src/HOL/Fun_Def.thy
 author blanchet Fri Mar 07 14:21:15 2014 +0100 (2014-03-07) changeset 55968 94242fa87638 parent 55466 786edc984c98 child 56248 67dc9549fa15 permissions -rw-r--r--
tuning
1 (*  Title:      HOL/Fun_Def.thy
2     Author:     Alexander Krauss, TU Muenchen
3 *)
5 header {* Function Definitions and Termination Proofs *}
7 theory Fun_Def
8 imports Partial_Function SAT
9 keywords "function" "termination" :: thy_goal and "fun" "fun_cases" :: thy_decl
10 begin
12 subsection {* Definitions with default value *}
14 definition
15   THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
16   "THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)"
18 lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)"
19   by (simp add: theI' THE_default_def)
21 lemma THE_default1_equality:
22     "\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> THE_default d P = a"
23   by (simp add: the1_equality THE_default_def)
25 lemma THE_default_none:
26     "\<not>(\<exists>!x. P x) \<Longrightarrow> THE_default d P = d"
30 lemma fundef_ex1_existence:
31   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
32   assumes ex1: "\<exists>!y. G x y"
33   shows "G x (f x)"
34   apply (simp only: f_def)
35   apply (rule THE_defaultI')
36   apply (rule ex1)
37   done
39 lemma fundef_ex1_uniqueness:
40   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
41   assumes ex1: "\<exists>!y. G x y"
42   assumes elm: "G x (h x)"
43   shows "h x = f x"
44   apply (simp only: f_def)
45   apply (rule THE_default1_equality [symmetric])
46    apply (rule ex1)
47   apply (rule elm)
48   done
50 lemma fundef_ex1_iff:
51   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
52   assumes ex1: "\<exists>!y. G x y"
53   shows "(G x y) = (f x = y)"
54   apply (auto simp:ex1 f_def THE_default1_equality)
55   apply (rule THE_defaultI')
56   apply (rule ex1)
57   done
59 lemma fundef_default_value:
60   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
61   assumes graph: "\<And>x y. G x y \<Longrightarrow> D x"
62   assumes "\<not> D x"
63   shows "f x = d x"
64 proof -
65   have "\<not>(\<exists>y. G x y)"
66   proof
67     assume "\<exists>y. G x y"
68     hence "D x" using graph ..
69     with `\<not> D x` show False ..
70   qed
71   hence "\<not>(\<exists>!y. G x y)" by blast
73   thus ?thesis
74     unfolding f_def
75     by (rule THE_default_none)
76 qed
78 definition in_rel_def[simp]:
79   "in_rel R x y == (x, y) \<in> R"
81 lemma wf_in_rel:
82   "wf R \<Longrightarrow> wfP (in_rel R)"
85 ML_file "Tools/Function/function_core.ML"
86 ML_file "Tools/Function/mutual.ML"
87 ML_file "Tools/Function/pattern_split.ML"
88 ML_file "Tools/Function/relation.ML"
89 ML_file "Tools/Function/function_elims.ML"
91 method_setup relation = {*
92   Args.term >> (fn t => fn ctxt => SIMPLE_METHOD' (Function_Relation.relation_infer_tac ctxt t))
93 *} "prove termination using a user-specified wellfounded relation"
95 ML_file "Tools/Function/function.ML"
96 ML_file "Tools/Function/pat_completeness.ML"
98 method_setup pat_completeness = {*
99   Scan.succeed (SIMPLE_METHOD' o Pat_Completeness.pat_completeness_tac)
100 *} "prove completeness of datatype patterns"
102 ML_file "Tools/Function/fun.ML"
103 ML_file "Tools/Function/induction_schema.ML"
105 method_setup induction_schema = {*
106   Scan.succeed (RAW_METHOD o Induction_Schema.induction_schema_tac)
107 *} "prove an induction principle"
109 setup {*
110   Function.setup
111   #> Function_Fun.setup
112 *}
114 subsection {* Measure Functions *}
116 inductive is_measure :: "('a \<Rightarrow> nat) \<Rightarrow> bool"
117 where is_measure_trivial: "is_measure f"
119 ML_file "Tools/Function/measure_functions.ML"
120 setup MeasureFunctions.setup
122 lemma measure_size[measure_function]: "is_measure size"
123 by (rule is_measure_trivial)
125 lemma measure_fst[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (fst p))"
126 by (rule is_measure_trivial)
127 lemma measure_snd[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (snd p))"
128 by (rule is_measure_trivial)
130 ML_file "Tools/Function/lexicographic_order.ML"
132 method_setup lexicographic_order = {*
133   Method.sections clasimp_modifiers >>
134   (K (SIMPLE_METHOD o Lexicographic_Order.lexicographic_order_tac false))
135 *} "termination prover for lexicographic orderings"
137 setup Lexicographic_Order.setup
140 subsection {* Congruence Rules *}
142 lemma let_cong [fundef_cong]:
143   "M = N \<Longrightarrow> (\<And>x. x = N \<Longrightarrow> f x = g x) \<Longrightarrow> Let M f = Let N g"
144   unfolding Let_def by blast
146 lemmas [fundef_cong] =
147   if_cong image_cong INT_cong UN_cong
148   bex_cong ball_cong imp_cong map_option_cong Option.bind_cong
150 lemma split_cong [fundef_cong]:
151   "(\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y) \<Longrightarrow> p = q
152     \<Longrightarrow> split f p = split g q"
153   by (auto simp: split_def)
155 lemma comp_cong [fundef_cong]:
156   "f (g x) = f' (g' x') \<Longrightarrow> (f o g) x = (f' o g') x'"
157   unfolding o_apply .
159 subsection {* Simp rules for termination proofs *}
161 lemma termination_basic_simps[termination_simp]:
162   "x < (y::nat) \<Longrightarrow> x < y + z"
163   "x < z \<Longrightarrow> x < y + z"
164   "x \<le> y \<Longrightarrow> x \<le> y + (z::nat)"
165   "x \<le> z \<Longrightarrow> x \<le> y + (z::nat)"
166   "x < y \<Longrightarrow> x \<le> (y::nat)"
167 by arith+
169 declare le_imp_less_Suc[termination_simp]
171 lemma prod_size_simp[termination_simp]:
172   "prod_size f g p = f (fst p) + g (snd p) + Suc 0"
173 by (induct p) auto
175 subsection {* Decomposition *}
177 lemma less_by_empty:
178   "A = {} \<Longrightarrow> A \<subseteq> B"
179 and  union_comp_emptyL:
180   "\<lbrakk> A O C = {}; B O C = {} \<rbrakk> \<Longrightarrow> (A \<union> B) O C = {}"
181 and union_comp_emptyR:
182   "\<lbrakk> A O B = {}; A O C = {} \<rbrakk> \<Longrightarrow> A O (B \<union> C) = {}"
183 and wf_no_loop:
184   "R O R = {} \<Longrightarrow> wf R"
185 by (auto simp add: wf_comp_self[of R])
188 subsection {* Reduction Pairs *}
190 definition
191   "reduction_pair P = (wf (fst P) \<and> fst P O snd P \<subseteq> fst P)"
193 lemma reduction_pairI[intro]: "wf R \<Longrightarrow> R O S \<subseteq> R \<Longrightarrow> reduction_pair (R, S)"
194 unfolding reduction_pair_def by auto
196 lemma reduction_pair_lemma:
197   assumes rp: "reduction_pair P"
198   assumes "R \<subseteq> fst P"
199   assumes "S \<subseteq> snd P"
200   assumes "wf S"
201   shows "wf (R \<union> S)"
202 proof -
203   from rp `S \<subseteq> snd P` have "wf (fst P)" "fst P O S \<subseteq> fst P"
204     unfolding reduction_pair_def by auto
205   with `wf S` have "wf (fst P \<union> S)"
206     by (auto intro: wf_union_compatible)
207   moreover from `R \<subseteq> fst P` have "R \<union> S \<subseteq> fst P \<union> S" by auto
208   ultimately show ?thesis by (rule wf_subset)
209 qed
211 definition
212   "rp_inv_image = (\<lambda>(R,S) f. (inv_image R f, inv_image S f))"
214 lemma rp_inv_image_rp:
215   "reduction_pair P \<Longrightarrow> reduction_pair (rp_inv_image P f)"
216   unfolding reduction_pair_def rp_inv_image_def split_def
217   by force
220 subsection {* Concrete orders for SCNP termination proofs *}
222 definition "pair_less = less_than <*lex*> less_than"
223 definition "pair_leq = pair_less^="
224 definition "max_strict = max_ext pair_less"
225 definition "max_weak = max_ext pair_leq \<union> {({}, {})}"
226 definition "min_strict = min_ext pair_less"
227 definition "min_weak = min_ext pair_leq \<union> {({}, {})}"
229 lemma wf_pair_less[simp]: "wf pair_less"
230   by (auto simp: pair_less_def)
232 text {* Introduction rules for @{text pair_less}/@{text pair_leq} *}
233 lemma pair_leqI1: "a < b \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq"
234   and pair_leqI2: "a \<le> b \<Longrightarrow> s \<le> t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq"
235   and pair_lessI1: "a < b  \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
236   and pair_lessI2: "a \<le> b \<Longrightarrow> s < t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
237   unfolding pair_leq_def pair_less_def by auto
239 text {* Introduction rules for max *}
240 lemma smax_emptyI:
241   "finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> ({}, Y) \<in> max_strict"
242   and smax_insertI:
243   "\<lbrakk>y \<in> Y; (x, y) \<in> pair_less; (X, Y) \<in> max_strict\<rbrakk> \<Longrightarrow> (insert x X, Y) \<in> max_strict"
244   and wmax_emptyI:
245   "finite X \<Longrightarrow> ({}, X) \<in> max_weak"
246   and wmax_insertI:
247   "\<lbrakk>y \<in> YS; (x, y) \<in> pair_leq; (XS, YS) \<in> max_weak\<rbrakk> \<Longrightarrow> (insert x XS, YS) \<in> max_weak"
248 unfolding max_strict_def max_weak_def by (auto elim!: max_ext.cases)
250 text {* Introduction rules for min *}
251 lemma smin_emptyI:
252   "X \<noteq> {} \<Longrightarrow> (X, {}) \<in> min_strict"
253   and smin_insertI:
254   "\<lbrakk>x \<in> XS; (x, y) \<in> pair_less; (XS, YS) \<in> min_strict\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_strict"
255   and wmin_emptyI:
256   "(X, {}) \<in> min_weak"
257   and wmin_insertI:
258   "\<lbrakk>x \<in> XS; (x, y) \<in> pair_leq; (XS, YS) \<in> min_weak\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_weak"
259 by (auto simp: min_strict_def min_weak_def min_ext_def)
261 text {* Reduction Pairs *}
263 lemma max_ext_compat:
264   assumes "R O S \<subseteq> R"
265   shows "max_ext R O (max_ext S \<union> {({},{})}) \<subseteq> max_ext R"
266 using assms
267 apply auto
268 apply (elim max_ext.cases)
269 apply rule
270 apply auto[3]
271 apply (drule_tac x=xa in meta_spec)
272 apply simp
273 apply (erule bexE)
274 apply (drule_tac x=xb in meta_spec)
275 by auto
277 lemma max_rpair_set: "reduction_pair (max_strict, max_weak)"
278   unfolding max_strict_def max_weak_def
279 apply (intro reduction_pairI max_ext_wf)
280 apply simp
281 apply (rule max_ext_compat)
282 by (auto simp: pair_less_def pair_leq_def)
284 lemma min_ext_compat:
285   assumes "R O S \<subseteq> R"
286   shows "min_ext R O  (min_ext S \<union> {({},{})}) \<subseteq> min_ext R"
287 using assms
288 apply (auto simp: min_ext_def)
289 apply (drule_tac x=ya in bspec, assumption)
290 apply (erule bexE)
291 apply (drule_tac x=xc in bspec)
292 apply assumption
293 by auto
295 lemma min_rpair_set: "reduction_pair (min_strict, min_weak)"
296   unfolding min_strict_def min_weak_def
297 apply (intro reduction_pairI min_ext_wf)
298 apply simp
299 apply (rule min_ext_compat)
300 by (auto simp: pair_less_def pair_leq_def)
303 subsection {* Tool setup *}
305 ML_file "Tools/Function/termination.ML"
306 ML_file "Tools/Function/scnp_solve.ML"
307 ML_file "Tools/Function/scnp_reconstruct.ML"
308 ML_file "Tools/Function/fun_cases.ML"
310 setup ScnpReconstruct.setup
312 ML_val -- "setup inactive"
313 {*
314   Context.theory_map (Function_Common.set_termination_prover
315     (ScnpReconstruct.decomp_scnp_tac [ScnpSolve.MAX, ScnpSolve.MIN, ScnpSolve.MS]))
316 *}
318 end