src/HOL/Wellfounded_Relations.thy
 author wenzelm Mon Nov 28 10:58:40 2005 +0100 (2005-11-28) changeset 18277 6c2b039b4847 parent 15352 cba05842bd7a child 19404 9bf2cdc9e8e8 permissions -rw-r--r--
added proof of measure_induct_rule;
1 (*  ID:   \$Id\$
2     Author:     Konrad Slind
3     Copyright   1995 TU Munich
4 *)
6 header {*Well-founded Relations*}
8 theory Wellfounded_Relations
9 imports Finite_Set
10 begin
12 text{*Derived WF relations such as inverse image, lexicographic product and
13 measure. The simple relational product, in which @{term "(x',y')"} precedes
14 @{term "(x,y)"} if @{term "x'<x"} and @{term "y'<y"}, is a subset of the
15 lexicographic product, and therefore does not need to be defined separately.*}
17 constdefs
18  less_than :: "(nat*nat)set"
19     "less_than == trancl pred_nat"
21  measure   :: "('a => nat) => ('a * 'a)set"
22     "measure == inv_image less_than"
24  lex_prod  :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
25                (infixr "<*lex*>" 80)
26     "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
28  finite_psubset  :: "('a set * 'a set) set"
29    --{* finite proper subset*}
30     "finite_psubset == {(A,B). A < B & finite B}"
32  same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
33     "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
34    --{*For @{text rec_def} declarations where the first n parameters
35        stay unchanged in the recursive call.
36        See @{text "Library/While_Combinator.thy"} for an application.*}
41 subsection{*Measure Functions make Wellfounded Relations*}
43 subsubsection{*`Less than' on the natural numbers*}
45 lemma wf_less_than [iff]: "wf less_than"
46 by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
48 lemma trans_less_than [iff]: "trans less_than"
49 by (simp add: less_than_def trans_trancl)
51 lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
52 by (simp add: less_than_def less_def)
54 lemma full_nat_induct:
55   assumes ih: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
56   shows "P n"
57 apply (rule wf_less_than [THEN wf_induct])
58 apply (rule ih, auto)
59 done
61 subsubsection{*The Inverse Image into a Wellfounded Relation is Wellfounded.*}
63 lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
64 apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
65 apply clarify
66 apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
67 prefer 2 apply (blast del: allE)
68 apply (erule allE)
69 apply (erule (1) notE impE)
70 apply blast
71 done
74 subsubsection{*Finally, All Measures are Wellfounded.*}
76 lemma wf_measure [iff]: "wf (measure f)"
77 apply (unfold measure_def)
78 apply (rule wf_less_than [THEN wf_inv_image])
79 done
81 lemma measure_induct_rule [case_names less]:
82   fixes f :: "'a \<Rightarrow> nat"
83   assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
84   shows "P a"
85 proof -
86   have "wf (measure f)" ..
87   then show ?thesis
88   proof induct
89     case (less x)
90     show ?case
91     proof (rule step)
92       fix y
93       assume "f y < f x"
94       then have "(y, x) \<in> measure f"
95         by (simp add: measure_def inv_image_def)
96       then show "P y" by (rule less)
97     qed
98   qed
99 qed
101 lemma measure_induct:
102   fixes f :: "'a \<Rightarrow> nat"
103   shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
104   by (rule measure_induct_rule [of f P a]) iprover
107 subsection{*Other Ways of Constructing Wellfounded Relations*}
109 text{*Wellfoundedness of lexicographic combinations*}
110 lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
111 apply (unfold wf_def lex_prod_def)
112 apply (rule allI, rule impI)
113 apply (simp (no_asm_use) only: split_paired_All)
114 apply (drule spec, erule mp)
115 apply (rule allI, rule impI)
116 apply (drule spec, erule mp, blast)
117 done
120 text{*Transitivity of WF combinators.*}
121 lemma trans_lex_prod [intro!]:
122     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
123 by (unfold trans_def lex_prod_def, blast)
126 subsubsection{*Wellfoundedness of proper subset on finite sets.*}
127 lemma wf_finite_psubset: "wf(finite_psubset)"
128 apply (unfold finite_psubset_def)
129 apply (rule wf_measure [THEN wf_subset])
130 apply (simp add: measure_def inv_image_def less_than_def less_def [symmetric])
131 apply (fast elim!: psubset_card_mono)
132 done
134 lemma trans_finite_psubset: "trans finite_psubset"
135 by (simp add: finite_psubset_def psubset_def trans_def, blast)
138 subsubsection{*Wellfoundedness of finite acyclic relations*}
140 text{*This proof belongs in this theory because it needs Finite.*}
142 lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
143 apply (erule finite_induct, blast)
144 apply (simp (no_asm_simp) only: split_tupled_all)
145 apply simp
146 done
148 lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
149 apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
150 apply (erule acyclic_converse [THEN iffD2])
151 done
153 lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
154 by (blast intro: finite_acyclic_wf wf_acyclic)
157 subsubsection{*Wellfoundedness of @{term same_fst}*}
159 lemma same_fstI [intro!]:
160      "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
161 by (simp add: same_fst_def)
163 lemma wf_same_fst:
164   assumes prem: "(!!x. P x ==> wf(R x))"
165   shows "wf(same_fst P R)"
166 apply (simp cong del: imp_cong add: wf_def same_fst_def)
167 apply (intro strip)
168 apply (rename_tac a b)
169 apply (case_tac "wf (R a)")
170  apply (erule_tac a = b in wf_induct, blast)
171 apply (blast intro: prem)
172 done
175 subsection{*Weakly decreasing sequences (w.r.t. some well-founded order)
176    stabilize.*}
178 text{*This material does not appear to be used any longer.*}
180 lemma lemma1: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
181 apply (induct_tac "k", simp_all)
182 apply (blast intro: rtrancl_trans)
183 done
185 lemma lemma2: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
186       ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
187 apply (erule wf_induct, clarify)
188 apply (case_tac "EX j. (f (m+j), f m) : r^+")
189  apply clarify
190  apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
191   apply clarify
192   apply (rule_tac x = "j+i" in exI)
194 apply (rule_tac x = 0 in exI, clarsimp)
195 apply (drule_tac i = m and k = k in lemma1)
196 apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
197 done
199 lemma wf_weak_decr_stable: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
200       ==> EX i. ALL k. f (i+k) = f i"
201 apply (drule_tac x = 0 in lemma2 [THEN spec], auto)
202 done
204 (* special case of the theorem above: <= *)
205 lemma weak_decr_stable:
206      "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
207 apply (rule_tac r = pred_nat in wf_weak_decr_stable)
208 apply (simp add: pred_nat_trancl_eq_le)
209 apply (intro wf_trancl wf_pred_nat)
210 done
213 ML
214 {*
215 val less_than_def = thm "less_than_def";
216 val measure_def = thm "measure_def";
217 val lex_prod_def = thm "lex_prod_def";
218 val finite_psubset_def = thm "finite_psubset_def";
220 val wf_less_than = thm "wf_less_than";
221 val trans_less_than = thm "trans_less_than";
222 val less_than_iff = thm "less_than_iff";
223 val full_nat_induct = thm "full_nat_induct";
224 val wf_inv_image = thm "wf_inv_image";
225 val wf_measure = thm "wf_measure";
226 val measure_induct = thm "measure_induct";
227 val wf_lex_prod = thm "wf_lex_prod";
228 val trans_lex_prod = thm "trans_lex_prod";
229 val wf_finite_psubset = thm "wf_finite_psubset";
230 val trans_finite_psubset = thm "trans_finite_psubset";
231 val finite_acyclic_wf = thm "finite_acyclic_wf";
232 val finite_acyclic_wf_converse = thm "finite_acyclic_wf_converse";
233 val wf_iff_acyclic_if_finite = thm "wf_iff_acyclic_if_finite";
234 val wf_weak_decr_stable = thm "wf_weak_decr_stable";
235 val weak_decr_stable = thm "weak_decr_stable";
236 val same_fstI = thm "same_fstI";
237 val wf_same_fst = thm "wf_same_fst";
238 *}
241 end