diff -r f04b33ce250f -r a4dc62a46ee4 ex/LList.ML --- a/ex/LList.ML Tue Oct 24 14:59:17 1995 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,880 +0,0 @@ -(* Title: HOL/llist - ID: $Id$ - Author: Lawrence C Paulson, Cambridge University Computer Laboratory - Copyright 1993 University of Cambridge - -SHOULD LListD_Fun_CONS_I, etc., be equations (for rewriting)? -*) - -open LList; - -(** Simplification **) - -val llist_ss = univ_ss addcongs [split_weak_cong, sum_case_weak_cong] - setloop split_tac [expand_split, expand_sum_case]; - -(*For adding _eqI rules to a simpset; we must remove Pair_eq because - it may turn an instance of reflexivity into a conjunction!*) -fun add_eqI ss = ss addsimps [range_eqI, image_eqI] - delsimps [Pair_eq]; - - -(*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)); -qed "llist_mono"; - - -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; -qed "llist_unfold"; - - -(*** Type checking by coinduction, using list_Fun - THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS! -***) - -goalw LList.thy [list_Fun_def] - "!!M. [| M : X; X <= list_Fun(A, X Un llist(A)) |] ==> M : llist(A)"; -by (etac llist.coinduct 1); -by (etac (subsetD RS CollectD) 1); -by (assume_tac 1); -qed "llist_coinduct"; - -goalw LList.thy [list_Fun_def, NIL_def] "NIL: list_Fun(A,X)"; -by (fast_tac set_cs 1); -qed "list_Fun_NIL_I"; - -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); -qed "list_Fun_CONS_I"; - -(*Utilise the "strong" part, i.e. gfp(f)*) -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); -qed "list_Fun_llist_I"; - -(*** 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); -qed "CONS_UN1"; - -(*UNUSED; obsolete? -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 (prod_ss setloop (split_tac [expand_split])) 1); -qed "split_UN1"; - -goal Sum.thy "sum_case(s,f,%y.UN z.g(y,z)) = (UN z.sum_case(s,f,%y. g(y,z)))"; -by (simp_tac (sum_ss setloop (split_tac [expand_sum_case])) 1); -qed "sum_case2_UN1"; -*) - -val prems = goalw LList.thy [CONS_def] - "[| M<=M'; N<=N' |] ==> CONS(M,N) <= CONS(M',N')"; -by (REPEAT (resolve_tac ([In1_mono,Scons_mono]@prems) 1)); -qed "CONS_mono"; - -val corec_fun_simps = [LList_corec_fun_def RS def_nat_rec_0, - LList_corec_fun_def RS def_nat_rec_Suc]; -val corec_fun_ss = llist_ss addsimps corec_fun_simps; - -(** The directions of the equality are proved separately **) - -goalw LList.thy [LList_corec_def] - "LList_corec(a,f) <= sum_case(%u.NIL, \ -\ split(%z w. CONS(z, LList_corec(w,f))), f(a))"; -by (rtac UN1_least 1); -by (res_inst_tac [("n","k")] natE 1); -by (ALLGOALS (asm_simp_tac corec_fun_ss)); -by (REPEAT (resolve_tac [allI, impI, subset_refl RS CONS_mono, UN1_upper] 1)); -qed "LList_corec_subset1"; - -goalw LList.thy [LList_corec_def] - "sum_case(%u.NIL, split(%z w. CONS(z, LList_corec(w,f))), f(a)) <= \ -\ LList_corec(a,f)"; -by (simp_tac (corec_fun_ss addsimps [CONS_UN1]) 1); -by (safe_tac set_cs); -by (ALLGOALS (res_inst_tac [("x","Suc(?k)")] UN1_I THEN' - asm_simp_tac corec_fun_ss)); -qed "LList_corec_subset2"; - -(*the recursion equation for LList_corec -- NOT SUITABLE FOR REWRITING!*) -goal LList.thy - "LList_corec(a,f) = sum_case(%u. NIL, \ -\ split(%z w. CONS(z, LList_corec(w,f))), f(a))"; -by (REPEAT (resolve_tac [equalityI, LList_corec_subset1, - LList_corec_subset2] 1)); -qed "LList_corec"; - -(*definitional version of same*) -val [rew] = goal LList.thy - "[| !!x. h(x) == LList_corec(x,f) |] ==> \ -\ h(a) = sum_case(%u.NIL, split(%z w. CONS(z, h(w))), f(a))"; -by (rewtac rew); -by (rtac LList_corec 1); -qed "def_LList_corec"; - -(*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); -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] - |> add_eqI) 1); -qed "LList_corec_type"; - -(*Lemma for the proof of llist_corec*) -goal LList.thy - "LList_corec(a, %z.sum_case(Inl, split(%v w.Inr()), f(z))) : \ -\ 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]) 1); -by (fast_tac (set_cs addSIs [list_Fun_CONS_I]) 1); -qed "LList_corec_type2"; - - -(**** llist equality as a gfp; the bisimulation principle ****) - -(*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; -qed "LListD_unfold"; - -goal LList.thy "!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.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 (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); -qed "LListD_implies_ntrunc_equality"; - -(*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); -qed "fst_image_LListD"; - -(*This inclusion justifies the use of coinduction to show M=N*) -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 (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 (etac (fst_imageI RS (fst_image_LListD RS subsetD)) 1); -qed "LListD_subset_diag"; - -(** 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)"; -by (etac LListD.coinduct 1); -by (etac (subsetD RS CollectD) 1); -by (assume_tac 1); -qed "LListD_coinduct"; - -goalw LList.thy [LListD_Fun_def,NIL_def] " : LListD_Fun(r,s)"; -by (fast_tac set_cs 1); -qed "LListD_Fun_NIL_I"; - -goalw LList.thy [LListD_Fun_def,CONS_def] - "!!x. [| x:A; :s |] ==> : LListD_Fun(diag(A),s)"; -by (fast_tac univ_cs 1); -qed "LListD_Fun_CONS_I"; - -(*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); -qed "LListD_Fun_LListD_I"; - - -(*This converse inclusion helps to strengthen LList_equalityI*) -goal LList.thy "diag(llist(A)) <= LListD(diag(A))"; -by (rtac subsetI 1); -by (etac LListD_coinduct 1); -by (rtac subsetI 1); -by (etac diagE 1); -by (etac 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]))); -qed "diag_subset_LListD"; - -goal LList.thy "LListD(diag(A)) = diag(llist(A))"; -by (REPEAT (resolve_tac [equalityI, LListD_subset_diag, - diag_subset_LListD] 1)); -qed "LListD_eq_diag"; - -goal LList.thy - "!!M N. M: llist(A) ==> : LListD_Fun(diag(A), X Un diag(llist(A)))"; -by (rtac (LListD_eq_diag RS subst) 1); -by (rtac LListD_Fun_LListD_I 1); -by (asm_simp_tac (HOL_ss addsimps [LListD_eq_diag, diagI]) 1); -qed "LListD_Fun_diag_I"; - - -(** 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 (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); -qed "LList_equalityI"; - - -(*** Finality of llist(A): Uniqueness of functions defined by corecursion ***) - -(*abstract proof using a bisimulation*) -val [prem1,prem2] = goal LList.thy - "[| !!x. h1(x) = sum_case(%u.NIL, split(%z w. CONS(z,h1(w))), f(x)); \ -\ !!x. h2(x) = sum_case(%u.NIL, split(%z w. CONS(z,h2(w))), f(x)) |]\ -\ ==> h1=h2"; -by (rtac ext 1); -(*next step avoids an unknown (and flexflex pair) in simplification*) -by (res_inst_tac [("A", "{u.True}"), - ("r", "range(%u. )")] LList_equalityI 1); -by (rtac rangeI 1); -by (safe_tac set_cs); -by (stac prem1 1); -by (stac prem2 1); -by (simp_tac (llist_ss addsimps [LListD_Fun_NIL_I, - CollectI RS LListD_Fun_CONS_I] - |> add_eqI) 1); -qed "LList_corec_unique"; - -val [prem] = goal LList.thy - "[| !!x. h(x) = sum_case(%u.NIL, split(%z w. CONS(z,h(w))), f(x)) |] \ -\ ==> h = (%x.LList_corec(x,f))"; -by (rtac (LList_corec RS (prem RS LList_corec_unique)) 1); -qed "equals_LList_corec"; - - -(** Obsolete LList_corec_unique proof: complete induction, not coinduction **) - -goalw LList.thy [CONS_def] "ntrunc(Suc(0), CONS(M,N)) = {}"; -by (rtac ntrunc_one_In1 1); -qed "ntrunc_one_CONS"; - -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); -qed "ntrunc_CONS"; - -val [prem1,prem2] = goal LList.thy - "[| !!x. h1(x) = sum_case(%u.NIL, split(%z w. CONS(z,h1(w))), f(x)); \ -\ !!x. h2(x) = sum_case(%u.NIL, split(%z w. CONS(z,h2(w))), f(x)) |]\ -\ ==> h1=h2"; -by (rtac (ntrunc_equality RS ext) 1); -by (res_inst_tac [("x", "x")] spec 1); -by (res_inst_tac [("n", "k")] less_induct 1); -by (rtac allI 1); -by (stac prem1 1); -by (stac prem2 1); -by (simp_tac (sum_ss setloop (split_tac [expand_split,expand_sum_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 - [ntrunc_0,ntrunc_one_CONS,ntrunc_CONS]))); -result(); - - -(*** Lconst -- defined directly using lfp, but equivalent to a LList_corec ***) - -goal LList.thy "mono(CONS(M))"; -by (REPEAT (ares_tac [monoI, subset_refl, CONS_mono] 1)); -qed "Lconst_fun_mono"; - -(* Lconst(M) = CONS(M,Lconst(M)) *) -bind_thm ("Lconst", (Lconst_fun_mono RS (Lconst_def RS def_lfp_Tarski))); - -(*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); -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)); -qed "Lconst_type"; - -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 (rtac Lconst 1); -qed "Lconst_eq_LList_corec"; - -(*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 (rtac (Lconst_fun_mono RS gfp_Tarski) 1); -qed "gfp_Lconst_eq_LList_corec"; - - -(*** Isomorphisms ***) - -goal LList.thy "inj(Rep_llist)"; -by (rtac inj_inverseI 1); -by (rtac Rep_llist_inverse 1); -qed "inj_Rep_llist"; - -goal LList.thy "inj_onto(Abs_llist,llist(range(Leaf)))"; -by (rtac inj_onto_inverseI 1); -by (etac Abs_llist_inverse 1); -qed "inj_onto_Abs_llist"; - -(** 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 (llist.intrs @ [rangeI, Rep_llist]) 1)); -qed "LCons_not_LNil"; - -bind_thm ("LNil_not_LCons", (LCons_not_LNil RS not_sym)); - -bind_thm ("LCons_neq_LNil", (LCons_not_LNil RS notE)); -val LNil_neq_LCons = sym RS LCons_neq_LNil; - -(** llist constructors **) - -goalw LList.thy [LNil_def] - "Rep_llist(LNil) = NIL"; -by (rtac (llist.NIL_I RS Abs_llist_inverse) 1); -qed "Rep_llist_LNil"; - -goalw LList.thy [LCons_def] - "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)); -qed "Rep_llist_LCons"; - -(** Injectiveness of CONS and LCons **) - -goalw LList.thy [CONS_def] "(CONS(M,N)=CONS(M',N')) = (M=M' & N=N')"; -by (fast_tac (HOL_cs addSEs [Scons_inject, make_elim In1_inject]) 1); -qed "CONS_CONS_eq"; - -bind_thm ("CONS_inject", (CONS_CONS_eq RS iffD1 RS conjE)); - - -(*For reasoning about abstract llist constructors*) -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]; - -goalw LList.thy [LCons_def] "(LCons(x,xs)=LCons(y,ys)) = (x=y & xs=ys)"; -by (fast_tac llist_cs 1); -qed "LCons_LCons_eq"; -bind_thm ("LCons_inject", (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 llist.elim) 1); -by (etac CONS_neq_NIL 1); -by (fast_tac llist_cs 1); -qed "CONS_D"; - - -(****** 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); \ -\ f(NIL)=g(NIL); \ -\ !!x l. [| x:A; l: llist(A) |] ==> \ -\ : \ -\ LListD_Fun(diag(A), (%u.)``llist(A) Un \ -\ diag(llist(A))) \ -\ |] ==> f(M) = g(M)"; -by (rtac LList_equalityI 1); -by (rtac (Mlist RS imageI) 1); -by (rtac subsetI 1); -by (etac imageE 1); -by (etac ssubst 1); -by (etac llist.elim 1); -by (etac ssubst 1); -by (stac NILcase 1); -by (rtac (gMlist RS LListD_Fun_diag_I) 1); -by (etac ssubst 1); -by (REPEAT (ares_tac [CONScase] 1)); -qed "LList_fun_equalityI"; - - -(*** The functional "Lmap" ***) - -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); -qed "Lmap_NIL"; - -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); -qed "Lmap_CONS"; - -(*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); -by (safe_tac set_cs); -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, - minor, imageI, UnI1] 1)); -qed "Lmap_type"; - -(*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)"; -by (rtac (major RS Lmap_type) 1); -by (etac imageI 1); -qed "Lmap_type2"; - -(** 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); -by (safe_tac set_cs); -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)); -qed "Lmap_compose"; - -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 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)); -qed "Lmap_ident"; - - -(*** Lappend -- its two arguments cause some complications! ***) - -goalw LList.thy [Lappend_def] "Lappend(NIL,NIL) = NIL"; -by (rtac (LList_corec RS trans) 1); -by (simp_tac List_case_ss 1); -qed "Lappend_NIL_NIL"; - -goalw LList.thy [Lappend_def] - "Lappend(NIL,CONS(N,N')) = CONS(N, Lappend(NIL,N'))"; -by (rtac (LList_corec RS trans) 1); -by (simp_tac List_case_ss 1); -qed "Lappend_NIL_CONS"; - -goalw LList.thy [Lappend_def] - "Lappend(CONS(M,M'), N) = CONS(M, Lappend(M',N))"; -by (rtac (LList_corec RS trans) 1); -by (simp_tac List_case_ss 1); -qed "Lappend_CONS"; - -val Lappend_ss = - 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); -by (ALLGOALS (asm_simp_tac Lappend_ss)); -qed "Lappend_NIL"; - -goal LList.thy "!!M. M: llist(A) ==> Lappend(M,NIL) = M"; -by (etac LList_fun_equalityI 1); -by (ALLGOALS (asm_simp_tac Lappend_ss)); -qed "Lappend_NIL2"; - -(** Alternative type-checking proofs for Lappend **) - -(*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)"; -by (res_inst_tac - [("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 [("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 [llist.NIL_I, list_Fun_NIL_I, list_Fun_CONS_I]))); -qed "Lappend_type"; - -(*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); -by (etac imageI 1); -by (rtac subsetI 1); -by (etac imageE 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); -qed "Lappend_type"; - -(**** Lazy lists as the type 'a llist -- strongly typed versions of above ****) - -(** llist_case: case analysis for 'a llist **) - -val Rep_llist_simps = - [List_case_NIL, List_case_CONS, - 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); -qed "llist_case_LNil"; - -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); -qed "llist_case_LCons"; - -(*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 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 refl 1); -qed "llistE"; - -(** llist_corec: corecursion for 'a llist **) - -goalw LList.thy [llist_corec_def,LNil_def,LCons_def] - "llist_corec(a,f) = sum_case(%u. LNil, \ -\ 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 (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);*) -qed "llist_corec"; - -(*definitional version of same*) -val [rew] = goal LList.thy - "[| !!x. h(x) == llist_corec(x,f) |] ==> \ -\ h(a) = sum_case(%u.LNil, split(%z w. LCons(z, h(w))), f(a))"; -by (rewtac rew); -by (rtac llist_corec 1); -qed "def_llist_corec"; - -(**** Proofs about type 'a llist functions ****) - -(*** Deriving llist_equalityI -- llist equality is a bisimulation ***) - -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); -qed "LListD_Fun_subset_Sigma_llist"; - -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); -qed "subset_Sigma_llist"; - -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"; -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); -qed "prod_fun_lemma"; - -goal LList.thy - "prod_fun(Rep_llist, Rep_llist) `` range(%x. ) = \ -\ diag(llist(range(Leaf)))"; -by (rtac equalityI 1); -by (fast_tac (univ_cs addIs [Rep_llist]) 1); -by (fast_tac (univ_cs addSEs [Abs_llist_inverse RS subst]) 1); -qed "prod_fun_range_eq_diag"; - -(** 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 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_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 diag_subset_Sigma 1); -qed "llist_equalityI"; - -(** 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); -qed "llistD_Fun_LNil_I"; - -val [prem] = goalw LList.thy [llistD_Fun_def,LCons_def] - ":r ==> : llistD_Fun(r)"; -by (rtac (rangeI RS LListD_Fun_CONS_I RS prod_fun_imageI) 1); -by (rtac (prem RS prod_fun_imageI) 1); -qed "llistD_Fun_LCons_I"; - -(*Utilise the "strong" part, i.e. gfp(f)*) -goalw LList.thy [llistD_Fun_def] - "!!l. : llistD_Fun(r Un range(%x.))"; -by (rtac (Rep_llist_inverse RS subst) 1); -by (rtac prod_fun_imageI 1); -by (rtac (image_Un RS ssubst) 1); -by (stac prod_fun_range_eq_diag 1); -by (rtac (Rep_llist RS LListD_Fun_diag_I) 1); -qed "llistD_Fun_range_I"; - -(*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. )) \ -\ |] ==> f(l) = (g(l :: 'a llist) :: 'b llist)"; -by (res_inst_tac [("r", "range(%u. )")] llist_equalityI 1); -by (rtac rangeI 1); -by (rtac subsetI 1); -by (etac rangeE 1); -by (etac ssubst 1); -by (res_inst_tac [("l", "u")] llistE 1); -by (etac ssubst 1); -by (stac prem1 1); -by (rtac llistD_Fun_range_I 1); -by (etac ssubst 1); -by (rtac prem2 1); -qed "llist_fun_equalityI"; - -(*simpset for llist bisimulations*) -val llistD_simps = [llist_case_LNil, llist_case_LCons, - llistD_Fun_LNil_I, llistD_Fun_LCons_I]; -(*Don't use llist_ss, as it does case splits!*) -val llistD_ss = univ_ss addsimps llistD_simps |> add_eqI; - - -(*** The functional "lmap" ***) - -goal LList.thy "lmap(f,LNil) = LNil"; -by (rtac (lmap_def RS def_llist_corec RS trans) 1); -by (simp_tac llistD_ss 1); -qed "lmap_LNil"; - -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); -qed "lmap_LCons"; - - -(** Two easy results about lmap **) - -goal LList.thy "lmap(f o g, l) = lmap(f, lmap(g, l))"; -by (res_inst_tac [("l","l")] llist_fun_equalityI 1); -by (ALLGOALS (simp_tac (llistD_ss addsimps [lmap_LNil, lmap_LCons]))); -qed "lmap_compose"; - -goal LList.thy "lmap(%x.x, l) = l"; -by (res_inst_tac [("l","l")] llist_fun_equalityI 1); -by (ALLGOALS (simp_tac (llistD_ss addsimps [lmap_LNil, lmap_LCons]))); -qed "lmap_ident"; - - -(*** iterates -- llist_fun_equalityI cannot be used! ***) - -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); -qed "iterates"; - -goal LList.thy "lmap(f, iterates(f,x)) = iterates(f,f(x))"; -by (res_inst_tac [("r", "range(%u.)")] - llist_equalityI 1); -by (rtac rangeI 1); -by (safe_tac set_cs); -by (res_inst_tac [("x1", "f(u)")] (iterates RS ssubst) 1); -by (res_inst_tac [("x1", "u")] (iterates RS ssubst) 1); -by (simp_tac (llistD_ss addsimps [lmap_LCons]) 1); -qed "lmap_iterates"; - -goal LList.thy "iterates(f,x) = LCons(x, lmap(f, iterates(f,x)))"; -by (rtac (lmap_iterates RS ssubst) 1); -by (rtac iterates 1); -qed "iterates_lmap"; - -(*** A rather complex proof about iterates -- cf Andy Pitts ***) - -(** Two lemmas about natrec(n,x,%m.g), which is essentially (g^n)(x) **) - -goal LList.thy - "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]))); -qed "fun_power_lmap"; - -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)); -qed "fun_power_Suc"; - -val Pair_cong = read_instantiate_sg (sign_of Prod.thy) - [("f","Pair")] (standard(refl RS cong RS cong)); - -(*The bisimulation consists of {} - for all u and all n::nat.*) -val [prem] = goal LList.thy - "(!!x. h(x) = LCons(x, lmap(f,h(x)))) ==> h = iterates(f)"; -by (rtac ext 1); -by (res_inst_tac [("r", - "UN u. range(%n. )")] - llist_equalityI 1); -by (REPEAT (resolve_tac [UN1_I, range_eqI, Pair_cong, nat_rec_0 RS sym] 1)); -by (safe_tac set_cs); -by (stac iterates 1); -by (stac prem 1); -by (stac fun_power_lmap 1); -by (stac fun_power_lmap 1); -by (rtac llistD_Fun_LCons_I 1); -by (rtac (lmap_iterates RS subst) 1); -by (stac fun_power_Suc 1); -by (stac fun_power_Suc 1); -by (rtac (UN1_I RS UnI1) 1); -by (rtac rangeI 1); -qed "iterates_equality"; - - -(*** lappend -- its two arguments cause some complications! ***) - -goalw LList.thy [lappend_def] "lappend(LNil,LNil) = LNil"; -by (rtac (llist_corec RS trans) 1); -by (simp_tac llistD_ss 1); -qed "lappend_LNil_LNil"; - -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); -qed "lappend_LNil_LCons"; - -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); -qed "lappend_LCons"; - -goal LList.thy "lappend(LNil,l) = l"; -by (res_inst_tac [("l","l")] llist_fun_equalityI 1); -by (ALLGOALS - (simp_tac (llistD_ss addsimps [lappend_LNil_LNil, lappend_LNil_LCons]))); -qed "lappend_LNil"; - -goal LList.thy "lappend(l,LNil) = l"; -by (res_inst_tac [("l","l")] llist_fun_equalityI 1); -by (ALLGOALS - (simp_tac (llistD_ss addsimps [lappend_LNil_LNil, lappend_LCons]))); -qed "lappend_LNil2"; - -(*The infinite first argument blocks the second*) -goal LList.thy "lappend(iterates(f,x), N) = iterates(f,x)"; -by (res_inst_tac [("r", "range(%u.)")] - llist_equalityI 1); -by (rtac rangeI 1); -by (safe_tac set_cs); -by (stac iterates 1); -by (simp_tac (llistD_ss addsimps [lappend_LCons]) 1); -qed "lappend_iterates"; - -(** Two proofs that lmap distributes over lappend **) - -(*Long proof requiring case analysis on both both arguments*) -goal LList.thy "lmap(f, lappend(l,n)) = lappend(lmap(f,l), lmap(f,n))"; -by (res_inst_tac - [("r", - "UN n. range(%l.)")] - llist_equalityI 1); -by (rtac UN1_I 1); -by (rtac rangeI 1); -by (safe_tac set_cs); -by (res_inst_tac [("l", "l")] llistE 1); -by (res_inst_tac [("l", "n")] llistE 1); -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 RS UnI1, rangeI])); -by (rtac range_eqI 1); -by (rtac (refl RS Pair_cong) 1); -by (stac lmap_LNil 1); -by (rtac refl 1); -qed "lmap_lappend_distrib"; - -(*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); -qed "lmap_lappend_distrib"; - -(*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); -by (simp_tac (llistD_ss addsimps [lappend_LCons]) 1); -qed "lappend_assoc";