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(* Title: HOL/Library/Coinductive_Lists.thy
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ID: $Id$
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Author: Lawrence C Paulson and Makarius
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*)
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header {* Potentially infinite lists as greatest fixed-point *}
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theory Coinductive_List
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imports Main
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begin
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subsection {* List constructors over the datatype universe *}
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definition
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"NIL = Datatype_Universe.In0 (Datatype_Universe.Numb 0)"
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"CONS M N = Datatype_Universe.In1 (Datatype_Universe.Scons M N)"
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lemma CONS_not_NIL [iff]: "CONS M N \<noteq> NIL"
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and NIL_not_CONS [iff]: "NIL \<noteq> CONS M N"
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and CONS_inject [iff]: "(CONS K M) = (CONS L N) = (K = L \<and> M = N)"
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by (simp_all add: NIL_def CONS_def)
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lemma CONS_mono: "M \<subseteq> M' \<Longrightarrow> N \<subseteq> N' \<Longrightarrow> CONS M N \<subseteq> CONS M' N'"
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by (simp add: CONS_def In1_mono Scons_mono)
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lemma CONS_UN1: "CONS M (\<Union>x. f x) = (\<Union>x. CONS M (f x))"
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-- {* A continuity result? *}
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by (simp add: CONS_def In1_UN1 Scons_UN1_y)
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definition
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"List_case c h = Datatype_Universe.Case (\<lambda>_. c) (Datatype_Universe.Split h)"
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lemma List_case_NIL [simp]: "List_case c h NIL = c"
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and List_case_CONS [simp]: "List_case c h (CONS M N) = h M N"
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by (simp_all add: List_case_def NIL_def CONS_def)
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subsection {* Corecursive lists *}
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consts
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LList :: "'a Datatype_Universe.item set \<Rightarrow> 'a Datatype_Universe.item set"
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coinductive "LList A"
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intros
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NIL [intro]: "NIL \<in> LList A"
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CONS [intro]: "a \<in> A \<Longrightarrow> M \<in> LList A \<Longrightarrow> CONS a M \<in> LList A"
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lemma LList_mono: "A \<subseteq> B \<Longrightarrow> LList A \<subseteq> LList B"
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-- {* This justifies using @{text LList} in other recursive type definitions. *}
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unfolding LList.defs by (blast intro!: gfp_mono)
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consts
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LList_corec_aux :: "nat \<Rightarrow> ('a \<Rightarrow> ('b Datatype_Universe.item \<times> 'a) option) \<Rightarrow>
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'a \<Rightarrow> 'b Datatype_Universe.item"
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primrec
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"LList_corec_aux 0 f x = {}"
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"LList_corec_aux (Suc k) f x =
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(case f x of
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None \<Rightarrow> NIL
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| Some (z, w) \<Rightarrow> CONS z (LList_corec_aux k f w))"
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definition
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"LList_corec a f = (\<Union>k. LList_corec_aux k f a)"
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text {*
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Note: the subsequent recursion equation for @{text LList_corec} may
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be used with the Simplifier, provided it operates in a non-strict
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fashion for case expressions (i.e.\ the usual @{text case}
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congruence rule needs to be present).
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*}
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lemma LList_corec:
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"LList_corec a f =
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(case f a of None \<Rightarrow> NIL | Some (z, w) \<Rightarrow> CONS z (LList_corec w f))"
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(is "?lhs = ?rhs")
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proof
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show "?lhs \<subseteq> ?rhs"
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apply (unfold LList_corec_def)
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apply (rule UN_least)
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apply (case_tac k)
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apply (simp_all (no_asm_simp) split: option.splits)
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apply (rule allI impI subset_refl [THEN CONS_mono] UNIV_I [THEN UN_upper])+
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done
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show "?rhs \<subseteq> ?lhs"
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apply (simp add: LList_corec_def split: option.splits)
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apply (simp add: CONS_UN1)
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apply safe
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apply (rule_tac a = "Suc ?k" in UN_I, simp, simp)+
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done
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qed
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lemma LList_corec_type: "LList_corec a f \<in> LList UNIV"
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proof -
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have "LList_corec a f \<in> {LList_corec a f | a. True}" by blast
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then show ?thesis
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proof coinduct
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case (LList L)
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then obtain x where L: "L = LList_corec x f" by blast
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show ?case
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proof (cases "f x")
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case None
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then have "LList_corec x f = NIL"
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by (simp add: LList_corec)
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with L have ?NIL by simp
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then show ?thesis ..
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next
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case (Some p)
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then have "LList_corec x f = CONS (fst p) (LList_corec (snd p) f)"
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by (simp add: split_def LList_corec)
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with L have ?CONS by auto
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then show ?thesis ..
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qed
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qed
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qed
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subsection {* Abstract type definition *}
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typedef 'a llist =
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"LList (range Datatype_Universe.Leaf) :: 'a Datatype_Universe.item set"
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proof
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show "NIL \<in> ?llist" ..
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qed
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lemma NIL_type: "NIL \<in> llist"
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unfolding llist_def by (rule LList.NIL)
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lemma CONS_type: "a \<in> range Datatype_Universe.Leaf \<Longrightarrow>
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M \<in> llist \<Longrightarrow> CONS a M \<in> llist"
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unfolding llist_def by (rule LList.CONS)
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lemma llistI: "x \<in> LList (range Datatype_Universe.Leaf) \<Longrightarrow> x \<in> llist"
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by (simp add: llist_def)
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lemma llistD: "x \<in> llist \<Longrightarrow> x \<in> LList (range Datatype_Universe.Leaf)"
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by (simp add: llist_def)
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lemma Rep_llist_UNIV: "Rep_llist x \<in> LList UNIV"
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proof -
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have "Rep_llist x \<in> llist" by (rule Rep_llist)
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then have "Rep_llist x \<in> LList (range Datatype_Universe.Leaf)"
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by (simp add: llist_def)
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also have "\<dots> \<subseteq> LList UNIV" by (rule LList_mono) simp
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finally show ?thesis .
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qed
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definition
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"LNil = Abs_llist NIL"
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"LCons x xs = Abs_llist (CONS (Datatype_Universe.Leaf x) (Rep_llist xs))"
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lemma LCons_not_LNil [iff]: "LCons x xs \<noteq> LNil"
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apply (simp add: LNil_def LCons_def)
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apply (subst Abs_llist_inject)
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apply (auto intro: NIL_type CONS_type Rep_llist)
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done
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lemma LNil_not_LCons [iff]: "LNil \<noteq> LCons x xs"
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by (rule LCons_not_LNil [symmetric])
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lemma LCons_inject [iff]: "(LCons x xs = LCons y ys) = (x = y \<and> xs = ys)"
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apply (simp add: LCons_def)
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apply (subst Abs_llist_inject)
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apply (auto simp add: Rep_llist_inject intro: CONS_type Rep_llist)
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done
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lemma Rep_llist_LNil: "Rep_llist LNil = NIL"
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by (simp add: LNil_def add: Abs_llist_inverse NIL_type)
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lemma Rep_llist_LCons: "Rep_llist (LCons x l) =
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CONS (Datatype_Universe.Leaf x) (Rep_llist l)"
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by (simp add: LCons_def Abs_llist_inverse CONS_type Rep_llist)
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lemma llist_cases [case_names LNil LCons, cases type: llist]:
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assumes LNil: "l = LNil \<Longrightarrow> C"
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and LCons: "\<And>x l'. l = LCons x l' \<Longrightarrow> C"
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shows C
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proof (cases l)
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case (Abs_llist L)
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from `L \<in> llist` have "L \<in> LList (range Datatype_Universe.Leaf)" by (rule llistD)
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then show ?thesis
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proof cases
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case NIL
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with Abs_llist have "l = LNil" by (simp add: LNil_def)
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with LNil show ?thesis .
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next
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case (CONS K a)
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then have "K \<in> llist" by (blast intro: llistI)
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then obtain l' where "K = Rep_llist l'" by cases
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with CONS and Abs_llist obtain x where "l = LCons x l'"
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by (auto simp add: LCons_def Abs_llist_inject)
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with LCons show ?thesis .
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qed
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qed
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definition
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"llist_case c d l =
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List_case c (\<lambda>x y. d (inv Datatype_Universe.Leaf x) (Abs_llist y)) (Rep_llist l)"
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translations
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"case p of LNil \<Rightarrow> a | LCons x l \<Rightarrow> b" \<rightleftharpoons> "llist_case a (\<lambda>x l. b) p"
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lemma llist_case_LNil [simp]: "llist_case c d LNil = c"
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by (simp add: llist_case_def LNil_def
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NIL_type Abs_llist_inverse)
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lemma llist_case_LCons [simp]: "llist_case c d (LCons M N) = d M N"
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by (simp add: llist_case_def LCons_def
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CONS_type Abs_llist_inverse Rep_llist Rep_llist_inverse inj_Leaf)
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definition
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"llist_corec a f =
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Abs_llist (LList_corec a
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(\<lambda>z.
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case f z of None \<Rightarrow> None
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| Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w)))"
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lemma LList_corec_type2:
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"LList_corec a
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(\<lambda>z. case f z of None \<Rightarrow> None
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| Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w)) \<in> llist"
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(is "?corec a \<in> _")
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proof (unfold llist_def)
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let "LList_corec a ?g" = "?corec a"
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have "?corec a \<in> {?corec x | x. True}" by blast
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then show "?corec a \<in> LList (range Datatype_Universe.Leaf)"
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proof coinduct
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case (LList L)
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then obtain x where L: "L = ?corec x" by blast
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show ?case
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proof (cases "f x")
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case None
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then have "?corec x = NIL"
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by (simp add: LList_corec)
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with L have ?NIL by simp
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then show ?thesis ..
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next
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case (Some p)
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then have "?corec x =
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CONS (Datatype_Universe.Leaf (fst p)) (?corec (snd p))"
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by (simp add: split_def LList_corec)
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with L have ?CONS by auto
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then show ?thesis ..
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qed
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qed
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qed
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lemma llist_corec:
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"llist_corec a f =
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(case f a of None \<Rightarrow> LNil | Some (z, w) \<Rightarrow> LCons z (llist_corec w f))"
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proof (cases "f a")
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case None
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then show ?thesis
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by (simp add: llist_corec_def LList_corec LNil_def)
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next
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case (Some p)
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let "?corec a" = "llist_corec a f"
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let "?rep_corec a" =
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"LList_corec a
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(\<lambda>z. case f z of None \<Rightarrow> None
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| Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w))"
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have "?corec a = Abs_llist (?rep_corec a)"
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by (simp only: llist_corec_def)
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also from Some have "?rep_corec a =
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CONS (Datatype_Universe.Leaf (fst p)) (?rep_corec (snd p))"
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by (simp add: split_def LList_corec)
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also have "?rep_corec (snd p) = Rep_llist (?corec (snd p))"
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by (simp only: llist_corec_def Abs_llist_inverse LList_corec_type2)
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finally have "?corec a = LCons (fst p) (?corec (snd p))"
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by (simp only: LCons_def)
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with Some show ?thesis by (simp add: split_def)
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qed
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subsection {* Equality as greatest fixed-point; the bisimulation principle. *}
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consts
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EqLList :: "('a Datatype_Universe.item \<times> 'a Datatype_Universe.item) set \<Rightarrow>
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('a Datatype_Universe.item \<times> 'a Datatype_Universe.item) set"
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coinductive "EqLList r"
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intros
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EqNIL: "(NIL, NIL) \<in> EqLList r"
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EqCONS: "(a, b) \<in> r \<Longrightarrow> (M, N) \<in> EqLList r \<Longrightarrow>
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(CONS a M, CONS b N) \<in> EqLList r"
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lemma EqLList_unfold:
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"EqLList r = dsum (diag {Datatype_Universe.Numb 0}) (dprod r (EqLList r))"
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by (fast intro!: EqLList.intros [unfolded NIL_def CONS_def]
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elim: EqLList.cases [unfolded NIL_def CONS_def])
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lemma EqLList_implies_ntrunc_equality:
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"(M, N) \<in> EqLList (diag A) \<Longrightarrow> ntrunc k M = ntrunc k N"
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apply (induct k fixing: M N rule: nat_less_induct)
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apply (erule EqLList.cases)
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apply (safe del: equalityI)
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apply (case_tac n)
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apply simp
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apply (rename_tac n')
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apply (case_tac n')
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apply (simp_all add: CONS_def less_Suc_eq)
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done
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lemma Domain_EqLList: "Domain (EqLList (diag A)) \<subseteq> LList A"
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apply (simp add: LList.defs NIL_def CONS_def)
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apply (rule gfp_upperbound)
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apply (subst EqLList_unfold)
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apply auto
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done
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lemma EqLList_diag: "EqLList (diag A) = diag (LList A)"
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(is "?lhs = ?rhs")
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proof
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show "?lhs \<subseteq> ?rhs"
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apply (rule subsetI)
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apply (rule_tac p = x in PairE)
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apply clarify
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apply (rule diag_eqI)
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apply (rule EqLList_implies_ntrunc_equality [THEN ntrunc_equality],
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assumption)
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apply (erule DomainI [THEN Domain_EqLList [THEN subsetD]])
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done
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show "?rhs \<subseteq> ?lhs"
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proof
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fix p assume "p \<in> diag (LList A)"
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then show "p \<in> EqLList (diag A)"
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proof coinduct
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case (EqLList q)
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then obtain L where L: "L \<in> LList A" and q: "q = (L, L)" ..
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from L show ?case
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proof cases
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case NIL with q have ?EqNIL by simp
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then show ?thesis ..
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next
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case CONS with q have ?EqCONS by (simp add: diagI)
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then show ?thesis ..
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qed
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qed
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qed
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qed
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lemma EqLList_diag_iff [iff]: "(p \<in> EqLList (diag A)) = (p \<in> diag (LList A))"
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by (simp only: EqLList_diag)
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text {*
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To show two LLists are equal, exhibit a bisimulation! (Also admits
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true equality.)
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*}
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lemma LList_equalityI
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[consumes 1, case_names EqLList, case_conclusion EqLList EqNIL EqCONS]:
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assumes r: "(M, N) \<in> r"
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356 |
and step: "\<And>p. p \<in> r \<Longrightarrow>
|
|
357 |
p = (NIL, NIL) \<or>
|
|
358 |
(\<exists>M N a b.
|
|
359 |
p = (CONS a M, CONS b N) \<and> (a, b) \<in> diag A \<and>
|
|
360 |
(M, N) \<in> r \<union> EqLList (diag A))"
|
|
361 |
shows "M = N"
|
|
362 |
proof -
|
|
363 |
from r have "(M, N) \<in> EqLList (diag A)"
|
|
364 |
proof coinduct
|
|
365 |
case EqLList
|
|
366 |
then show ?case by (rule step)
|
|
367 |
qed
|
|
368 |
then show ?thesis by auto
|
|
369 |
qed
|
|
370 |
|
|
371 |
lemma LList_fun_equalityI
|
|
372 |
[consumes 1, case_names NIL_type NIL CONS, case_conclusion CONS EqNIL EqCONS]:
|
|
373 |
assumes M: "M \<in> LList A"
|
|
374 |
and fun_NIL: "g NIL \<in> LList A" "f NIL = g NIL"
|
|
375 |
and fun_CONS: "\<And>x l. x \<in> A \<Longrightarrow> l \<in> LList A \<Longrightarrow>
|
|
376 |
(f (CONS x l), g (CONS x l)) = (NIL, NIL) \<or>
|
|
377 |
(\<exists>M N a b.
|
|
378 |
(f (CONS x l), g (CONS x l)) = (CONS a M, CONS b N) \<and>
|
|
379 |
(a, b) \<in> diag A \<and>
|
|
380 |
(M, N) \<in> {(f u, g u) | u. u \<in> LList A} \<union> diag (LList A))"
|
|
381 |
(is "\<And>x l. _ \<Longrightarrow> _ \<Longrightarrow> ?fun_CONS x l")
|
|
382 |
shows "f M = g M"
|
|
383 |
proof -
|
|
384 |
let ?bisim = "{(f L, g L) | L. L \<in> LList A}"
|
|
385 |
have "(f M, g M) \<in> ?bisim" using M by blast
|
|
386 |
then show ?thesis
|
|
387 |
proof (coinduct taking: A rule: LList_equalityI)
|
|
388 |
case (EqLList q)
|
|
389 |
then obtain L where q: "q = (f L, g L)" and L: "L \<in> LList A" by blast
|
|
390 |
from L show ?case
|
|
391 |
proof (cases L)
|
|
392 |
case NIL
|
|
393 |
with fun_NIL and q have "q \<in> diag (LList A)" by auto
|
|
394 |
then have "q \<in> EqLList (diag A)" ..
|
|
395 |
then show ?thesis by cases simp_all
|
|
396 |
next
|
|
397 |
case (CONS K a)
|
|
398 |
from fun_CONS and `a \<in> A` `K \<in> LList A`
|
|
399 |
have "?fun_CONS a K" (is "?NIL \<or> ?CONS") .
|
|
400 |
then show ?thesis
|
|
401 |
proof
|
|
402 |
assume ?NIL
|
|
403 |
with q CONS have "q \<in> diag (LList A)" by auto
|
|
404 |
then have "q \<in> EqLList (diag A)" ..
|
|
405 |
then show ?thesis by cases simp_all
|
|
406 |
next
|
|
407 |
assume ?CONS
|
|
408 |
with CONS obtain a b M N where
|
|
409 |
fg: "(f L, g L) = (CONS a M, CONS b N)"
|
|
410 |
and ab: "(a, b) \<in> diag A"
|
|
411 |
and MN: "(M, N) \<in> ?bisim \<union> diag (LList A)"
|
|
412 |
by blast
|
|
413 |
from MN show ?thesis
|
|
414 |
proof
|
|
415 |
assume "(M, N) \<in> ?bisim"
|
|
416 |
with q fg ab show ?thesis by simp
|
|
417 |
next
|
|
418 |
assume "(M, N) \<in> diag (LList A)"
|
|
419 |
then have "(M, N) \<in> EqLList (diag A)" ..
|
|
420 |
with q fg ab show ?thesis by simp
|
|
421 |
qed
|
|
422 |
qed
|
|
423 |
qed
|
|
424 |
qed
|
|
425 |
qed
|
|
426 |
|
|
427 |
text {*
|
|
428 |
Finality of @{text "llist A"}: Uniqueness of functions defined by corecursion.
|
|
429 |
*}
|
|
430 |
|
|
431 |
lemma equals_LList_corec:
|
|
432 |
assumes h: "\<And>x. h x =
|
|
433 |
(case f x of None \<Rightarrow> NIL | Some (z, w) \<Rightarrow> CONS z (h w))"
|
|
434 |
shows "h x = (\<lambda>x. LList_corec x f) x"
|
|
435 |
proof -
|
|
436 |
def h' \<equiv> "\<lambda>x. LList_corec x f"
|
|
437 |
then have h': "\<And>x. h' x =
|
|
438 |
(case f x of None \<Rightarrow> NIL | Some (z, w) \<Rightarrow> CONS z (h' w))"
|
18730
|
439 |
unfolding h'_def by (simp add: LList_corec)
|
18400
|
440 |
have "(h x, h' x) \<in> {(h u, h' u) | u. True}" by blast
|
|
441 |
then show "h x = h' x"
|
|
442 |
proof (coinduct rule: LList_equalityI [where A = UNIV])
|
|
443 |
case (EqLList q)
|
|
444 |
then obtain x where q: "q = (h x, h' x)" by blast
|
|
445 |
show ?case
|
|
446 |
proof (cases "f x")
|
|
447 |
case None
|
|
448 |
with h h' q have ?EqNIL by simp
|
|
449 |
then show ?thesis ..
|
|
450 |
next
|
|
451 |
case (Some p)
|
|
452 |
with h h' q have "q =
|
|
453 |
(CONS (fst p) (h (snd p)), CONS (fst p) (h' (snd p)))"
|
|
454 |
by (simp add: split_def)
|
|
455 |
then have ?EqCONS by (auto iff: diag_iff)
|
|
456 |
then show ?thesis ..
|
|
457 |
qed
|
|
458 |
qed
|
|
459 |
qed
|
|
460 |
|
|
461 |
|
|
462 |
lemma llist_equalityI
|
|
463 |
[consumes 1, case_names Eqllist, case_conclusion Eqllist EqLNil EqLCons]:
|
|
464 |
assumes r: "(l1, l2) \<in> r"
|
|
465 |
and step: "\<And>q. q \<in> r \<Longrightarrow>
|
|
466 |
q = (LNil, LNil) \<or>
|
|
467 |
(\<exists>l1 l2 a b.
|
|
468 |
q = (LCons a l1, LCons b l2) \<and> a = b \<and>
|
|
469 |
((l1, l2) \<in> r \<or> l1 = l2))"
|
|
470 |
(is "\<And>q. _ \<Longrightarrow> ?EqLNil q \<or> ?EqLCons q")
|
|
471 |
shows "l1 = l2"
|
|
472 |
proof -
|
|
473 |
def M \<equiv> "Rep_llist l1" and N \<equiv> "Rep_llist l2"
|
|
474 |
with r have "(M, N) \<in> {(Rep_llist l1, Rep_llist l2) | l1 l2. (l1, l2) \<in> r}"
|
|
475 |
by blast
|
|
476 |
then have "M = N"
|
|
477 |
proof (coinduct rule: LList_equalityI [where A = UNIV])
|
|
478 |
case (EqLList q)
|
|
479 |
then obtain l1 l2 where
|
|
480 |
q: "q = (Rep_llist l1, Rep_llist l2)" and r: "(l1, l2) \<in> r"
|
|
481 |
by auto
|
|
482 |
from step [OF r] show ?case
|
|
483 |
proof
|
|
484 |
assume "?EqLNil (l1, l2)"
|
|
485 |
with q have ?EqNIL by (simp add: Rep_llist_LNil)
|
|
486 |
then show ?thesis ..
|
|
487 |
next
|
|
488 |
assume "?EqLCons (l1, l2)"
|
|
489 |
with q have ?EqCONS
|
|
490 |
by (force simp add: Rep_llist_LCons EqLList_diag intro: Rep_llist_UNIV)
|
|
491 |
then show ?thesis ..
|
|
492 |
qed
|
|
493 |
qed
|
|
494 |
then show ?thesis by (simp add: M_def N_def Rep_llist_inject)
|
|
495 |
qed
|
|
496 |
|
|
497 |
lemma llist_fun_equalityI
|
|
498 |
[case_names LNil LCons, case_conclusion LCons EqLNil EqLCons]:
|
|
499 |
assumes fun_LNil: "f LNil = g LNil"
|
|
500 |
and fun_LCons: "\<And>x l.
|
|
501 |
(f (LCons x l), g (LCons x l)) = (LNil, LNil) \<or>
|
|
502 |
(\<exists>l1 l2 a b.
|
|
503 |
(f (LCons x l), g (LCons x l)) = (LCons a l1, LCons b l2) \<and>
|
|
504 |
a = b \<and> ((l1, l2) \<in> {(f u, g u) | u. True} \<or> l1 = l2))"
|
|
505 |
(is "\<And>x l. ?fun_LCons x l")
|
|
506 |
shows "f l = g l"
|
|
507 |
proof -
|
|
508 |
have "(f l, g l) \<in> {(f l, g l) | l. True}" by blast
|
|
509 |
then show ?thesis
|
|
510 |
proof (coinduct rule: llist_equalityI)
|
|
511 |
case (Eqllist q)
|
|
512 |
then obtain l where q: "q = (f l, g l)" by blast
|
|
513 |
show ?case
|
|
514 |
proof (cases l)
|
|
515 |
case LNil
|
|
516 |
with fun_LNil and q have "q = (g LNil, g LNil)" by simp
|
|
517 |
then show ?thesis by (cases "g LNil") simp_all
|
|
518 |
next
|
|
519 |
case (LCons x l')
|
|
520 |
with `?fun_LCons x l'` q LCons show ?thesis by blast
|
|
521 |
qed
|
|
522 |
qed
|
|
523 |
qed
|
|
524 |
|
|
525 |
|
|
526 |
subsection {* Derived operations -- both on the set and abstract type *}
|
|
527 |
|
|
528 |
subsubsection {* @{text Lconst} *}
|
|
529 |
|
19086
|
530 |
definition
|
18400
|
531 |
"Lconst M \<equiv> lfp (\<lambda>N. CONS M N)"
|
|
532 |
|
|
533 |
lemma Lconst_fun_mono: "mono (CONS M)"
|
|
534 |
by (simp add: monoI CONS_mono)
|
|
535 |
|
|
536 |
lemma Lconst: "Lconst M = CONS M (Lconst M)"
|
|
537 |
by (rule Lconst_def [THEN def_lfp_unfold]) (rule Lconst_fun_mono)
|
|
538 |
|
|
539 |
lemma Lconst_type:
|
|
540 |
assumes "M \<in> A"
|
|
541 |
shows "Lconst M \<in> LList A"
|
|
542 |
proof -
|
|
543 |
have "Lconst M \<in> {Lconst M}" by simp
|
|
544 |
then show ?thesis
|
|
545 |
proof coinduct
|
|
546 |
case (LList N)
|
|
547 |
then have "N = Lconst M" by simp
|
|
548 |
also have "\<dots> = CONS M (Lconst M)" by (rule Lconst)
|
|
549 |
finally have ?CONS using `M \<in> A` by simp
|
|
550 |
then show ?case ..
|
|
551 |
qed
|
|
552 |
qed
|
|
553 |
|
|
554 |
lemma Lconst_eq_LList_corec: "Lconst M = LList_corec M (\<lambda>x. Some (x, x))"
|
|
555 |
apply (rule equals_LList_corec)
|
|
556 |
apply simp
|
|
557 |
apply (rule Lconst)
|
|
558 |
done
|
|
559 |
|
|
560 |
lemma gfp_Lconst_eq_LList_corec:
|
|
561 |
"gfp (\<lambda>N. CONS M N) = LList_corec M (\<lambda>x. Some(x, x))"
|
|
562 |
apply (rule equals_LList_corec)
|
|
563 |
apply simp
|
|
564 |
apply (rule Lconst_fun_mono [THEN gfp_unfold])
|
|
565 |
done
|
|
566 |
|
|
567 |
|
|
568 |
subsubsection {* @{text Lmap} and @{text lmap} *}
|
|
569 |
|
19086
|
570 |
definition
|
|
571 |
"Lmap f M = LList_corec M (List_case None (\<lambda>x M'. Some (f x, M')))"
|
|
572 |
"lmap f l = llist_corec l
|
18400
|
573 |
(\<lambda>z.
|
|
574 |
case z of LNil \<Rightarrow> None
|
|
575 |
| LCons y z \<Rightarrow> Some (f y, z))"
|
|
576 |
|
|
577 |
lemma Lmap_NIL [simp]: "Lmap f NIL = NIL"
|
|
578 |
and Lmap_CONS [simp]: "Lmap f (CONS M N) = CONS (f M) (Lmap f N)"
|
|
579 |
by (simp_all add: Lmap_def LList_corec)
|
|
580 |
|
|
581 |
lemma Lmap_type:
|
|
582 |
assumes M: "M \<in> LList A"
|
|
583 |
and f: "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B"
|
|
584 |
shows "Lmap f M \<in> LList B"
|
|
585 |
proof -
|
|
586 |
from M have "Lmap f M \<in> {Lmap f N | N. N \<in> LList A}" by blast
|
|
587 |
then show ?thesis
|
|
588 |
proof coinduct
|
|
589 |
case (LList L)
|
|
590 |
then obtain N where L: "L = Lmap f N" and N: "N \<in> LList A" by blast
|
|
591 |
from N show ?case
|
|
592 |
proof cases
|
|
593 |
case NIL
|
|
594 |
with L have ?NIL by simp
|
|
595 |
then show ?thesis ..
|
|
596 |
next
|
|
597 |
case (CONS K a)
|
|
598 |
with f L have ?CONS by auto
|
|
599 |
then show ?thesis ..
|
|
600 |
qed
|
|
601 |
qed
|
|
602 |
qed
|
|
603 |
|
|
604 |
lemma Lmap_compose:
|
|
605 |
assumes M: "M \<in> LList A"
|
|
606 |
shows "Lmap (f o g) M = Lmap f (Lmap g M)" (is "?lhs M = ?rhs M")
|
|
607 |
proof -
|
|
608 |
have "(?lhs M, ?rhs M) \<in> {(?lhs N, ?rhs N) | N. N \<in> LList A}"
|
|
609 |
using M by blast
|
|
610 |
then show ?thesis
|
|
611 |
proof (coinduct taking: "range (\<lambda>N :: 'a Datatype_Universe.item. N)"
|
|
612 |
rule: LList_equalityI)
|
|
613 |
case (EqLList q)
|
|
614 |
then obtain N where q: "q = (?lhs N, ?rhs N)" and N: "N \<in> LList A" by blast
|
|
615 |
from N show ?case
|
|
616 |
proof cases
|
|
617 |
case NIL
|
|
618 |
with q have ?EqNIL by simp
|
|
619 |
then show ?thesis ..
|
|
620 |
next
|
|
621 |
case CONS
|
|
622 |
with q have ?EqCONS by auto
|
|
623 |
then show ?thesis ..
|
|
624 |
qed
|
|
625 |
qed
|
|
626 |
qed
|
|
627 |
|
|
628 |
lemma Lmap_ident:
|
|
629 |
assumes M: "M \<in> LList A"
|
|
630 |
shows "Lmap (\<lambda>x. x) M = M" (is "?lmap M = _")
|
|
631 |
proof -
|
|
632 |
have "(?lmap M, M) \<in> {(?lmap N, N) | N. N \<in> LList A}" using M by blast
|
|
633 |
then show ?thesis
|
|
634 |
proof (coinduct taking: "range (\<lambda>N :: 'a Datatype_Universe.item. N)"
|
|
635 |
rule: LList_equalityI)
|
|
636 |
case (EqLList q)
|
|
637 |
then obtain N where q: "q = (?lmap N, N)" and N: "N \<in> LList A" by blast
|
|
638 |
from N show ?case
|
|
639 |
proof cases
|
|
640 |
case NIL
|
|
641 |
with q have ?EqNIL by simp
|
|
642 |
then show ?thesis ..
|
|
643 |
next
|
|
644 |
case CONS
|
|
645 |
with q have ?EqCONS by auto
|
|
646 |
then show ?thesis ..
|
|
647 |
qed
|
|
648 |
qed
|
|
649 |
qed
|
|
650 |
|
|
651 |
lemma lmap_LNil [simp]: "lmap f LNil = LNil"
|
|
652 |
and lmap_LCons [simp]: "lmap f (LCons M N) = LCons (f M) (lmap f N)"
|
|
653 |
by (simp_all add: lmap_def llist_corec)
|
|
654 |
|
|
655 |
lemma lmap_compose [simp]: "lmap (f o g) l = lmap f (lmap g l)"
|
|
656 |
by (coinduct _ _ l rule: llist_fun_equalityI) auto
|
|
657 |
|
|
658 |
lemma lmap_ident [simp]: "lmap (\<lambda>x. x) l = l"
|
|
659 |
by (coinduct _ _ l rule: llist_fun_equalityI) auto
|
|
660 |
|
|
661 |
|
|
662 |
|
|
663 |
subsubsection {* @{text Lappend} *}
|
|
664 |
|
19086
|
665 |
definition
|
|
666 |
"Lappend M N = LList_corec (M, N)
|
18400
|
667 |
(split (List_case
|
|
668 |
(List_case None (\<lambda>N1 N2. Some (N1, (NIL, N2))))
|
|
669 |
(\<lambda>M1 M2 N. Some (M1, (M2, N)))))"
|
19086
|
670 |
"lappend l n = llist_corec (l, n)
|
18400
|
671 |
(split (llist_case
|
|
672 |
(llist_case None (\<lambda>n1 n2. Some (n1, (LNil, n2))))
|
|
673 |
(\<lambda>l1 l2 n. Some (l1, (l2, n)))))"
|
|
674 |
|
|
675 |
lemma Lappend_NIL_NIL [simp]:
|
|
676 |
"Lappend NIL NIL = NIL"
|
|
677 |
and Lappend_NIL_CONS [simp]:
|
|
678 |
"Lappend NIL (CONS N N') = CONS N (Lappend NIL N')"
|
|
679 |
and Lappend_CONS [simp]:
|
|
680 |
"Lappend (CONS M M') N = CONS M (Lappend M' N)"
|
|
681 |
by (simp_all add: Lappend_def LList_corec)
|
|
682 |
|
|
683 |
lemma Lappend_NIL [simp]: "M \<in> LList A \<Longrightarrow> Lappend NIL M = M"
|
|
684 |
by (erule LList_fun_equalityI) auto
|
|
685 |
|
|
686 |
lemma Lappend_NIL2: "M \<in> LList A \<Longrightarrow> Lappend M NIL = M"
|
|
687 |
by (erule LList_fun_equalityI) auto
|
|
688 |
|
|
689 |
lemma Lappend_type:
|
|
690 |
assumes M: "M \<in> LList A" and N: "N \<in> LList A"
|
|
691 |
shows "Lappend M N \<in> LList A"
|
|
692 |
proof -
|
|
693 |
have "Lappend M N \<in> {Lappend u v | u v. u \<in> LList A \<and> v \<in> LList A}"
|
|
694 |
using M N by blast
|
|
695 |
then show ?thesis
|
|
696 |
proof coinduct
|
|
697 |
case (LList L)
|
|
698 |
then obtain M N where L: "L = Lappend M N"
|
|
699 |
and M: "M \<in> LList A" and N: "N \<in> LList A"
|
|
700 |
by blast
|
|
701 |
from M show ?case
|
|
702 |
proof cases
|
|
703 |
case NIL
|
|
704 |
from N show ?thesis
|
|
705 |
proof cases
|
|
706 |
case NIL
|
|
707 |
with L and `M = NIL` have ?NIL by simp
|
|
708 |
then show ?thesis ..
|
|
709 |
next
|
|
710 |
case CONS
|
|
711 |
with L and `M = NIL` have ?CONS by simp
|
|
712 |
then show ?thesis ..
|
|
713 |
qed
|
|
714 |
next
|
|
715 |
case CONS
|
|
716 |
with L N have ?CONS by auto
|
|
717 |
then show ?thesis ..
|
|
718 |
qed
|
|
719 |
qed
|
|
720 |
qed
|
|
721 |
|
|
722 |
lemma lappend_LNil_LNil [simp]: "lappend LNil LNil = LNil"
|
|
723 |
and lappend_LNil_LCons [simp]: "lappend LNil (LCons l l') = LCons l (lappend LNil l')"
|
|
724 |
and lappend_LCons [simp]: "lappend (LCons l l') m = LCons l (lappend l' m)"
|
|
725 |
by (simp_all add: lappend_def llist_corec)
|
|
726 |
|
|
727 |
lemma lappend_LNil1 [simp]: "lappend LNil l = l"
|
|
728 |
by (coinduct _ _ l rule: llist_fun_equalityI) auto
|
|
729 |
|
|
730 |
lemma lappend_LNil2 [simp]: "lappend l LNil = l"
|
|
731 |
by (coinduct _ _ l rule: llist_fun_equalityI) auto
|
|
732 |
|
|
733 |
lemma lappend_assoc: "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)"
|
|
734 |
by (coinduct _ _ l1 rule: llist_fun_equalityI) auto
|
|
735 |
|
|
736 |
lemma lmap_lappend_distrib: "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"
|
|
737 |
by (coinduct _ _ l rule: llist_fun_equalityI) auto
|
|
738 |
|
|
739 |
|
|
740 |
subsection{* iterates *}
|
|
741 |
|
|
742 |
text {* @{text llist_fun_equalityI} cannot be used here! *}
|
|
743 |
|
19086
|
744 |
definition
|
18400
|
745 |
iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist"
|
19086
|
746 |
"iterates f a = llist_corec a (\<lambda>x. Some (x, f x))"
|
18400
|
747 |
|
|
748 |
lemma iterates: "iterates f x = LCons x (iterates f (f x))"
|
|
749 |
apply (unfold iterates_def)
|
|
750 |
apply (subst llist_corec)
|
|
751 |
apply simp
|
|
752 |
done
|
|
753 |
|
|
754 |
lemma lmap_iterates: "lmap f (iterates f x) = iterates f (f x)"
|
|
755 |
proof -
|
|
756 |
have "(lmap f (iterates f x), iterates f (f x)) \<in>
|
|
757 |
{(lmap f (iterates f u), iterates f (f u)) | u. True}" by blast
|
|
758 |
then show ?thesis
|
|
759 |
proof (coinduct rule: llist_equalityI)
|
|
760 |
case (Eqllist q)
|
|
761 |
then obtain x where q: "q = (lmap f (iterates f x), iterates f (f x))"
|
|
762 |
by blast
|
|
763 |
also have "iterates f (f x) = LCons (f x) (iterates f (f (f x)))"
|
|
764 |
by (subst iterates) rule
|
|
765 |
also have "iterates f x = LCons x (iterates f (f x))"
|
|
766 |
by (subst iterates) rule
|
|
767 |
finally have ?EqLCons by auto
|
|
768 |
then show ?case ..
|
|
769 |
qed
|
|
770 |
qed
|
|
771 |
|
|
772 |
lemma iterates_lmap: "iterates f x = LCons x (lmap f (iterates f x))"
|
|
773 |
by (subst lmap_iterates) (rule iterates)
|
|
774 |
|
|
775 |
|
|
776 |
subsection{* A rather complex proof about iterates -- cf.\ Andy Pitts *}
|
|
777 |
|
|
778 |
lemma funpow_lmap:
|
|
779 |
fixes f :: "'a \<Rightarrow> 'a"
|
|
780 |
shows "(lmap f ^ n) (LCons b l) = LCons ((f ^ n) b) ((lmap f ^ n) l)"
|
|
781 |
by (induct n) simp_all
|
|
782 |
|
|
783 |
|
|
784 |
lemma iterates_equality:
|
|
785 |
assumes h: "\<And>x. h x = LCons x (lmap f (h x))"
|
|
786 |
shows "h = iterates f"
|
|
787 |
proof
|
|
788 |
fix x
|
|
789 |
have "(h x, iterates f x) \<in>
|
|
790 |
{((lmap f ^ n) (h u), (lmap f ^ n) (iterates f u)) | u n. True}"
|
|
791 |
proof -
|
|
792 |
have "(h x, iterates f x) = ((lmap f ^ 0) (h x), (lmap f ^ 0) (iterates f x))"
|
|
793 |
by simp
|
|
794 |
then show ?thesis by blast
|
|
795 |
qed
|
|
796 |
then show "h x = iterates f x"
|
|
797 |
proof (coinduct rule: llist_equalityI)
|
|
798 |
case (Eqllist q)
|
|
799 |
then obtain u n where "q = ((lmap f ^ n) (h u), (lmap f ^ n) (iterates f u))"
|
|
800 |
(is "_ = (?q1, ?q2)")
|
|
801 |
by auto
|
|
802 |
also have "?q1 = LCons ((f ^ n) u) ((lmap f ^ Suc n) (h u))"
|
|
803 |
proof -
|
|
804 |
have "?q1 = (lmap f ^ n) (LCons u (lmap f (h u)))"
|
|
805 |
by (subst h) rule
|
|
806 |
also have "\<dots> = LCons ((f ^ n) u) ((lmap f ^ n) (lmap f (h u)))"
|
|
807 |
by (rule funpow_lmap)
|
|
808 |
also have "(lmap f ^ n) (lmap f (h u)) = (lmap f ^ Suc n) (h u)"
|
|
809 |
by (simp add: funpow_swap1)
|
|
810 |
finally show ?thesis .
|
|
811 |
qed
|
|
812 |
also have "?q2 = LCons ((f ^ n) u) ((lmap f ^ Suc n) (iterates f u))"
|
|
813 |
proof -
|
|
814 |
have "?q2 = (lmap f ^ n) (LCons u (iterates f (f u)))"
|
|
815 |
by (subst iterates) rule
|
|
816 |
also have "\<dots> = LCons ((f ^ n) u) ((lmap f ^ n) (iterates f (f u)))"
|
|
817 |
by (rule funpow_lmap)
|
|
818 |
also have "(lmap f ^ n) (iterates f (f u)) = (lmap f ^ Suc n) (iterates f u)"
|
|
819 |
by (simp add: lmap_iterates funpow_swap1)
|
|
820 |
finally show ?thesis .
|
|
821 |
qed
|
|
822 |
finally have ?EqLCons by (auto simp del: funpow.simps)
|
|
823 |
then show ?case ..
|
|
824 |
qed
|
|
825 |
qed
|
|
826 |
|
|
827 |
lemma lappend_iterates: "lappend (iterates f x) l = iterates f x"
|
|
828 |
proof -
|
|
829 |
have "(lappend (iterates f x) l, iterates f x) \<in>
|
|
830 |
{(lappend (iterates f u) l, iterates f u) | u. True}" by blast
|
|
831 |
then show ?thesis
|
|
832 |
proof (coinduct rule: llist_equalityI)
|
|
833 |
case (Eqllist q)
|
|
834 |
then obtain x where "q = (lappend (iterates f x) l, iterates f x)" by blast
|
|
835 |
also have "iterates f x = LCons x (iterates f (f x))" by (rule iterates)
|
|
836 |
finally have ?EqLCons by auto
|
|
837 |
then show ?case ..
|
|
838 |
qed
|
|
839 |
qed
|
|
840 |
|
|
841 |
end
|