
wenzelm@20324: (*  Title:      HOL/FunDef.thy
wenzelm@20324:     Author:     Alexander Krauss, TU Muenchen
wenzelm@22816: *)
wenzelm@20324: 
krauss@29125: header {* Function Definitions and Termination Proofs *}
wenzelm@20324: 
krauss@19564: theory FunDef
krauss@26748: imports Wellfounded
wenzelm@22816: uses
krauss@29125:   "Tools/prop_logic.ML"
krauss@29125:   "Tools/sat_solver.ML"
krauss@33099:   ("Tools/Function/function_lib.ML")
krauss@33099:   ("Tools/Function/function_common.ML")
haftmann@31775:   ("Tools/Function/context_tree.ML")
krauss@33099:   ("Tools/Function/function_core.ML")
haftmann@31775:   ("Tools/Function/sum_tree.ML")
haftmann@31775:   ("Tools/Function/mutual.ML")
haftmann@31775:   ("Tools/Function/pattern_split.ML")
krauss@33099:   ("Tools/Function/function.ML")
krauss@33100:   ("Tools/Function/relation.ML")
haftmann@31775:   ("Tools/Function/measure_functions.ML")
haftmann@31775:   ("Tools/Function/lexicographic_order.ML")
krauss@33083:   ("Tools/Function/pat_completeness.ML")
krauss@33098:   ("Tools/Function/fun.ML")
krauss@33471:   ("Tools/Function/induction_schema.ML")
haftmann@31775:   ("Tools/Function/termination.ML")
haftmann@31775:   ("Tools/Function/scnp_solve.ML")
haftmann@31775:   ("Tools/Function/scnp_reconstruct.ML")
krauss@19564: begin
krauss@19564: 
krauss@29125: subsection {* Definitions with default value. *}
krauss@20536: 
krauss@20536: definition
wenzelm@21404:   THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
krauss@20536:   "THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)"
krauss@20536: 
krauss@20536: lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)"
wenzelm@22816:   by (simp add: theI' THE_default_def)
krauss@20536: 
wenzelm@22816: lemma THE_default1_equality:
wenzelm@22816:     "\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> THE_default d P = a"
wenzelm@22816:   by (simp add: the1_equality THE_default_def)
krauss@20536: 
krauss@20536: lemma THE_default_none:
wenzelm@22816:     "\<not>(\<exists>!x. P x) \<Longrightarrow> THE_default d P = d"
wenzelm@22816:   by (simp add:THE_default_def)
krauss@20536: 
krauss@20536: 
krauss@19564: lemma fundef_ex1_existence:
wenzelm@22816:   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
wenzelm@22816:   assumes ex1: "\<exists>!y. G x y"
wenzelm@22816:   shows "G x (f x)"
wenzelm@22816:   apply (simp only: f_def)
wenzelm@22816:   apply (rule THE_defaultI')
wenzelm@22816:   apply (rule ex1)
wenzelm@22816:   done
krauss@21051: 
krauss@19564: lemma fundef_ex1_uniqueness:
wenzelm@22816:   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
wenzelm@22816:   assumes ex1: "\<exists>!y. G x y"
wenzelm@22816:   assumes elm: "G x (h x)"
wenzelm@22816:   shows "h x = f x"
wenzelm@22816:   apply (simp only: f_def)
wenzelm@22816:   apply (rule THE_default1_equality [symmetric])
wenzelm@22816:    apply (rule ex1)
wenzelm@22816:   apply (rule elm)
wenzelm@22816:   done
krauss@19564: 
krauss@19564: lemma fundef_ex1_iff:
wenzelm@22816:   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
wenzelm@22816:   assumes ex1: "\<exists>!y. G x y"
wenzelm@22816:   shows "(G x y) = (f x = y)"
krauss@20536:   apply (auto simp:ex1 f_def THE_default1_equality)
wenzelm@22816:   apply (rule THE_defaultI')
wenzelm@22816:   apply (rule ex1)
wenzelm@22816:   done
krauss@19564: 
krauss@20654: lemma fundef_default_value:
wenzelm@22816:   assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
wenzelm@22816:   assumes graph: "\<And>x y. G x y \<Longrightarrow> D x"
wenzelm@22816:   assumes "\<not> D x"
wenzelm@22816:   shows "f x = d x"
krauss@20654: proof -
krauss@21051:   have "\<not>(\<exists>y. G x y)"
krauss@20654:   proof
krauss@21512:     assume "\<exists>y. G x y"
krauss@21512:     hence "D x" using graph ..
krauss@21512:     with `\<not> D x` show False ..
krauss@20654:   qed
krauss@21051:   hence "\<not>(\<exists>!y. G x y)" by blast
wenzelm@22816: 
krauss@20654:   thus ?thesis
krauss@20654:     unfolding f_def
krauss@20654:     by (rule THE_default_none)
krauss@20654: qed
krauss@20654: 
berghofe@23739: definition in_rel_def[simp]:
berghofe@23739:   "in_rel R x y == (x, y) \<in> R"
berghofe@23739: 
berghofe@23739: lemma wf_in_rel:
berghofe@23739:   "wf R \<Longrightarrow> wfP (in_rel R)"
berghofe@23739:   by (simp add: wfP_def)
berghofe@23739: 
krauss@33099: use "Tools/Function/function_lib.ML"
krauss@33099: use "Tools/Function/function_common.ML"
haftmann@31775: use "Tools/Function/context_tree.ML"
krauss@33099: use "Tools/Function/function_core.ML"
haftmann@31775: use "Tools/Function/sum_tree.ML"
haftmann@31775: use "Tools/Function/mutual.ML"
haftmann@31775: use "Tools/Function/pattern_split.ML"
krauss@33100: use "Tools/Function/relation.ML"
krauss@33099: use "Tools/Function/function.ML"
krauss@33083: use "Tools/Function/pat_completeness.ML"
krauss@33098: use "Tools/Function/fun.ML"
krauss@33471: use "Tools/Function/induction_schema.ML"
krauss@19564: 
krauss@25567: setup {* 
krauss@33099:   Function.setup
krauss@33083:   #> Pat_Completeness.setup
krauss@33098:   #> Function_Fun.setup
krauss@33471:   #> Induction_Schema.setup
krauss@25567: *}
krauss@19770: 
krauss@29125: subsection {* Measure Functions *}
krauss@29125: 
krauss@29125: inductive is_measure :: "('a \<Rightarrow> nat) \<Rightarrow> bool"
krauss@29125: where is_measure_trivial: "is_measure f"
krauss@29125: 
haftmann@31775: use "Tools/Function/measure_functions.ML"
krauss@29125: setup MeasureFunctions.setup
krauss@29125: 
krauss@29125: lemma measure_size[measure_function]: "is_measure size"
krauss@29125: by (rule is_measure_trivial)
krauss@29125: 
krauss@29125: lemma measure_fst[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (fst p))"
krauss@29125: by (rule is_measure_trivial)
krauss@29125: lemma measure_snd[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (snd p))"
krauss@29125: by (rule is_measure_trivial)
krauss@29125: 
haftmann@31775: use "Tools/Function/lexicographic_order.ML"
krauss@33099: setup Lexicographic_Order.setup 
krauss@29125: 
krauss@29125: 
krauss@29125: subsection {* Congruence Rules *}
krauss@29125: 
haftmann@22838: lemma let_cong [fundef_cong]:
haftmann@22838:   "M = N \<Longrightarrow> (\<And>x. x = N \<Longrightarrow> f x = g x) \<Longrightarrow> Let M f = Let N g"
wenzelm@22816:   unfolding Let_def by blast
krauss@22622: 
wenzelm@22816: lemmas [fundef_cong] =
haftmann@22838:   if_cong image_cong INT_cong UN_cong
haftmann@22838:   bex_cong ball_cong imp_cong
krauss@19564: 
wenzelm@22816: lemma split_cong [fundef_cong]:
haftmann@22838:   "(\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y) \<Longrightarrow> p = q
wenzelm@22816:     \<Longrightarrow> split f p = split g q"
wenzelm@22816:   by (auto simp: split_def)
krauss@19934: 
wenzelm@22816: lemma comp_cong [fundef_cong]:
haftmann@22838:   "f (g x) = f' (g' x') \<Longrightarrow> (f o g) x = (f' o g') x'"
wenzelm@22816:   unfolding o_apply .
krauss@19934: 
krauss@29125: subsection {* Simp rules for termination proofs *}
krauss@26875: 
krauss@26749: lemma termination_basic_simps[termination_simp]:
krauss@26749:   "x < (y::nat) \<Longrightarrow> x < y + z" 
krauss@26749:   "x < z \<Longrightarrow> x < y + z"
krauss@26875:   "x \<le> y \<Longrightarrow> x \<le> y + (z::nat)"
krauss@26875:   "x \<le> z \<Longrightarrow> x \<le> y + (z::nat)"
krauss@26875:   "x < y \<Longrightarrow> x \<le> (y::nat)"
krauss@26749: by arith+
krauss@26749: 
krauss@26875: declare le_imp_less_Suc[termination_simp]
krauss@26875: 
krauss@26875: lemma prod_size_simp[termination_simp]:
krauss@26875:   "prod_size f g p = f (fst p) + g (snd p) + Suc 0"
krauss@26875: by (induct p) auto
krauss@26875: 
krauss@29125: subsection {* Decomposition *}
krauss@29125: 
krauss@29125: lemma less_by_empty: 
krauss@29125:   "A = {} \<Longrightarrow> A \<subseteq> B"
krauss@29125: and  union_comp_emptyL:
krauss@29125:   "\<lbrakk> A O C = {}; B O C = {} \<rbrakk> \<Longrightarrow> (A \<union> B) O C = {}"
krauss@29125: and union_comp_emptyR:
krauss@29125:   "\<lbrakk> A O B = {}; A O C = {} \<rbrakk> \<Longrightarrow> A O (B \<union> C) = {}"
krauss@29125: and wf_no_loop: 
krauss@29125:   "R O R = {} \<Longrightarrow> wf R"
krauss@29125: by (auto simp add: wf_comp_self[of R])
krauss@29125: 
krauss@29125: 
krauss@29125: subsection {* Reduction Pairs *}
krauss@29125: 
krauss@29125: definition
krauss@32235:   "reduction_pair P = (wf (fst P) \<and> fst P O snd P \<subseteq> fst P)"
krauss@29125: 
krauss@32235: lemma reduction_pairI[intro]: "wf R \<Longrightarrow> R O S \<subseteq> R \<Longrightarrow> reduction_pair (R, S)"
krauss@29125: unfolding reduction_pair_def by auto
krauss@29125: 
krauss@29125: lemma reduction_pair_lemma:
krauss@29125:   assumes rp: "reduction_pair P"
krauss@29125:   assumes "R \<subseteq> fst P"
krauss@29125:   assumes "S \<subseteq> snd P"
krauss@29125:   assumes "wf S"
krauss@29125:   shows "wf (R \<union> S)"
krauss@29125: proof -
krauss@32235:   from rp `S \<subseteq> snd P` have "wf (fst P)" "fst P O S \<subseteq> fst P"
krauss@29125:     unfolding reduction_pair_def by auto
krauss@29125:   with `wf S` have "wf (fst P \<union> S)" 
krauss@29125:     by (auto intro: wf_union_compatible)
krauss@29125:   moreover from `R \<subseteq> fst P` have "R \<union> S \<subseteq> fst P \<union> S" by auto
krauss@29125:   ultimately show ?thesis by (rule wf_subset) 
krauss@29125: qed
krauss@29125: 
krauss@29125: definition
krauss@29125:   "rp_inv_image = (\<lambda>(R,S) f. (inv_image R f, inv_image S f))"
krauss@29125: 
krauss@29125: lemma rp_inv_image_rp:
krauss@29125:   "reduction_pair P \<Longrightarrow> reduction_pair (rp_inv_image P f)"
krauss@29125:   unfolding reduction_pair_def rp_inv_image_def split_def
krauss@29125:   by force
krauss@29125: 
krauss@29125: 
krauss@29125: subsection {* Concrete orders for SCNP termination proofs *}
krauss@29125: 
krauss@29125: definition "pair_less = less_than <*lex*> less_than"
haftmann@30428: definition [code del]: "pair_leq = pair_less^="
krauss@29125: definition "max_strict = max_ext pair_less"
haftmann@30428: definition [code del]: "max_weak = max_ext pair_leq \<union> {({}, {})}"
haftmann@30428: definition [code del]: "min_strict = min_ext pair_less"
haftmann@30446: definition [code del]: "min_weak = min_ext pair_leq \<union> {({}, {})}"
krauss@29125: 
krauss@29125: lemma wf_pair_less[simp]: "wf pair_less"
krauss@29125:   by (auto simp: pair_less_def)
krauss@29125: 
wenzelm@29127: text {* Introduction rules for @{text pair_less}/@{text pair_leq} *}
krauss@29125: lemma pair_leqI1: "a < b \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq"
krauss@29125:   and pair_leqI2: "a \<le> b \<Longrightarrow> s \<le> t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq"
krauss@29125:   and pair_lessI1: "a < b  \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
krauss@29125:   and pair_lessI2: "a \<le> b \<Longrightarrow> s < t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
krauss@29125:   unfolding pair_leq_def pair_less_def by auto
krauss@29125: 
krauss@29125: text {* Introduction rules for max *}
krauss@29125: lemma smax_emptyI: 
krauss@29125:   "finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> ({}, Y) \<in> max_strict" 
krauss@29125:   and smax_insertI: 
krauss@29125:   "\<lbrakk>y \<in> Y; (x, y) \<in> pair_less; (X, Y) \<in> max_strict\<rbrakk> \<Longrightarrow> (insert x X, Y) \<in> max_strict"
krauss@29125:   and wmax_emptyI: 
krauss@29125:   "finite X \<Longrightarrow> ({}, X) \<in> max_weak" 
krauss@29125:   and wmax_insertI:
krauss@29125:   "\<lbrakk>y \<in> YS; (x, y) \<in> pair_leq; (XS, YS) \<in> max_weak\<rbrakk> \<Longrightarrow> (insert x XS, YS) \<in> max_weak" 
krauss@29125: unfolding max_strict_def max_weak_def by (auto elim!: max_ext.cases)
krauss@29125: 
krauss@29125: text {* Introduction rules for min *}
krauss@29125: lemma smin_emptyI: 
krauss@29125:   "X \<noteq> {} \<Longrightarrow> (X, {}) \<in> min_strict" 
krauss@29125:   and smin_insertI: 
krauss@29125:   "\<lbrakk>x \<in> XS; (x, y) \<in> pair_less; (XS, YS) \<in> min_strict\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_strict"
krauss@29125:   and wmin_emptyI: 
krauss@29125:   "(X, {}) \<in> min_weak" 
krauss@29125:   and wmin_insertI: 
krauss@29125:   "\<lbrakk>x \<in> XS; (x, y) \<in> pair_leq; (XS, YS) \<in> min_weak\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_weak" 
krauss@29125: by (auto simp: min_strict_def min_weak_def min_ext_def)
krauss@29125: 
krauss@29125: text {* Reduction Pairs *}
krauss@29125: 
krauss@29125: lemma max_ext_compat: 
krauss@32235:   assumes "R O S \<subseteq> R"
krauss@32235:   shows "max_ext R O (max_ext S \<union> {({},{})}) \<subseteq> max_ext R"
krauss@29125: using assms 
krauss@29125: apply auto
krauss@29125: apply (elim max_ext.cases)
krauss@29125: apply rule
krauss@29125: apply auto[3]
krauss@29125: apply (drule_tac x=xa in meta_spec)
krauss@29125: apply simp
krauss@29125: apply (erule bexE)
krauss@29125: apply (drule_tac x=xb in meta_spec)
krauss@29125: by auto
krauss@29125: 
krauss@29125: lemma max_rpair_set: "reduction_pair (max_strict, max_weak)"
krauss@29125:   unfolding max_strict_def max_weak_def 
krauss@29125: apply (intro reduction_pairI max_ext_wf)
krauss@29125: apply simp
krauss@29125: apply (rule max_ext_compat)
krauss@29125: by (auto simp: pair_less_def pair_leq_def)
krauss@29125: 
krauss@29125: lemma min_ext_compat: 
krauss@32235:   assumes "R O S \<subseteq> R"
krauss@32235:   shows "min_ext R O  (min_ext S \<union> {({},{})}) \<subseteq> min_ext R"
krauss@29125: using assms 
krauss@29125: apply (auto simp: min_ext_def)
krauss@29125: apply (drule_tac x=ya in bspec, assumption)
krauss@29125: apply (erule bexE)
krauss@29125: apply (drule_tac x=xc in bspec)
krauss@29125: apply assumption
krauss@29125: by auto
krauss@29125: 
krauss@29125: lemma min_rpair_set: "reduction_pair (min_strict, min_weak)"
krauss@29125:   unfolding min_strict_def min_weak_def 
krauss@29125: apply (intro reduction_pairI min_ext_wf)
krauss@29125: apply simp
krauss@29125: apply (rule min_ext_compat)
krauss@29125: by (auto simp: pair_less_def pair_leq_def)
krauss@29125: 
krauss@29125: 
krauss@29125: subsection {* Tool setup *}
krauss@29125: 
haftmann@31775: use "Tools/Function/termination.ML"
haftmann@31775: use "Tools/Function/scnp_solve.ML"
haftmann@31775: use "Tools/Function/scnp_reconstruct.ML"
krauss@29125: 
krauss@29125: setup {* ScnpReconstruct.setup *}
wenzelm@30480: 
wenzelm@30480: ML_val -- "setup inactive"
wenzelm@30480: {*
krauss@36521:   Context.theory_map (Function_Common.set_termination_prover
krauss@36521:     (ScnpReconstruct.decomp_scnp_tac [ScnpSolve.MAX, ScnpSolve.MIN, ScnpSolve.MS]))
krauss@29125: *}
krauss@26875: 
krauss@19564: end




