1(* Title: HOL/Fun_Def.thy 2 Author: Alexander Krauss, TU Muenchen 3*) 4 5section \<open>Function Definitions and Termination Proofs\<close> 6 7theory Fun_Def 8 imports Basic_BNF_LFPs Partial_Function SAT 9 keywords 10 "function" "termination" :: thy_goal and 11 "fun" "fun_cases" :: thy_decl 12begin 13 14subsection \<open>Definitions with default value\<close> 15 16definition THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" 17 where "THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)" 18 19lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)" 20 by (simp add: theI' THE_default_def) 21 22lemma THE_default1_equality: "\<exists>!x. P x \<Longrightarrow> P a \<Longrightarrow> THE_default d P = a" 23 by (simp add: the1_equality THE_default_def) 24 25lemma THE_default_none: "\<not> (\<exists>!x. P x) \<Longrightarrow> THE_default d P = d" 26 by (simp add: THE_default_def) 27 28 29lemma fundef_ex1_existence: 30 assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 31 assumes ex1: "\<exists>!y. G x y" 32 shows "G x (f x)" 33 apply (simp only: f_def) 34 apply (rule THE_defaultI') 35 apply (rule ex1) 36 done 37 38lemma fundef_ex1_uniqueness: 39 assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 40 assumes ex1: "\<exists>!y. G x y" 41 assumes elm: "G x (h x)" 42 shows "h x = f x" 43 apply (simp only: f_def) 44 apply (rule THE_default1_equality [symmetric]) 45 apply (rule ex1) 46 apply (rule elm) 47 done 48 49lemma fundef_ex1_iff: 50 assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 51 assumes ex1: "\<exists>!y. G x y" 52 shows "(G x y) = (f x = y)" 53 apply (auto simp:ex1 f_def THE_default1_equality) 54 apply (rule THE_defaultI') 55 apply (rule ex1) 56 done 57 58lemma fundef_default_value: 59 assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))" 60 assumes graph: "\<And>x y. G x y \<Longrightarrow> D x" 61 assumes "\<not> D x" 62 shows "f x = d x" 63proof - 64 have "\<not>(\<exists>y. G x y)" 65 proof 66 assume "\<exists>y. G x y" 67 then have "D x" using graph .. 68 with \<open>\<not> D x\<close> show False .. 69 qed 70 then have "\<not>(\<exists>!y. G x y)" by blast 71 then show ?thesis 72 unfolding f_def by (rule THE_default_none) 73qed 74 75definition in_rel_def[simp]: "in_rel R x y \<equiv> (x, y) \<in> R" 76 77lemma wf_in_rel: "wf R \<Longrightarrow> wfP (in_rel R)" 78 by (simp add: wfP_def) 79 80ML_file "Tools/Function/function_core.ML" 81ML_file "Tools/Function/mutual.ML" 82ML_file "Tools/Function/pattern_split.ML" 83ML_file "Tools/Function/relation.ML" 84ML_file "Tools/Function/function_elims.ML" 85 86method_setup relation = \<open> 87 Args.term >> (fn t => fn ctxt => SIMPLE_METHOD' (Function_Relation.relation_infer_tac ctxt t)) 88\<close> "prove termination using a user-specified wellfounded relation" 89 90ML_file "Tools/Function/function.ML" 91ML_file "Tools/Function/pat_completeness.ML" 92 93method_setup pat_completeness = \<open> 94 Scan.succeed (SIMPLE_METHOD' o Pat_Completeness.pat_completeness_tac) 95\<close> "prove completeness of (co)datatype patterns" 96 97ML_file "Tools/Function/fun.ML" 98ML_file "Tools/Function/induction_schema.ML" 99 100method_setup induction_schema = \<open> 101 Scan.succeed (Method.CONTEXT_TACTIC oo Induction_Schema.induction_schema_tac) 102\<close> "prove an induction principle" 103 104 105subsection \<open>Measure functions\<close> 106 107inductive is_measure :: "('a \<Rightarrow> nat) \<Rightarrow> bool" 108 where is_measure_trivial: "is_measure f" 109 110named_theorems measure_function "rules that guide the heuristic generation of measure functions" 111ML_file "Tools/Function/measure_functions.ML" 112 113lemma measure_size[measure_function]: "is_measure size" 114 by (rule is_measure_trivial) 115 116lemma measure_fst[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (fst p))" 117 by (rule is_measure_trivial) 118 119lemma measure_snd[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (snd p))" 120 by (rule is_measure_trivial) 121 122ML_file "Tools/Function/lexicographic_order.ML" 123 124method_setup lexicographic_order = \<open> 125 Method.sections clasimp_modifiers >> 126 (K (SIMPLE_METHOD o Lexicographic_Order.lexicographic_order_tac false)) 127\<close> "termination prover for lexicographic orderings" 128 129 130subsection \<open>Congruence rules\<close> 131 132lemma let_cong [fundef_cong]: "M = N \<Longrightarrow> (\<And>x. x = N \<Longrightarrow> f x = g x) \<Longrightarrow> Let M f = Let N g" 133 unfolding Let_def by blast 134 135lemmas [fundef_cong] = 136 if_cong image_cong 137 bex_cong ball_cong imp_cong map_option_cong Option.bind_cong 138 139lemma split_cong [fundef_cong]: 140 "(\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y) \<Longrightarrow> p = q \<Longrightarrow> case_prod f p = case_prod g q" 141 by (auto simp: split_def) 142 143lemma comp_cong [fundef_cong]: "f (g x) = f' (g' x') \<Longrightarrow> (f \<circ> g) x = (f' \<circ> g') x'" 144 by (simp only: o_apply) 145 146 147subsection \<open>Simp rules for termination proofs\<close> 148 149declare 150 trans_less_add1[termination_simp] 151 trans_less_add2[termination_simp] 152 trans_le_add1[termination_simp] 153 trans_le_add2[termination_simp] 154 less_imp_le_nat[termination_simp] 155 le_imp_less_Suc[termination_simp] 156 157lemma size_prod_simp[termination_simp]: "size_prod f g p = f (fst p) + g (snd p) + Suc 0" 158 by (induct p) auto 159 160 161subsection \<open>Decomposition\<close> 162 163lemma less_by_empty: "A = {} \<Longrightarrow> A \<subseteq> B" 164 and union_comp_emptyL: "A O C = {} \<Longrightarrow> B O C = {} \<Longrightarrow> (A \<union> B) O C = {}" 165 and union_comp_emptyR: "A O B = {} \<Longrightarrow> A O C = {} \<Longrightarrow> A O (B \<union> C) = {}" 166 and wf_no_loop: "R O R = {} \<Longrightarrow> wf R" 167 by (auto simp add: wf_comp_self [of R]) 168 169 170subsection \<open>Reduction pairs\<close> 171 172definition "reduction_pair P \<longleftrightarrow> wf (fst P) \<and> fst P O snd P \<subseteq> fst P" 173 174lemma reduction_pairI[intro]: "wf R \<Longrightarrow> R O S \<subseteq> R \<Longrightarrow> reduction_pair (R, S)" 175 by (auto simp: reduction_pair_def) 176 177lemma reduction_pair_lemma: 178 assumes rp: "reduction_pair P" 179 assumes "R \<subseteq> fst P" 180 assumes "S \<subseteq> snd P" 181 assumes "wf S" 182 shows "wf (R \<union> S)" 183proof - 184 from rp \<open>S \<subseteq> snd P\<close> have "wf (fst P)" "fst P O S \<subseteq> fst P" 185 unfolding reduction_pair_def by auto 186 with \<open>wf S\<close> have "wf (fst P \<union> S)" 187 by (auto intro: wf_union_compatible) 188 moreover from \<open>R \<subseteq> fst P\<close> have "R \<union> S \<subseteq> fst P \<union> S" by auto 189 ultimately show ?thesis by (rule wf_subset) 190qed 191 192definition "rp_inv_image = (\<lambda>(R,S) f. (inv_image R f, inv_image S f))" 193 194lemma rp_inv_image_rp: "reduction_pair P \<Longrightarrow> reduction_pair (rp_inv_image P f)" 195 unfolding reduction_pair_def rp_inv_image_def split_def by force 196 197 198subsection \<open>Concrete orders for SCNP termination proofs\<close> 199 200definition "pair_less = less_than <*lex*> less_than" 201definition "pair_leq = pair_less\<^sup>=" 202definition "max_strict = max_ext pair_less" 203definition "max_weak = max_ext pair_leq \<union> {({}, {})}" 204definition "min_strict = min_ext pair_less" 205definition "min_weak = min_ext pair_leq \<union> {({}, {})}" 206 207lemma wf_pair_less[simp]: "wf pair_less" 208 by (auto simp: pair_less_def) 209 210text \<open>Introduction rules for \<open>pair_less\<close>/\<open>pair_leq\<close>\<close> 211lemma pair_leqI1: "a < b \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq" 212 and pair_leqI2: "a \<le> b \<Longrightarrow> s \<le> t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq" 213 and pair_lessI1: "a < b \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less" 214 and pair_lessI2: "a \<le> b \<Longrightarrow> s < t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less" 215 by (auto simp: pair_leq_def pair_less_def) 216 217text \<open>Introduction rules for max\<close> 218lemma smax_emptyI: "finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> ({}, Y) \<in> max_strict" 219 and smax_insertI: 220 "y \<in> Y \<Longrightarrow> (x, y) \<in> pair_less \<Longrightarrow> (X, Y) \<in> max_strict \<Longrightarrow> (insert x X, Y) \<in> max_strict" 221 and wmax_emptyI: "finite X \<Longrightarrow> ({}, X) \<in> max_weak" 222 and wmax_insertI: 223 "y \<in> YS \<Longrightarrow> (x, y) \<in> pair_leq \<Longrightarrow> (XS, YS) \<in> max_weak \<Longrightarrow> (insert x XS, YS) \<in> max_weak" 224 by (auto simp: max_strict_def max_weak_def elim!: max_ext.cases) 225 226text \<open>Introduction rules for min\<close> 227lemma smin_emptyI: "X \<noteq> {} \<Longrightarrow> (X, {}) \<in> min_strict" 228 and smin_insertI: 229 "x \<in> XS \<Longrightarrow> (x, y) \<in> pair_less \<Longrightarrow> (XS, YS) \<in> min_strict \<Longrightarrow> (XS, insert y YS) \<in> min_strict" 230 and wmin_emptyI: "(X, {}) \<in> min_weak" 231 and wmin_insertI: 232 "x \<in> XS \<Longrightarrow> (x, y) \<in> pair_leq \<Longrightarrow> (XS, YS) \<in> min_weak \<Longrightarrow> (XS, insert y YS) \<in> min_weak" 233 by (auto simp: min_strict_def min_weak_def min_ext_def) 234 235text \<open>Reduction Pairs.\<close> 236 237lemma max_ext_compat: 238 assumes "R O S \<subseteq> R" 239 shows "max_ext R O (max_ext S \<union> {({}, {})}) \<subseteq> max_ext R" 240 using assms 241 apply auto 242 apply (elim max_ext.cases) 243 apply rule 244 apply auto[3] 245 apply (drule_tac x=xa in meta_spec) 246 apply simp 247 apply (erule bexE) 248 apply (drule_tac x=xb in meta_spec) 249 apply auto 250 done 251 252lemma max_rpair_set: "reduction_pair (max_strict, max_weak)" 253 unfolding max_strict_def max_weak_def 254 apply (intro reduction_pairI max_ext_wf) 255 apply simp 256 apply (rule max_ext_compat) 257 apply (auto simp: pair_less_def pair_leq_def) 258 done 259 260lemma min_ext_compat: 261 assumes "R O S \<subseteq> R" 262 shows "min_ext R O (min_ext S \<union> {({},{})}) \<subseteq> min_ext R" 263 using assms 264 apply (auto simp: min_ext_def) 265 apply (drule_tac x=ya in bspec, assumption) 266 apply (erule bexE) 267 apply (drule_tac x=xc in bspec) 268 apply assumption 269 apply auto 270 done 271 272lemma min_rpair_set: "reduction_pair (min_strict, min_weak)" 273 unfolding min_strict_def min_weak_def 274 apply (intro reduction_pairI min_ext_wf) 275 apply simp 276 apply (rule min_ext_compat) 277 apply (auto simp: pair_less_def pair_leq_def) 278 done 279 280 281subsection \<open>Yet another induction principle on the natural numbers\<close> 282 283lemma nat_descend_induct [case_names base descend]: 284 fixes P :: "nat \<Rightarrow> bool" 285 assumes H1: "\<And>k. k > n \<Longrightarrow> P k" 286 assumes H2: "\<And>k. k \<le> n \<Longrightarrow> (\<And>i. i > k \<Longrightarrow> P i) \<Longrightarrow> P k" 287 shows "P m" 288 using assms by induction_schema (force intro!: wf_measure [of "\<lambda>k. Suc n - k"])+ 289 290 291subsection \<open>Tool setup\<close> 292 293ML_file "Tools/Function/termination.ML" 294ML_file "Tools/Function/scnp_solve.ML" 295ML_file "Tools/Function/scnp_reconstruct.ML" 296ML_file "Tools/Function/fun_cases.ML" 297 298ML_val \<comment> \<open>setup inactive\<close> 299\<open> 300 Context.theory_map (Function_Common.set_termination_prover 301 (K (ScnpReconstruct.decomp_scnp_tac [ScnpSolve.MAX, ScnpSolve.MIN, ScnpSolve.MS]))) 302\<close> 303 304end 305