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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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Lemma "fundef_default_value" uses predicate instead of set.
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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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use "Tools/Function/function.ML" 
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use "Tools/Function/pat_completeness.ML" 
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use "Tools/Function/fun.ML" 
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use "Tools/Function/induction_schema.ML" 
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setup {* 
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Function.setup 
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#> Pat_Completeness.setup 
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#> Function_Fun.setup 
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#> Induction_Schema.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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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" 

230 
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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29127  235 
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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31775  308 
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 *} 
30480  313 

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ML_val  "setup inactive" 

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{* 

36521  316 
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 