Theory Fun_Def

(*  Title:      HOL/Fun_Def.thy
    Author:     Alexander Krauss, TU Muenchen

section ‹Function Definitions and Termination Proofs›

theory Fun_Def
  imports Basic_BNF_LFPs Partial_Function SAT
    "function" "termination" :: thy_goal_defn and
    "fun" "fun_cases" :: thy_defn

subsection ‹Definitions with default value›

definition THE_default :: "'a  ('a  bool)  'a"
  where "THE_default d P = (if (∃!x. P x) then (THE x. P x) else d)"

lemma THE_defaultI': "∃!x. P x  P (THE_default d P)"
  by (simp add: theI' THE_default_def)

lemma THE_default1_equality: "∃!x. P x  P a  THE_default d P = a"
  by (simp add: the1_equality THE_default_def)

lemma THE_default_none: "¬ (∃!x. P x)  THE_default d P = d"
  by (simp add: THE_default_def)

lemma fundef_ex1_existence:
  assumes f_def: "f  (λx::'a. THE_default (d x) (λy. G x y))"
  assumes ex1: "∃!y. G x y"
  shows "G x (f x)"
  apply (simp only: f_def)
  apply (rule THE_defaultI')
  apply (rule ex1)

lemma fundef_ex1_uniqueness:
  assumes f_def: "f  (λx::'a. THE_default (d x) (λy. G x y))"
  assumes ex1: "∃!y. G x y"
  assumes elm: "G x (h x)"
  shows "h x = f x"
  by (auto simp add: f_def ex1 elm THE_default1_equality[symmetric])

lemma fundef_ex1_iff:
  assumes f_def: "f  (λx::'a. THE_default (d x) (λy. G x y))"
  assumes ex1: "∃!y. G x y"
  shows "(G x y) = (f x = y)"
  by (auto simp add: ex1 f_def THE_default1_equality THE_defaultI')

lemma fundef_default_value:
  assumes f_def: "f  (λx::'a. THE_default (d x) (λy. G x y))"
  assumes graph: "x y. G x y  D x"
  assumes "¬ D x"
  shows "f x = d x"
proof -
  have "¬(y. G x y)"
    assume "y. G x y"
    then have "D x" using graph ..
    with ¬ D x show False ..
  then have "¬(∃!y. G x y)" by blast
  then show ?thesis
    unfolding f_def by (rule THE_default_none)

definition in_rel_def[simp]: "in_rel R x y  (x, y)  R"

lemma wf_in_rel: "wf R  wfP (in_rel R)"
  by (simp add: wfP_def)

ML_file ‹Tools/Function/function_core.ML›
ML_file ‹Tools/Function/mutual.ML›
ML_file ‹Tools/Function/pattern_split.ML›
ML_file ‹Tools/Function/relation.ML›
ML_file ‹Tools/Function/function_elims.ML›

method_setup relation = Args.term >> (fn t => fn ctxt => SIMPLE_METHOD' (Function_Relation.relation_infer_tac ctxt t)) "prove termination using a user-specified wellfounded relation"

ML_file ‹Tools/Function/function.ML›
ML_file ‹Tools/Function/pat_completeness.ML›

method_setup pat_completeness = Scan.succeed (SIMPLE_METHOD' o Pat_Completeness.pat_completeness_tac) "prove completeness of (co)datatype patterns"

ML_file ‹Tools/Function/fun.ML›
ML_file ‹Tools/Function/induction_schema.ML›

method_setup induction_schema = Scan.succeed (CONTEXT_TACTIC oo Induction_Schema.induction_schema_tac) "prove an induction principle"

subsection ‹Measure functions›

inductive is_measure :: "('a  nat)  bool"
  where is_measure_trivial: "is_measure f"

named_theorems measure_function "rules that guide the heuristic generation of measure functions"
ML_file ‹Tools/Function/measure_functions.ML›

lemma measure_size[measure_function]: "is_measure size"
  by (rule is_measure_trivial)

lemma measure_fst[measure_function]: "is_measure f  is_measure (λp. f (fst p))"
  by (rule is_measure_trivial)

lemma measure_snd[measure_function]: "is_measure f  is_measure (λp. f (snd p))"
  by (rule is_measure_trivial)

ML_file ‹Tools/Function/lexicographic_order.ML›

method_setup lexicographic_order = Method.sections clasimp_modifiers >>
  (K (SIMPLE_METHOD o Lexicographic_Order.lexicographic_order_tac false)) "termination prover for lexicographic orderings"

subsection ‹Congruence rules›

lemma let_cong [fundef_cong]: "M = N  (x. x = N  f x = g x)  Let M f = Let N g"
  unfolding Let_def by blast

lemmas [fundef_cong] =
  if_cong image_cong
  bex_cong ball_cong imp_cong map_option_cong Option.bind_cong

lemma split_cong [fundef_cong]:
  "(x y. (x, y) = q  f x y = g x y)  p = q  case_prod f p = case_prod g q"
  by (auto simp: split_def)

lemma comp_cong [fundef_cong]: "f (g x) = f' (g' x')  (f  g) x = (f'  g') x'"
  by (simp only: o_apply)

subsection ‹Simp rules for termination proofs›


lemma size_prod_simp[termination_simp]: "size_prod f g p = f (fst p) + g (snd p) + Suc 0"
  by (induct p) auto

subsection ‹Decomposition›

lemma less_by_empty: "A = {}  A  B"
  and union_comp_emptyL: "A O C = {}  B O C = {}  (A  B) O C = {}"
  and union_comp_emptyR: "A O B = {}  A O C = {}  A O (B  C) = {}"
  and wf_no_loop: "R O R = {}  wf R"
  by (auto simp add: wf_comp_self [of R])

subsection ‹Reduction pairs›

definition "reduction_pair P  wf (fst P)  fst P O snd P  fst P"

lemma reduction_pairI[intro]: "wf R  R O S  R  reduction_pair (R, S)"
  by (auto simp: reduction_pair_def)

lemma reduction_pair_lemma:
  assumes rp: "reduction_pair P"
  assumes "R  fst P"
  assumes "S  snd P"
  assumes "wf S"
  shows "wf (R  S)"
proof -
  from rp S  snd P have "wf (fst P)" "fst P O S  fst P"
    unfolding reduction_pair_def by auto
  with wf S have "wf (fst P  S)"
    by (auto intro: wf_union_compatible)
  moreover from R  fst P have "R  S  fst P  S" by auto
  ultimately show ?thesis by (rule wf_subset)

definition "rp_inv_image = (λ(R,S) f. (inv_image R f, inv_image S f))"

lemma rp_inv_image_rp: "reduction_pair P  reduction_pair (rp_inv_image P f)"
  unfolding reduction_pair_def rp_inv_image_def split_def by force

subsection ‹Concrete orders for SCNP termination proofs›

definition "pair_less = less_than <*lex*> less_than"
definition "pair_leq = pair_less="
definition "max_strict = max_ext pair_less"
definition "max_weak = max_ext pair_leq  {({}, {})}"
definition "min_strict = min_ext pair_less"
definition "min_weak = min_ext pair_leq  {({}, {})}"

lemma wf_pair_less[simp]: "wf pair_less"
  by (auto simp: pair_less_def)

lemma total_pair_less [iff]: "total_on A pair_less" and trans_pair_less [iff]: "trans pair_less"
  by (auto simp: total_on_def pair_less_def)

text ‹Introduction rules for pair_less›/pair_leq›
lemma pair_leqI1: "a < b  ((a, s), (b, t))  pair_leq"
  and pair_leqI2: "a  b  s  t  ((a, s), (b, t))  pair_leq"
  and pair_lessI1: "a < b   ((a, s), (b, t))  pair_less"
  and pair_lessI2: "a  b  s < t  ((a, s), (b, t))  pair_less"
  by (auto simp: pair_leq_def pair_less_def)

lemma pair_less_iff1 [simp]: "((x,y), (x,z))  pair_less  y<z"
  by (simp add: pair_less_def)

text ‹Introduction rules for max›
lemma smax_emptyI: "finite Y  Y  {}  ({}, Y)  max_strict"
  and smax_insertI:
    "y  Y  (x, y)  pair_less  (X, Y)  max_strict  (insert x X, Y)  max_strict"
  and wmax_emptyI: "finite X  ({}, X)  max_weak"
  and wmax_insertI:
    "y  YS  (x, y)  pair_leq  (XS, YS)  max_weak  (insert x XS, YS)  max_weak"
  by (auto simp: max_strict_def max_weak_def elim!: max_ext.cases)

text ‹Introduction rules for min›
lemma smin_emptyI: "X  {}  (X, {})  min_strict"
  and smin_insertI:
    "x  XS  (x, y)  pair_less  (XS, YS)  min_strict  (XS, insert y YS)  min_strict"
  and wmin_emptyI: "(X, {})  min_weak"
  and wmin_insertI:
    "x  XS  (x, y)  pair_leq  (XS, YS)  min_weak  (XS, insert y YS)  min_weak"
  by (auto simp: min_strict_def min_weak_def min_ext_def)

text ‹Reduction Pairs.›

lemma max_ext_compat:
  assumes "R O S  R"
  shows "max_ext R O (max_ext S  {({}, {})})  max_ext R"
proof -
  have "X Y Z. (X, Y)  max_ext R  (Y, Z)  max_ext S  (X, Z)  max_ext R"
  proof -
    fix X Y Z
    assume "(X,Y)max_ext R"
      "(Y, Z)max_ext S"
    then have *: "finite X" "finite Y" "finite Z" "Y{}" "Z{}"
      "(x. xX  yY. (x, y)R)"
      "(y. yY  zZ. (y, z)S)"
      by (auto elim: max_ext.cases)
    moreover have "x. xX  zZ. (x, z)R"
    proof -
      fix x
      assume "xX"
      then obtain y where 1: "yY" "(x, y)R"
        using * by auto
      then obtain z where "zZ" "(y, z)S"
        using * by auto
      then show "zZ. (x, z)R"
        using assms 1 by (auto elim: max_ext.cases)
    ultimately show "(X,Z)max_ext R"
      by auto
  then show "max_ext R O (max_ext S  {({}, {})})  max_ext R"
    by auto

lemma max_rpair_set: "reduction_pair (max_strict, max_weak)"
  unfolding max_strict_def max_weak_def
  apply (intro reduction_pairI max_ext_wf)
   apply simp
  apply (rule max_ext_compat)
  apply (auto simp: pair_less_def pair_leq_def)

lemma min_ext_compat:
  assumes "R O S  R"
  shows "min_ext R O (min_ext S  {({},{})})  min_ext R"
proof -
  have "X Y Z z. yY. xX. (x, y)  R  zZ. yY. (y, z)  S
   z  Z  xX. (x, z)  R"
  proof -
    fix X Y Z z
    assume *: "yY. xX. (x, y)  R"
      "zZ. yY. (y, z)  S"
    then obtain y' where 1: "y'Y" "(y', z)  S"
      by auto
    then obtain x' where 2: "x'X" "(x', y')  R"
      using * by auto
    show "xX. (x, z)  R"
      using 1 2 assms by auto
  then show ?thesis
    using assms by (auto simp: min_ext_def)

lemma min_rpair_set: "reduction_pair (min_strict, min_weak)"
  unfolding min_strict_def min_weak_def
  apply (intro reduction_pairI min_ext_wf)
   apply simp
  apply (rule min_ext_compat)
  apply (auto simp: pair_less_def pair_leq_def)

subsection ‹Yet more induction principles on the natural numbers›

lemma nat_descend_induct [case_names base descend]:
  fixes P :: "nat  bool"
  assumes H1: "k. k > n  P k"
  assumes H2: "k. k  n  (i. i > k  P i)  P k"
  shows "P m"
  using assms by induction_schema (force intro!: wf_measure [of "λk. Suc n - k"])+

lemma induct_nat_012[case_names 0 1 ge2]:
  "P 0  P (Suc 0)  (n. P n  P (Suc n)  P (Suc (Suc n)))  P n"
proof (induction_schema, pat_completeness)
  show "wf (Wellfounded.measure id)"
    by simp
qed auto

subsection ‹Tool setup›

ML_file ‹Tools/Function/termination.ML›
ML_file ‹Tools/Function/scnp_solve.ML›
ML_file ‹Tools/Function/scnp_reconstruct.ML›
ML_file ‹Tools/Function/fun_cases.ML›

ML_val ― ‹setup inactive›
Context.theory_map (Function_Common.set_termination_prover
    (K (ScnpReconstruct.decomp_scnp_tac [ScnpSolve.MAX, ScnpSolve.MIN, ScnpSolve.MS])))