src/HOL/FunDef.thy
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more standard method setup;
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(*  Title:      HOL/FunDef.thy
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    Author:     Alexander Krauss, TU Muenchen
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*)
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header {* Function Definitions and Termination Proofs *}
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theory FunDef
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imports Partial_Function Wellfounded
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keywords "function" "termination" :: thy_goal and "fun" :: thy_decl
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uses
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  "Tools/prop_logic.ML"
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  "Tools/sat_solver.ML"
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  ("Tools/Function/function_common.ML")
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  ("Tools/Function/context_tree.ML")
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  ("Tools/Function/function_core.ML")
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  ("Tools/Function/sum_tree.ML")
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  ("Tools/Function/mutual.ML")
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  ("Tools/Function/pattern_split.ML")
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  ("Tools/Function/function.ML")
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  ("Tools/Function/relation.ML")
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  ("Tools/Function/measure_functions.ML")
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  ("Tools/Function/lexicographic_order.ML")
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  ("Tools/Function/pat_completeness.ML")
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  ("Tools/Function/fun.ML")
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  ("Tools/Function/induction_schema.ML")
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  ("Tools/Function/termination.ML")
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  ("Tools/Function/scnp_solve.ML")
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  ("Tools/Function/scnp_reconstruct.ML")
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begin
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subsection {* Definitions with default value. *}
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definition
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  THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
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  "THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)"
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lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)"
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  by (simp add: theI' THE_default_def)
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lemma THE_default1_equality:
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    "\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> THE_default d P = a"
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  by (simp add: the1_equality THE_default_def)
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lemma THE_default_none:
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    "\<not>(\<exists>!x. P x) \<Longrightarrow> THE_default d P = d"
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  by (simp add:THE_default_def)
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lemma fundef_ex1_existence:
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  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
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  assumes ex1: "\<exists>!y. G x y"
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  shows "G x (f x)"
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  apply (simp only: f_def)
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  apply (rule THE_defaultI')
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  apply (rule ex1)
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  done
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lemma fundef_ex1_uniqueness:
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  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
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  assumes ex1: "\<exists>!y. G x y"
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  assumes elm: "G x (h x)"
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  shows "h x = f x"
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  apply (simp only: f_def)
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  apply (rule THE_default1_equality [symmetric])
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   apply (rule ex1)
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  apply (rule elm)
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  done
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lemma fundef_ex1_iff:
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  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
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  assumes ex1: "\<exists>!y. G x y"
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  shows "(G x y) = (f x = y)"
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  apply (auto simp:ex1 f_def THE_default1_equality)
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  apply (rule THE_defaultI')
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  apply (rule ex1)
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  done
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lemma fundef_default_value:
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  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
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  assumes graph: "\<And>x y. G x y \<Longrightarrow> D x"
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  assumes "\<not> D x"
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  shows "f x = d x"
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proof -
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  have "\<not>(\<exists>y. G x y)"
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  proof
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    assume "\<exists>y. G x y"
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    hence "D x" using graph ..
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    with `\<not> D x` show False ..
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  qed
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  hence "\<not>(\<exists>!y. G x y)" by blast
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  thus ?thesis
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    unfolding f_def
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    by (rule THE_default_none)
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qed
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definition in_rel_def[simp]:
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  "in_rel R x y == (x, y) \<in> R"
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lemma wf_in_rel:
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  "wf R \<Longrightarrow> wfP (in_rel R)"
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  by (simp add: wfP_def)
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use "Tools/Function/function_common.ML"
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use "Tools/Function/context_tree.ML"
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use "Tools/Function/function_core.ML"
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use "Tools/Function/sum_tree.ML"
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use "Tools/Function/mutual.ML"
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use "Tools/Function/pattern_split.ML"
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use "Tools/Function/relation.ML"
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method_setup relation = {*
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  Args.term >> (fn t => fn ctxt => SIMPLE_METHOD' (Function_Relation.relation_infer_tac ctxt t))
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*} "prove termination using a user-specified wellfounded relation"
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use "Tools/Function/function.ML"
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use "Tools/Function/pat_completeness.ML"
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method_setup pat_completeness = {*
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  Scan.succeed (SIMPLE_METHOD' o Pat_Completeness.pat_completeness_tac)
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*} "prove completeness of datatype patterns"
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use "Tools/Function/fun.ML"
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use "Tools/Function/induction_schema.ML"
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method_setup induction_schema = {*
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  Scan.succeed (RAW_METHOD o Induction_Schema.induction_schema_tac)
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*} "prove an induction principle"
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setup {*
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  Function.setup
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  #> Function_Fun.setup
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*}
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subsection {* Measure Functions *}
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inductive is_measure :: "('a \<Rightarrow> nat) \<Rightarrow> bool"
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where is_measure_trivial: "is_measure f"
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use "Tools/Function/measure_functions.ML"
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setup MeasureFunctions.setup
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lemma measure_size[measure_function]: "is_measure size"
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by (rule is_measure_trivial)
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lemma measure_fst[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (fst p))"
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by (rule is_measure_trivial)
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lemma measure_snd[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (snd p))"
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by (rule is_measure_trivial)
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use "Tools/Function/lexicographic_order.ML"
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method_setup lexicographic_order = {*
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  Method.sections clasimp_modifiers >>
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  (K (SIMPLE_METHOD o Lexicographic_Order.lexicographic_order_tac false))
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*} "termination prover for lexicographic orderings"
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setup Lexicographic_Order.setup
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subsection {* Congruence Rules *}
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lemma let_cong [fundef_cong]:
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  "M = N \<Longrightarrow> (\<And>x. x = N \<Longrightarrow> f x = g x) \<Longrightarrow> Let M f = Let N g"
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  unfolding Let_def by blast
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lemmas [fundef_cong] =
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  if_cong image_cong INT_cong UN_cong
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  bex_cong ball_cong imp_cong Option.map_cong Option.bind_cong
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lemma split_cong [fundef_cong]:
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  "(\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y) \<Longrightarrow> p = q
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    \<Longrightarrow> split f p = split g q"
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  by (auto simp: split_def)
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lemma comp_cong [fundef_cong]:
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  "f (g x) = f' (g' x') \<Longrightarrow> (f o g) x = (f' o g') x'"
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  unfolding o_apply .
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subsection {* Simp rules for termination proofs *}
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lemma termination_basic_simps[termination_simp]:
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  "x < (y::nat) \<Longrightarrow> x < y + z"
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  "x < z \<Longrightarrow> x < y + z"
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  "x \<le> y \<Longrightarrow> x \<le> y + (z::nat)"
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  "x \<le> z \<Longrightarrow> x \<le> y + (z::nat)"
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  "x < y \<Longrightarrow> x \<le> (y::nat)"
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by arith+
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declare le_imp_less_Suc[termination_simp]
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lemma prod_size_simp[termination_simp]:
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  "prod_size f g p = f (fst p) + g (snd p) + Suc 0"
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by (induct p) auto
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subsection {* Decomposition *}
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lemma less_by_empty:
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  "A = {} \<Longrightarrow> A \<subseteq> B"
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and  union_comp_emptyL:
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  "\<lbrakk> A O C = {}; B O C = {} \<rbrakk> \<Longrightarrow> (A \<union> B) O C = {}"
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and union_comp_emptyR:
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  "\<lbrakk> A O B = {}; A O C = {} \<rbrakk> \<Longrightarrow> A O (B \<union> C) = {}"
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and wf_no_loop:
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  "R O R = {} \<Longrightarrow> wf R"
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by (auto simp add: wf_comp_self[of R])
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subsection {* Reduction Pairs *}
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definition
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  "reduction_pair P = (wf (fst P) \<and> fst P O snd P \<subseteq> fst P)"
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lemma reduction_pairI[intro]: "wf R \<Longrightarrow> R O S \<subseteq> R \<Longrightarrow> reduction_pair (R, S)"
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unfolding reduction_pair_def by auto
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lemma reduction_pair_lemma:
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  assumes rp: "reduction_pair P"
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  assumes "R \<subseteq> fst P"
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  assumes "S \<subseteq> snd P"
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  assumes "wf S"
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  shows "wf (R \<union> S)"
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proof -
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  from rp `S \<subseteq> snd P` have "wf (fst P)" "fst P O S \<subseteq> fst P"
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    unfolding reduction_pair_def by auto
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  with `wf S` have "wf (fst P \<union> S)"
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    by (auto intro: wf_union_compatible)
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  moreover from `R \<subseteq> fst P` have "R \<union> S \<subseteq> fst P \<union> S" by auto
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  ultimately show ?thesis by (rule wf_subset)
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qed
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definition
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  "rp_inv_image = (\<lambda>(R,S) f. (inv_image R f, inv_image S f))"
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lemma rp_inv_image_rp:
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  "reduction_pair P \<Longrightarrow> reduction_pair (rp_inv_image P f)"
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  unfolding reduction_pair_def rp_inv_image_def split_def
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  by force
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subsection {* Concrete orders for SCNP termination proofs *}
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definition "pair_less = less_than <*lex*> less_than"
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definition "pair_leq = pair_less^="
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definition "max_strict = max_ext pair_less"
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definition "max_weak = max_ext pair_leq \<union> {({}, {})}"
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definition "min_strict = min_ext pair_less"
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definition "min_weak = min_ext pair_leq \<union> {({}, {})}"
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lemma wf_pair_less[simp]: "wf pair_less"
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  by (auto simp: pair_less_def)
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text {* Introduction rules for @{text pair_less}/@{text pair_leq} *}
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lemma pair_leqI1: "a < b \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq"
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  and pair_leqI2: "a \<le> b \<Longrightarrow> s \<le> t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq"
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  and pair_lessI1: "a < b  \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
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  and pair_lessI2: "a \<le> b \<Longrightarrow> s < t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
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  unfolding pair_leq_def pair_less_def by auto
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text {* Introduction rules for max *}
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lemma smax_emptyI:
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  "finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> ({}, Y) \<in> max_strict"
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  and smax_insertI:
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  "\<lbrakk>y \<in> Y; (x, y) \<in> pair_less; (X, Y) \<in> max_strict\<rbrakk> \<Longrightarrow> (insert x X, Y) \<in> max_strict"
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  and wmax_emptyI:
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  "finite X \<Longrightarrow> ({}, X) \<in> max_weak"
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  and wmax_insertI:
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  "\<lbrakk>y \<in> YS; (x, y) \<in> pair_leq; (XS, YS) \<in> max_weak\<rbrakk> \<Longrightarrow> (insert x XS, YS) \<in> max_weak"
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unfolding max_strict_def max_weak_def by (auto elim!: max_ext.cases)
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text {* Introduction rules for min *}
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lemma smin_emptyI:
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  "X \<noteq> {} \<Longrightarrow> (X, {}) \<in> min_strict"
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  and smin_insertI:
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  "\<lbrakk>x \<in> XS; (x, y) \<in> pair_less; (XS, YS) \<in> min_strict\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_strict"
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  and wmin_emptyI:
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  "(X, {}) \<in> min_weak"
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  and wmin_insertI:
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  "\<lbrakk>x \<in> XS; (x, y) \<in> pair_leq; (XS, YS) \<in> min_weak\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_weak"
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by (auto simp: min_strict_def min_weak_def min_ext_def)
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text {* Reduction Pairs *}
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lemma max_ext_compat:
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  assumes "R O S \<subseteq> R"
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  shows "max_ext R O (max_ext S \<union> {({},{})}) \<subseteq> max_ext R"
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using assms
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apply auto
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apply (elim max_ext.cases)
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apply rule
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apply auto[3]
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apply (drule_tac x=xa in meta_spec)
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apply simp
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apply (erule bexE)
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apply (drule_tac x=xb in meta_spec)
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by auto
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lemma max_rpair_set: "reduction_pair (max_strict, max_weak)"
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  unfolding max_strict_def max_weak_def
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apply (intro reduction_pairI max_ext_wf)
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apply simp
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apply (rule max_ext_compat)
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by (auto simp: pair_less_def pair_leq_def)
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lemma min_ext_compat:
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  assumes "R O S \<subseteq> R"
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  shows "min_ext R O  (min_ext S \<union> {({},{})}) \<subseteq> min_ext R"
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using assms
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apply (auto simp: min_ext_def)
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apply (drule_tac x=ya in bspec, assumption)
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apply (erule bexE)
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apply (drule_tac x=xc in bspec)
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apply assumption
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by auto
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lemma min_rpair_set: "reduction_pair (min_strict, min_weak)"
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  unfolding min_strict_def min_weak_def
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apply (intro reduction_pairI min_ext_wf)
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apply simp
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apply (rule min_ext_compat)
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by (auto simp: pair_less_def pair_leq_def)
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subsection {* Tool setup *}
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use "Tools/Function/termination.ML"
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use "Tools/Function/scnp_solve.ML"
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use "Tools/Function/scnp_reconstruct.ML"
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setup {* ScnpReconstruct.setup *}
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ML_val -- "setup inactive"
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{*
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  Context.theory_map (Function_Common.set_termination_prover
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    (ScnpReconstruct.decomp_scnp_tac [ScnpSolve.MAX, ScnpSolve.MIN, ScnpSolve.MS]))
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*}
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end