--- a/LList.ML Thu Aug 25 10:47:33 1994 +0200
+++ b/LList.ML Thu Aug 25 11:01:45 1994 +0200
@@ -3,11 +3,7 @@
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
-For llist.thy.
-
SHOULD LListD_Fun_CONS_I, etc., be equations (for rewriting)?
-
-TOO LONG! needs splitting up
*)
open LList;
@@ -23,55 +19,46 @@
delsimps [Pair_eq];
-(** the llist functional **)
-
-val LList_unfold = rewrite_rule [List_Fun_def]
- (List_Fun_mono RS (LList_def RS def_gfp_Tarski));
+(*This justifies using llist in other recursive type definitions*)
+goalw LList.thy llist.defs "!!A B. A<=B ==> llist(A) <= llist(B)";
+by (rtac gfp_mono 1);
+by (REPEAT (ares_tac basic_monos 1));
+val llist_mono = result();
-(*This justifies using LList in other recursive type definitions*)
-goalw LList.thy [LList_def] "!!A B. A<=B ==> LList(A) <= LList(B)";
-by (rtac gfp_mono 1);
-by (etac List_Fun_mono2 1);
-val LList_mono = result();
-(*Elimination is case analysis, not induction.*)
-val [major,prem1,prem2] = goalw LList.thy [NIL_def,CONS_def]
- "[| L : LList(A); \
-\ L=NIL ==> P; \
-\ !!M N. [| M:A; N: LList(A); L=CONS(M,N) |] ==> P \
-\ |] ==> P";
-by (rtac (major RS (LList_unfold RS equalityD1 RS subsetD RS usumE)) 1);
-by (etac uprodE 2);
-by (rtac prem2 2);
-by (rtac prem1 1);
-by (REPEAT (ares_tac [refl] 1
- ORELSE eresolve_tac [singletonE,ssubst] 1));
-val LListE = result();
+goal LList.thy "llist(A) = {Numb(0)} <+> (A <*> llist(A))";
+let val rew = rewrite_rule [NIL_def, CONS_def] in
+by (fast_tac (univ_cs addSIs (equalityI :: map rew llist.intrs)
+ addEs [rew llist.elim]) 1)
+end;
+val llist_unfold = result();
-(*** Type checking by co-induction, using List_Fun ***)
-
-val prems = goalw LList.thy [LList_def]
- "[| M : X; X <= List_Fun(A, X Un LList(A)) |] ==> M : LList(A)";
-by (REPEAT (resolve_tac (prems@[List_Fun_mono RS coinduct]) 1));
-val LList_coinduct = result();
-
-(** Rules to prove the 2nd premise of LList_coinduct **)
+(*** Type checking by coinduction, using list_Fun
+ THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS!
+***)
-goalw LList.thy [List_Fun_def,NIL_def] "NIL: List_Fun(A,X)";
-by (resolve_tac [singletonI RS usum_In0I] 1);
-val List_Fun_NIL_I = result();
+goalw LList.thy [list_Fun_def]
+ "!!M. [| M : X; X <= list_Fun(A, X Un llist(A)) |] ==> M : llist(A)";
+be llist.coinduct 1;
+be (subsetD RS CollectD) 1;
+ba 1;
+val llist_coinduct = result();
-goalw LList.thy [List_Fun_def,CONS_def]
- "!!M N. [| M: A; N: X |] ==> CONS(M,N) : List_Fun(A,X)";
-by (REPEAT (ares_tac [uprodI RS usum_In1I] 1));
-val List_Fun_CONS_I = result();
+goalw LList.thy [list_Fun_def, NIL_def] "NIL: list_Fun(A,X)";
+by (fast_tac set_cs 1);
+val list_Fun_NIL_I = result();
+
+goalw LList.thy [list_Fun_def,CONS_def]
+ "!!M N. [| M: A; N: X |] ==> CONS(M,N) : list_Fun(A,X)";
+by (fast_tac set_cs 1);
+val list_Fun_CONS_I = result();
(*Utilise the "strong" part, i.e. gfp(f)*)
-goalw LList.thy [LList_def]
- "!!M N. M: LList(A) ==> M : List_Fun(A, X Un LList(A))";
-by (etac (List_Fun_mono RS gfp_fun_UnI2) 1);
-val List_Fun_LList_I = result();
+goalw LList.thy (llist.defs @ [list_Fun_def])
+ "!!M N. M: llist(A) ==> M : list_Fun(A, X Un llist(A))";
+by (etac (llist.mono RS gfp_fun_UnI2) 1);
+val list_Fun_llist_I = result();
(*** LList_corec satisfies the desired recurion equation ***)
@@ -137,119 +124,140 @@
(*A typical use of co-induction to show membership in the gfp.
Bisimulation is range(%x. LList_corec(x,f)) *)
-goal LList.thy "LList_corec(a,f) : LList({u.True})";
-by (res_inst_tac [("X", "range(%x.LList_corec(x,?g))")] LList_coinduct 1);
+goal LList.thy "LList_corec(a,f) : llist({u.True})";
+by (res_inst_tac [("X", "range(%x.LList_corec(x,?g))")] llist_coinduct 1);
by (rtac rangeI 1);
by (safe_tac set_cs);
by (stac LList_corec 1);
-by (simp_tac (llist_ss addsimps [List_Fun_NIL_I,List_Fun_CONS_I, CollectI]
+by (simp_tac (llist_ss addsimps [list_Fun_NIL_I, list_Fun_CONS_I, CollectI]
|> add_eqI) 1);
val LList_corec_type = result();
(*Lemma for the proof of llist_corec*)
goal LList.thy
"LList_corec(a, %z.sum_case(Inl, split(%v w.Inr(<Leaf(v),w>)), f(z))) : \
-\ LList(range(Leaf))";
-by (res_inst_tac [("X", "range(%x.LList_corec(x,?g))")] LList_coinduct 1);
+\ llist(range(Leaf))";
+by (res_inst_tac [("X", "range(%x.LList_corec(x,?g))")] llist_coinduct 1);
by (rtac rangeI 1);
by (safe_tac set_cs);
by (stac LList_corec 1);
-by (asm_simp_tac (llist_ss addsimps [List_Fun_NIL_I, Pair_eq]) 1);
-by (fast_tac (set_cs addSIs [List_Fun_CONS_I]) 1);
+by (asm_simp_tac (llist_ss addsimps [list_Fun_NIL_I]) 1);
+by (fast_tac (set_cs addSIs [list_Fun_CONS_I]) 1);
val LList_corec_type2 = result();
-(**** LList equality as a gfp; the bisimulation principle ****)
+
+(**** llist equality as a gfp; the bisimulation principle ****)
-goalw LList.thy [LListD_Fun_def] "mono(LListD_Fun(r))";
-by (REPEAT (ares_tac [monoI, subset_refl, dsum_mono, dprod_mono] 1));
-val LListD_Fun_mono = result();
-
-val LListD_unfold = rewrite_rule [LListD_Fun_def]
- (LListD_Fun_mono RS (LListD_def RS def_gfp_Tarski));
+(*This theorem is actually used, unlike the many similar ones in ZF*)
+goal LList.thy "LListD(r) = diag({Numb(0)}) <++> (r <**> LListD(r))";
+let val rew = rewrite_rule [NIL_def, CONS_def] in
+by (fast_tac (univ_cs addSIs (equalityI :: map rew LListD.intrs)
+ addEs [rew LListD.elim]) 1)
+end;
+val LListD_unfold = result();
goal LList.thy "!M N. <M,N> : LListD(diag(A)) --> ntrunc(k,M) = ntrunc(k,N)";
by (res_inst_tac [("n", "k")] less_induct 1);
by (safe_tac set_cs);
-by (etac (LListD_unfold RS equalityD1 RS subsetD RS dsumE) 1);
-by (safe_tac (set_cs addSEs [Pair_inject, dprodE, diagE]));
+by (etac LListD.elim 1);
+by (safe_tac (prod_cs addSEs [diagE]));
by (res_inst_tac [("n", "n")] natE 1);
by (asm_simp_tac (univ_ss addsimps [ntrunc_0]) 1);
-by (res_inst_tac [("n", "xb")] natE 1);
-by (asm_simp_tac (univ_ss addsimps [ntrunc_one_In1]) 1);
-by (asm_simp_tac (univ_ss addsimps [ntrunc_In1, ntrunc_Scons]) 1);
+by (rename_tac "n'" 1);
+by (res_inst_tac [("n", "n'")] natE 1);
+by (asm_simp_tac (univ_ss addsimps [CONS_def, ntrunc_one_In1]) 1);
+by (asm_simp_tac (univ_ss addsimps [CONS_def, ntrunc_In1, ntrunc_Scons]) 1);
val LListD_implies_ntrunc_equality = result();
-goalw LList.thy [LList_def,List_Fun_def] "fst``LListD(diag(A)) <= LList(A)";
+(*The domain of the LListD relation*)
+goalw LList.thy (llist.defs @ [NIL_def, CONS_def])
+ "fst``LListD(diag(A)) <= llist(A)";
by (rtac gfp_upperbound 1);
+(*avoids unfolding LListD on the rhs*)
by (res_inst_tac [("P", "%x. fst``x <= ?B")] (LListD_unfold RS ssubst) 1);
by (simp_tac fst_image_ss 1);
+by (fast_tac univ_cs 1);
val fst_image_LListD = result();
(*This inclusion justifies the use of coinduction to show M=N*)
-goal LList.thy "LListD(diag(A)) <= diag(LList(A))";
+goal LList.thy "LListD(diag(A)) <= diag(llist(A))";
by (rtac subsetI 1);
by (res_inst_tac [("p","x")] PairE 1);
by (safe_tac HOL_cs);
-by (res_inst_tac [("s","xa")] subst 1);
+by (rtac diag_eqI 1);
by (rtac (LListD_implies_ntrunc_equality RS spec RS spec RS mp RS
ntrunc_equality) 1);
by (assume_tac 1);
-by (rtac diagI 1);
by (etac (fst_imageI RS (fst_image_LListD RS subsetD)) 1);
val LListD_subset_diag = result();
-(*This converse inclusion helps to strengthen LList_equalityI*)
-goalw LList.thy [LListD_def] "diag(LList(A)) <= LListD(diag(A))";
-by (rtac gfp_upperbound 1);
+(** Coinduction, using LListD_Fun
+ THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS!
+ **)
+
+goalw LList.thy [LListD_Fun_def]
+ "!!M. [| M : X; X <= LListD_Fun(r, X Un LListD(r)) |] ==> M : LListD(r)";
+be LListD.coinduct 1;
+be (subsetD RS CollectD) 1;
+ba 1;
+val LListD_coinduct = result();
+
+goalw LList.thy [LListD_Fun_def,NIL_def] "<NIL,NIL> : LListD_Fun(r,s)";
+by (fast_tac set_cs 1);
+val LListD_Fun_NIL_I = result();
+
+goalw LList.thy [LListD_Fun_def,CONS_def]
+ "!!x. [| x:A; <M,N>:s |] ==> <CONS(x,M), CONS(x,N)> : LListD_Fun(diag(A),s)";
+by (fast_tac univ_cs 1);
+val LListD_Fun_CONS_I = result();
+
+(*Utilise the "strong" part, i.e. gfp(f)*)
+goalw LList.thy (LListD.defs @ [LListD_Fun_def])
+ "!!M N. M: LListD(r) ==> M : LListD_Fun(r, X Un LListD(r))";
+by (etac (LListD.mono RS gfp_fun_UnI2) 1);
+val LListD_Fun_LListD_I = result();
+
+
+(*This converse inclusion helps to strengthen llist_equalityI*)
+goal LList.thy "diag(llist(A)) <= LListD(diag(A))";
by (rtac subsetI 1);
-by (etac diagE 1);
-by (etac ssubst 1);
-by (etac (LList_unfold RS equalityD1 RS subsetD RS usumE) 1);
-by (rewtac LListD_Fun_def);
-by (ALLGOALS (fast_tac univ_cs));
+by (etac LListD_coinduct 1);
+by (rtac subsetI 1);
+by (eresolve_tac [diagE] 1);
+by (eresolve_tac [ssubst] 1);
+by (eresolve_tac [llist.elim] 1);
+by (ALLGOALS
+ (asm_simp_tac (llist_ss addsimps [diagI, LListD_Fun_NIL_I,
+ LListD_Fun_CONS_I])));
val diag_subset_LListD = result();
-goal LList.thy "LListD(diag(A)) = diag(LList(A))";
+goal LList.thy "LListD(diag(A)) = diag(llist(A))";
by (REPEAT (resolve_tac [equalityI, LListD_subset_diag,
diag_subset_LListD] 1));
val LListD_eq_diag = result();
+goal LList.thy
+ "!!M N. M: llist(A) ==> <M,M> : LListD_Fun(diag(A), X Un diag(llist(A)))";
+by (rtac (LListD_eq_diag RS subst) 1);
+br LListD_Fun_LListD_I 1;
+by (asm_simp_tac (HOL_ss addsimps [LListD_eq_diag, diagI]) 1);
+val LListD_Fun_diag_I = result();
+
+
(** To show two LLists are equal, exhibit a bisimulation!
[also admits true equality]
Replace "A" by some particular set, like {x.True}??? *)
goal LList.thy
- "!!r. [| <M,N> : r; r <= LListD_Fun(diag(A), r Un diag(LList(A))) \
+ "!!r. [| <M,N> : r; r <= LListD_Fun(diag(A), r Un diag(llist(A))) \
\ |] ==> M=N";
-by (rtac (rewrite_rule [LListD_def]
- (LListD_subset_diag RS subsetD RS diagE)) 1);
-by (etac (LListD_Fun_mono RS coinduct) 1);
-by (etac (rewrite_rule [LListD_def] LListD_eq_diag RS ssubst) 1);
-by (safe_tac univ_cs);
-val LList_equalityI = result();
-
-(** Rules to prove the 2nd premise of LList_equalityI **)
-
-goalw LList.thy [LListD_Fun_def,NIL_def] "<NIL,NIL> : LListD_Fun(r,s)";
-by (rtac (singletonI RS diagI RS dsum_In0I) 1);
-val LListD_Fun_NIL_I = result();
-
-val prems = goalw LList.thy [LListD_Fun_def,CONS_def]
- "[| x:A; <M,N>:s |] ==> <CONS(x,M), CONS(x,N)> : LListD_Fun(diag(A),s)";
-by (rtac (dprodI RS dsum_In1I) 1);
-by (REPEAT (resolve_tac (diagI::prems) 1));
-val LListD_Fun_CONS_I = result();
-
-(*Utilise the "strong" part, i.e. gfp(f)*)
-goal LList.thy
- "!!M N. M: LList(A) ==> <M,M> : LListD_Fun(diag(A), X Un diag(LList(A)))";
-br (rewrite_rule [LListD_def] LListD_eq_diag RS subst) 1;
-br (LListD_Fun_mono RS gfp_fun_UnI2) 1;
-br (rewrite_rule [LListD_def] LListD_eq_diag RS ssubst) 1;
-be diagI 1;
-val LListD_Fun_diag_I = result();
+by (rtac (LListD_subset_diag RS subsetD RS diagE) 1);
+by (etac LListD_coinduct 1);
+by (asm_simp_tac (HOL_ss addsimps [LListD_eq_diag]) 1);
+by (safe_tac prod_cs);
+val llist_equalityI = result();
-(*** Finality of LList(A): Uniqueness of functions defined by corecursion ***)
+(*** Finality of llist(A): Uniqueness of functions defined by corecursion ***)
(*abstract proof using a bisimulation*)
val [prem1,prem2] = goal LList.thy
@@ -259,7 +267,7 @@
by (rtac ext 1);
(*next step avoids an unknown (and flexflex pair) in simplification*)
by (res_inst_tac [("A", "{u.True}"),
- ("r", "range(%u. <h1(u),h2(u)>)")] LList_equalityI 1);
+ ("r", "range(%u. <h1(u),h2(u)>)")] llist_equalityI 1);
by (rtac rangeI 1);
by (safe_tac set_cs);
by (stac prem1 1);
@@ -317,11 +325,11 @@
(*A typical use of co-induction to show membership in the gfp.
The containing set is simply the singleton {Lconst(M)}. *)
-goal LList.thy "!!M A. M:A ==> Lconst(M): LList(A)";
-by (rtac (singletonI RS LList_coinduct) 1);
+goal LList.thy "!!M A. M:A ==> Lconst(M): llist(A)";
+by (rtac (singletonI RS llist_coinduct) 1);
by (safe_tac set_cs);
by (res_inst_tac [("P", "%u. u: ?A")] (Lconst RS ssubst) 1);
-by (REPEAT (ares_tac [List_Fun_CONS_I, singletonI, UnI1] 1));
+by (REPEAT (ares_tac [list_Fun_CONS_I, singletonI, UnI1] 1));
val Lconst_type = result();
goal LList.thy "Lconst(M) = LList_corec(M, %x.Inr(<x,x>))";
@@ -338,39 +346,23 @@
val gfp_Lconst_eq_LList_corec = result();
-(** Introduction rules for LList constructors **)
-
-(* c : {Numb(0)} <+> A <*> LList(A) ==> c : LList(A) *)
-val LListI = LList_unfold RS equalityD2 RS subsetD;
-
-(*This justifies the type definition: LList(A) is nonempty.*)
-goalw LList.thy [NIL_def] "NIL: LList(A)";
-by (rtac (singletonI RS usum_In0I RS LListI) 1);
-val NIL_LListI = result();
-
-val prems = goalw LList.thy [CONS_def]
- "[| M: A; N: LList(A) |] ==> CONS(M,N) : LList(A)";
-by (rtac (uprodI RS usum_In1I RS LListI) 1);
-by (REPEAT (resolve_tac prems 1));
-val CONS_LListI = result();
-
(*** Isomorphisms ***)
-goal LList.thy "inj(Rep_LList)";
+goal LList.thy "inj(Rep_llist)";
by (rtac inj_inverseI 1);
-by (rtac Rep_LList_inverse 1);
-val inj_Rep_LList = result();
+by (rtac Rep_llist_inverse 1);
+val inj_Rep_llist = result();
-goal LList.thy "inj_onto(Abs_LList,LList(range(Leaf)))";
+goal LList.thy "inj_onto(Abs_llist,llist(range(Leaf)))";
by (rtac inj_onto_inverseI 1);
-by (etac Abs_LList_inverse 1);
-val inj_onto_Abs_LList = result();
+by (etac Abs_llist_inverse 1);
+val inj_onto_Abs_llist = result();
(** Distinctness of constructors **)
goalw LList.thy [LNil_def,LCons_def] "~ LCons(x,xs) = LNil";
-by (rtac (CONS_not_NIL RS (inj_onto_Abs_LList RS inj_onto_contraD)) 1);
-by (REPEAT (resolve_tac [rangeI, NIL_LListI, CONS_LListI, Rep_LList] 1));
+by (rtac (CONS_not_NIL RS (inj_onto_Abs_llist RS inj_onto_contraD)) 1);
+by (REPEAT (resolve_tac (llist.intrs @ [rangeI, Rep_llist]) 1));
val LCons_not_LNil = result();
val LNil_not_LCons = standard (LCons_not_LNil RS not_sym);
@@ -381,15 +373,15 @@
(** llist constructors **)
goalw LList.thy [LNil_def]
- "Rep_LList(LNil) = NIL";
-by (rtac (NIL_LListI RS Abs_LList_inverse) 1);
-val Rep_LList_LNil = result();
+ "Rep_llist(LNil) = NIL";
+by (rtac (llist.NIL_I RS Abs_llist_inverse) 1);
+val Rep_llist_LNil = result();
goalw LList.thy [LCons_def]
- "Rep_LList(LCons(x,l)) = CONS(Leaf(x),Rep_LList(l))";
-by (REPEAT (resolve_tac [CONS_LListI RS Abs_LList_inverse,
- rangeI, Rep_LList] 1));
-val Rep_LList_LCons = result();
+ "Rep_llist(LCons(x,l)) = CONS(Leaf(x),Rep_llist(l))";
+by (REPEAT (resolve_tac [llist.CONS_I RS Abs_llist_inverse,
+ rangeI, Rep_llist] 1));
+val Rep_llist_LCons = result();
(** Injectiveness of CONS and LCons **)
@@ -401,49 +393,49 @@
(*For reasoning about abstract llist constructors*)
-val LList_cs = set_cs addIs [Rep_LList, NIL_LListI, CONS_LListI]
+val llist_cs = set_cs addIs [Rep_llist]@llist.intrs
addSEs [CONS_neq_NIL,NIL_neq_CONS,CONS_inject]
- addSDs [inj_onto_Abs_LList RS inj_ontoD,
- inj_Rep_LList RS injD, Leaf_inject];
+ addSDs [inj_onto_Abs_llist RS inj_ontoD,
+ inj_Rep_llist RS injD, Leaf_inject];
goalw LList.thy [LCons_def] "(LCons(x,xs)=LCons(y,ys)) = (x=y & xs=ys)";
-by (fast_tac LList_cs 1);
+by (fast_tac llist_cs 1);
val LCons_LCons_eq = result();
val LCons_inject = standard (LCons_LCons_eq RS iffD1 RS conjE);
-val [major] = goal LList.thy "CONS(M,N): LList(A) ==> M: A & N: LList(A)";
-by (rtac (major RS LListE) 1);
+val [major] = goal LList.thy "CONS(M,N): llist(A) ==> M: A & N: llist(A)";
+by (rtac (major RS llist.elim) 1);
by (etac CONS_neq_NIL 1);
-by (fast_tac LList_cs 1);
+by (fast_tac llist_cs 1);
val CONS_D = result();
-(****** Reasoning about LList(A) ******)
+(****** Reasoning about llist(A) ******)
(*Don't use llist_ss, as it does case splits!*)
val List_case_ss = univ_ss addsimps [List_case_NIL, List_case_CONS];
(*A special case of list_equality for functions over lazy lists*)
-val [MList,gMList,NILcase,CONScase] = goal LList.thy
- "[| M: LList(A); g(NIL): LList(A); \
+val [Mlist,gMlist,NILcase,CONScase] = goal LList.thy
+ "[| M: llist(A); g(NIL): llist(A); \
\ f(NIL)=g(NIL); \
-\ !!x l. [| x:A; l: LList(A) |] ==> \
+\ !!x l. [| x:A; l: llist(A) |] ==> \
\ <f(CONS(x,l)),g(CONS(x,l))> : \
-\ LListD_Fun(diag(A), (%u.<f(u),g(u)>)``LList(A) Un \
-\ diag(LList(A))) \
+\ LListD_Fun(diag(A), (%u.<f(u),g(u)>)``llist(A) Un \
+\ diag(llist(A))) \
\ |] ==> f(M) = g(M)";
-by (rtac LList_equalityI 1);
-br (MList RS imageI) 1;
+by (rtac llist_equalityI 1);
+br (Mlist RS imageI) 1;
by (rtac subsetI 1);
by (etac imageE 1);
by (etac ssubst 1);
-by (etac LListE 1);
+by (etac llist.elim 1);
by (etac ssubst 1);
by (stac NILcase 1);
-br (gMList RS LListD_Fun_diag_I) 1;
+br (gMlist RS LListD_Fun_diag_I) 1;
by (etac ssubst 1);
by (REPEAT (ares_tac [CONScase] 1));
-val LList_fun_equalityI = result();
+val llist_fun_equalityI = result();
(*** The functional "Lmap" ***)
@@ -460,17 +452,17 @@
(*Another type-checking proof by coinduction*)
val [major,minor] = goal LList.thy
- "[| M: LList(A); !!x. x:A ==> f(x):B |] ==> Lmap(f,M): LList(B)";
-by (rtac (major RS imageI RS LList_coinduct) 1);
+ "[| M: llist(A); !!x. x:A ==> f(x):B |] ==> Lmap(f,M): llist(B)";
+by (rtac (major RS imageI RS llist_coinduct) 1);
by (safe_tac set_cs);
-by (etac LListE 1);
+by (etac llist.elim 1);
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS])));
-by (REPEAT (ares_tac [List_Fun_NIL_I, List_Fun_CONS_I,
+by (REPEAT (ares_tac [list_Fun_NIL_I, list_Fun_CONS_I,
minor, imageI, UnI1] 1));
val Lmap_type = result();
(*This type checking rule synthesises a sufficiently large set for f*)
-val [major] = goal LList.thy "M: LList(A) ==> Lmap(f,M): LList(f``A)";
+val [major] = goal LList.thy "M: llist(A) ==> Lmap(f,M): llist(f``A)";
by (rtac (major RS Lmap_type) 1);
by (etac imageI 1);
val Lmap_type2 = result();
@@ -478,19 +470,19 @@
(** Two easy results about Lmap **)
val [prem] = goalw LList.thy [o_def]
- "M: LList(A) ==> Lmap(f o g, M) = Lmap(f, Lmap(g, M))";
-by (rtac (prem RS imageI RS LList_equalityI) 1);
+ "M: llist(A) ==> Lmap(f o g, M) = Lmap(f, Lmap(g, M))";
+by (rtac (prem RS imageI RS llist_equalityI) 1);
by (safe_tac set_cs);
-by (etac LListE 1);
+by (etac llist.elim 1);
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS])));
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, UnI1,
rangeI RS LListD_Fun_CONS_I] 1));
val Lmap_compose = result();
-val [prem] = goal LList.thy "M: LList(A) ==> Lmap(%x.x, M) = M";
-by (rtac (prem RS imageI RS LList_equalityI) 1);
+val [prem] = goal LList.thy "M: llist(A) ==> Lmap(%x.x, M) = M";
+by (rtac (prem RS imageI RS llist_equalityI) 1);
by (safe_tac set_cs);
-by (etac LListE 1);
+by (etac llist.elim 1);
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS])));
by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI RS UnI1,
rangeI RS LListD_Fun_CONS_I] 1));
@@ -517,17 +509,17 @@
val Lappend_CONS = result();
val Lappend_ss =
- List_case_ss addsimps [NIL_LListI, Lappend_NIL_NIL, Lappend_NIL_CONS,
+ List_case_ss addsimps [llist.NIL_I, Lappend_NIL_NIL, Lappend_NIL_CONS,
Lappend_CONS, LListD_Fun_CONS_I]
|> add_eqI;
-goal LList.thy "!!M. M: LList(A) ==> Lappend(NIL,M) = M";
-by (etac LList_fun_equalityI 1);
+goal LList.thy "!!M. M: llist(A) ==> Lappend(NIL,M) = M";
+by (etac llist_fun_equalityI 1);
by (ALLGOALS (asm_simp_tac Lappend_ss));
val Lappend_NIL = result();
-goal LList.thy "!!M. M: LList(A) ==> Lappend(M,NIL) = M";
-by (etac LList_fun_equalityI 1);
+goal LList.thy "!!M. M: llist(A) ==> Lappend(M,NIL) = M";
+by (etac llist_fun_equalityI 1);
by (ALLGOALS (asm_simp_tac Lappend_ss));
val Lappend_NIL2 = result();
@@ -535,62 +527,63 @@
(*weak co-induction: bisimulation and case analysis on both variables*)
goal LList.thy
- "!!M N. [| M: LList(A); N: LList(A) |] ==> Lappend(M,N): LList(A)";
+ "!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend(M,N): llist(A)";
by (res_inst_tac
- [("X", "UN u:LList(A). UN v: LList(A). {Lappend(u,v)}")] LList_coinduct 1);
+ [("X", "UN u:llist(A). UN v: llist(A). {Lappend(u,v)}")] llist_coinduct 1);
by (fast_tac set_cs 1);
by (safe_tac set_cs);
-by (eres_inst_tac [("L", "u")] LListE 1);
-by (eres_inst_tac [("L", "v")] LListE 1);
+by (eres_inst_tac [("a", "u")] llist.elim 1);
+by (eres_inst_tac [("a", "v")] llist.elim 1);
by (ALLGOALS
(asm_simp_tac Lappend_ss THEN'
- fast_tac (set_cs addSIs [NIL_LListI,List_Fun_NIL_I,List_Fun_CONS_I]) ));
+ fast_tac (set_cs addSIs [llist.NIL_I, list_Fun_NIL_I, list_Fun_CONS_I])));
val Lappend_type = result();
(*strong co-induction: bisimulation and case analysis on one variable*)
goal LList.thy
- "!!M N. [| M: LList(A); N: LList(A) |] ==> Lappend(M,N): LList(A)";
-by (res_inst_tac [("X", "(%u.Lappend(u,N))``LList(A)")] LList_coinduct 1);
+ "!!M N. [| M: llist(A); N: llist(A) |] ==> Lappend(M,N): llist(A)";
+by (res_inst_tac [("X", "(%u.Lappend(u,N))``llist(A)")] llist_coinduct 1);
be imageI 1;
br subsetI 1;
be imageE 1;
-by (eres_inst_tac [("L", "u")] LListE 1);
-by (asm_simp_tac (Lappend_ss addsimps [Lappend_NIL, List_Fun_LList_I]) 1);
+by (eres_inst_tac [("a", "u")] llist.elim 1);
+by (asm_simp_tac (Lappend_ss addsimps [Lappend_NIL, list_Fun_llist_I]) 1);
by (asm_simp_tac Lappend_ss 1);
-by (fast_tac (set_cs addSIs [List_Fun_CONS_I]) 1);
+by (fast_tac (set_cs addSIs [list_Fun_CONS_I]) 1);
val Lappend_type = result();
(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****)
(** llist_case: case analysis for 'a llist **)
-val Rep_LList_simps =
+val Rep_llist_simps =
[List_case_NIL, List_case_CONS,
- Abs_LList_inverse, Rep_LList_inverse, NIL_LListI, CONS_LListI,
- Rep_LList, rangeI, inj_Leaf, Inv_f_f];
-val Rep_LList_ss = llist_ss addsimps Rep_LList_simps;
+ Abs_llist_inverse, Rep_llist_inverse,
+ Rep_llist, rangeI, inj_Leaf, Inv_f_f]
+ @ llist.intrs;
+val Rep_llist_ss = llist_ss addsimps Rep_llist_simps;
goalw LList.thy [llist_case_def,LNil_def] "llist_case(c, d, LNil) = c";
-by (simp_tac Rep_LList_ss 1);
+by (simp_tac Rep_llist_ss 1);
val llist_case_LNil = result();
goalw LList.thy [llist_case_def,LCons_def]
"llist_case(c, d, LCons(M,N)) = d(M,N)";
-by (simp_tac Rep_LList_ss 1);
+by (simp_tac Rep_llist_ss 1);
val llist_case_LCons = result();
(*Elimination is case analysis, not induction.*)
val [prem1,prem2] = goalw LList.thy [NIL_def,CONS_def]
"[| l=LNil ==> P; !!x l'. l=LCons(x,l') ==> P \
\ |] ==> P";
-by (rtac (Rep_LList RS LListE) 1);
-by (rtac (inj_Rep_LList RS injD RS prem1) 1);
-by (stac Rep_LList_LNil 1);
+by (rtac (Rep_llist RS llist.elim) 1);
+by (rtac (inj_Rep_llist RS injD RS prem1) 1);
+by (stac Rep_llist_LNil 1);
by (assume_tac 1);
by (etac rangeE 1);
-by (rtac (inj_Rep_LList RS injD RS prem2) 1);
-by (asm_simp_tac (HOL_ss addsimps [Rep_LList_LCons]) 1);
-by (etac (Abs_LList_inverse RS ssubst) 1);
+by (rtac (inj_Rep_llist RS injD RS prem2) 1);
+by (asm_simp_tac (HOL_ss addsimps [Rep_llist_LCons]) 1);
+by (etac (Abs_llist_inverse RS ssubst) 1);
by (rtac refl 1);
val llistE = result();
@@ -601,11 +594,11 @@
\ split(%z w. LCons(z, llist_corec(w,f))), f(a))";
by (stac LList_corec 1);
by (res_inst_tac [("s","f(a)")] sumE 1);
-by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_LList_inverse]) 1);
+by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_llist_inverse]) 1);
by (res_inst_tac [("p","y")] PairE 1);
-by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_LList_inverse]) 1);
+by (asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_llist_inverse]) 1);
(*FIXME: correct case splits usd to be found automatically:
-by (ASM_SIMP_TAC(llist_ss addsimps [LList_corec_type2,Abs_LList_inverse]) 1);*)
+by (ASM_SIMP_TAC(llist_ss addsimps [LList_corec_type2,Abs_llist_inverse]) 1);*)
val llist_corec = result();
(*definitional version of same*)
@@ -620,53 +613,53 @@
(*** Deriving llist_equalityI -- llist equality is a bisimulation ***)
-val prems = goalw LList.thy [LListD_Fun_def]
- "r <= Sigma(LList(A), %x.LList(A)) ==> \
-\ LListD_Fun(diag(A),r) <= Sigma(LList(A), %x.LList(A))";
-by (stac LList_unfold 1);
-by (cut_facts_tac prems 1);
+goalw LList.thy [LListD_Fun_def]
+ "!!r A. r <= Sigma(llist(A), %x.llist(A)) ==> \
+\ LListD_Fun(diag(A),r) <= Sigma(llist(A), %x.llist(A))";
+by (stac llist_unfold 1);
+by (simp_tac (HOL_ss addsimps [NIL_def, CONS_def]) 1);
by (fast_tac univ_cs 1);
-val LListD_Fun_subset_Sigma_LList = result();
+val LListD_Fun_subset_Sigma_llist = result();
goal LList.thy
- "prod_fun(Rep_LList,Rep_LList) `` r <= \
-\ Sigma(LList(range(Leaf)), %x.LList(range(Leaf)))";
-by (fast_tac (prod_cs addIs [Rep_LList]) 1);
-val subset_Sigma_LList = result();
+ "prod_fun(Rep_llist,Rep_llist) `` r <= \
+\ Sigma(llist(range(Leaf)), %x.llist(range(Leaf)))";
+by (fast_tac (prod_cs addIs [Rep_llist]) 1);
+val subset_Sigma_llist = result();
val [prem] = goal LList.thy
- "r <= Sigma(LList(range(Leaf)), %x.LList(range(Leaf))) ==> \
-\ prod_fun(Rep_LList o Abs_LList, Rep_LList o Abs_LList) `` r <= r";
+ "r <= Sigma(llist(range(Leaf)), %x.llist(range(Leaf))) ==> \
+\ prod_fun(Rep_llist o Abs_llist, Rep_llist o Abs_llist) `` r <= r";
by (safe_tac prod_cs);
by (rtac (prem RS subsetD RS SigmaE2) 1);
by (assume_tac 1);
-by (asm_simp_tac (HOL_ss addsimps [o_def,prod_fun,Abs_LList_inverse]) 1);
+by (asm_simp_tac (HOL_ss addsimps [o_def,prod_fun,Abs_llist_inverse]) 1);
val prod_fun_lemma = result();
goal LList.thy
- "prod_fun(Rep_LList, Rep_LList) `` range(%x. <x, x>) = \
-\ diag(LList(range(Leaf)))";
+ "prod_fun(Rep_llist, Rep_llist) `` range(%x. <x, x>) = \
+\ diag(llist(range(Leaf)))";
br equalityI 1;
-by (fast_tac (univ_cs addIs [Rep_LList]) 1);
-by (fast_tac (univ_cs addSEs [Abs_LList_inverse RS subst]) 1);
+by (fast_tac (univ_cs addIs [Rep_llist]) 1);
+by (fast_tac (univ_cs addSEs [Abs_llist_inverse RS subst]) 1);
val prod_fun_range_eq_diag = result();
(** To show two llists are equal, exhibit a bisimulation!
[also admits true equality] **)
val [prem1,prem2] = goalw LList.thy [llistD_Fun_def]
"[| <l1,l2> : r; r <= llistD_Fun(r Un range(%x.<x,x>)) |] ==> l1=l2";
-by (rtac (inj_Rep_LList RS injD) 1);
-by (res_inst_tac [("r", "prod_fun(Rep_LList,Rep_LList)``r"),
+by (rtac (inj_Rep_llist RS injD) 1);
+by (res_inst_tac [("r", "prod_fun(Rep_llist,Rep_llist)``r"),
("A", "range(Leaf)")]
- LList_equalityI 1);
+ llist_equalityI 1);
by (rtac (prem1 RS prod_fun_imageI) 1);
by (rtac (prem2 RS image_mono RS subset_trans) 1);
by (rtac (image_compose RS subst) 1);
by (rtac (prod_fun_compose RS subst) 1);
by (rtac (image_Un RS ssubst) 1);
by (stac prod_fun_range_eq_diag 1);
-by (rtac (LListD_Fun_subset_Sigma_LList RS prod_fun_lemma) 1);
-by (rtac (subset_Sigma_LList RS Un_least) 1);
+by (rtac (LListD_Fun_subset_Sigma_llist RS prod_fun_lemma) 1);
+by (rtac (subset_Sigma_llist RS Un_least) 1);
by (rtac diag_subset_Sigma 1);
val llist_equalityI = result();
@@ -684,11 +677,11 @@
(*Utilise the "strong" part, i.e. gfp(f)*)
goalw LList.thy [llistD_Fun_def]
"!!l. <l,l> : llistD_Fun(r Un range(%x.<x,x>))";
-br (Rep_LList_inverse RS subst) 1;
+br (Rep_llist_inverse RS subst) 1;
br prod_fun_imageI 1;
by (rtac (image_Un RS ssubst) 1);
by (stac prod_fun_range_eq_diag 1);
-br (Rep_LList RS LListD_Fun_diag_I) 1;
+br (Rep_llist RS LListD_Fun_diag_I) 1;
val llistD_Fun_range_I = result();
(*A special case of list_equality for functions over lazy lists*)