src/HOL/Fun_Def.thy
 author wenzelm Tue Sep 26 20:54:40 2017 +0200 (21 months ago) changeset 66695 91500c024c7f parent 64591 240a39af9ec4 child 67443 3abf6a722518 permissions -rw-r--r--
tuned;
```     1 (*  Title:      HOL/Fun_Def.thy
```
```     2     Author:     Alexander Krauss, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Function Definitions and Termination Proofs\<close>
```
```     6
```
```     7 theory Fun_Def
```
```     8   imports Basic_BNF_LFPs Partial_Function SAT
```
```     9   keywords
```
```    10     "function" "termination" :: thy_goal and
```
```    11     "fun" "fun_cases" :: thy_decl
```
```    12 begin
```
```    13
```
```    14 subsection \<open>Definitions with default value\<close>
```
```    15
```
```    16 definition 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
```
```    19 lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)"
```
```    20   by (simp add: theI' THE_default_def)
```
```    21
```
```    22 lemma 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
```
```    25 lemma THE_default_none: "\<not> (\<exists>!x. P x) \<Longrightarrow> THE_default d P = d"
```
```    26   by (simp add: THE_default_def)
```
```    27
```
```    28
```
```    29 lemma 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
```
```    38 lemma 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
```
```    49 lemma 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
```
```    58 lemma 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"
```
```    63 proof -
```
```    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)
```
```    73 qed
```
```    74
```
```    75 definition in_rel_def[simp]: "in_rel R x y \<equiv> (x, y) \<in> R"
```
```    76
```
```    77 lemma wf_in_rel: "wf R \<Longrightarrow> wfP (in_rel R)"
```
```    78   by (simp add: wfP_def)
```
```    79
```
```    80 ML_file "Tools/Function/function_core.ML"
```
```    81 ML_file "Tools/Function/mutual.ML"
```
```    82 ML_file "Tools/Function/pattern_split.ML"
```
```    83 ML_file "Tools/Function/relation.ML"
```
```    84 ML_file "Tools/Function/function_elims.ML"
```
```    85
```
```    86 method_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
```
```    90 ML_file "Tools/Function/function.ML"
```
```    91 ML_file "Tools/Function/pat_completeness.ML"
```
```    92
```
```    93 method_setup pat_completeness = \<open>
```
```    94   Scan.succeed (SIMPLE_METHOD' o Pat_Completeness.pat_completeness_tac)
```
```    95 \<close> "prove completeness of (co)datatype patterns"
```
```    96
```
```    97 ML_file "Tools/Function/fun.ML"
```
```    98 ML_file "Tools/Function/induction_schema.ML"
```
```    99
```
```   100 method_setup induction_schema = \<open>
```
```   101   Scan.succeed (Method.CONTEXT_TACTIC oo Induction_Schema.induction_schema_tac)
```
```   102 \<close> "prove an induction principle"
```
```   103
```
```   104
```
```   105 subsection \<open>Measure functions\<close>
```
```   106
```
```   107 inductive is_measure :: "('a \<Rightarrow> nat) \<Rightarrow> bool"
```
```   108   where is_measure_trivial: "is_measure f"
```
```   109
```
```   110 named_theorems measure_function "rules that guide the heuristic generation of measure functions"
```
```   111 ML_file "Tools/Function/measure_functions.ML"
```
```   112
```
```   113 lemma measure_size[measure_function]: "is_measure size"
```
```   114   by (rule is_measure_trivial)
```
```   115
```
```   116 lemma measure_fst[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (fst p))"
```
```   117   by (rule is_measure_trivial)
```
```   118
```
```   119 lemma measure_snd[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (snd p))"
```
```   120   by (rule is_measure_trivial)
```
```   121
```
```   122 ML_file "Tools/Function/lexicographic_order.ML"
```
```   123
```
```   124 method_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
```
```   130 subsection \<open>Congruence rules\<close>
```
```   131
```
```   132 lemma 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
```
```   135 lemmas [fundef_cong] =
```
```   136   if_cong image_cong INF_cong SUP_cong
```
```   137   bex_cong ball_cong imp_cong map_option_cong Option.bind_cong
```
```   138
```
```   139 lemma 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
```
```   143 lemma 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
```
```   147 subsection \<open>Simp rules for termination proofs\<close>
```
```   148
```
```   149 declare
```
```   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
```
```   157 lemma 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
```
```   161 subsection \<open>Decomposition\<close>
```
```   162
```
```   163 lemma 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
```
```   170 subsection \<open>Reduction pairs\<close>
```
```   171
```
```   172 definition "reduction_pair P \<longleftrightarrow> wf (fst P) \<and> fst P O snd P \<subseteq> fst P"
```
```   173
```
```   174 lemma reduction_pairI[intro]: "wf R \<Longrightarrow> R O S \<subseteq> R \<Longrightarrow> reduction_pair (R, S)"
```
```   175   by (auto simp: reduction_pair_def)
```
```   176
```
```   177 lemma 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)"
```
```   183 proof -
```
```   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)
```
```   190 qed
```
```   191
```
```   192 definition "rp_inv_image = (\<lambda>(R,S) f. (inv_image R f, inv_image S f))"
```
```   193
```
```   194 lemma 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
```
```   198 subsection \<open>Concrete orders for SCNP termination proofs\<close>
```
```   199
```
```   200 definition "pair_less = less_than <*lex*> less_than"
```
```   201 definition "pair_leq = pair_less^="
```
```   202 definition "max_strict = max_ext pair_less"
```
```   203 definition "max_weak = max_ext pair_leq \<union> {({}, {})}"
```
```   204 definition "min_strict = min_ext pair_less"
```
```   205 definition "min_weak = min_ext pair_leq \<union> {({}, {})}"
```
```   206
```
```   207 lemma wf_pair_less[simp]: "wf pair_less"
```
```   208   by (auto simp: pair_less_def)
```
```   209
```
```   210 text \<open>Introduction rules for \<open>pair_less\<close>/\<open>pair_leq\<close>\<close>
```
```   211 lemma 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
```
```   217 text \<open>Introduction rules for max\<close>
```
```   218 lemma 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
```
```   226 text \<open>Introduction rules for min\<close>
```
```   227 lemma 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
```
```   235 text \<open>Reduction Pairs.\<close>
```
```   236
```
```   237 lemma 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
```
```   252 lemma 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
```
```   260 lemma 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
```
```   272 lemma 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
```
```   281 subsection \<open>Yet another induction principle on the natural numbers\<close>
```
```   282
```
```   283 lemma 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
```
```   291 subsection \<open>Tool setup\<close>
```
```   292
```
```   293 ML_file "Tools/Function/termination.ML"
```
```   294 ML_file "Tools/Function/scnp_solve.ML"
```
```   295 ML_file "Tools/Function/scnp_reconstruct.ML"
```
```   296 ML_file "Tools/Function/fun_cases.ML"
```
```   297
```
```   298 ML_val \<comment> "setup inactive"
```
```   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
```
```   304 end
```