. wfrec(r,x,H))"; by (rtac (wfr RS wf_trancl RS wftrec RS ssubst) 1); by (rtac trans_trancl 1); by (rtac (refl RS H_cong) 1); by (rtac (vimage_pair_mono RS restrict_lam_eq) 1); by (etac r_into_trancl 1); by (rtac subset_refl 1); val wfrec = result(); (*This form avoids giant explosions in proofs. NOTE USE OF == *) val rew::prems = goal WF.thy "[| !!x. h(x)==wfrec(r,x,H); wf(r) |] ==> \ \ h(a) = H(a, lam x: r-``{a}. h(x))"; by (rewtac rew); by (REPEAT (resolve_tac (prems@[wfrec]) 1)); val def_wfrec = result(); val prems = goal WF.thy "[| wf(r); a:A; field(r)<=A; \ \ !!x u. [| x: A; u: Pi(r-``{x}, B) |] ==> H(x,u) : B(x) \ \ |] ==> wfrec(r,a,H) : B(a)"; by (res_inst_tac [("a","a")] wf_induct2 1); by (rtac (wfrec RS ssubst) 4); by (REPEAT (ares_tac (prems@[lam_type]) 1 ORELSE eresolve_tac [spec RS mp, underD] 1)); val wfrec_type = result(); val prems = goalw WF.thy [wfrec_def,wftrec_def,the_recfun_def,is_recfun_def] "[| r=r'; !!x u. H(x,u)=H'(x,u); a=a' |] \ \ ==> wfrec(r,a,H)=wfrec(r',a',H')"; by (EVERY1 (map rtac (prems RL [subst]))); by (SIMP_TAC (wf_ss addrews (prems RL [sym])) 1); val wfrec_cong = result();