author | lcp |
Tue, 26 Jul 1994 13:44:42 +0200 | |
changeset 485 | 5e00a676a211 |
parent 443 | 10884e64c241 |
child 494 | 3686157a5a51 |
permissions | -rw-r--r-- |
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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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(** Equivalences between wf and wf_on **) |
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goalw WF.thy [wf_def, wf_on_def] "!!A r. wf(r) ==> wf[A](r)"; |
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by (fast_tac ZF_cs 1); |
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val wf_imp_wf_on = result(); |
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goalw WF.thy [wf_def, wf_on_def] "!!r. wf[field(r)](r) ==> wf(r)"; |
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by (fast_tac ZF_cs 1); |
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val wf_on_field_imp_wf = result(); |
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goal WF.thy "wf(r) <-> wf[field(r)](r)"; |
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by (fast_tac (ZF_cs addSEs [wf_imp_wf_on, wf_on_field_imp_wf]) 1); |
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val wf_iff_wf_on_field = result(); |
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goalw WF.thy [wf_on_def, wf_def] "!!A B r. [| wf[A](r); B<=A |] ==> wf[B](r)"; |
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by (fast_tac ZF_cs 1); |
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val wf_on_subset_A = result(); |
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goalw WF.thy [wf_on_def, wf_def] "!!A r s. [| wf[A](r); s<=r |] ==> wf[A](s)"; |
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by (fast_tac ZF_cs 1); |
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val wf_on_subset_r = result(); |
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(** Introduction rules for wf_on **) |
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(*If every non-empty subset of A has an r-minimal element then wf[A](r).*) |
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val [prem] = goalw WF.thy [wf_on_def, wf_def] |
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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[A](r)"; |
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by (rtac (equals0I RS disjCI RS allI) 1); |
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by (res_inst_tac [ ("Z", "Z") ] prem 1); |
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by (ALLGOALS (fast_tac ZF_cs)); |
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val wf_onI = result(); |
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(*If r allows well-founded induction over A then wf[A](r) |
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Premise is equivalent to |
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!!B. ALL x:A. (ALL y. <y,x>: r --> y:B) --> x:B ==> A<=B *) |
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val [prem] = goal WF.thy |
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"[| !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A \ |
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\ |] ==> y:B |] \ |
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\ ==> wf[A](r)"; |
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by (rtac wf_onI 1); |
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by (res_inst_tac [ ("c", "u") ] (prem RS DiffE) 1); |
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by (contr_tac 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 wf_onI2 = 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 wfI*) |
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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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goal ZF.thy "!!r A. field(r Int A*A) <= A"; |
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by (fast_tac ZF_cs 1); |
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val field_Int_square = result(); |
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val wfr::amem::prems = goalw WF.thy [wf_on_def] |
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"[| wf[A](r); a:A; \ |
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\ !!x.[| x: A; ALL y:A. <y,x>: r --> P(y) |] ==> P(x) \ |
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\ |] ==> P(a)"; |
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by (rtac ([wfr, amem, field_Int_square] MRS wf_induct2) 1); |
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by (REPEAT (ares_tac prems 1)); |
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by (fast_tac ZF_cs 1); |
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val wf_on_induct = result(); |
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fun wf_on_ind_tac a prems i = |
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EVERY [res_inst_tac [("a",a)] wf_on_induct i, |
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rename_last_tac a ["1"] (i+2), |
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REPEAT (ares_tac prems i)]; |
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(*If r allows well-founded induction then wf(r)*) |
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val [subs,indhyp] = goal WF.thy |
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"[| field(r)<=A; \ |
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\ !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B; y:A \ |
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\ |] ==> y:B |] \ |
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\ ==> wf(r)"; |
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by (rtac ([wf_onI2, subs] MRS (wf_on_subset_A RS wf_on_field_imp_wf)) 1); |
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by (REPEAT (ares_tac [indhyp] 1)); |
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val wfI = result(); |
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(*** Properties of well-founded relations ***) |
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goal WF.thy "!!r. wf(r) ==> <a,a> ~: r"; |
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by (wf_ind_tac "a" [] 1); |
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by (fast_tac ZF_cs 1); |
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val wf_not_refl = result(); |
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goal WF.thy "!!r. [| wf(r); <a,x>:r; <x,a>:r |] ==> P"; |
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by (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> P" 1); |
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by (wf_ind_tac "a" [] 2); |
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by (fast_tac ZF_cs 2); |
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by (fast_tac FOL_cs 1); |
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val wf_asym = result(); |
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goal WF.thy "!!r. [| wf[A](r); a: A |] ==> <a,a> ~: r"; |
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by (wf_on_ind_tac "a" [] 1); |
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by (fast_tac ZF_cs 1); |
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val wf_on_not_refl = result(); |
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goal WF.thy "!!r. [| wf[A](r); <a,b>:r; <b,a>:r; a:A; b:A |] ==> P"; |
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by (subgoal_tac "ALL y:A. <a,y>:r --> <y,a>:r --> P" 1); |
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by (wf_on_ind_tac "a" [] 2); |
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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 wf_on_asym = result(); |
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(*Needed to prove well_ordI. Could also reason that wf[A](r) means |
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wf(r Int A*A); thus wf( (r Int A*A)^+ ) and use wf_not_refl *) |
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goal WF.thy |
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"!!r. [| wf[A](r); <a,b>:r; <b,c>:r; <c,a>:r; a:A; b:A; c:A |] ==> P"; |
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by (subgoal_tac |
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"ALL y:A. ALL z:A. <a,y>:r --> <y,z>:r --> <z,a>:r --> P" 1); |
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by (wf_on_ind_tac "a" [] 2); |
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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 wf_on_chain3 = result(); |
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(*retains the universal formula for later use!*) |
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val bchain_tac = EVERY' [rtac (bspec RS mp), assume_tac, assume_tac ]; |
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(*transitive closure of a WF relation is WF provided A is downwards closed*) |
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val [wfr,subs] = goal WF.thy |
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"[| wf[A](r); r-``A <= A |] ==> wf[A](r^+)"; |
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by (rtac wf_onI2 1); |
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by (bchain_tac 1); |
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by (eres_inst_tac [("a","y")] (wfr RS wf_on_induct) 1); |
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by (rtac (impI RS ballI) 1); |
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by (etac tranclE 1); |
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by (etac (bspec RS mp) 1 THEN assume_tac 1); |
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by (fast_tac ZF_cs 1); |
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by (cut_facts_tac [subs] 1); |
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(*astar_tac is slightly faster*) |
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by (best_tac ZF_cs 1); |
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val wf_on_trancl = result(); |
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goal WF.thy "!!r. wf(r) ==> wf(r^+)"; |
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by (asm_full_simp_tac (ZF_ss addsimps [wf_iff_wf_on_field]) 1); |
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by (rtac (trancl_type RS field_rel_subset RSN (2, wf_on_subset_A)) 1); |
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by (etac wf_on_trancl 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 ]))); |
256 |
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 | 287 |
by (rtac is_recfun_type 1); |
288 |
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]))); |
0 | 291 |
val unfold_the_recfun = result(); |
292 |
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293 |
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294 |
(*** Unfolding wftrec ***) |
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295 |
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296 |
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))"; |
0 | 306 |
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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308 |
(ZF_ss addsimps [vimage_singleton_iff RS iff_sym, the_recfun_cut]))); |
0 | 309 |
val wftrec = result(); |
310 |
||
311 |
(** Removal of the premise trans(r) **) |
|
312 |
||
313 |
(*NOT SUITABLE FOR REWRITING since it is recursive!*) |
|
314 |
val [wfr] = goalw WF.thy [wfrec_def] |
|
315 |
"wf(r) ==> wfrec(r,a,H) = H(a, lam x:r-``{a}. wfrec(r,x,H))"; |
|
316 |
by (rtac (wfr RS wf_trancl RS wftrec RS ssubst) 1); |
|
317 |
by (rtac trans_trancl 1); |
|
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset
|
318 |
by (rtac (vimage_pair_mono RS restrict_lam_eq RS subst_context) 1); |
0 | 319 |
by (etac r_into_trancl 1); |
320 |
by (rtac subset_refl 1); |
|
321 |
val wfrec = result(); |
|
322 |
||
323 |
(*This form avoids giant explosions in proofs. NOTE USE OF == *) |
|
324 |
val rew::prems = goal WF.thy |
|
325 |
"[| !!x. h(x)==wfrec(r,x,H); wf(r) |] ==> \ |
|
326 |
\ h(a) = H(a, lam x: r-``{a}. h(x))"; |
|
327 |
by (rewtac rew); |
|
328 |
by (REPEAT (resolve_tac (prems@[wfrec]) 1)); |
|
329 |
val def_wfrec = result(); |
|
330 |
||
331 |
val prems = goal WF.thy |
|
332 |
"[| wf(r); a:A; field(r)<=A; \ |
|
333 |
\ !!x u. [| x: A; u: Pi(r-``{x}, B) |] ==> H(x,u) : B(x) \ |
|
334 |
\ |] ==> wfrec(r,a,H) : B(a)"; |
|
335 |
by (res_inst_tac [("a","a")] wf_induct2 1); |
|
336 |
by (rtac (wfrec RS ssubst) 4); |
|
337 |
by (REPEAT (ares_tac (prems@[lam_type]) 1 |
|
338 |
ORELSE eresolve_tac [spec RS mp, underD] 1)); |
|
339 |
val wfrec_type = result(); |
|
435 | 340 |
|
341 |
||
342 |
goalw WF.thy [wf_on_def, wfrec_on_def] |
|
343 |
"!!A r. [| wf[A](r); a: A |] ==> \ |
|
344 |
\ wfrec[A](r,a,H) = H(a, lam x: (r-``{a}) Int A. wfrec[A](r,x,H))"; |
|
437 | 345 |
by (etac (wfrec RS trans) 1); |
435 | 346 |
by (asm_simp_tac (ZF_ss addsimps [vimage_Int_square, cons_subset_iff]) 1); |
347 |
val wfrec_on = result(); |
|
348 |