src/HOL/Wfrec.thy
 author wenzelm Fri Jan 31 14:33:02 2014 +0100 (2014-01-31) changeset 55210 d1e3b708d74b parent 55017 2df6ad1dbd66 child 58184 db1381d811ab permissions -rw-r--r--
1 (*  Title:      HOL/Wfrec.thy
2     Author:     Tobias Nipkow
3     Author:     Lawrence C Paulson
4     Author:     Konrad Slind
5 *)
7 header {* Well-Founded Recursion Combinator *}
9 theory Wfrec
10 imports Wellfounded
11 begin
13 inductive
14   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
15   for R :: "('a * 'a) set"
16   and F :: "('a => 'b) => 'a => 'b"
17 where
18   wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
19             wfrec_rel R F x (F g x)"
21 definition
22   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b" where
23   "cut f r x == (%y. if (y,x):r then f y else undefined)"
25 definition
26   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool" where
27   "adm_wf R F == ALL f g x.
28      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
30 definition
31   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where
32   "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
34 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
35 by (simp add: fun_eq_iff cut_def)
37 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
38 by (simp add: cut_def)
40 text{*Inductive characterization of wfrec combinator; for details see:
41 John Harrison, "Inductive definitions: automation and application"*}
43 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
45 apply (erule_tac a=x in wf_induct)
46 apply (rule ex1I)
47 apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
48 apply (fast dest!: theI')
49 apply (erule wfrec_rel.cases, simp)
50 apply (erule allE, erule allE, erule allE, erule mp)
51 apply (blast intro: the_equality [symmetric])
52 done
54 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
56 apply (intro strip)
57 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
58 apply (rule refl)
59 done
61 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
62 apply (simp add: wfrec_def)
63 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
64 apply (rule wfrec_rel.wfrecI)
65 apply (intro strip)
66 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
67 done
70 text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}
71 lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"
72 apply auto
73 apply (blast intro: wfrec)
74 done
77 subsection {* Wellfoundedness of @{text same_fst} *}
79 definition
80  same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
81 where
82     "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
83    --{*For @{text rec_def} declarations where the first n parameters
84        stay unchanged in the recursive call. *}
86 lemma same_fstI [intro!]:
87      "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
88 by (simp add: same_fst_def)
90 lemma wf_same_fst:
91   assumes prem: "(!!x. P x ==> wf(R x))"
92   shows "wf(same_fst P R)"
93 apply (simp cong del: imp_cong add: wf_def same_fst_def)
94 apply (intro strip)
95 apply (rename_tac a b)
96 apply (case_tac "wf (R a)")
97  apply (erule_tac a = b in wf_induct, blast)
98 apply (blast intro: prem)
99 done
101 end