src/HOL/WF.ML
 author nipkow Tue Apr 08 10:48:42 1997 +0200 (1997-04-08) changeset 2919 953a47dc0519 parent 2637 e9b203f854ae child 2935 998cb95fdd43 permissions -rw-r--r--
Dep. on Provers/nat_transitive
 clasohm@1475 ` 1` ```(* Title: HOL/wf.ML ``` clasohm@923 ` 2` ``` ID: \$Id\$ ``` clasohm@1475 ` 3` ``` Author: Tobias Nipkow, with minor changes by Konrad Slind ``` clasohm@1475 ` 4` ``` Copyright 1992 University of Cambridge/1995 TU Munich ``` clasohm@923 ` 5` clasohm@1475 ` 6` ```For WF.thy. Wellfoundedness, induction, and recursion ``` clasohm@923 ` 7` ```*) ``` clasohm@923 ` 8` clasohm@923 ` 9` ```open WF; ``` clasohm@923 ` 10` nipkow@950 ` 11` ```val H_cong = read_instantiate [("f","H")] (standard(refl RS cong RS cong)); ``` clasohm@923 ` 12` ```val H_cong1 = refl RS H_cong; ``` clasohm@923 ` 13` clasohm@923 ` 14` ```(*Restriction to domain A. If r is well-founded over A then wf(r)*) ``` clasohm@923 ` 15` ```val [prem1,prem2] = goalw WF.thy [wf_def] ``` paulson@1642 ` 16` ``` "[| r <= A Times A; \ ``` clasohm@972 ` 17` ```\ !!x P. [| ! x. (! y. (y,x) : r --> P(y)) --> P(x); x:A |] ==> P(x) |] \ ``` clasohm@923 ` 18` ```\ ==> wf(r)"; ``` clasohm@923 ` 19` ```by (strip_tac 1); ``` clasohm@923 ` 20` ```by (rtac allE 1); ``` clasohm@923 ` 21` ```by (assume_tac 1); ``` berghofe@1786 ` 22` ```by (best_tac (!claset addSEs [prem1 RS subsetD RS SigmaE2] addIs [prem2]) 1); ``` clasohm@923 ` 23` ```qed "wfI"; ``` clasohm@923 ` 24` clasohm@923 ` 25` ```val major::prems = goalw WF.thy [wf_def] ``` clasohm@923 ` 26` ``` "[| wf(r); \ ``` clasohm@972 ` 27` ```\ !!x.[| ! y. (y,x): r --> P(y) |] ==> P(x) \ ``` clasohm@923 ` 28` ```\ |] ==> P(a)"; ``` clasohm@923 ` 29` ```by (rtac (major RS spec RS mp RS spec) 1); ``` berghofe@1760 ` 30` ```by (fast_tac (!claset addEs prems) 1); ``` clasohm@923 ` 31` ```qed "wf_induct"; ``` clasohm@923 ` 32` clasohm@923 ` 33` ```(*Perform induction on i, then prove the wf(r) subgoal using prems. *) ``` clasohm@923 ` 34` ```fun wf_ind_tac a prems i = ``` clasohm@923 ` 35` ``` EVERY [res_inst_tac [("a",a)] wf_induct i, ``` clasohm@1465 ` 36` ``` rename_last_tac a ["1"] (i+1), ``` clasohm@1465 ` 37` ``` ares_tac prems i]; ``` clasohm@923 ` 38` clasohm@972 ` 39` ```val prems = goal WF.thy "[| wf(r); (a,x):r; (x,a):r |] ==> P"; ``` clasohm@972 ` 40` ```by (subgoal_tac "! x. (a,x):r --> (x,a):r --> P" 1); ``` berghofe@1760 ` 41` ```by (fast_tac (!claset addIs prems) 1); ``` clasohm@923 ` 42` ```by (wf_ind_tac "a" prems 1); ``` berghofe@1760 ` 43` ```by (Fast_tac 1); ``` clasohm@923 ` 44` ```qed "wf_asym"; ``` clasohm@923 ` 45` clasohm@972 ` 46` ```val prems = goal WF.thy "[| wf(r); (a,a): r |] ==> P"; ``` clasohm@923 ` 47` ```by (rtac wf_asym 1); ``` clasohm@923 ` 48` ```by (REPEAT (resolve_tac prems 1)); ``` paulson@1618 ` 49` ```qed "wf_irrefl"; ``` clasohm@923 ` 50` clasohm@1475 ` 51` ```(*transitive closure of a wf relation is wf! *) ``` clasohm@923 ` 52` ```val [prem] = goal WF.thy "wf(r) ==> wf(r^+)"; ``` clasohm@923 ` 53` ```by (rewtac wf_def); ``` clasohm@923 ` 54` ```by (strip_tac 1); ``` clasohm@923 ` 55` ```(*must retain the universal formula for later use!*) ``` clasohm@923 ` 56` ```by (rtac allE 1 THEN assume_tac 1); ``` clasohm@923 ` 57` ```by (etac mp 1); ``` clasohm@923 ` 58` ```by (res_inst_tac [("a","x")] (prem RS wf_induct) 1); ``` clasohm@923 ` 59` ```by (rtac (impI RS allI) 1); ``` clasohm@923 ` 60` ```by (etac tranclE 1); ``` berghofe@1760 ` 61` ```by (Fast_tac 1); ``` berghofe@1760 ` 62` ```by (Fast_tac 1); ``` clasohm@923 ` 63` ```qed "wf_trancl"; ``` clasohm@923 ` 64` clasohm@923 ` 65` clasohm@923 ` 66` ```(** cut **) ``` clasohm@923 ` 67` clasohm@923 ` 68` ```(*This rewrite rule works upon formulae; thus it requires explicit use of ``` clasohm@923 ` 69` ``` H_cong to expose the equality*) ``` clasohm@923 ` 70` ```goalw WF.thy [cut_def] ``` clasohm@972 ` 71` ``` "(cut f r x = cut g r x) = (!y. (y,x):r --> f(y)=g(y))"; ``` paulson@1552 ` 72` ```by (simp_tac (HOL_ss addsimps [expand_fun_eq] ``` clasohm@1475 ` 73` ``` setloop (split_tac [expand_if])) 1); ``` clasohm@1475 ` 74` ```qed "cuts_eq"; ``` clasohm@923 ` 75` clasohm@972 ` 76` ```goalw WF.thy [cut_def] "!!x. (x,a):r ==> (cut f r a)(x) = f(x)"; ``` paulson@1552 ` 77` ```by (asm_simp_tac HOL_ss 1); ``` clasohm@923 ` 78` ```qed "cut_apply"; ``` clasohm@923 ` 79` clasohm@923 ` 80` ```(*** is_recfun ***) ``` clasohm@923 ` 81` clasohm@923 ` 82` ```goalw WF.thy [is_recfun_def,cut_def] ``` clasohm@1475 ` 83` ``` "!!f. [| is_recfun r H a f; ~(b,a):r |] ==> f(b) = (@z.True)"; ``` clasohm@923 ` 84` ```by (etac ssubst 1); ``` paulson@1552 ` 85` ```by (asm_simp_tac HOL_ss 1); ``` clasohm@923 ` 86` ```qed "is_recfun_undef"; ``` clasohm@923 ` 87` clasohm@923 ` 88` ```(*** NOTE! some simplifications need a different finish_tac!! ***) ``` clasohm@923 ` 89` ```fun indhyp_tac hyps = ``` clasohm@923 ` 90` ``` (cut_facts_tac hyps THEN' ``` clasohm@923 ` 91` ``` DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE' ``` clasohm@1465 ` 92` ``` eresolve_tac [transD, mp, allE])); ``` oheimb@2637 ` 93` ```val wf_super_ss = HOL_ss addSolver indhyp_tac; ``` clasohm@923 ` 94` clasohm@923 ` 95` ```val prems = goalw WF.thy [is_recfun_def,cut_def] ``` clasohm@1475 ` 96` ``` "[| wf(r); trans(r); is_recfun r H a f; is_recfun r H b g |] ==> \ ``` clasohm@972 ` 97` ``` \ (x,a):r --> (x,b):r --> f(x)=g(x)"; ``` clasohm@923 ` 98` ```by (cut_facts_tac prems 1); ``` clasohm@923 ` 99` ```by (etac wf_induct 1); ``` clasohm@923 ` 100` ```by (REPEAT (rtac impI 1 ORELSE etac ssubst 1)); ``` clasohm@923 ` 101` ```by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1); ``` nipkow@1485 ` 102` ```qed_spec_mp "is_recfun_equal"; ``` clasohm@923 ` 103` clasohm@923 ` 104` clasohm@923 ` 105` ```val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def] ``` clasohm@923 ` 106` ``` "[| wf(r); trans(r); \ ``` clasohm@1475 ` 107` ```\ is_recfun r H a f; is_recfun r H b g; (b,a):r |] ==> \ ``` clasohm@923 ` 108` ```\ cut f r b = g"; ``` clasohm@923 ` 109` ```val gundef = recgb RS is_recfun_undef ``` clasohm@923 ` 110` ```and fisg = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal))); ``` clasohm@923 ` 111` ```by (cut_facts_tac prems 1); ``` clasohm@923 ` 112` ```by (rtac ext 1); ``` clasohm@923 ` 113` ```by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg] ``` clasohm@923 ` 114` ``` setloop (split_tac [expand_if])) 1); ``` clasohm@923 ` 115` ```qed "is_recfun_cut"; ``` clasohm@923 ` 116` clasohm@923 ` 117` ```(*** Main Existence Lemma -- Basic Properties of the_recfun ***) ``` clasohm@923 ` 118` clasohm@923 ` 119` ```val prems = goalw WF.thy [the_recfun_def] ``` clasohm@1475 ` 120` ``` "is_recfun r H a f ==> is_recfun r H a (the_recfun r H a)"; ``` clasohm@1475 ` 121` ```by (res_inst_tac [("P", "is_recfun r H a")] selectI 1); ``` clasohm@923 ` 122` ```by (resolve_tac prems 1); ``` clasohm@923 ` 123` ```qed "is_the_recfun"; ``` clasohm@923 ` 124` clasohm@923 ` 125` ```val prems = goal WF.thy ``` clasohm@1475 ` 126` ``` "[| wf(r); trans(r) |] ==> is_recfun r H a (the_recfun r H a)"; ``` clasohm@1475 ` 127` ``` by (cut_facts_tac prems 1); ``` clasohm@1475 ` 128` ``` by (wf_ind_tac "a" prems 1); ``` clasohm@1475 ` 129` ``` by (res_inst_tac [("f","cut (%y. H (the_recfun r H y) y) r a1")] ``` clasohm@1475 ` 130` ``` is_the_recfun 1); ``` paulson@1552 ` 131` ``` by (rewtac is_recfun_def); ``` paulson@2031 ` 132` ``` by (stac cuts_eq 1); ``` clasohm@1475 ` 133` ``` by (rtac allI 1); ``` clasohm@1475 ` 134` ``` by (rtac impI 1); ``` clasohm@1475 ` 135` ``` by (res_inst_tac [("f1","H"),("x","y")](arg_cong RS fun_cong) 1); ``` clasohm@1475 ` 136` ``` by (subgoal_tac ``` clasohm@1475 ` 137` ``` "the_recfun r H y = cut(%x. H(cut(the_recfun r H y) r x) x) r y" 1); ``` clasohm@1475 ` 138` ``` by (etac allE 2); ``` clasohm@1475 ` 139` ``` by (dtac impE 2); ``` clasohm@1475 ` 140` ``` by (atac 2); ``` clasohm@1475 ` 141` ``` by (atac 3); ``` clasohm@1475 ` 142` ``` by (atac 2); ``` clasohm@1475 ` 143` ``` by (etac ssubst 1); ``` clasohm@1475 ` 144` ``` by (simp_tac (HOL_ss addsimps [cuts_eq]) 1); ``` clasohm@1475 ` 145` ``` by (rtac allI 1); ``` clasohm@1475 ` 146` ``` by (rtac impI 1); ``` clasohm@1475 ` 147` ``` by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1); ``` clasohm@1475 ` 148` ``` by (res_inst_tac [("f1","H"),("x","ya")](arg_cong RS fun_cong) 1); ``` clasohm@1475 ` 149` ``` by (fold_tac [is_recfun_def]); ``` clasohm@1475 ` 150` ``` by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1); ``` clasohm@923 ` 151` ```qed "unfold_the_recfun"; ``` clasohm@923 ` 152` clasohm@1475 ` 153` ```val unwind1_the_recfun = rewrite_rule[is_recfun_def] unfold_the_recfun; ``` clasohm@923 ` 154` clasohm@1475 ` 155` ```(*--------------Old proof----------------------------------------------------- ``` clasohm@923 ` 156` ```val prems = goal WF.thy ``` clasohm@1475 ` 157` ``` "[| wf(r); trans(r) |] ==> is_recfun r H a (the_recfun r H a)"; ``` clasohm@1475 ` 158` ```by (cut_facts_tac prems 1); ``` clasohm@1475 ` 159` ```by (wf_ind_tac "a" prems 1); ``` clasohm@1475 ` 160` ```by (res_inst_tac [("f", "cut (%y. wftrec r H y) r a1")] is_the_recfun 1); ``` clasohm@1475 ` 161` ```by (rewrite_goals_tac [is_recfun_def, wftrec_def]); ``` paulson@2031 ` 162` ```by (stac cuts_eq 1); ``` clasohm@1475 ` 163` ```(*Applying the substitution: must keep the quantified assumption!!*) ``` clasohm@1475 ` 164` ```by (EVERY1 [strip_tac, rtac H_cong1, rtac allE, atac, ``` clasohm@1475 ` 165` ``` etac (mp RS ssubst), atac]); ``` clasohm@1475 ` 166` ```by (fold_tac [is_recfun_def]); ``` clasohm@1475 ` 167` ```by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1); ``` clasohm@1475 ` 168` ```qed "unfold_the_recfun"; ``` clasohm@1475 ` 169` ```---------------------------------------------------------------------------*) ``` clasohm@923 ` 170` clasohm@923 ` 171` ```(** Removal of the premise trans(r) **) ``` clasohm@1475 ` 172` ```val th = rewrite_rule[is_recfun_def] ``` clasohm@1475 ` 173` ``` (trans_trancl RSN (2,(wf_trancl RS unfold_the_recfun))); ``` clasohm@923 ` 174` clasohm@923 ` 175` ```goalw WF.thy [wfrec_def] ``` clasohm@1475 ` 176` ``` "!!r. wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"; ``` clasohm@1475 ` 177` ```by (rtac H_cong 1); ``` clasohm@1475 ` 178` ```by (rtac refl 2); ``` clasohm@1475 ` 179` ```by (simp_tac (HOL_ss addsimps [cuts_eq]) 1); ``` clasohm@1475 ` 180` ```by (rtac allI 1); ``` clasohm@1475 ` 181` ```by (rtac impI 1); ``` clasohm@1475 ` 182` ```by (simp_tac(HOL_ss addsimps [wfrec_def]) 1); ``` clasohm@1475 ` 183` ```by (res_inst_tac [("a1","a")] (th RS ssubst) 1); ``` clasohm@1475 ` 184` ```by (atac 1); ``` clasohm@1475 ` 185` ```by (forward_tac[wf_trancl] 1); ``` clasohm@1475 ` 186` ```by (forward_tac[r_into_trancl] 1); ``` clasohm@1475 ` 187` ```by (asm_simp_tac (HOL_ss addsimps [cut_apply]) 1); ``` clasohm@1475 ` 188` ```by (rtac H_cong 1); (*expose the equality of cuts*) ``` clasohm@1475 ` 189` ```by (rtac refl 2); ``` clasohm@1475 ` 190` ```by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1); ``` clasohm@1475 ` 191` ```by (strip_tac 1); ``` nipkow@1485 ` 192` ```by (res_inst_tac [("r","r^+")] is_recfun_equal 1); ``` clasohm@1475 ` 193` ```by (atac 1); ``` clasohm@1475 ` 194` ```by (rtac trans_trancl 1); ``` clasohm@1475 ` 195` ```by (rtac unfold_the_recfun 1); ``` clasohm@1475 ` 196` ```by (atac 1); ``` clasohm@1475 ` 197` ```by (rtac trans_trancl 1); ``` clasohm@1475 ` 198` ```by (rtac unfold_the_recfun 1); ``` clasohm@1475 ` 199` ```by (atac 1); ``` clasohm@1475 ` 200` ```by (rtac trans_trancl 1); ``` clasohm@1475 ` 201` ```by (rtac transD 1); ``` clasohm@1475 ` 202` ```by (rtac trans_trancl 1); ``` clasohm@1475 ` 203` ```by (forw_inst_tac [("a","ya")] r_into_trancl 1); ``` clasohm@1475 ` 204` ```by (atac 1); ``` clasohm@1475 ` 205` ```by (atac 1); ``` clasohm@1475 ` 206` ```by (forw_inst_tac [("a","ya")] r_into_trancl 1); ``` clasohm@1475 ` 207` ```by (atac 1); ``` clasohm@1475 ` 208` ```qed "wfrec"; ``` clasohm@1475 ` 209` clasohm@1475 ` 210` ```(*--------------Old proof----------------------------------------------------- ``` clasohm@1475 ` 211` ```goalw WF.thy [wfrec_def] ``` clasohm@1475 ` 212` ``` "!!r. wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"; ``` clasohm@923 ` 213` ```by (etac (wf_trancl RS wftrec RS ssubst) 1); ``` clasohm@923 ` 214` ```by (rtac trans_trancl 1); ``` clasohm@923 ` 215` ```by (rtac (refl RS H_cong) 1); (*expose the equality of cuts*) ``` clasohm@1475 ` 216` ```by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1); ``` clasohm@923 ` 217` ```qed "wfrec"; ``` clasohm@1475 ` 218` ```---------------------------------------------------------------------------*) ``` clasohm@923 ` 219` clasohm@1475 ` 220` ```(*--------------------------------------------------------------------------- ``` clasohm@1475 ` 221` ``` * This form avoids giant explosions in proofs. NOTE USE OF == ``` clasohm@1475 ` 222` ``` *---------------------------------------------------------------------------*) ``` clasohm@923 ` 223` ```val rew::prems = goal WF.thy ``` clasohm@1475 ` 224` ``` "[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a"; ``` clasohm@923 ` 225` ```by (rewtac rew); ``` clasohm@923 ` 226` ```by (REPEAT (resolve_tac (prems@[wfrec]) 1)); ``` clasohm@923 ` 227` ```qed "def_wfrec"; ``` clasohm@1475 ` 228`