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src/ZF/WF.ML

author | lcp |

Tue, 26 Jul 1994 13:44:42 +0200 | |

changeset 485 | 5e00a676a211 |

parent 443 | 10884e64c241 |

child 494 | 3686157a5a51 |

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

Axiom of choice, cardinality results, etc.

(* 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; (*** Well-founded relations ***) (** Equivalences between wf and wf_on **) goalw WF.thy [wf_def, wf_on_def] "!!A r. wf(r) ==> wf[A](r)"; by (fast_tac ZF_cs 1); val wf_imp_wf_on = result(); goalw WF.thy [wf_def, wf_on_def] "!!r. wf[field(r)](r) ==> wf(r)"; by (fast_tac ZF_cs 1); val wf_on_field_imp_wf = result(); goal WF.thy "wf(r) <-> wf[field(r)](r)"; by (fast_tac (ZF_cs addSEs [wf_imp_wf_on, wf_on_field_imp_wf]) 1); val wf_iff_wf_on_field = result(); goalw WF.thy [wf_on_def, wf_def] "!!A B r. [| wf[A](r); B<=A |] ==> wf[B](r)"; by (fast_tac ZF_cs 1); val wf_on_subset_A = result(); goalw WF.thy [wf_on_def, wf_def] "!!A r s. [| wf[A](r); s<=r |] ==> wf[A](s)"; by (fast_tac ZF_cs 1); val wf_on_subset_r = result(); (** Introduction rules for wf_on **) (*If every non-empty subset of A has an r-minimal element then wf[A](r).*) val [prem] = goalw WF.thy [wf_on_def, wf_def] "[| !!Z u. [| Z<=A; u:Z; ALL x:Z. EX y:Z. <y,x>:r |] ==> False |] \ \ ==> wf[A](r)"; by (rtac (equals0I RS disjCI RS allI) 1); by (res_inst_tac [ ("Z", "Z") ] prem 1); by (ALLGOALS (fast_tac ZF_cs)); val wf_onI = result(); (*If r allows well-founded induction over A then wf[A](r) Premise is equivalent to !!B. ALL x:A. (ALL y. <y,x>: r --> y:B) --> x:B ==> A<=B *) val [prem] = goal WF.thy "[| !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A \ \ |] ==> y:B |] \ \ ==> wf[A](r)"; by (rtac wf_onI 1); by (res_inst_tac [ ("c", "u") ] (prem RS DiffE) 1); by (contr_tac 3); by (fast_tac ZF_cs 2); by (fast_tac ZF_cs 1); val wf_onI2 = 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 wfI*) 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(); goal ZF.thy "!!r A. field(r Int A*A) <= A"; by (fast_tac ZF_cs 1); val field_Int_square = result(); val wfr::amem::prems = goalw WF.thy [wf_on_def] "[| wf[A](r); a:A; \ \ !!x.[| x: A; ALL y:A. <y,x>: r --> P(y) |] ==> P(x) \ \ |] ==> P(a)"; by (rtac ([wfr, amem, field_Int_square] MRS wf_induct2) 1); by (REPEAT (ares_tac prems 1)); by (fast_tac ZF_cs 1); val wf_on_induct = result(); fun wf_on_ind_tac a prems i = EVERY [res_inst_tac [("a",a)] wf_on_induct i, rename_last_tac a ["1"] (i+2), REPEAT (ares_tac prems i)]; (*If r allows well-founded induction then wf(r)*) val [subs,indhyp] = goal WF.thy "[| field(r)<=A; \ \ !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A \ \ |] ==> y:B |] \ \ ==> wf(r)"; by (rtac ([wf_onI2, subs] MRS (wf_on_subset_A RS wf_on_field_imp_wf)) 1); by (REPEAT (ares_tac [indhyp] 1)); val wfI = result(); (*** Properties of well-founded relations ***) goal WF.thy "!!r. wf(r) ==> <a,a> ~: r"; by (wf_ind_tac "a" [] 1); by (fast_tac ZF_cs 1); val wf_not_refl = result(); goal WF.thy "!!r. [| wf(r); <a,x>:r; <x,a>:r |] ==> P"; by (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> P" 1); by (wf_ind_tac "a" [] 2); by (fast_tac ZF_cs 2); by (fast_tac FOL_cs 1); val wf_asym = result(); goal WF.thy "!!r. [| wf[A](r); a: A |] ==> <a,a> ~: r"; by (wf_on_ind_tac "a" [] 1); by (fast_tac ZF_cs 1); val wf_on_not_refl = result(); goal WF.thy "!!r. [| wf[A](r); <a,b>:r; <b,a>:r; a:A; b:A |] ==> P"; by (subgoal_tac "ALL y:A. <a,y>:r --> <y,a>:r --> P" 1); by (wf_on_ind_tac "a" [] 2); by (fast_tac ZF_cs 2); by (fast_tac ZF_cs 1); val wf_on_asym = result(); (*Needed to prove well_ordI. Could also reason that wf[A](r) means wf(r Int A*A); thus wf( (r Int A*A)^+ ) and use wf_not_refl *) goal WF.thy "!!r. [| wf[A](r); <a,b>:r; <b,c>:r; <c,a>:r; a:A; b:A; c:A |] ==> P"; by (subgoal_tac "ALL y:A. ALL z:A. <a,y>:r --> <y,z>:r --> <z,a>:r --> P" 1); by (wf_on_ind_tac "a" [] 2); by (fast_tac ZF_cs 2); by (fast_tac ZF_cs 1); val wf_on_chain3 = result(); (*retains the universal formula for later use!*) val bchain_tac = EVERY' [rtac (bspec RS mp), assume_tac, assume_tac ]; (*transitive closure of a WF relation is WF provided A is downwards closed*) val [wfr,subs] = goal WF.thy "[| wf[A](r); r-``A <= A |] ==> wf[A](r^+)"; by (rtac wf_onI2 1); by (bchain_tac 1); by (eres_inst_tac [("a","y")] (wfr RS wf_on_induct) 1); by (rtac (impI RS ballI) 1); by (etac tranclE 1); by (etac (bspec RS mp) 1 THEN assume_tac 1); by (fast_tac ZF_cs 1); by (cut_facts_tac [subs] 1); (*astar_tac is slightly faster*) by (best_tac ZF_cs 1); val wf_on_trancl = result(); goal WF.thy "!!r. wf(r) ==> wf(r^+)"; by (asm_full_simp_tac (ZF_ss addsimps [wf_iff_wf_on_field]) 1); by (rtac (trancl_type RS field_rel_subset RSN (2, wf_on_subset_A)) 1); by (etac wf_on_trancl 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 = resolve_tac (TrueI::refl::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 solver!! ***) val wf_super_ss = ZF_ss setsolver 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 addsimps [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 addsimps [ [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] 1)); by (fold_tac [is_recfun_def]); by (rtac (consI1 RS restrict_type RSN (2,fun_extension) RS subst_context) 1); by (rtac is_recfun_type 1); by (ALLGOALS (asm_simp_tac (wf_super_ss addsimps [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!*) goalw WF.thy [wftrec_def] "!!r. [| 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 (ZF_ss addsimps [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 (vimage_pair_mono RS restrict_lam_eq RS subst_context) 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(); goalw WF.thy [wf_on_def, wfrec_on_def] "!!A r. [| wf[A](r); a: A |] ==> \ \ wfrec[A](r,a,H) = H(a, lam x: (r-``{a}) Int A. wfrec[A](r,x,H))"; by (etac (wfrec RS trans) 1); by (asm_simp_tac (ZF_ss addsimps [vimage_Int_square, cons_subset_iff]) 1); val wfrec_on = result();