--- a/src/HOL/Library/Coinductive_List.thy Sat Mar 13 16:44:12 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,856 +0,0 @@
-(* Title: HOL/Library/Coinductive_Lists.thy
- Author: Lawrence C Paulson and Makarius
-*)
-
-header {* Potentially infinite lists as greatest fixed-point *}
-
-theory Coinductive_List
-imports List Main
-begin
-
-subsection {* List constructors over the datatype universe *}
-
-definition "NIL = Datatype.In0 (Datatype.Numb 0)"
-definition "CONS M N = Datatype.In1 (Datatype.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)
-
-definition "List_case c h = Datatype.Case (\<lambda>_. c) (Datatype.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 *}
-
-coinductive_set LList for A
-where 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:
- assumes subset: "A \<subseteq> B"
- shows "LList A \<subseteq> LList B"
- -- {* This justifies using @{text LList} in other recursive type definitions. *}
-proof
- fix x
- assume "x \<in> LList A"
- then show "x \<in> LList B"
- proof coinduct
- case LList
- then show ?case using subset
- by cases blast+
- qed
-qed
-
-primrec
- LList_corec_aux :: "nat \<Rightarrow> ('a \<Rightarrow> ('b Datatype.item \<times> 'a) option) \<Rightarrow>
- 'a \<Rightarrow> 'b Datatype.item" where
- "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))"
-
-definition "LList_corec a f = (\<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 "\<exists>x. LList_corec a f = LList_corec x f" 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: LList_corec split: prod.split)
- with L have ?CONS by auto
- then show ?thesis ..
- qed
- qed
-qed
-
-
-subsection {* Abstract type definition *}
-
-typedef 'a llist = "LList (range Datatype.Leaf) :: 'a Datatype.item set"
-proof
- show "NIL \<in> ?llist" ..
-qed
-
-lemma NIL_type: "NIL \<in> llist"
- unfolding llist_def by (rule LList.NIL)
-
-lemma CONS_type: "a \<in> range Datatype.Leaf \<Longrightarrow>
- M \<in> llist \<Longrightarrow> CONS a M \<in> llist"
- unfolding llist_def by (rule LList.CONS)
-
-lemma llistI: "x \<in> LList (range Datatype.Leaf) \<Longrightarrow> x \<in> llist"
- by (simp add: llist_def)
-
-lemma llistD: "x \<in> llist \<Longrightarrow> x \<in> LList (range Datatype.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.Leaf)"
- by (simp add: llist_def)
- also have "\<dots> \<subseteq> LList UNIV" by (rule LList_mono) simp
- finally show ?thesis .
-qed
-
-definition "LNil = Abs_llist NIL"
-definition "LCons x xs = Abs_llist (CONS (Datatype.Leaf x) (Rep_llist xs))"
-
-code_datatype LNil LCons
-
-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.Leaf x) (Rep_llist l)"
- by (simp add: LCons_def Abs_llist_inverse CONS_type Rep_llist)
-
-lemma llist_cases [cases type: llist]:
- obtains
- (LNil) "l = LNil"
- | (LCons) x l' where "l = LCons x l'"
-proof (cases l)
- case (Abs_llist L)
- from `L \<in> llist` have "L \<in> LList (range Datatype.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 a K)
- 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
-
-
-definition
- [code del]: "llist_case c d l =
- List_case c (\<lambda>x y. d (inv Datatype.Leaf x) (Abs_llist y)) (Rep_llist l)"
-
-
-syntax (* FIXME? *)
- LNil :: logic
- LCons :: logic
-translations
- "case p of XCONST LNil \<Rightarrow> a | XCONST LCons x l \<Rightarrow> b" \<rightleftharpoons> "CONST llist_case a (\<lambda>x l. b) p"
-
-lemma llist_case_LNil [simp, code]: "llist_case c d LNil = c"
- by (simp add: llist_case_def LNil_def
- NIL_type Abs_llist_inverse)
-
-lemma llist_case_LCons [simp, code]: "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)
-
-lemma llist_case_cert:
- assumes "CASE \<equiv> llist_case c d"
- shows "(CASE LNil \<equiv> c) &&& (CASE (LCons M N) \<equiv> d M N)"
- using assms by simp_all
-
-setup {*
- Code.add_case @{thm llist_case_cert}
-*}
-
-definition
- [code del]: "llist_corec a f =
- Abs_llist (LList_corec a
- (\<lambda>z.
- case f z of None \<Rightarrow> None
- | Some (v, w) \<Rightarrow> Some (Datatype.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.Leaf v, w)) \<in> llist"
- (is "?corec a \<in> _")
-proof (unfold llist_def)
- let "LList_corec a ?g" = "?corec a"
- have "\<exists>x. ?corec a = ?corec x" by blast
- then show "?corec a \<in> LList (range Datatype.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.Leaf (fst p)) (?corec (snd p))"
- by (simp add: LList_corec split: prod.split)
- with L have ?CONS by auto
- then show ?thesis ..
- qed
- qed
-qed
-
-lemma llist_corec [code, nitpick_simp]:
- "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.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.Leaf (fst p)) (?rep_corec (snd p))"
- by (simp add: LList_corec split: prod.split)
- 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 split: prod.split)
-qed
-
-
-subsection {* Equality as greatest fixed-point -- the bisimulation principle *}
-
-coinductive_set EqLList for r
-where 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 (Id_on {Datatype.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 (Id_on A) \<Longrightarrow> ntrunc k M = ntrunc k N"
- apply (induct k arbitrary: 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 (Id_on A)) \<subseteq> LList A"
- apply (rule subsetI)
- apply (erule LList.coinduct)
- apply (subst (asm) EqLList_unfold)
- apply (auto simp add: NIL_def CONS_def)
- done
-
-lemma EqLList_Id_on: "EqLList (Id_on A) = Id_on (LList A)"
- (is "?lhs = ?rhs")
-proof
- show "?lhs \<subseteq> ?rhs"
- apply (rule subsetI)
- apply (rule_tac p = x in PairE)
- apply clarify
- apply (rule Id_on_eqI)
- apply (rule EqLList_implies_ntrunc_equality [THEN ntrunc_equality],
- assumption)
- apply (erule DomainI [THEN Domain_EqLList [THEN subsetD]])
- done
- {
- fix M N assume "(M, N) \<in> Id_on (LList A)"
- then have "(M, N) \<in> EqLList (Id_on A)"
- proof coinduct
- case (EqLList M N)
- then obtain L where L: "L \<in> LList A" and MN: "M = L" "N = L" by blast
- from L show ?case
- proof cases
- case NIL with MN have ?EqNIL by simp
- then show ?thesis ..
- next
- case CONS with MN have ?EqCONS by (simp add: Id_onI)
- then show ?thesis ..
- qed
- qed
- }
- then show "?rhs \<subseteq> ?lhs" by auto
-qed
-
-lemma EqLList_Id_on_iff [iff]: "(p \<in> EqLList (Id_on A)) = (p \<in> Id_on (LList A))"
- by (simp only: EqLList_Id_on)
-
-
-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>M N. (M, N) \<in> r \<Longrightarrow>
- M = NIL \<and> N = NIL \<or>
- (\<exists>a b M' N'.
- M = CONS a M' \<and> N = CONS b N' \<and> (a, b) \<in> Id_on A \<and>
- ((M', N') \<in> r \<or> (M', N') \<in> EqLList (Id_on A)))"
- shows "M = N"
-proof -
- from r have "(M, N) \<in> EqLList (Id_on 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> Id_on A \<and>
- (M, N) \<in> {(f u, g u) | u. u \<in> LList A} \<union> Id_on (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 M N)
- then obtain L where MN: "M = f L" "N = g L" and L: "L \<in> LList A" by blast
- from L show ?case
- proof (cases L)
- case NIL
- with fun_NIL and MN have "(M, N) \<in> Id_on (LList A)" by auto
- then have "(M, N) \<in> EqLList (Id_on A)" ..
- then show ?thesis by cases simp_all
- next
- case (CONS a K)
- 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 MN CONS have "(M, N) \<in> Id_on (LList A)" by auto
- then have "(M, N) \<in> EqLList (Id_on 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> Id_on A"
- and M'N': "(M', N') \<in> ?bisim \<union> Id_on (LList A)"
- by blast
- from M'N' show ?thesis
- proof
- assume "(M', N') \<in> ?bisim"
- with MN fg ab show ?thesis by simp
- next
- assume "(M', N') \<in> Id_on (LList A)"
- then have "(M', N') \<in> EqLList (Id_on A)" ..
- with MN 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))"
- unfolding h'_def by (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 taking: UNIV rule: LList_equalityI)
- case (EqLList M N)
- then obtain x where MN: "M = h x" "N = h' x" by blast
- show ?case
- proof (cases "f x")
- case None
- with h h' MN have ?EqNIL by simp
- then show ?thesis ..
- next
- case (Some p)
- with h h' MN have "M = CONS (fst p) (h (snd p))"
- and "N = CONS (fst p) (h' (snd p))"
- by (simp_all split: prod.split)
- then have ?EqCONS by (auto iff: Id_on_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 taking: UNIV rule: LList_equalityI)
- case (EqLList M N)
- then obtain l1 l2 where
- MN: "M = Rep_llist l1" "N = Rep_llist l2" and r: "(l1, l2) \<in> r"
- by auto
- from step [OF r] show ?case
- proof
- assume "?EqLNil (l1, l2)"
- with MN have ?EqNIL by (simp add: Rep_llist_LNil)
- then show ?thesis ..
- next
- assume "?EqLCons (l1, l2)"
- with MN have ?EqCONS
- by (force simp add: Rep_llist_LCons EqLList_Id_on 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} *}
-
-definition "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 (id 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} *}
-
-definition
- "Lmap f M = LList_corec M (List_case None (\<lambda>x M'. Some (f x, M')))"
-definition
- "lmap f l = 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. N)" rule: LList_equalityI)
- case (EqLList L M)
- then obtain N where LM: "L = ?lhs N" "M = ?rhs N" and N: "N \<in> LList A" by blast
- from N show ?case
- proof cases
- case NIL
- with LM have ?EqNIL by simp
- then show ?thesis ..
- next
- case CONS
- with LM 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. N)" rule: LList_equalityI)
- case (EqLList L M)
- then obtain N where LM: "L = ?lmap N" "M = N" and N: "N \<in> LList A" by blast
- from N show ?case
- proof cases
- case NIL
- with LM have ?EqNIL by simp
- then show ?thesis ..
- next
- case CONS
- with LM have ?EqCONS by auto
- then show ?thesis ..
- qed
- qed
-qed
-
-lemma lmap_LNil [simp, nitpick_simp]: "lmap f LNil = LNil"
- and lmap_LCons [simp, nitpick_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} *}
-
-definition
- "Lappend M N = 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)))))"
-definition
- "lappend l n = 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, nitpick_simp]: "lappend LNil LNil = LNil"
- and lappend_LNil_LCons [simp, nitpick_simp]: "lappend LNil (LCons l l') = LCons l (lappend LNil l')"
- and lappend_LCons [simp, nitpick_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! *}
-
-definition
- iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" where
- "iterates f a = llist_corec a (\<lambda>x. Some (x, f x))"
-
-lemma iterates [nitpick_simp]: "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
-
-setup {*
- Nitpick.register_codatatype @{typ "'a llist"} @{const_name llist_case}
- (map dest_Const [@{term LNil}, @{term LCons}])
-*}
-
-end