6 Some general lemmas |
6 Some general lemmas |
7 *) |
7 *) |
8 |
8 |
9 goal Sorting.thy "!x. mset (xs@ys) x = mset xs x + mset ys x"; |
9 goal Sorting.thy "!x. mset (xs@ys) x = mset xs x + mset ys x"; |
10 by (list.induct_tac "xs" 1); |
10 by (list.induct_tac "xs" 1); |
11 by (ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
11 by (ALLGOALS(asm_simp_tac (!simpset addsplits [expand_if]))); |
12 qed "mset_append"; |
12 qed "mset_append"; |
13 |
13 |
14 goal Sorting.thy "!x. mset [x:xs. ~p(x)] x + mset [x:xs. p(x)] x = \ |
14 goal Sorting.thy "!x. mset [x:xs. ~p(x)] x + mset [x:xs. p(x)] x = \ |
15 \ mset xs x"; |
15 \ mset xs x"; |
16 by (list.induct_tac "xs" 1); |
16 by (list.induct_tac "xs" 1); |
17 by (ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
17 by (ALLGOALS(asm_simp_tac (!simpset addsplits [expand_if]))); |
18 qed "mset_compl_add"; |
18 qed "mset_compl_add"; |
19 |
19 |
20 Addsimps [mset_append, mset_compl_add]; |
20 Addsimps [mset_append, mset_compl_add]; |
21 |
21 |
22 goal Sorting.thy "set xs = {x. mset xs x ~= 0}"; |
22 goal Sorting.thy "set xs = {x. mset xs x ~= 0}"; |
23 by (list.induct_tac "xs" 1); |
23 by (list.induct_tac "xs" 1); |
24 by (ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
24 by (ALLGOALS(asm_simp_tac (!simpset addsplits [expand_if]))); |
25 by (Fast_tac 1); |
25 by (Fast_tac 1); |
26 qed "set_via_mset"; |
26 qed "set_via_mset"; |
27 |
27 |
28 (* Equivalence of two definitions of `sorted' *) |
28 (* Equivalence of two definitions of `sorted' *) |
29 |
29 |
30 val prems = goalw Sorting.thy [transf_def] |
30 val prems = goalw Sorting.thy [transf_def] |
31 "transf(le) ==> sorted1 le xs = sorted le xs"; |
31 "transf(le) ==> sorted1 le xs = sorted le xs"; |
32 by (list.induct_tac "xs" 1); |
32 by (list.induct_tac "xs" 1); |
33 by (ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_list_case])))); |
33 by (ALLGOALS(asm_simp_tac (!simpset addsplits [expand_list_case]))); |
34 by (cut_facts_tac prems 1); |
34 by (cut_facts_tac prems 1); |
35 by (Fast_tac 1); |
35 by (Fast_tac 1); |
36 qed "sorted1_is_sorted"; |
36 qed "sorted1_is_sorted"; |