author | wenzelm |
Thu, 09 Sep 1999 12:25:44 +0200 | |
changeset 7530 | 505f6f8e9dcf |
parent 7527 | 9e2dddd8b81f |
child 7565 | bfa85f429629 |
permissions | -rw-r--r-- |
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
1 |
(* Title: HOL/Isar_examples/MultisetOrder.thy |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
2 |
ID: $Id$ |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
3 |
Author: Markus Wenzel |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
4 |
|
7442 | 5 |
Wellfoundedness proof for the multiset order. |
6 |
||
7 |
Original tactic script by Tobias Nipkow (see also |
|
8 |
HOL/Induct/Multiset). Pen-and-paper proof by Wilfried Buchholz. |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
9 |
*) |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
10 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
11 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
12 |
theory MultisetOrder = Multiset:; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
13 |
|
7527 | 14 |
|
15 |
lemma less_add: "(N, M0 + {#a#}) : mult1 r ==> |
|
16 |
(EX M. (M, M0) : mult1 r & N = M + {#a#}) | |
|
17 |
(EX K. (ALL b. elem K b --> (b, a) : r) & N = M0 + K)" |
|
18 |
(concl is "?case1 (mult1 r) | ?case2"); |
|
19 |
proof (unfold mult1_def); |
|
20 |
let ?r = "%K a. ALL b. elem K b --> (b, a) : r"; |
|
21 |
let ?R = "%N M. EX a M0 K. M = M0 + {#a#} & N = M0 + K & ?r K a"; |
|
22 |
let ?case1 = "?case1 {(N, M). ?R N M}"; |
|
23 |
||
24 |
assume "(N, M0 + {#a#}) : {(N, M). ?R N M}"; |
|
25 |
hence "EX a' M0' K. M0 + {#a#} = M0' + {#a'#} & N = M0' + K & ?r K a'"; by simp; |
|
26 |
thus "?case1 | ?case2"; |
|
27 |
proof (elim exE conjE); |
|
28 |
fix a' M0' K; assume N: "N = M0' + K" and r: "?r K a'"; |
|
29 |
assume "M0 + {#a#} = M0' + {#a'#}"; |
|
30 |
hence "M0 = M0' & a = a' | (EX K'. M0 = K' + {#a'#} & M0' = K' + {#a#})"; |
|
31 |
by (simp only: add_eq_conv_ex); |
|
32 |
thus ?thesis; |
|
33 |
proof (elim disjE conjE exE); |
|
34 |
assume "M0 = M0'" "a = a'"; |
|
35 |
with N r; have "?r K a & N = M0 + K"; by simp; |
|
36 |
hence ?case2; ..; thus ?thesis; ..; |
|
37 |
next; |
|
38 |
fix K'; |
|
39 |
assume "M0' = K' + {#a#}"; |
|
40 |
with N; have n: "N = K' + K + {#a#}"; by (simp add: union_ac); |
|
41 |
||
42 |
assume "M0 = K' + {#a'#}"; |
|
43 |
with r; have "?R (K' + K) M0"; by simp blast; |
|
44 |
with n; have ?case1; by simp; thus ?thesis; ..; |
|
45 |
qed; |
|
46 |
qed; |
|
47 |
qed; |
|
48 |
||
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
49 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
50 |
lemma all_accessible: "wf r ==> ALL M. M : acc (mult1 r)"; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
51 |
proof; |
7480 | 52 |
let ?R = "mult1 r"; |
53 |
let ?W = "acc ?R"; |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
54 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
55 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
56 |
{{; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
57 |
fix M M0 a; |
7480 | 58 |
assume wf_hyp: "ALL b. (b, a) : r --> (ALL M:?W. M + {#b#} : ?W)" |
59 |
and M0: "M0 : ?W" |
|
60 |
and acc_hyp: "ALL M. (M, M0) : ?R --> M + {#a#} : ?W"; |
|
61 |
have "M0 + {#a#} : ?W"; |
|
7442 | 62 |
proof (rule accI [of "M0 + {#a#}"]); |
7480 | 63 |
fix N; assume "(N, M0 + {#a#}) : ?R"; |
64 |
hence "((EX M. (M, M0) : ?R & N = M + {#a#}) | |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
65 |
(EX K. (ALL b. elem K b --> (b, a) : r) & N = M0 + K))"; |
7527 | 66 |
by (rule less_add); |
7480 | 67 |
thus "N : ?W"; |
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
68 |
proof (elim exE disjE conjE); |
7480 | 69 |
fix M; assume "(M, M0) : ?R" and N: "N = M + {#a#}"; |
70 |
from acc_hyp; have "(M, M0) : ?R --> M + {#a#} : ?W"; ..; |
|
71 |
hence "M + {#a#} : ?W"; ..; |
|
72 |
thus "N : ?W"; by (simp only: N); |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
73 |
next; |
7451 | 74 |
fix K; |
75 |
assume N: "N = M0 + K"; |
|
76 |
assume "ALL b. elem K b --> (b, a) : r"; |
|
7480 | 77 |
have "?this --> M0 + K : ?W" (is "?P K"); |
7527 | 78 |
proof (induct K rule: multiset_induct); |
7480 | 79 |
from M0; have "M0 + {#} : ?W"; by simp; |
80 |
thus "?P {#}"; ..; |
|
7442 | 81 |
|
7480 | 82 |
fix K x; assume hyp: "?P K"; |
83 |
show "?P (K + {#x#})"; |
|
7442 | 84 |
proof; |
7451 | 85 |
assume a: "ALL b. elem (K + {#x#}) b --> (b, a) : r"; |
7442 | 86 |
hence "(x, a) : r"; by simp; |
7480 | 87 |
with wf_hyp [RS spec]; have b: "ALL M:?W. M + {#x#} : ?W"; ..; |
7442 | 88 |
|
7480 | 89 |
from a hyp; have "M0 + K : ?W"; by simp; |
90 |
with b; have "(M0 + K) + {#x#} : ?W"; ..; |
|
91 |
thus "M0 + (K + {#x#}) : ?W"; by (simp only: union_assoc); |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
92 |
qed; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
93 |
qed; |
7480 | 94 |
hence "M0 + K : ?W"; ..; |
95 |
thus "N : ?W"; by (simp only: N); |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
96 |
qed; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
97 |
qed; |
7442 | 98 |
}}; note tedious_reasoning = this; |
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
99 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
100 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
101 |
assume wf: "wf r"; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
102 |
fix M; |
7480 | 103 |
show "M : ?W"; |
7527 | 104 |
proof (induct M rule: multiset_induct); |
7480 | 105 |
show "{#} : ?W"; |
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
106 |
proof (rule accI); |
7480 | 107 |
fix b; assume "(b, {#}) : ?R"; |
108 |
with not_less_empty; show "b : ?W"; by contradiction; |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
109 |
qed; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
110 |
|
7480 | 111 |
fix M a; assume "M : ?W"; |
112 |
from wf; have "ALL M:?W. M + {#a#} : ?W"; |
|
7442 | 113 |
proof (rule wf_induct [of r]); |
7480 | 114 |
fix a; assume "ALL b. (b, a) : r --> (ALL M:?W. M + {#b#} : ?W)"; |
115 |
show "ALL M:?W. M + {#a#} : ?W"; |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
116 |
proof; |
7480 | 117 |
fix M; assume "M : ?W"; |
118 |
thus "M + {#a#} : ?W"; by (rule acc_induct) (rule tedious_reasoning); |
|
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
119 |
qed; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
120 |
qed; |
7480 | 121 |
thus "M + {#a#} : ?W"; ..; |
7432
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
122 |
qed; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
123 |
qed; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
124 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
125 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
126 |
theorem wf_mult1: "wf r ==> wf (mult1 r)"; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
127 |
by (rule acc_wfI, rule all_accessible); |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
128 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
129 |
theorem wf_mult: "wf r ==> wf (mult r)"; |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
130 |
by (unfold mult_def, rule wf_trancl, rule wf_mult1); |
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
131 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
132 |
|
c32a0fd117a0
Wellfoundedness proof for the multiset order (preliminary version).
wenzelm
parents:
diff
changeset
|
133 |
end; |