14 (*The intended introduction rule*) |
14 (*The intended introduction rule*) |
15 val prems = goal Acc.thy |
15 val prems = goal Acc.thy |
16 "[| !!b. <b,a>:r ==> b: acc(r) |] ==> a: acc(r)"; |
16 "[| !!b. <b,a>:r ==> b: acc(r) |] ==> a: acc(r)"; |
17 by (fast_tac (set_cs addIs (prems @ |
17 by (fast_tac (set_cs addIs (prems @ |
18 map (rewrite_rule [pred_def]) acc.intrs)) 1); |
18 map (rewrite_rule [pred_def]) acc.intrs)) 1); |
19 val accI = result(); |
19 qed "accI"; |
20 |
20 |
21 goal Acc.thy "!!a b r. [| b: acc(r); <a,b>: r |] ==> a: acc(r)"; |
21 goal Acc.thy "!!a b r. [| b: acc(r); <a,b>: r |] ==> a: acc(r)"; |
22 by (etac acc.elim 1); |
22 by (etac acc.elim 1); |
23 by (rewtac pred_def); |
23 by (rewtac pred_def); |
24 by (fast_tac set_cs 1); |
24 by (fast_tac set_cs 1); |
25 val acc_downward = result(); |
25 qed "acc_downward"; |
26 |
26 |
27 val [major,indhyp] = goal Acc.thy |
27 val [major,indhyp] = goal Acc.thy |
28 "[| a : acc(r); \ |
28 "[| a : acc(r); \ |
29 \ !!x. [| x: acc(r); ALL y. <y,x>:r --> P(y) |] ==> P(x) \ |
29 \ !!x. [| x: acc(r); ALL y. <y,x>:r --> P(y) |] ==> P(x) \ |
30 \ |] ==> P(a)"; |
30 \ |] ==> P(a)"; |
32 by (rtac indhyp 1); |
32 by (rtac indhyp 1); |
33 by (resolve_tac acc.intrs 1); |
33 by (resolve_tac acc.intrs 1); |
34 by (rewtac pred_def); |
34 by (rewtac pred_def); |
35 by (fast_tac set_cs 2); |
35 by (fast_tac set_cs 2); |
36 be (Int_lower1 RS Pow_mono RS subsetD) 1; |
36 be (Int_lower1 RS Pow_mono RS subsetD) 1; |
37 val acc_induct = result(); |
37 qed "acc_induct"; |
38 |
38 |
39 |
39 |
40 val [major] = goal Acc.thy "r <= Sigma(acc(r), %u. acc(r)) ==> wf(r)"; |
40 val [major] = goal Acc.thy "r <= Sigma(acc(r), %u. acc(r)) ==> wf(r)"; |
41 by (rtac (major RS wfI) 1); |
41 by (rtac (major RS wfI) 1); |
42 by (etac acc.induct 1); |
42 by (etac acc.induct 1); |
43 by (rewtac pred_def); |
43 by (rewtac pred_def); |
44 by (fast_tac set_cs 1); |
44 by (fast_tac set_cs 1); |
45 val acc_wfI = result(); |
45 qed "acc_wfI"; |
46 |
46 |
47 val [major] = goal Acc.thy "wf(r) ==> ALL x. <x,y>: r | <y,x>:r --> y: acc(r)"; |
47 val [major] = goal Acc.thy "wf(r) ==> ALL x. <x,y>: r | <y,x>:r --> y: acc(r)"; |
48 by (rtac (major RS wf_induct) 1); |
48 by (rtac (major RS wf_induct) 1); |
49 br (impI RS allI) 1; |
49 br (impI RS allI) 1; |
50 br accI 1; |
50 br accI 1; |
51 by (fast_tac set_cs 1); |
51 by (fast_tac set_cs 1); |
52 val acc_wfD_lemma = result(); |
52 qed "acc_wfD_lemma"; |
53 |
53 |
54 val [major] = goal Acc.thy "wf(r) ==> r <= Sigma(acc(r), %u. acc(r))"; |
54 val [major] = goal Acc.thy "wf(r) ==> r <= Sigma(acc(r), %u. acc(r))"; |
55 by (rtac subsetI 1); |
55 by (rtac subsetI 1); |
56 by (res_inst_tac [("p", "x")] PairE 1); |
56 by (res_inst_tac [("p", "x")] PairE 1); |
57 by (fast_tac (set_cs addSIs [SigmaI, |
57 by (fast_tac (set_cs addSIs [SigmaI, |
58 major RS acc_wfD_lemma RS spec RS mp]) 1); |
58 major RS acc_wfD_lemma RS spec RS mp]) 1); |
59 val acc_wfD = result(); |
59 qed "acc_wfD"; |
60 |
60 |
61 goal Acc.thy "wf(r) = (r <= Sigma(acc(r), %u. acc(r)))"; |
61 goal Acc.thy "wf(r) = (r <= Sigma(acc(r), %u. acc(r)))"; |
62 by (EVERY1 [rtac iffI, etac acc_wfD, etac acc_wfI]); |
62 by (EVERY1 [rtac iffI, etac acc_wfD, etac acc_wfI]); |
63 val wf_acc_iff = result(); |
63 qed "wf_acc_iff"; |