src/ZF/wf.ML
author paulson
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(*  Title: 	ZF/wf.ML
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    ID:         $Id$
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    Author: 	Tobias Nipkow and Lawrence C Paulson
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    Copyright   1992  University of Cambridge
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For wf.thy.  Well-founded Recursion
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Derived first for transitive relations, and finally for arbitrary WF relations
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via wf_trancl and trans_trancl.
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It is difficult to derive this general case directly, using r^+ instead of
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r.  In is_recfun, the two occurrences of the relation must have the same
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form.  Inserting r^+ in the_recfun or wftrec yields a recursion rule with
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r^+ -`` {a} instead of r-``{a}.  This recursion rule is stronger in
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principle, but harder to use, especially to prove wfrec_eclose_eq in
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epsilon.ML.  Expanding out the definition of wftrec in wfrec would yield
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a mess.
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*)
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open WF;
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(*** Well-founded relations ***)
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(*Are these two theorems at all useful??*)
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(*If every subset of field(r) possesses an r-minimal element then wf(r).
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  Seems impossible to prove this for domain(r) or range(r) instead...
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  Consider in particular finite wf relations!*)
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val [prem1,prem2] = goalw WF.thy [wf_def]
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    "[| field(r)<=A;  \
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\       !!Z u. [| Z<=A;  u:Z;  ALL x:Z. EX y:Z. <y,x>:r |] ==> False |] \
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\    ==>  wf(r)";
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by (rtac (equals0I RS disjCI RS allI) 1);
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by (rtac prem2 1);
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by (res_inst_tac [ ("B1", "Z") ] (prem1 RS (Int_lower1 RS subset_trans)) 1);
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by (fast_tac ZF_cs 1);
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by (fast_tac ZF_cs 1);
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val wfI = result();
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(*If r allows well-founded induction then wf(r)*)
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val [prem1,prem2] = goal WF.thy
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    "[| field(r)<=A;  \
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\       !!B. ALL x:A. (ALL y. <y,x>: r --> y:B) --> x:B ==> A<=B |]  \
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\    ==>  wf(r)";
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by (rtac (prem1 RS wfI) 1);
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by (res_inst_tac [ ("B", "A-Z") ] (prem2 RS subsetCE) 1);
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by (fast_tac ZF_cs 3);
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by (fast_tac ZF_cs 2);
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by (fast_tac ZF_cs 1);
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val wfI2 = result();
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(** Well-founded Induction **)
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(*Consider the least z in domain(r) Un {a} such that P(z) does not hold...*)
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val major::prems = goalw WF.thy [wf_def]
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    "[| wf(r);          \
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\       !!x.[| ALL y. <y,x>: r --> P(y) |] ==> P(x) \
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\    |]  ==>  P(a)";
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by (res_inst_tac [ ("x", "{z:domain(r) Un {a}. ~P(z)}") ]  (major RS allE) 1);
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by (etac disjE 1);
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by (rtac classical 1);
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by (etac equals0D 1);
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by (etac (singletonI RS UnI2 RS CollectI) 1);
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by (etac bexE 1);
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by (etac CollectE 1);
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by (etac swap 1);
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by (resolve_tac prems 1);
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by (fast_tac ZF_cs 1);
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val wf_induct = result();
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(*Perform induction on i, then prove the wf(r) subgoal using prems. *)
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fun wf_ind_tac a prems i = 
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    EVERY [res_inst_tac [("a",a)] wf_induct i,
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	   rename_last_tac a ["1"] (i+1),
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	   ares_tac prems i];
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(*The form of this rule is designed to match wfI2*)
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val wfr::amem::prems = goal WF.thy
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    "[| wf(r);  a:A;  field(r)<=A;  \
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\       !!x.[| x: A;  ALL y. <y,x>: r --> P(y) |] ==> P(x) \
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\    |]  ==>  P(a)";
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by (rtac (amem RS rev_mp) 1);
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by (wf_ind_tac "a" [wfr] 1);
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by (rtac impI 1);
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by (eresolve_tac prems 1);
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by (fast_tac (ZF_cs addIs (prems RL [subsetD])) 1);
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val wf_induct2 = result();
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val prems = goal WF.thy "[| wf(r);  <a,x>:r;  <x,a>:r |] ==> False";
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by (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> False" 1);
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by (wf_ind_tac "a" prems 2);
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by (fast_tac ZF_cs 2);
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by (fast_tac (FOL_cs addIs prems) 1);
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val wf_anti_sym = result();
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(*transitive closure of a WF relation is WF!*)
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val [prem] = goal WF.thy "wf(r) ==> wf(r^+)";
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by (rtac (trancl_type RS field_rel_subset RS wfI2) 1);
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by (rtac subsetI 1);
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(*must retain the universal formula for later use!*)
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by (rtac (bspec RS mp) 1 THEN assume_tac 1 THEN assume_tac 1);
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by (eres_inst_tac [("a","x")] (prem RS wf_induct2) 1);
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by (rtac subset_refl 1);
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by (rtac (impI RS allI) 1);
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by (etac tranclE 1);
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by (etac (bspec RS mp) 1);
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by (etac fieldI1 1);
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by (fast_tac ZF_cs 1);
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by (fast_tac ZF_cs 1);
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val wf_trancl = result();
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(** r-``{a} is the set of everything under a in r **)
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val underI = standard (vimage_singleton_iff RS iffD2);
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val underD = standard (vimage_singleton_iff RS iffD1);
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(** is_recfun **)
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val [major] = goalw WF.thy [is_recfun_def]
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    "is_recfun(r,a,H,f) ==> f: r-``{a} -> range(f)";
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by (rtac (major RS ssubst) 1);
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by (rtac (lamI RS rangeI RS lam_type) 1);
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by (assume_tac 1);
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val is_recfun_type = result();
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val [isrec,rel] = goalw WF.thy [is_recfun_def]
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    "[| is_recfun(r,a,H,f); <x,a>:r |] ==> f`x = H(x, restrict(f,r-``{x}))";
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by (res_inst_tac [("P", "%x.?t(x) = ?u::i")] (isrec RS ssubst) 1);
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by (rtac (rel RS underI RS beta) 1);
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val apply_recfun = result();
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(*eresolve_tac transD solves <a,b>:r using transitivity AT MOST ONCE
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  spec RS mp  instantiates induction hypotheses*)
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fun indhyp_tac hyps =
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    resolve_tac (TrueI::refl::hyps) ORELSE' 
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    (cut_facts_tac hyps THEN'
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       DEPTH_SOLVE_1 o (ares_tac [TrueI, ballI] ORELSE'
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		        eresolve_tac [underD, transD, spec RS mp]));
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(*** NOTE! some simplifications need a different solver!! ***)
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val wf_super_ss = ZF_ss setsolver indhyp_tac;
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val prems = goalw WF.thy [is_recfun_def]
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    "[| wf(r);  trans(r);  is_recfun(r,a,H,f);  is_recfun(r,b,H,g) |] ==> \
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\    <x,a>:r --> <x,b>:r --> f`x=g`x";
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by (cut_facts_tac prems 1);
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by (wf_ind_tac "x" prems 1);
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by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
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by (rewtac restrict_def);
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by (asm_simp_tac (wf_super_ss addsimps [vimage_singleton_iff]) 1);
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val is_recfun_equal_lemma = result();
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val is_recfun_equal = standard (is_recfun_equal_lemma RS mp RS mp);
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val prems as [wfr,transr,recf,recg,_] = goal WF.thy
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    "[| wf(r);  trans(r);       \
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\       is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  <b,a>:r |] ==> \
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\    restrict(f, r-``{b}) = g";
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by (cut_facts_tac prems 1);
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by (rtac (consI1 RS restrict_type RS fun_extension) 1);
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by (etac is_recfun_type 1);
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by (ALLGOALS
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    (asm_simp_tac (wf_super_ss addsimps
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		   [ [wfr,transr,recf,recg] MRS is_recfun_equal ])));
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val is_recfun_cut = result();
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(*** Main Existence Lemma ***)
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val prems = goal WF.thy
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    "[| wf(r); trans(r); is_recfun(r,a,H,f); is_recfun(r,a,H,g) |]  ==>  f=g";
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by (cut_facts_tac prems 1);
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by (rtac fun_extension 1);
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by (REPEAT (ares_tac [is_recfun_equal] 1
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     ORELSE eresolve_tac [is_recfun_type,underD] 1));
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val is_recfun_functional = result();
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(*If some f satisfies is_recfun(r,a,H,-) then so does the_recfun(r,a,H) *)
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val prems = goalw WF.thy [the_recfun_def]
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    "[| is_recfun(r,a,H,f);  wf(r);  trans(r) |]  \
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\    ==> is_recfun(r, a, H, the_recfun(r,a,H))";
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by (rtac (ex1I RS theI) 1);
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by (REPEAT (ares_tac (prems@[is_recfun_functional]) 1));
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val is_the_recfun = result();
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val prems = goal WF.thy
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    "[| wf(r);  trans(r) |] ==> is_recfun(r, a, H, the_recfun(r,a,H))";
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by (cut_facts_tac prems 1);
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by (wf_ind_tac "a" prems 1);
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by (res_inst_tac [("f", "lam y: r-``{a1}. wftrec(r,y,H)")] is_the_recfun 1);
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by (REPEAT (assume_tac 2));
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by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
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(*Applying the substitution: must keep the quantified assumption!!*)
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by (REPEAT (dtac underD 1 ORELSE resolve_tac [refl, lam_cong] 1));
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by (fold_tac [is_recfun_def]);
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by (rtac (consI1 RS restrict_type RSN (2,fun_extension) RS subst_context) 1);
0
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by (rtac is_recfun_type 1);
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by (ALLGOALS
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    (asm_simp_tac
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     (wf_super_ss addsimps [underI RS beta, apply_recfun, is_recfun_cut])));
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val unfold_the_recfun = result();
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(*** Unfolding wftrec ***)
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val prems = goal WF.thy
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    "[| wf(r);  trans(r);  <b,a>:r |] ==> \
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\    restrict(the_recfun(r,a,H), r-``{b}) = the_recfun(r,b,H)";
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by (REPEAT (ares_tac (prems @ [is_recfun_cut, unfold_the_recfun]) 1));
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val the_recfun_cut = result();
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(*NOT SUITABLE FOR REWRITING since it is recursive!*)
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goalw WF.thy [wftrec_def]
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    "!!r. [| wf(r);  trans(r) |] ==> \
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\         wftrec(r,a,H) = H(a, lam x: r-``{a}. wftrec(r,x,H))";
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by (rtac (rewrite_rule [is_recfun_def] unfold_the_recfun RS ssubst) 1);
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by (ALLGOALS (asm_simp_tac
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	(ZF_ss addsimps [vimage_singleton_iff RS iff_sym, the_recfun_cut])));
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val wftrec = result();
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(** Removal of the premise trans(r) **)
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(*NOT SUITABLE FOR REWRITING since it is recursive!*)
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val [wfr] = goalw WF.thy [wfrec_def]
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    "wf(r) ==> wfrec(r,a,H) = H(a, lam x:r-``{a}. wfrec(r,x,H))";
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by (rtac (wfr RS wf_trancl RS wftrec RS ssubst) 1);
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by (rtac trans_trancl 1);
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by (rtac (vimage_pair_mono RS restrict_lam_eq RS subst_context) 1);
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by (etac r_into_trancl 1);
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by (rtac subset_refl 1);
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val wfrec = result();
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(*This form avoids giant explosions in proofs.  NOTE USE OF == *)
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val rew::prems = goal WF.thy
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    "[| !!x. h(x)==wfrec(r,x,H);  wf(r) |] ==> \
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\    h(a) = H(a, lam x: r-``{a}. h(x))";
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by (rewtac rew);
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by (REPEAT (resolve_tac (prems@[wfrec]) 1));
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val def_wfrec = result();
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val prems = goal WF.thy
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    "[| wf(r);  a:A;  field(r)<=A;  \
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\       !!x u. [| x: A;  u: Pi(r-``{x}, B) |] ==> H(x,u) : B(x)   \
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\    |] ==> wfrec(r,a,H) : B(a)";
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by (res_inst_tac [("a","a")] wf_induct2 1);
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by (rtac (wfrec RS ssubst) 4);
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by (REPEAT (ares_tac (prems@[lam_type]) 1
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     ORELSE eresolve_tac [spec RS mp, underD] 1));
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val wfrec_type = result();