diff -r 142f1eb707b4 -r befa4e9f7c90 llist.ML --- a/llist.ML Thu Sep 16 14:29:14 1993 +0200 +++ b/llist.ML Wed Sep 22 15:43:05 1993 +0200 @@ -5,7 +5,9 @@ For llist.thy. -SHOULD LListD_Fun_CONS_I, etc., be equations (for rewriting) +SHOULD LListD_Fun_CONS_I, etc., be equations (for rewriting)? + +TOO LONG! needs splitting up *) open LList; @@ -14,7 +16,8 @@ val llist_simps = [case_Inl, case_Inr]; val llist_ss = univ_ss addsimps llist_simps - setloop (split_tac [expand_split,expand_case]); + addcongs [split_weak_cong, case_weak_cong] + setloop (split_tac [expand_split, expand_case]); (** the llist functional **) @@ -45,16 +48,10 @@ (*** Type checking by co-induction, using List_Fun ***) val prems = goalw LList.thy [LList_def] - "[| M: X; X <= List_Fun(A,X) |] ==> M: LList(A)"; -by (REPEAT (resolve_tac (prems@[coinduct]) 1)); + "[| 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(); -(*stronger version*) -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@[coinduct2,List_Fun_mono]) 1)); -val LList_coinduct2 = result(); - (** Rules to prove the 2nd premise of LList_coinduct **) goalw LList.thy [List_Fun_def,NIL_def] "NIL: List_Fun(A,X)"; @@ -66,19 +63,25 @@ by (REPEAT (ares_tac [uprodI RS usum_In1I] 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(); + (*** LList_corec satisfies the desired recurion equation ***) (*A continuity result?*) goalw LList.thy [CONS_def] "CONS(M, UN x.f(x)) = (UN x. CONS(M, f(x)))"; -by(simp_tac (univ_ss addsimps [In1_UN1, Scons_UN1_y]) 1); +by (simp_tac (univ_ss addsimps [In1_UN1, Scons_UN1_y]) 1); val CONS_UN1 = result(); goal Prod.thy "split(p, %x y.UN z.f(x,y,z)) = (UN z. split(p, %x y.f(x,y,z)))"; -by(simp_tac (pair_ss setloop (split_tac [expand_split])) 1); +by (simp_tac (pair_ss setloop (split_tac [expand_split])) 1); val split_UN1 = result(); goal Sum.thy "case(s, f, %y. UN z.g(y,z)) = (UN z. case(s, f, %y. g(y,z)))"; -by(simp_tac (sum_ss setloop (split_tac [expand_case])) 1); +by (simp_tac (sum_ss setloop (split_tac [expand_case])) 1); val case2_UN1 = result(); val prems = goalw LList.thy [CONS_def] @@ -97,7 +100,7 @@ \ %v. split(v, %z w. CONS(z, LList_corec(w,f))))"; by (rtac UN1_least 1); by (nat_ind_tac "k" 1); -by(ALLGOALS(simp_tac corec_fun_ss)); +by (ALLGOALS(simp_tac corec_fun_ss)); by (REPEAT (resolve_tac [allI, impI, subset_refl RS CONS_mono, UN1_upper] 1)); val LList_corec_subset1 = result(); @@ -133,8 +136,8 @@ 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, range_eqI]) 1); +by (simp_tac (llist_ss addsimps [List_Fun_NIL_I,List_Fun_CONS_I, + CollectI, range_eqI]) 1); (* 6.7 vs 3.4 !!! by (ASM_SIMP_TAC (llist_ss addsimps [List_Fun_NIL_I,List_Fun_CONS_I, CollectI, rangeI]) 1); @@ -151,8 +154,8 @@ by (stac LList_corec 1); (*nested "case"; requires an explicit split*) by (res_inst_tac [("s", "f(xa)")] sumE 1); -by(asm_simp_tac (univ_ss addsimps (llist_simps@[List_Fun_NIL_I])) 1); -by(asm_simp_tac (univ_ss addsimps (llist_simps@[List_Fun_CONS_I, range_eqI]) +by (asm_simp_tac (univ_ss addsimps (llist_simps@[List_Fun_NIL_I])) 1); +by (asm_simp_tac (univ_ss addsimps (llist_simps@[List_Fun_CONS_I, range_eqI]) setloop (split_tac [expand_split])) 1); (* FIXME: can the selection of the case split be automated? by (ASM_SIMP_TAC (llist_ss addsimps [List_Fun_CONS_I, rangeI]) 1);*) @@ -162,10 +165,10 @@ 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_Fun_mono = result(); val LListD_unfold = rewrite_rule [LListD_Fun_def] - (LListD_fun_mono RS (LListD_def RS def_gfp_Tarski)); + (LListD_Fun_mono RS (LListD_def RS def_gfp_Tarski)); goal LList.thy "!M N. : LListD(diag(A)) --> ntrunc(k,M) = ntrunc(k,N)"; by (res_inst_tac [("n", "k")] less_induct 1); @@ -173,16 +176,16 @@ by (etac (LListD_unfold RS equalityD1 RS subsetD RS dsumE) 1); by (safe_tac (set_cs addSEs [Pair_inject, dprodE, diagE])); by (res_inst_tac [("n", "n")] natE 1); -by(asm_simp_tac (univ_ss addsimps [ntrunc_0]) 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 (asm_simp_tac (univ_ss addsimps [ntrunc_one_In1]) 1); +by (asm_simp_tac (univ_ss addsimps [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)"; by (rtac gfp_upperbound 1); by (res_inst_tac [("P", "%x. fst``x <= ?B")] (LListD_unfold RS ssubst) 1); -by(simp_tac fst_image_ss 1); +by (simp_tac fst_image_ss 1); val fst_image_LListD = result(); (*This inclusion justifies the use of coinduction to show M=N*) @@ -215,27 +218,19 @@ diag_subset_LListD] 1)); val LListD_eq_diag = result(); -(** To show two LLists are equal, exhibit a bisimulation! **) -(* Replace "A" by some particular set, like {x.True}??? *) -val prems = goal LList.thy - "[| : r; r <= LListD_Fun(diag(A), r) |] ==> M=N"; +(** 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. [| : 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 (REPEAT (resolve_tac (prems@[coinduct]) 1)); +by (etac (LListD_Fun_mono RS coinduct) 1); +by (etac (rewrite_rule [LListD_def] LListD_eq_diag RS ssubst) 1); by (safe_tac (set_cs addSEs [Pair_inject])); val LList_equalityI = result(); -(*Stronger notion of bisimulation -- also admits true equality*) -val prems = goal LList.thy - "[| : 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 (rtac coinduct2 1); -by (stac (rewrite_rule [LListD_def] LListD_eq_diag) 2); -by (REPEAT (resolve_tac (prems@[LListD_fun_mono]) 1)); -by (safe_tac (set_cs addSEs [Pair_inject])); -val LList_equalityI2 = result(); - (** Rules to prove the 2nd premise of LList_equalityI **) goalw LList.thy [LListD_Fun_def,NIL_def] " : LListD_Fun(r,s)"; @@ -248,6 +243,15 @@ 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) ==> : 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(); + (*** Finality of LList(A): Uniqueness of functions defined by corecursion ***) @@ -264,8 +268,8 @@ by (safe_tac set_cs); by (stac prem1 1); by (stac prem2 1); -by(simp_tac (llist_ss addsimps [LListD_Fun_NIL_I, range_eqI, - CollectI RS LListD_Fun_CONS_I]) 1); +by (simp_tac (llist_ss addsimps [LListD_Fun_NIL_I, range_eqI, + CollectI RS LListD_Fun_CONS_I]) 1); (* 9.5 vs 9.2/4.1/4.3 by (ASM_SIMP_TAC (llist_ss addsimps [LListD_Fun_NIL_I, rangeI, CollectI RS LListD_Fun_CONS_I]) 1);*) @@ -286,7 +290,7 @@ goalw LList.thy [CONS_def] "ntrunc(Suc(Suc(k)), CONS(M,N)) = CONS (ntrunc(k,M), ntrunc(k,N))"; -by(simp_tac (HOL_ss addsimps [ntrunc_Scons,ntrunc_In1]) 1); +by (simp_tac (HOL_ss addsimps [ntrunc_Scons,ntrunc_In1]) 1); val ntrunc_CONS = result(); val [prem1,prem2] = goal LList.thy @@ -299,11 +303,11 @@ by (rtac allI 1); by (stac prem1 1); by (stac prem2 1); -by(simp_tac (sum_ss setloop (split_tac [expand_split,expand_case])) 1); +by (simp_tac (sum_ss setloop (split_tac [expand_split,expand_case])) 1); by (strip_tac 1); by (res_inst_tac [("n", "n")] natE 1); by (res_inst_tac [("n", "xc")] natE 2); -by(ALLGOALS(asm_simp_tac(nat_ss addsimps +by (ALLGOALS(asm_simp_tac(nat_ss addsimps [ntrunc_0,ntrunc_one_CONS,ntrunc_CONS]))); val LList_corec_unique = result(); @@ -323,19 +327,19 @@ 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] 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())"; by (rtac (equals_LList_corec RS fun_cong) 1); -by(simp_tac sum_ss 1); +by (simp_tac sum_ss 1); by (rtac Lconst 1); val Lconst_eq_LList_corec = result(); (*Thus we could have used gfp in the definition of Lconst*) goal LList.thy "gfp(%N. CONS(M, N)) = LList_corec(M, %x.Inr())"; by (rtac (equals_LList_corec RS fun_cong) 1); -by(simp_tac sum_ss 1); +by (simp_tac sum_ss 1); by (rtac (Lconst_fun_mono RS gfp_Tarski) 1); val gfp_Lconst_eq_LList_corec = result(); @@ -431,10 +435,10 @@ \ f(NIL)=g(NIL); \ \ !!x l. [| x:A; l: LList(A) |] ==> \ \ : \ -\ LListD_Fun(diag(A), (%u.)``LList(A)) Un \ -\ diag(LList(A)) \ +\ LListD_Fun(diag(A), (%u.)``LList(A) Un \ +\ diag(LList(A))) \ \ |] ==> f(M) = g(M)"; -by (rtac LList_equalityI2 1); +by (rtac LList_equalityI 1); br (MList RS imageI) 1; by (rtac subsetI 1); by (etac imageE 1); @@ -442,7 +446,7 @@ by (etac LListE 1); by (etac ssubst 1); by (stac NILcase 1); -br (gMList RS diagI RS UnI2) 1; +br (gMList RS LListD_Fun_diag_I) 1; by (etac ssubst 1); by (REPEAT (ares_tac [CONScase] 1)); val LList_fun_equalityI = result(); @@ -452,12 +456,12 @@ goal LList.thy "Lmap(f,NIL) = NIL"; by (rtac (Lmap_def RS def_LList_corec RS trans) 1); -by(simp_tac List_case_ss 1); +by (simp_tac List_case_ss 1); val Lmap_NIL = result(); goal LList.thy "Lmap(f, CONS(M,N)) = CONS(f(M), Lmap(f,N))"; by (rtac (Lmap_def RS def_LList_corec RS trans) 1); -by(simp_tac List_case_ss 1); +by (simp_tac List_case_ss 1); val Lmap_CONS = result(); (*Another type-checking proof by coinduction*) @@ -466,8 +470,9 @@ by (rtac (major RS imageI RS LList_coinduct) 1); by (safe_tac set_cs); by (etac LListE 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, minor, imageI] 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, + minor, imageI, UnI1] 1)); val Lmap_type = result(); (*This type checking rule synthesises a sufficiently large set for f*) @@ -484,18 +489,18 @@ by (stac o_def 1); by (safe_tac set_cs); by (etac LListE 1); -by(ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS]))); -by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, - rangeI RS LListD_Fun_CONS_I] 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); by (safe_tac set_cs); by (etac LListE 1); -by(ALLGOALS (asm_simp_tac (HOL_ss addsimps [Lmap_NIL,Lmap_CONS]))); -by (REPEAT (ares_tac [LListD_Fun_NIL_I, imageI, - rangeI RS LListD_Fun_CONS_I] 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)); val Lmap_ident = result(); @@ -504,7 +509,7 @@ goalw LList.thy [Lappend_def] "Lappend(NIL,NIL) = NIL"; by (rtac (LList_corec RS trans) 1); (* takes 2.4(3.4 w NORM) vs 0.9 w/o NORM terms *) -by(simp_tac List_case_ss 1); +by (simp_tac List_case_ss 1); (*by (SIMP_TAC List_case_ss 1);*) val Lappend_NIL_NIL = result(); @@ -512,7 +517,7 @@ "Lappend(NIL,CONS(N,N')) = CONS(N, Lappend(NIL,N'))"; by (rtac (LList_corec RS trans) 1); (* takes 5(7 w NORM) vs 2.1 w/o NORM terms *) -by(simp_tac List_case_ss 1); +by (simp_tac List_case_ss 1); (*by (SIMP_TAC List_case_ss 1);*) val Lappend_NIL_CONS = result(); @@ -520,7 +525,7 @@ "Lappend(CONS(M,M'), N) = CONS(M, Lappend(M',N))"; by (rtac (LList_corec RS trans) 1); (* takes 4.9(6.7) vs 2.2 w/o NORM terms *) -by(simp_tac List_case_ss 1); +by (simp_tac List_case_ss 1); (*by (SIMP_TAC List_case_ss 1);*) val Lappend_CONS = result(); @@ -563,12 +568,12 @@ (*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_coinduct2 1); -fe imageI; +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]) 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); val Lappend_type = result(); @@ -584,12 +589,12 @@ val Rep_LList_ss = llist_ss addsimps Rep_LList_simps; goalw LList.thy [llist_case_def,LNil_def] "llist_case(LNil, c, d) = 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(LCons(M,N), c, d) = 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.*) @@ -602,7 +607,7 @@ 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 (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(); @@ -613,12 +618,12 @@ "llist_corec(a,f) = case(f(a), %u. LNil, \ \ %v. split(v, %z w. LCons(z, llist_corec(w,f))))"; 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(res_inst_tac [("p","y")] PairE 1); -by(asm_simp_tac (llist_ss addsimps [LList_corec_type2,Abs_LList_inverse]) 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 (res_inst_tac [("p","y")] PairE 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*) @@ -653,44 +658,38 @@ by (safe_tac (set_cs addSEs [prod_fun_imageE])); 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(); -(** To show two llists are equal, exhibit a bisimulation! **) +goal LList.thy + "prod_fun(Rep_LList, Rep_LList) `` range(%x. ) = \ +\ diag(LList(range(Leaf)))"; +br equalityI 1; +by (fast_tac (set_cs addIs [diagI, Rep_LList] + addSEs [prod_fun_imageE, Pair_inject]) 1); +by (fast_tac (set_cs addIs [prod_fun_imageI, rangeI] + addSEs [diagE, 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] - "[| : r; r <= llistD_Fun(r) |] ==> l1=l2"; + "[| : r; r <= llistD_Fun(r Un range(%x.)) |] ==> l1=l2"; by (rtac (inj_Rep_LList RS injD) 1); -by (res_inst_tac [("r", "prod_fun(Rep_LList,Rep_LList)``r")] +by (res_inst_tac [("r", "prod_fun(Rep_LList,Rep_LList)``r"), + ("A", "range(Leaf)")] 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 (subset_Sigma_LList RS LListD_Fun_subset_Sigma_LList RS - prod_fun_lemma) 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 diag_subset_Sigma 1); val llist_equalityI = result(); - -(*Stronger notion of bisimulation -- also admits true equality*) -val [prem1,prem2] = goalw LList.thy [llistD_Fun_def] - "[| : r; r <= llistD_Fun(r) Un range(%x.) |] ==> l1=l2"; -by (rtac (inj_Rep_LList RS injD) 1); -by (res_inst_tac [("r", "prod_fun(Rep_LList,Rep_LList)``r"), - ("A", "range(Leaf)")] - LList_equalityI2 1); -by (rtac (prem1 RS prod_fun_imageI) 1); -by (rtac (prem2 RS image_mono RS subset_trans) 1); -by (rtac (image_Un RS ssubst) 1); -by (rtac Un_least 1); -by (rtac (image_compose RS subst) 1); -by (rtac (prod_fun_compose RS subst) 1); -by (rtac (subset_Sigma_LList RS LListD_Fun_subset_Sigma_LList RS - prod_fun_lemma RS subset_trans) 1); -by (rtac Un_upper1 1); -by (fast_tac (set_cs addSEs [prod_fun_imageE, Pair_inject] - addIs [diagI,Rep_LList]) 1); -val llist_equalityI2 = result(); - (** Rules to prove the 2nd premise of llist_equalityI **) goalw LList.thy [llistD_Fun_def,LNil_def] " : llistD_Fun(r)"; by (rtac (LListD_Fun_NIL_I RS prod_fun_imageI) 1); @@ -702,14 +701,23 @@ by (rtac (prem RS prod_fun_imageI) 1); val llistD_Fun_LCons_I = result(); +(*Utilise the "strong" part, i.e. gfp(f)*) +goalw LList.thy [llistD_Fun_def] + "!!l. : llistD_Fun(r Un range(%x.))"; +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; +val llistD_Fun_range_I = result(); (*A special case of list_equality for functions over lazy lists*) val [prem1,prem2] = goal LList.thy "[| f(LNil)=g(LNil); \ \ !!x l. : \ -\ llistD_Fun(range(%u. )) Un range(%v. ) \ +\ llistD_Fun(range(%u. ) Un range(%v. )) \ \ |] ==> f(l) = g(l :: 'a llist) :: 'b llist"; -by (res_inst_tac [("r", "range(%u. )")] llist_equalityI2 1); +by (res_inst_tac [("r", "range(%u. )")] llist_equalityI 1); by (rtac rangeI 1); by (rtac subsetI 1); by (etac rangeE 1); @@ -717,7 +725,7 @@ by (res_inst_tac [("l", "u")] llistE 1); by (etac ssubst 1); by (stac prem1 1); -by (fast_tac set_cs 1); +by (rtac llistD_Fun_range_I 1); by (etac ssubst 1); by (rtac prem2 1); val llist_fun_equalityI = result(); @@ -732,12 +740,12 @@ goal LList.thy "lmap(f,LNil) = LNil"; by (rtac (lmap_def RS def_llist_corec RS trans) 1); -by(simp_tac llistD_ss 1); +by (simp_tac llistD_ss 1); val lmap_LNil = result(); goal LList.thy "lmap(f, LCons(M,N)) = LCons(f(M), lmap(f,N))"; by (rtac (lmap_def RS def_llist_corec RS trans) 1); -by(simp_tac llistD_ss 1); +by (simp_tac llistD_ss 1); val lmap_LCons = result(); @@ -758,7 +766,7 @@ goal LList.thy "iterates(f,x) = LCons(x, iterates(f,f(x)))"; by (rtac (iterates_def RS def_llist_corec RS trans) 1); -by(simp_tac sum_ss 1); +by (simp_tac sum_ss 1); val iterates = result(); goal LList.thy "lmap(f, iterates(f,x)) = iterates(f,f(x))"; @@ -784,12 +792,12 @@ "nat_rec(n, LCons(b, l), %m. lmap(f)) = \ \ LCons(nat_rec(n, b, %m. f), nat_rec(n, l, %m. lmap(f)))"; by (nat_ind_tac "n" 1); -by(ALLGOALS (asm_simp_tac (nat_ss addsimps [lmap_LCons]))); +by (ALLGOALS (asm_simp_tac (nat_ss addsimps [lmap_LCons]))); val fun_power_lmap = result(); goal Nat.thy "nat_rec(n, g(x), %m. g) = nat_rec(Suc(n), x, %m. g)"; by (nat_ind_tac "n" 1); -by(ALLGOALS (asm_simp_tac nat_ss)); +by (ALLGOALS (asm_simp_tac nat_ss)); val fun_power_Suc = result(); val Pair_cong = read_instantiate_sg (sign_of Prod.thy) @@ -814,7 +822,7 @@ by (rtac (lmap_iterates RS subst) 1); by (stac fun_power_Suc 1); by (stac fun_power_Suc 1); -br UN1_I 1; +br (UN1_I RS UnI1) 1; br rangeI 1; val iterates_equality = result(); @@ -823,20 +831,20 @@ goalw LList.thy [lappend_def] "lappend(LNil,LNil) = LNil"; by (rtac (llist_corec RS trans) 1); -by(simp_tac llistD_ss 1); +by (simp_tac llistD_ss 1); val lappend_LNil_LNil = result(); goalw LList.thy [lappend_def] "lappend(LNil,LCons(l,l')) = LCons(l, lappend(LNil,l'))"; by (rtac (llist_corec RS trans) 1); -by(simp_tac llistD_ss 1); +by (simp_tac llistD_ss 1); (* 3.3(5.7) vs 1.3 !by (SIMP_TAC llistD_ss 1);*) val lappend_LNil_LCons = result(); goalw LList.thy [lappend_def] "lappend(LCons(l,l'), N) = LCons(l, lappend(l',N))"; by (rtac (llist_corec RS trans) 1); -by(simp_tac llistD_ss 1); +by (simp_tac llistD_ss 1); (* 5(5.5) vs 1.3 !by (SIMP_TAC llistD_ss 1);*) val lappend_LCons = result(); @@ -877,21 +885,21 @@ by (ALLGOALS (asm_simp_tac (llistD_ss addsimps [lappend_LNil_LNil,lappend_LCons,lappend_LNil_LCons, lmap_LNil,lmap_LCons]))); -by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I, rangeI])); +by (REPEAT_SOME (ares_tac [llistD_Fun_LCons_I, UN1_I RS UnI1, rangeI])); by (rtac range_eqI 1); by (rtac (refl RS Pair_cong) 1); by (stac lmap_LNil 1); by (rtac refl 1); val lmap_lappend_distrib = result(); -(*Shorter proof of the theorem above using llist_equalityI2*) +(*Shorter proof of theorem above using llist_equalityI as strong coinduction*) goal LList.thy "lmap(f, lappend(l,n)) = lappend(lmap(f,l), lmap(f,n))"; by (res_inst_tac [("l","l")] llist_fun_equalityI 1); by (simp_tac (llistD_ss addsimps [lappend_LNil, lmap_LNil])1); by (simp_tac (llistD_ss addsimps [lappend_LCons, lmap_LCons]) 1); val lmap_lappend_distrib = result(); -(*Without llist_equalityI2, three case analyses might be needed*) +(*Without strong coinduction, three case analyses might be needed*) goal LList.thy "lappend(lappend(l1,l2) ,l3) = lappend(l1, lappend(l2,l3))"; by (res_inst_tac [("l","l1")] llist_fun_equalityI 1); by (simp_tac (llistD_ss addsimps [lappend_LNil])1);