# HG changeset patch
# User wenzelm
# Date 1134498756 -3600
# Node ID 6cc32c77d402bd510d5a0a9b04e33bfe7c7c53d9
# Parent  651438736fa1078bb55340edea9eea329c1ef828
Potentially infinite lists as greatest fixed-point.

diff -r 651438736fa1 -r 6cc32c77d402 src/HOL/Library/Coinductive_List.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Coinductive_List.thy	Tue Dec 13 19:32:36 2005 +0100
@@ -0,0 +1,860 @@
+(*  Title:      HOL/Library/Coinductive_Lists.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson and Makarius
+*)
+
+header {* Potentially infinite lists as greatest fixed-point *}
+
+theory Coinductive_List
+imports Main
+begin
+
+subsection {* List constructors over the datatype universe *}
+
+constdefs
+  NIL :: "'a Datatype_Universe.item"
+  "NIL \<equiv> Datatype_Universe.In0 (Datatype_Universe.Numb 0)"
+  CONS :: "'a Datatype_Universe.item \<Rightarrow> 'a Datatype_Universe.item
+    \<Rightarrow> 'a Datatype_Universe.item"
+  "CONS M N \<equiv> Datatype_Universe.In1 (Datatype_Universe.Scons M N)"
+
+lemma CONS_not_NIL [iff]: "CONS M N \<noteq> NIL"
+  and NIL_not_CONS [iff]: "NIL \<noteq> CONS M N"
+  and CONS_inject [iff]: "(CONS K M) = (CONS L N) = (K = L \<and> M = N)"
+  by (simp_all add: NIL_def CONS_def)
+
+lemma CONS_mono: "M \<subseteq> M' \<Longrightarrow> N \<subseteq> N' \<Longrightarrow> CONS M N \<subseteq> CONS M' N'"
+  by (simp add: CONS_def In1_mono Scons_mono)
+
+lemma CONS_UN1: "CONS M (\<Union>x. f x) = (\<Union>x. CONS M (f x))"
+    -- {* A continuity result? *}
+  by (simp add: CONS_def In1_UN1 Scons_UN1_y)
+
+constdefs
+  List_case where
+  "List_case c h \<equiv> Datatype_Universe.Case (\<lambda>_. c) (Datatype_Universe.Split h)"
+
+lemma List_case_NIL [simp]: "List_case c h NIL = c"
+  and List_case_CONS [simp]: "List_case c h (CONS M N) = h M N"
+  by (simp_all add: List_case_def NIL_def CONS_def)
+
+
+subsection {* Corecursive lists *}
+
+consts
+  LList  :: "'a Datatype_Universe.item set \<Rightarrow> 'a Datatype_Universe.item set"
+
+coinductive "LList A"
+  intros
+    NIL [intro]:  "NIL \<in> LList A"
+    CONS [intro]: "a \<in> A \<Longrightarrow> M \<in> LList A \<Longrightarrow> CONS a M \<in> LList A"
+
+lemma LList_mono: "A \<subseteq> B \<Longrightarrow> LList A \<subseteq> LList B"
+    -- {* This justifies using @{text LList} in other recursive type definitions. *}
+  by (unfold LList.defs) (blast intro!: gfp_mono)
+
+consts
+  LList_corec_aux :: "nat \<Rightarrow> ('a \<Rightarrow> ('b Datatype_Universe.item \<times> 'a) option) \<Rightarrow>
+    'a \<Rightarrow> 'b Datatype_Universe.item"
+primrec
+  "LList_corec_aux 0 f x = {}"
+  "LList_corec_aux (Suc k) f x =
+    (case f x of
+      None \<Rightarrow> NIL
+    | Some (z, w) \<Rightarrow> CONS z (LList_corec_aux k f w))"
+
+constdefs
+  LList_corec :: "'a \<Rightarrow> ('a \<Rightarrow> ('b Datatype_Universe.item \<times> 'a) option) \<Rightarrow>
+    'b Datatype_Universe.item"
+  "LList_corec a f \<equiv> \<Union>k. LList_corec_aux k f a"
+
+text {*
+  Note: the subsequent recursion equation for @{text LList_corec} may
+  be used with the Simplifier, provided it operates in a non-strict
+  fashion for case expressions (i.e.\ the usual @{text case}
+  congruence rule needs to be present).
+*}
+
+lemma LList_corec:
+  "LList_corec a f =
+    (case f a of None \<Rightarrow> NIL | Some (z, w) \<Rightarrow> CONS z (LList_corec w f))"
+  (is "?lhs = ?rhs")
+proof
+  show "?lhs \<subseteq> ?rhs"
+    apply (unfold LList_corec_def)
+    apply (rule UN_least)
+    apply (case_tac k)
+     apply (simp_all (no_asm_simp) split: option.splits)
+    apply (rule allI impI subset_refl [THEN CONS_mono] UNIV_I [THEN UN_upper])+
+    done
+  show "?rhs \<subseteq> ?lhs"
+    apply (simp add: LList_corec_def split: option.splits)
+    apply (simp add: CONS_UN1)
+    apply safe
+     apply (rule_tac a = "Suc ?k" in UN_I, simp, simp)+
+    done
+qed
+
+lemma LList_corec_type: "LList_corec a f \<in> LList UNIV"
+proof -
+  have "LList_corec a f \<in> {LList_corec a f | a. True}" by blast
+  then show ?thesis
+  proof coinduct
+    case (LList L)
+    then obtain x where L: "L = LList_corec x f" by blast
+    show ?case
+    proof (cases "f x")
+      case None
+      then have "LList_corec x f = NIL"
+        by (simp add: LList_corec)
+      with L have ?NIL by simp
+      then show ?thesis ..
+    next
+      case (Some p)
+      then have "LList_corec x f = CONS (fst p) (LList_corec (snd p) f)"
+        by (simp add: split_def LList_corec)
+      with L have ?CONS by auto
+      then show ?thesis ..
+    qed
+  qed
+qed
+
+
+subsection {* Abstract type definition *}
+
+typedef 'a llist =
+  "LList (range Datatype_Universe.Leaf) :: 'a Datatype_Universe.item set"
+proof
+  show "NIL \<in> ?llist" ..
+qed
+
+lemma NIL_type: "NIL \<in> llist"
+  by (unfold llist_def) (rule LList.NIL)
+
+lemma CONS_type: "a \<in> range Datatype_Universe.Leaf \<Longrightarrow>
+    M \<in> llist \<Longrightarrow> CONS a M \<in> llist"
+  by (unfold llist_def) (rule LList.CONS)
+
+lemma llistI: "x \<in> LList (range Datatype_Universe.Leaf) \<Longrightarrow> x \<in> llist"
+  by (simp add: llist_def)
+
+lemma llistD: "x \<in> llist \<Longrightarrow> x \<in> LList (range Datatype_Universe.Leaf)"
+  by (simp add: llist_def)
+
+lemma Rep_llist_UNIV: "Rep_llist x \<in> LList UNIV"
+proof -
+  have "Rep_llist x \<in> llist" by (rule Rep_llist)
+  then have "Rep_llist x \<in> LList (range Datatype_Universe.Leaf)"
+    by (simp add: llist_def)
+  also have "\<dots> \<subseteq> LList UNIV" by (rule LList_mono) simp
+  finally show ?thesis .
+qed
+
+constdefs
+  LNil :: "'a llist"
+  "LNil \<equiv> Abs_llist NIL"
+
+  LCons :: "'a \<Rightarrow> 'a llist \<Rightarrow> 'a llist"
+  "LCons x xs \<equiv> Abs_llist (CONS (Datatype_Universe.Leaf x) (Rep_llist xs))"
+
+lemma LCons_not_LNil [iff]: "LCons x xs \<noteq> LNil"
+  apply (simp add: LNil_def LCons_def)
+  apply (subst Abs_llist_inject)
+    apply (auto intro: NIL_type CONS_type Rep_llist)
+  done
+
+lemma LNil_not_LCons [iff]: "LNil \<noteq> LCons x xs"
+  by (rule LCons_not_LNil [symmetric])
+
+lemma LCons_inject [iff]: "(LCons x xs = LCons y ys) = (x = y \<and> xs = ys)"
+  apply (simp add: LCons_def)
+  apply (subst Abs_llist_inject)
+    apply (auto simp add: Rep_llist_inject intro: CONS_type Rep_llist)
+  done
+
+lemma Rep_llist_LNil: "Rep_llist LNil = NIL"
+  by (simp add: LNil_def add: Abs_llist_inverse NIL_type)
+
+lemma Rep_llist_LCons: "Rep_llist (LCons x l) =
+    CONS (Datatype_Universe.Leaf x) (Rep_llist l)"
+  by (simp add: LCons_def Abs_llist_inverse CONS_type Rep_llist)
+
+lemma llist_cases [case_names LNil LCons, cases type: llist]:
+  assumes LNil: "l = LNil \<Longrightarrow> C"
+    and LCons: "\<And>x l'. l = LCons x l' \<Longrightarrow> C"
+  shows C
+proof (cases l)
+  case (Abs_llist L)
+  from `L \<in> llist` have "L \<in> LList (range Datatype_Universe.Leaf)" by (rule llistD)
+  then show ?thesis
+  proof cases
+    case NIL
+    with Abs_llist have "l = LNil" by (simp add: LNil_def)
+    with LNil show ?thesis .
+  next
+    case (CONS K a)
+    then have "K \<in> llist" by (blast intro: llistI)
+    then obtain l' where "K = Rep_llist l'" by cases
+    with CONS and Abs_llist obtain x where "l = LCons x l'"
+      by (auto simp add: LCons_def Abs_llist_inject)
+    with LCons show ?thesis .
+  qed
+qed
+
+
+constdefs
+  llist_case :: "'b \<Rightarrow> ('a \<Rightarrow> 'a llist \<Rightarrow> 'b) \<Rightarrow> 'a llist \<Rightarrow> 'b"
+  "llist_case c d l \<equiv>
+    List_case c (\<lambda>x y. d (inv Datatype_Universe.Leaf x) (Abs_llist y)) (Rep_llist l)"
+
+translations
+  "case p of LNil \<Rightarrow> a | LCons x l \<Rightarrow> b" \<rightleftharpoons> "llist_case a (\<lambda>x l. b) p"
+
+lemma llist_case_LNil [simp]: "llist_case c d LNil = c"
+  by (simp add: llist_case_def LNil_def
+    NIL_type Abs_llist_inverse)
+
+lemma llist_case_LCons [simp]: "llist_case c d (LCons M N) = d M N"
+  by (simp add: llist_case_def LCons_def
+    CONS_type Abs_llist_inverse Rep_llist Rep_llist_inverse inj_Leaf)
+
+
+constdefs
+  llist_corec :: "'a \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'a) option) \<Rightarrow> 'b llist"
+  "llist_corec a f \<equiv>
+    Abs_llist (LList_corec a
+      (\<lambda>z.
+        case f z of None \<Rightarrow> None
+        | Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w)))"
+
+lemma LList_corec_type2:
+  "LList_corec a
+    (\<lambda>z. case f z of None \<Rightarrow> None
+      | Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w)) \<in> llist"
+  (is "?corec a \<in> _")
+proof (unfold llist_def)
+  let "LList_corec a ?g" = "?corec a"
+  have "?corec a \<in> {?corec x | x. True}" by blast
+  then show "?corec a \<in> LList (range Datatype_Universe.Leaf)"
+  proof coinduct
+    case (LList L)
+    then obtain x where L: "L = ?corec x" by blast
+    show ?case
+    proof (cases "f x")
+      case None
+      then have "?corec x = NIL"
+        by (simp add: LList_corec)
+      with L have ?NIL by simp
+      then show ?thesis ..
+    next
+      case (Some p)
+      then have "?corec x =
+          CONS (Datatype_Universe.Leaf (fst p)) (?corec (snd p))"
+        by (simp add: split_def LList_corec)
+      with L have ?CONS by auto
+      then show ?thesis ..
+    qed
+  qed
+qed
+
+lemma llist_corec:
+  "llist_corec a f =
+    (case f a of None \<Rightarrow> LNil | Some (z, w) \<Rightarrow> LCons z (llist_corec w f))"
+proof (cases "f a")
+  case None
+  then show ?thesis
+    by (simp add: llist_corec_def LList_corec LNil_def)
+next
+  case (Some p)
+
+  let "?corec a" = "llist_corec a f"
+  let "?rep_corec a" =
+    "LList_corec a
+      (\<lambda>z. case f z of None \<Rightarrow> None
+        | Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w))"
+
+  have "?corec a = Abs_llist (?rep_corec a)"
+    by (simp only: llist_corec_def)
+  also from Some have "?rep_corec a =
+      CONS (Datatype_Universe.Leaf (fst p)) (?rep_corec (snd p))"
+    by (simp add: split_def LList_corec)
+  also have "?rep_corec (snd p) = Rep_llist (?corec (snd p))"
+    by (simp only: llist_corec_def Abs_llist_inverse LList_corec_type2)
+  finally have "?corec a = LCons (fst p) (?corec (snd p))"
+    by (simp only: LCons_def)
+  with Some show ?thesis by (simp add: split_def)
+qed
+
+
+subsection {* Equality as greatest fixed-point; the bisimulation principle. *}
+
+consts
+  EqLList :: "('a Datatype_Universe.item \<times> 'a Datatype_Universe.item) set \<Rightarrow>
+    ('a Datatype_Universe.item \<times> 'a Datatype_Universe.item) set"
+
+coinductive "EqLList r"
+  intros
+    EqNIL: "(NIL, NIL) \<in> EqLList r"
+    EqCONS: "(a, b) \<in> r \<Longrightarrow> (M, N) \<in> EqLList r \<Longrightarrow>
+      (CONS a M, CONS b N) \<in> EqLList r"
+
+lemma EqLList_unfold:
+    "EqLList r = dsum (diag {Datatype_Universe.Numb 0}) (dprod r (EqLList r))"
+  by (fast intro!: EqLList.intros [unfolded NIL_def CONS_def]
+           elim: EqLList.cases [unfolded NIL_def CONS_def])
+
+lemma EqLList_implies_ntrunc_equality:
+    "(M, N) \<in> EqLList (diag A) \<Longrightarrow> ntrunc k M = ntrunc k N"
+  apply (induct k fixing: M N rule: nat_less_induct)
+  apply (erule EqLList.cases)
+   apply (safe del: equalityI)
+  apply (case_tac n)
+   apply simp
+  apply (rename_tac n')
+  apply (case_tac n')
+   apply (simp_all add: CONS_def less_Suc_eq)
+  done
+
+lemma Domain_EqLList: "Domain (EqLList (diag A)) \<subseteq> LList A"
+  apply (simp add: LList.defs NIL_def CONS_def)
+  apply (rule gfp_upperbound)
+  apply (subst EqLList_unfold)
+  apply auto
+  done
+
+lemma EqLList_diag: "EqLList (diag A) = diag (LList A)"
+  (is "?lhs = ?rhs")
+proof
+  show "?lhs \<subseteq> ?rhs"
+    apply (rule subsetI)
+    apply (rule_tac p = x in PairE)
+    apply clarify
+    apply (rule diag_eqI)
+     apply (rule EqLList_implies_ntrunc_equality [THEN ntrunc_equality],
+       assumption)
+    apply (erule DomainI [THEN Domain_EqLList [THEN subsetD]])
+    done
+  show "?rhs \<subseteq> ?lhs"
+  proof
+    fix p assume "p \<in> diag (LList A)"
+    then show "p \<in> EqLList (diag A)"
+    proof coinduct
+      case (EqLList q)
+      then obtain L where L: "L \<in> LList A" and q: "q = (L, L)" ..
+      from L show ?case
+      proof cases
+        case NIL with q have ?EqNIL by simp
+        then show ?thesis ..
+      next
+        case CONS with q have ?EqCONS by (simp add: diagI)
+        then show ?thesis ..
+      qed
+    qed
+  qed
+qed
+
+lemma EqLList_diag_iff [iff]: "(p \<in> EqLList (diag A)) = (p \<in> diag (LList A))"
+  by (simp only: EqLList_diag)
+
+
+text {*
+  To show two LLists are equal, exhibit a bisimulation!  (Also admits
+  true equality.)
+*}
+
+lemma LList_equalityI
+  [consumes 1, case_names EqLList, case_conclusion EqLList EqNIL EqCONS]:
+  assumes r: "(M, N) \<in> r"
+    and step: "\<And>p. p \<in> r \<Longrightarrow>
+      p = (NIL, NIL) \<or>
+        (\<exists>M N a b.
+          p = (CONS a M, CONS b N) \<and> (a, b) \<in> diag A \<and>
+            (M, N) \<in> r \<union> EqLList (diag A))"
+  shows "M = N"
+proof -
+  from r have "(M, N) \<in> EqLList (diag A)"
+  proof coinduct
+    case EqLList
+    then show ?case by (rule step)
+  qed
+  then show ?thesis by auto
+qed
+
+lemma LList_fun_equalityI
+  [consumes 1, case_names NIL_type NIL CONS, case_conclusion CONS EqNIL EqCONS]:
+  assumes M: "M \<in> LList A"
+    and fun_NIL: "g NIL \<in> LList A"  "f NIL = g NIL"
+    and fun_CONS: "\<And>x l. x \<in> A \<Longrightarrow> l \<in> LList A \<Longrightarrow>
+            (f (CONS x l), g (CONS x l)) = (NIL, NIL) \<or>
+            (\<exists>M N a b.
+              (f (CONS x l), g (CONS x l)) = (CONS a M, CONS b N) \<and>
+                (a, b) \<in> diag A \<and>
+                (M, N) \<in> {(f u, g u) | u. u \<in> LList A} \<union> diag (LList A))"
+      (is "\<And>x l. _ \<Longrightarrow> _ \<Longrightarrow> ?fun_CONS x l")
+  shows "f M = g M"
+proof -
+  let ?bisim = "{(f L, g L) | L. L \<in> LList A}"
+  have "(f M, g M) \<in> ?bisim" using M by blast
+  then show ?thesis
+  proof (coinduct taking: A rule: LList_equalityI)
+    case (EqLList q)
+    then obtain L where q: "q = (f L, g L)" and L: "L \<in> LList A" by blast
+    from L show ?case
+    proof (cases L)
+      case NIL
+      with fun_NIL and q have "q \<in> diag (LList A)" by auto
+      then have "q \<in> EqLList (diag A)" ..
+      then show ?thesis by cases simp_all
+    next
+      case (CONS K a)
+      from fun_CONS and `a \<in> A` `K \<in> LList A`
+      have "?fun_CONS a K" (is "?NIL \<or> ?CONS") .
+      then show ?thesis
+      proof
+        assume ?NIL
+        with q CONS have "q \<in> diag (LList A)" by auto
+        then have "q \<in> EqLList (diag A)" ..
+        then show ?thesis by cases simp_all
+      next
+        assume ?CONS
+        with CONS obtain a b M N where
+            fg: "(f L, g L) = (CONS a M, CONS b N)"
+          and ab: "(a, b) \<in> diag A"
+          and MN: "(M, N) \<in> ?bisim \<union> diag (LList A)"
+          by blast
+        from MN show ?thesis
+        proof
+          assume "(M, N) \<in> ?bisim"
+          with q fg ab show ?thesis by simp
+        next
+          assume "(M, N) \<in> diag (LList A)"
+          then have "(M, N) \<in> EqLList (diag A)" ..
+          with q fg ab show ?thesis by simp
+        qed
+      qed
+    qed
+  qed
+qed
+
+text {*
+  Finality of @{text "llist A"}: Uniqueness of functions defined by corecursion.
+*}
+
+lemma equals_LList_corec:
+  assumes h: "\<And>x. h x =
+    (case f x of None \<Rightarrow> NIL | Some (z, w) \<Rightarrow> CONS z (h w))"
+  shows "h x = (\<lambda>x. LList_corec x f) x"
+proof -
+  def h' \<equiv> "\<lambda>x. LList_corec x f"
+  then have h': "\<And>x. h' x =
+      (case f x of None \<Rightarrow> NIL | Some (z, w) \<Rightarrow> CONS z (h' w))"
+    by (unfold h'_def) (simp add: LList_corec)
+  have "(h x, h' x) \<in> {(h u, h' u) | u. True}" by blast
+  then show "h x = h' x"
+  proof (coinduct rule: LList_equalityI [where A = UNIV])
+    case (EqLList q)
+    then obtain x where q: "q = (h x, h' x)" by blast
+    show ?case
+    proof (cases "f x")
+      case None
+      with h h' q have ?EqNIL by simp
+      then show ?thesis ..
+    next
+      case (Some p)
+      with h h' q have "q =
+          (CONS (fst p) (h (snd p)), CONS (fst p) (h' (snd p)))"
+        by (simp add: split_def)
+      then have ?EqCONS by (auto iff: diag_iff)
+      then show ?thesis ..
+    qed
+  qed
+qed
+
+
+lemma llist_equalityI
+  [consumes 1, case_names Eqllist, case_conclusion Eqllist EqLNil EqLCons]:
+  assumes r: "(l1, l2) \<in> r"
+    and step: "\<And>q. q \<in> r \<Longrightarrow>
+      q = (LNil, LNil) \<or>
+        (\<exists>l1 l2 a b.
+          q = (LCons a l1, LCons b l2) \<and> a = b \<and>
+            ((l1, l2) \<in> r \<or> l1 = l2))"
+      (is "\<And>q. _ \<Longrightarrow> ?EqLNil q \<or> ?EqLCons q")
+  shows "l1 = l2"
+proof -
+  def M \<equiv> "Rep_llist l1" and N \<equiv> "Rep_llist l2"
+  with r have "(M, N) \<in> {(Rep_llist l1, Rep_llist l2) | l1 l2. (l1, l2) \<in> r}"
+    by blast
+  then have "M = N"
+  proof (coinduct rule: LList_equalityI [where A = UNIV])
+    case (EqLList q)
+    then obtain l1 l2 where
+        q: "q = (Rep_llist l1, Rep_llist l2)" and r: "(l1, l2) \<in> r"
+      by auto
+    from step [OF r] show ?case
+    proof
+      assume "?EqLNil (l1, l2)"
+      with q have ?EqNIL by (simp add: Rep_llist_LNil)
+      then show ?thesis ..
+    next
+      assume "?EqLCons (l1, l2)"
+      with q have ?EqCONS
+        by (force simp add: Rep_llist_LCons EqLList_diag intro: Rep_llist_UNIV)
+      then show ?thesis ..
+    qed
+  qed
+  then show ?thesis by (simp add: M_def N_def Rep_llist_inject)
+qed
+
+lemma llist_fun_equalityI
+  [case_names LNil LCons, case_conclusion LCons EqLNil EqLCons]:
+  assumes fun_LNil: "f LNil = g LNil"
+    and fun_LCons: "\<And>x l.
+      (f (LCons x l), g (LCons x l)) = (LNil, LNil) \<or>
+        (\<exists>l1 l2 a b.
+          (f (LCons x l), g (LCons x l)) = (LCons a l1, LCons b l2) \<and>
+            a = b \<and> ((l1, l2) \<in> {(f u, g u) | u. True} \<or> l1 = l2))"
+      (is "\<And>x l. ?fun_LCons x l")
+  shows "f l = g l"
+proof -
+  have "(f l, g l) \<in> {(f l, g l) | l. True}" by blast
+  then show ?thesis
+  proof (coinduct rule: llist_equalityI)
+    case (Eqllist q)
+    then obtain l where q: "q = (f l, g l)" by blast
+    show ?case
+    proof (cases l)
+      case LNil
+      with fun_LNil and q have "q = (g LNil, g LNil)" by simp
+      then show ?thesis by (cases "g LNil") simp_all
+    next
+      case (LCons x l')
+      with `?fun_LCons x l'` q LCons show ?thesis by blast
+    qed
+  qed
+qed
+
+
+subsection {* Derived operations -- both on the set and abstract type *}
+
+subsubsection {* @{text Lconst} *}
+
+constdefs
+  Lconst where
+  "Lconst M \<equiv> lfp (\<lambda>N. CONS M N)"
+
+lemma Lconst_fun_mono: "mono (CONS M)"
+  by (simp add: monoI CONS_mono)
+
+lemma Lconst: "Lconst M = CONS M (Lconst M)"
+  by (rule Lconst_def [THEN def_lfp_unfold]) (rule Lconst_fun_mono)
+
+lemma Lconst_type:
+  assumes "M \<in> A"
+  shows "Lconst M \<in> LList A"
+proof -
+  have "Lconst M \<in> {Lconst M}" by simp
+  then show ?thesis
+  proof coinduct
+    case (LList N)
+    then have "N = Lconst M" by simp
+    also have "\<dots> = CONS M (Lconst M)" by (rule Lconst)
+    finally have ?CONS using `M \<in> A` by simp
+    then show ?case ..
+  qed
+qed
+
+lemma Lconst_eq_LList_corec: "Lconst M = LList_corec M (\<lambda>x. Some (x, x))"
+  apply (rule equals_LList_corec)
+  apply simp
+  apply (rule Lconst)
+  done
+
+lemma gfp_Lconst_eq_LList_corec:
+    "gfp (\<lambda>N. CONS M N) = LList_corec M (\<lambda>x. Some(x, x))"
+  apply (rule equals_LList_corec)
+  apply simp
+  apply (rule Lconst_fun_mono [THEN gfp_unfold])
+  done
+
+
+subsubsection {* @{text Lmap} and @{text lmap} *}
+
+constdefs
+  Lmap where
+  "Lmap f M \<equiv> LList_corec M (List_case None (\<lambda>x M'. Some (f x, M')))"
+
+  lmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a llist \<Rightarrow> 'b llist"
+  "lmap f l \<equiv> llist_corec l
+    (\<lambda>z.
+      case z of LNil \<Rightarrow> None
+      | LCons y z \<Rightarrow> Some (f y, z))"
+
+lemma Lmap_NIL [simp]: "Lmap f NIL = NIL"
+  and Lmap_CONS [simp]: "Lmap f (CONS M N) = CONS (f M) (Lmap f N)"
+  by (simp_all add: Lmap_def LList_corec)
+
+lemma Lmap_type:
+  assumes M: "M \<in> LList A"
+    and f: "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B"
+  shows "Lmap f M \<in> LList B"
+proof -
+  from M have "Lmap f M \<in> {Lmap f N | N. N \<in> LList A}" by blast
+  then show ?thesis
+  proof coinduct
+    case (LList L)
+    then obtain N where L: "L = Lmap f N" and N: "N \<in> LList A" by blast
+    from N show ?case
+    proof cases
+      case NIL
+      with L have ?NIL by simp
+      then show ?thesis ..
+    next
+      case (CONS K a)
+      with f L have ?CONS by auto
+      then show ?thesis ..
+    qed
+  qed
+qed
+
+lemma Lmap_compose:
+  assumes M: "M \<in> LList A"
+  shows "Lmap (f o g) M = Lmap f (Lmap g M)"  (is "?lhs M = ?rhs M")
+proof -
+  have "(?lhs M, ?rhs M) \<in> {(?lhs N, ?rhs N) | N. N \<in> LList A}"
+    using M by blast
+  then show ?thesis
+  proof (coinduct taking: "range (\<lambda>N :: 'a Datatype_Universe.item. N)"
+      rule: LList_equalityI)
+    case (EqLList q)
+    then obtain N where q: "q = (?lhs N, ?rhs N)" and N: "N \<in> LList A" by blast
+    from N show ?case
+    proof cases
+      case NIL
+      with q have ?EqNIL by simp
+      then show ?thesis ..
+    next
+      case CONS
+      with q have ?EqCONS by auto
+      then show ?thesis ..
+    qed
+  qed
+qed
+
+lemma Lmap_ident:
+  assumes M: "M \<in> LList A"
+  shows "Lmap (\<lambda>x. x) M = M"  (is "?lmap M = _")
+proof -
+  have "(?lmap M, M) \<in> {(?lmap N, N) | N. N \<in> LList A}" using M by blast
+  then show ?thesis
+  proof (coinduct taking: "range (\<lambda>N :: 'a Datatype_Universe.item. N)"
+      rule: LList_equalityI)
+    case (EqLList q)
+    then obtain N where q: "q = (?lmap N, N)" and N: "N \<in> LList A" by blast
+    from N show ?case
+    proof cases
+      case NIL
+      with q have ?EqNIL by simp
+      then show ?thesis ..
+    next
+      case CONS
+      with q have ?EqCONS by auto
+      then show ?thesis ..
+    qed
+  qed
+qed
+
+lemma lmap_LNil [simp]: "lmap f LNil = LNil"
+  and lmap_LCons [simp]: "lmap f (LCons M N) = LCons (f M) (lmap f N)"
+  by (simp_all add: lmap_def llist_corec)
+
+lemma lmap_compose [simp]: "lmap (f o g) l = lmap f (lmap g l)"
+  by (coinduct _ _ l rule: llist_fun_equalityI) auto
+
+lemma lmap_ident [simp]: "lmap (\<lambda>x. x) l = l"
+  by (coinduct _ _ l rule: llist_fun_equalityI) auto
+
+
+
+subsubsection {* @{text Lappend} *}
+
+constdefs
+  Lappend where
+  "Lappend M N \<equiv> LList_corec (M, N)
+    (split (List_case
+        (List_case None (\<lambda>N1 N2. Some (N1, (NIL, N2))))
+        (\<lambda>M1 M2 N. Some (M1, (M2, N)))))"
+
+  lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist"
+  "lappend l n \<equiv> llist_corec (l, n)
+    (split (llist_case
+        (llist_case None (\<lambda>n1 n2. Some (n1, (LNil, n2))))
+        (\<lambda>l1 l2 n. Some (l1, (l2, n)))))"
+
+lemma Lappend_NIL_NIL [simp]:
+    "Lappend NIL NIL = NIL"
+  and Lappend_NIL_CONS [simp]:
+    "Lappend NIL (CONS N N') = CONS N (Lappend NIL N')"
+  and Lappend_CONS [simp]:
+    "Lappend (CONS M M') N = CONS M (Lappend M' N)"
+  by (simp_all add: Lappend_def LList_corec)
+
+lemma Lappend_NIL [simp]: "M \<in> LList A \<Longrightarrow> Lappend NIL M = M"
+  by (erule LList_fun_equalityI) auto
+
+lemma Lappend_NIL2: "M \<in> LList A \<Longrightarrow> Lappend M NIL = M"
+  by (erule LList_fun_equalityI) auto
+
+lemma Lappend_type:
+  assumes M: "M \<in> LList A" and N: "N \<in> LList A"
+  shows "Lappend M N \<in> LList A"
+proof -
+  have "Lappend M N \<in> {Lappend u v | u v. u \<in> LList A \<and> v \<in> LList A}"
+    using M N by blast
+  then show ?thesis
+  proof coinduct
+    case (LList L)
+    then obtain M N where L: "L = Lappend M N"
+        and M: "M \<in> LList A" and N: "N \<in> LList A"
+      by blast
+    from M show ?case
+    proof cases
+      case NIL
+      from N show ?thesis
+      proof cases
+        case NIL
+        with L and `M = NIL` have ?NIL by simp
+        then show ?thesis ..
+      next
+        case CONS
+        with L and `M = NIL` have ?CONS by simp
+        then show ?thesis ..
+      qed
+    next
+      case CONS
+      with L N have ?CONS by auto
+      then show ?thesis ..
+    qed
+  qed
+qed
+
+lemma lappend_LNil_LNil [simp]: "lappend LNil LNil = LNil"
+  and lappend_LNil_LCons [simp]: "lappend LNil (LCons l l') = LCons l (lappend LNil l')"
+  and lappend_LCons [simp]: "lappend (LCons l l') m = LCons l (lappend l' m)"
+  by (simp_all add: lappend_def llist_corec)
+
+lemma lappend_LNil1 [simp]: "lappend LNil l = l"
+  by (coinduct _ _ l rule: llist_fun_equalityI) auto
+
+lemma lappend_LNil2 [simp]: "lappend l LNil = l"
+  by (coinduct _ _ l rule: llist_fun_equalityI) auto
+
+lemma lappend_assoc: "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)"
+  by (coinduct _ _ l1 rule: llist_fun_equalityI) auto
+
+lemma lmap_lappend_distrib: "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"
+  by (coinduct _ _ l rule: llist_fun_equalityI) auto
+
+
+subsection{* iterates *}
+
+text {* @{text llist_fun_equalityI} cannot be used here! *}
+
+constdefs
+  iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist"
+  "iterates f a \<equiv> llist_corec a (\<lambda>x. Some (x, f x))"
+
+lemma iterates: "iterates f x = LCons x (iterates f (f x))"
+  apply (unfold iterates_def)
+  apply (subst llist_corec)
+  apply simp
+  done
+
+lemma lmap_iterates: "lmap f (iterates f x) = iterates f (f x)"
+proof -
+  have "(lmap f (iterates f x), iterates f (f x)) \<in>
+    {(lmap f (iterates f u), iterates f (f u)) | u. True}" by blast
+  then show ?thesis
+  proof (coinduct rule: llist_equalityI)
+    case (Eqllist q)
+    then obtain x where q: "q = (lmap f (iterates f x), iterates f (f x))"
+      by blast
+    also have "iterates f (f x) = LCons (f x) (iterates f (f (f x)))"
+      by (subst iterates) rule
+    also have "iterates f x = LCons x (iterates f (f x))"
+      by (subst iterates) rule
+    finally have ?EqLCons by auto
+    then show ?case ..
+  qed
+qed
+
+lemma iterates_lmap: "iterates f x = LCons x (lmap f (iterates f x))"
+  by (subst lmap_iterates) (rule iterates)
+
+
+subsection{* A rather complex proof about iterates -- cf.\ Andy Pitts *}
+
+lemma funpow_lmap:
+  fixes f :: "'a \<Rightarrow> 'a"
+  shows "(lmap f ^ n) (LCons b l) = LCons ((f ^ n) b) ((lmap f ^ n) l)"
+  by (induct n) simp_all
+
+
+lemma iterates_equality:
+  assumes h: "\<And>x. h x = LCons x (lmap f (h x))"
+  shows "h = iterates f"
+proof
+  fix x
+  have "(h x, iterates f x) \<in>
+      {((lmap f ^ n) (h u), (lmap f ^ n) (iterates f u)) | u n. True}"
+  proof -
+    have "(h x, iterates f x) = ((lmap f ^ 0) (h x), (lmap f ^ 0) (iterates f x))"
+      by simp
+    then show ?thesis by blast
+  qed
+  then show "h x = iterates f x"
+  proof (coinduct rule: llist_equalityI)
+    case (Eqllist q)
+    then obtain u n where "q = ((lmap f ^ n) (h u), (lmap f ^ n) (iterates f u))"
+        (is "_ = (?q1, ?q2)")
+      by auto
+    also have "?q1 = LCons ((f ^ n) u) ((lmap f ^ Suc n) (h u))"
+    proof -
+      have "?q1 = (lmap f ^ n) (LCons u (lmap f (h u)))"
+        by (subst h) rule
+      also have "\<dots> = LCons ((f ^ n) u) ((lmap f ^ n) (lmap f (h u)))"
+        by (rule funpow_lmap)
+      also have "(lmap f ^ n) (lmap f (h u)) = (lmap f ^ Suc n) (h u)"
+        by (simp add: funpow_swap1)
+      finally show ?thesis .
+    qed
+    also have "?q2 = LCons ((f ^ n) u) ((lmap f ^ Suc n) (iterates f u))"
+    proof -
+      have "?q2 = (lmap f ^ n) (LCons u (iterates f (f u)))"
+        by (subst iterates) rule
+      also have "\<dots> = LCons ((f ^ n) u) ((lmap f ^ n) (iterates f (f u)))"
+        by (rule funpow_lmap)
+      also have "(lmap f ^ n) (iterates f (f u)) = (lmap f ^ Suc n) (iterates f u)"
+        by (simp add: lmap_iterates funpow_swap1)
+      finally show ?thesis .
+    qed
+    finally have ?EqLCons by (auto simp del: funpow.simps)
+    then show ?case ..
+  qed
+qed
+
+lemma lappend_iterates: "lappend (iterates f x) l = iterates f x"
+proof -
+  have "(lappend (iterates f x) l, iterates f x) \<in>
+    {(lappend (iterates f u) l, iterates f u) | u. True}" by blast
+  then show ?thesis
+  proof (coinduct rule: llist_equalityI)
+    case (Eqllist q)
+    then obtain x where "q = (lappend (iterates f x) l, iterates f x)" by blast
+    also have "iterates f x = LCons x (iterates f (f x))" by (rule iterates)
+    finally have ?EqLCons by auto
+    then show ?case ..
+  qed
+qed
+
+end