
diff -r c00df7765656 -r 5c900a821a7c src/ZF/ex/LList.thy
--- a/src/ZF/ex/LList.thy	Sat Feb 02 13:26:51 2002 +0100
+++ b/src/ZF/ex/LList.thy	Mon Feb 04 13:16:54 2002 +0100
@@ -14,43 +14,242 @@
 a typing rule for it, based on some notion of "productivity..."
 *)
 
-LList = Main +
+theory LList = Main:
 
 consts
-  llist  :: i=>i
+  llist  :: "i=>i";
 
 codatatype
-  "llist(A)" = LNil | LCons ("a \\<in> A", "l \\<in> llist(A)")
+  "llist(A)" = LNil | LCons ("a \<in> A", "l \<in> llist(A)")
 
 
 (*Coinductive definition of equality*)
 consts
-  lleq :: i=>i
+  lleq :: "i=>i"
 
 (*Previously used <*> in the domain and variant pairs as elements.  But
   standard pairs work just as well.  To use variant pairs, must change prefix
   a q/Q to the Sigma, Pair and converse rules.*)
 coinductive
   domains "lleq(A)" <= "llist(A) * llist(A)"
-  intrs
-    LNil  "<LNil, LNil> \\<in> lleq(A)"
-    LCons "[| a \\<in> A; <l,l'>: lleq(A) |] ==> <LCons(a,l), LCons(a,l')>: lleq(A)"
-  type_intrs  "llist.intrs"
+  intros
+    LNil:  "<LNil, LNil> \<in> lleq(A)"
+    LCons: "[| a \<in> A; <l,l'> \<in> lleq(A) |] 
+            ==> <LCons(a,l), LCons(a,l')> \<in> lleq(A)"
+  type_intros  llist.intros
 
 
 (*Lazy list functions; flip is not definitional!*)
 consts
-  lconst   :: i => i
-  flip     :: i => i
+  lconst   :: "i => i"
+  flip     :: "i => i"
 
 defs
-  lconst_def  "lconst(a) == lfp(univ(a), %l. LCons(a,l))"
+  lconst_def:  "lconst(a) == lfp(univ(a), %l. LCons(a,l))"
+
+axioms
+  flip_LNil:   "flip(LNil) = LNil"
+
+  flip_LCons:  "[| x \<in> bool; l \<in> llist(bool) |] 
+                ==> flip(LCons(x,l)) = LCons(not(x), flip(l))"
+
+
+(*These commands cause classical reasoning to regard the subset relation
+  as primitive, not reducing it to membership*)
+  
+declare subsetI [rule del]
+        subsetCE [rule del]
+
+declare subset_refl [intro!] 
+        cons_subsetI [intro!] 
+        subset_consI [intro!]
+        Union_least [intro!]
+        UN_least [intro!]
+        Un_least [intro!]
+	Inter_greatest [intro!]
+        Int_greatest [intro!]
+        RepFun_subset [intro!]
+	Un_upper1 [intro!]
+        Un_upper2 [intro!]
+        Int_lower1 [intro!]
+        Int_lower2 [intro!]
+
+(*An elimination rule, for type-checking*)
+inductive_cases LConsE: "LCons(a,l) \<in> llist(A)"
+
+(*Proving freeness results*)
+lemma LCons_iff: "LCons(a,l)=LCons(a',l') <-> a=a' & l=l'"
+by auto
+
+lemma LNil_LCons_iff: "~ LNil=LCons(a,l)"
+by auto
+
+(*
+lemma llist_unfold: "llist(A) = {0} <+> (A <*> llist(A))";
+let open llist  val rew = rewrite_rule con_defs in  
+by (fast_tac (claset() addSIs (subsetI ::map rew intros) addEs [rew elim]) 1)
+end
+done
+*)
+
+(*** Lemmas to justify using "llist" in other recursive type definitions ***)
+
+lemma llist_mono: "A \<subseteq> B ==> llist(A) \<subseteq> llist(B)"
+apply (unfold llist.defs )
+apply (rule gfp_mono)
+apply (rule llist.bnd_mono)
+apply (assumption | rule quniv_mono basic_monos)+
+done
+
+(** Closure of quniv(A) under llist -- why so complex?  Its a gfp... **)
+
+declare QPair_Int_Vset_subset_UN [THEN subset_trans, intro!]
+        QPair_subset_univ [intro!]
+        empty_subsetI [intro!]
+        one_in_quniv [THEN qunivD, intro!]
+declare qunivD [dest!]
+declare Ord_in_Ord [elim!]
+
+lemma llist_quniv_lemma [rule_format]:
+     "Ord(i) ==> \<forall>l \<in> llist(quniv(A)). l Int Vset(i) \<subseteq> univ(eclose(A))"
+apply (erule trans_induct)
+apply (rule ballI)
+apply (erule llist.cases)
+apply (simp_all add: QInl_def QInr_def llist.con_defs)
+(*LCons case: I simply can't get rid of the deepen_tac*)
+apply (tactic "deepen_tac (claset() addIs [Ord_trans] addIs [Int_lower1 RS subset_trans]) 2 1")
+done
+
+lemma llist_quniv: "llist(quniv(A)) \<subseteq> quniv(A)"
+apply (rule qunivI [THEN subsetI])
+apply (rule Int_Vset_subset)
+apply (assumption | rule llist_quniv_lemma)+
+done
+
+lemmas llist_subset_quniv =
+       subset_trans [OF llist_mono llist_quniv]
+
+
+(*** Lazy List Equality: lleq ***)
+
+declare QPair_Int_Vset_subset_UN [THEN subset_trans, intro!] 
+        QPair_mono [intro!]
+
+declare Ord_in_Ord [elim!] 
+
+(*Lemma for proving finality.  Unfold the lazy list; use induction hypothesis*)
+lemma lleq_Int_Vset_subset [rule_format]:
+     "Ord(i) ==> \<forall>l l'. <l,l'> \<in> lleq(A) --> l Int Vset(i) \<subseteq> l'"
+apply (erule trans_induct)
+apply (intro allI impI)
+apply (erule lleq.cases)
+apply (unfold QInr_def llist.con_defs)
+apply safe
+apply (fast elim!: Ord_trans bspec [elim_format])
+done
 
-rules
-  flip_LNil   "flip(LNil) = LNil"
+(*lleq(A) is a symmetric relation because qconverse(lleq(A)) is a fixedpoint*)
+lemma lleq_symmetric: "<l,l'> \<in> lleq(A) ==> <l',l> \<in> lleq(A)"
+apply (erule lleq.coinduct [OF converseI]) 
+apply (rule lleq.dom_subset [THEN converse_type])
+apply safe
+apply (erule lleq.cases)
+apply blast+
+done
+
+lemma lleq_implies_equal: "<l,l'> \<in> lleq(A) ==> l=l'"
+apply (rule equalityI)
+apply (assumption | rule lleq_Int_Vset_subset [THEN Int_Vset_subset] | 
+       erule lleq_symmetric)+
+done
+
+lemma equal_llist_implies_leq:
+     "[| l=l';  l \<in> llist(A) |] ==> <l,l'> \<in> lleq(A)"
+apply (rule_tac X = "{<l,l>. l \<in> llist (A) }" in lleq.coinduct)
+apply blast
+apply safe
+apply (erule_tac a="l" in llist.cases)
+apply fast+
+done
+
+
+(*** Lazy List Functions ***)
+
+(*Examples of coinduction for type-checking and to prove llist equations*)
+
+(*** lconst -- defined directly using lfp, but equivalent to a LList_corec ***)
+
+lemma lconst_fun_bnd_mono: "bnd_mono(univ(a), %l. LCons(a,l))"
+apply (unfold llist.con_defs )
+apply (rule bnd_monoI)
+apply (blast intro: A_subset_univ QInr_subset_univ)
+apply (blast intro: subset_refl QInr_mono QPair_mono)
+done
+
+(* lconst(a) = LCons(a,lconst(a)) *)
+lemmas lconst = def_lfp_unfold [OF lconst_def lconst_fun_bnd_mono]
+lemmas lconst_subset = lconst_def [THEN def_lfp_subset]
+lemmas member_subset_Union_eclose = arg_into_eclose [THEN Union_upper]
+
+lemma lconst_in_quniv: "a \<in> A ==> lconst(a) \<in> quniv(A)"
+apply (rule lconst_subset [THEN subset_trans, THEN qunivI])
+apply (erule arg_into_eclose [THEN eclose_subset, THEN univ_mono])
+done
 
-  flip_LCons  "[| x \\<in> bool; l \\<in> llist(bool) |] ==> 
-              flip(LCons(x,l)) = LCons(not(x), flip(l))"
+lemma lconst_type: "a \<in> A ==> lconst(a): llist(A)"
+apply (rule singletonI [THEN llist.coinduct])
+apply (erule lconst_in_quniv [THEN singleton_subsetI])
+apply (fast intro!: lconst)
+done
+
+(*** flip --- equations merely assumed; certain consequences proved ***)
+
+declare flip_LNil [simp] 
+        flip_LCons [simp] 
+        not_type [simp]
+
+lemma bool_Int_subset_univ: "b \<in> bool ==> b Int X \<subseteq> univ(eclose(A))"
+by (fast intro: Int_lower1 [THEN subset_trans] elim!: boolE)
+
+declare not_type [intro!]
+declare bool_Int_subset_univ [intro]
+
+(*Reasoning borrowed from lleq.ML; a similar proof works for all
+  "productive" functions -- cf Coquand's "Infinite Objects in Type Theory".*)
+lemma flip_llist_quniv_lemma [rule_format]:
+     "Ord(i) ==> \<forall>l \<in> llist(bool). flip(l) Int Vset(i) \<subseteq> univ(eclose(bool))"
+apply (erule trans_induct)
+apply (rule ballI)
+apply (erule llist.cases)
+apply (simp_all)
+apply (simp_all add: QInl_def QInr_def llist.con_defs)
+(*LCons case: I simply can't get rid of the deepen_tac*)
+apply (tactic "deepen_tac (claset() addIs [Ord_trans] addIs [Int_lower1 RS subset_trans]) 2 1")
+done
+
+lemma flip_in_quniv: "l \<in> llist(bool) ==> flip(l) \<in> quniv(bool)"
+apply (rule flip_llist_quniv_lemma [THEN Int_Vset_subset, THEN qunivI])
+apply assumption+
+done
+
+lemma flip_type: "l \<in> llist(bool) ==> flip(l): llist(bool)"
+apply (rule_tac X = "{flip (l) . l \<in> llist (bool) }" in llist.coinduct)
+apply blast
+apply (fast intro!: flip_in_quniv)
+apply (erule RepFunE)
+apply (erule_tac a="la" in llist.cases)
+apply auto
+done
+
+lemma flip_flip: "l \<in> llist(bool) ==> flip(flip(l)) = l"
+apply (rule_tac X1 = "{<flip (flip (l)),l> . l \<in> llist (bool) }" in 
+       lleq.coinduct [THEN lleq_implies_equal])
+apply blast
+apply (fast intro!: flip_type)
+apply (erule RepFunE)
+apply (erule_tac a="la" in llist.cases)
+apply (simp_all add: flip_type not_not)
+done
 
 end
 




