src/HOL/Wellfounded_Relations.thy
 author haftmann Mon Aug 14 13:46:06 2006 +0200 (2006-08-14) changeset 20380 14f9f2a1caa6 parent 19769 c40ce2de2020 child 22261 9e185f78e7d4 permissions -rw-r--r--
simplified code generator setup
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
73 lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
74   by (auto simp:inv_image_def)
76 subsubsection{*Finally, All Measures are Wellfounded.*}
78 lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
79   by (simp add:measure_def)
81 lemma wf_measure [iff]: "wf (measure f)"
82 apply (unfold measure_def)
83 apply (rule wf_less_than [THEN wf_inv_image])
84 done
86 lemma measure_induct_rule [case_names less]:
87   fixes f :: "'a \<Rightarrow> nat"
88   assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
89   shows "P a"
90 proof -
91   have "wf (measure f)" ..
92   then show ?thesis
93   proof induct
94     case (less x)
95     show ?case
96     proof (rule step)
97       fix y
98       assume "f y < f x"
99       hence "(y, x) \<in> measure f" by simp
100       thus "P y" by (rule less)
101     qed
102   qed
103 qed
105 lemma measure_induct:
106   fixes f :: "'a \<Rightarrow> nat"
107   shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
108   by (rule measure_induct_rule [of f P a]) iprover
111 subsection{*Other Ways of Constructing Wellfounded Relations*}
113 text{*Wellfoundedness of lexicographic combinations*}
114 lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
115 apply (unfold wf_def lex_prod_def)
116 apply (rule allI, rule impI)
117 apply (simp (no_asm_use) only: split_paired_All)
118 apply (drule spec, erule mp)
119 apply (rule allI, rule impI)
120 apply (drule spec, erule mp, blast)
121 done
123 lemma in_lex_prod[simp]:
124   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
125   by (auto simp:lex_prod_def)
127 text{*Transitivity of WF combinators.*}
128 lemma trans_lex_prod [intro!]:
129     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
130 by (unfold trans_def lex_prod_def, blast)
132 subsubsection{*Wellfoundedness of proper subset on finite sets.*}
133 lemma wf_finite_psubset: "wf(finite_psubset)"
134 apply (unfold finite_psubset_def)
135 apply (rule wf_measure [THEN wf_subset])
136 apply (simp add: measure_def inv_image_def less_than_def less_def [symmetric])
137 apply (fast elim!: psubset_card_mono)
138 done
140 lemma trans_finite_psubset: "trans finite_psubset"
141 by (simp add: finite_psubset_def psubset_def trans_def, blast)
144 subsubsection{*Wellfoundedness of finite acyclic relations*}
146 text{*This proof belongs in this theory because it needs Finite.*}
148 lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
149 apply (erule finite_induct, blast)
150 apply (simp (no_asm_simp) only: split_tupled_all)
151 apply simp
152 done
154 lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
155 apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
156 apply (erule acyclic_converse [THEN iffD2])
157 done
159 lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
160 by (blast intro: finite_acyclic_wf wf_acyclic)
163 subsubsection{*Wellfoundedness of @{term same_fst}*}
165 lemma same_fstI [intro!]:
166      "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
167 by (simp add: same_fst_def)
169 lemma wf_same_fst:
170   assumes prem: "(!!x. P x ==> wf(R x))"
171   shows "wf(same_fst P R)"
172 apply (simp cong del: imp_cong add: wf_def same_fst_def)
173 apply (intro strip)
174 apply (rename_tac a b)
175 apply (case_tac "wf (R a)")
176  apply (erule_tac a = b in wf_induct, blast)
177 apply (blast intro: prem)
178 done
181 subsection{*Weakly decreasing sequences (w.r.t. some well-founded order)
182    stabilize.*}
184 text{*This material does not appear to be used any longer.*}
186 lemma lemma1: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
187 apply (induct_tac "k", simp_all)
188 apply (blast intro: rtrancl_trans)
189 done
191 lemma lemma2: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
192       ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
193 apply (erule wf_induct, clarify)
194 apply (case_tac "EX j. (f (m+j), f m) : r^+")
195  apply clarify
196  apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
197   apply clarify
198   apply (rule_tac x = "j+i" in exI)
200 apply (rule_tac x = 0 in exI, clarsimp)
201 apply (drule_tac i = m and k = k in lemma1)
202 apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
203 done
205 lemma wf_weak_decr_stable: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
206       ==> EX i. ALL k. f (i+k) = f i"
207 apply (drule_tac x = 0 in lemma2 [THEN spec], auto)
208 done
210 (* special case of the theorem above: <= *)
211 lemma weak_decr_stable:
212      "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
213 apply (rule_tac r = pred_nat in wf_weak_decr_stable)
214 apply (simp add: pred_nat_trancl_eq_le)
215 apply (intro wf_trancl wf_pred_nat)
216 done
219 ML
220 {*
221 val less_than_def = thm "less_than_def";
222 val measure_def = thm "measure_def";
223 val lex_prod_def = thm "lex_prod_def";
224 val finite_psubset_def = thm "finite_psubset_def";
226 val wf_less_than = thm "wf_less_than";
227 val trans_less_than = thm "trans_less_than";
228 val less_than_iff = thm "less_than_iff";
229 val full_nat_induct = thm "full_nat_induct";
230 val wf_inv_image = thm "wf_inv_image";
231 val wf_measure = thm "wf_measure";
232 val measure_induct = thm "measure_induct";
233 val wf_lex_prod = thm "wf_lex_prod";
234 val trans_lex_prod = thm "trans_lex_prod";
235 val wf_finite_psubset = thm "wf_finite_psubset";
236 val trans_finite_psubset = thm "trans_finite_psubset";
237 val finite_acyclic_wf = thm "finite_acyclic_wf";
238 val finite_acyclic_wf_converse = thm "finite_acyclic_wf_converse";
239 val wf_iff_acyclic_if_finite = thm "wf_iff_acyclic_if_finite";
240 val wf_weak_decr_stable = thm "wf_weak_decr_stable";
241 val weak_decr_stable = thm "weak_decr_stable";
242 val same_fstI = thm "same_fstI";
243 val wf_same_fst = thm "wf_same_fst";
244 *}
247 end