--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/ex/Limit.ML Mon Oct 16 14:56:24 1995 +0100
@@ -0,0 +1,2876 @@
+(* Title: ZF/ex/Limit
+ ID: $Id$
+ Author: Sten Agerholm
+
+The inverse limit construction.
+*)
+
+open Limit;
+
+(*----------------------------------------------------------------------*)
+(* Useful goal commands. *)
+(*----------------------------------------------------------------------*)
+
+val brr = fn thl => fn n => by(REPEAT(ares_tac thl n));
+val trr = fn thl => fn n => (REPEAT(ares_tac thl n));
+fun rotate n i = EVERY(replicate n (etac revcut_rl i));
+
+(*----------------------------------------------------------------------*)
+(* Preliminary theorems. *)
+(*----------------------------------------------------------------------*)
+
+val theI2 = prove_goal ZF.thy (* From Larry *)
+ "[| EX! x. P(x); !!x. P(x) ==> Q(x) |] ==> Q(THE x.P(x))"
+ (fn prems => [ resolve_tac prems 1,
+ rtac theI 1,
+ resolve_tac prems 1 ]);
+
+(*----------------------------------------------------------------------*)
+(* Basic results. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [set_def]
+ "x:fst(D) ==> x:set(D)";
+by(resolve_tac prems 1);
+val set_I = result();
+
+val prems = goalw Limit.thy [rel_def]
+ "<x,y>:snd(D) ==> rel(D,x,y)";
+by(resolve_tac prems 1);
+val rel_I = result();
+
+val prems = goalw Limit.thy [rel_def]
+ "!!z. rel(D,x,y) ==> <x,y>:snd(D)";
+by (assume_tac 1);
+val rel_E = result();
+
+(*----------------------------------------------------------------------*)
+(* I/E/D rules for po and cpo. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [po_def]
+ "[|po(D); x:set(D)|] ==> rel(D,x,x)";
+br(hd prems RS conjunct1 RS bspec)1;
+by(resolve_tac prems 1);
+val po_refl = result();
+
+val [po,xy,yz,x,y,z] = goalw Limit.thy [po_def]
+ "[|po(D); rel(D,x,y); rel(D,y,z); x:set(D); \
+\ y:set(D); z:set(D)|] ==> rel(D,x,z)";
+br(po RS conjunct2 RS conjunct1 RS bspec RS bspec
+ RS bspec RS mp RS mp)1;
+by (rtac x 1);
+by (rtac y 1);
+by (rtac z 1);
+by (rtac xy 1);
+by (rtac yz 1);
+val po_trans = result();
+
+val prems = goalw Limit.thy [po_def]
+ "[|po(D); rel(D,x,y); rel(D,y,x); x:set(D); y:set(D)|] ==> x = y";
+br(hd prems RS conjunct2 RS conjunct2 RS bspec RS bspec RS mp RS mp)1;
+by(REPEAT(resolve_tac prems 1));
+val po_antisym = result();
+
+val prems = goalw Limit.thy [po_def]
+ "[| !!x. x:set(D) ==> rel(D,x,x); \
+\ !!x y z. [| rel(D,x,y); rel(D,y,z); x:set(D); y:set(D); z:set(D)|] ==> \
+\ rel(D,x,z); \
+\ !!x y. [| rel(D,x,y); rel(D,y,x); x:set(D); y:set(D)|] ==> x=y |] ==> \
+\ po(D)";
+by(safe_tac lemmas_cs);
+brr prems 1;
+val poI = result();
+
+val prems = goalw Limit.thy [cpo_def]
+ "[| po(D); !!X. chain(D,X) ==> islub(D,X,x(D,X))|] ==> cpo(D)";
+by(safe_tac (lemmas_cs addSIs [exI]));
+brr prems 1;
+val cpoI = result();
+
+val [cpo] = goalw Limit.thy [cpo_def] "cpo(D) ==> po(D)";
+br(cpo RS conjunct1)1;
+val cpo_po = result();
+
+val prems = goal Limit.thy
+ "[|cpo(D); x:set(D)|] ==> rel(D,x,x)";
+by (rtac po_refl 1);
+by(REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
+val cpo_refl = result();
+
+val prems = goal Limit.thy
+ "[|cpo(D); rel(D,x,y); rel(D,y,z); x:set(D); \
+\ y:set(D); z:set(D)|] ==> rel(D,x,z)";
+by (rtac po_trans 1);
+by(REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
+val cpo_trans = result();
+
+val prems = goal Limit.thy
+ "[|cpo(D); rel(D,x,y); rel(D,y,x); x:set(D); y:set(D)|] ==> x = y";
+by (rtac po_antisym 1);
+by(REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
+val cpo_antisym = result();
+
+val [cpo,chain,ex] = goalw Limit.thy [cpo_def] (* cpo_islub *)
+ "[|cpo(D); chain(D,X);!!x. islub(D,X,x) ==> R|] ==> R";
+br(chain RS (cpo RS conjunct2 RS spec RS mp) RS exE)1;
+brr[ex]1; (* above theorem would loop *)
+val cpo_islub = result();
+
+(*----------------------------------------------------------------------*)
+(* Theorems about isub and islub. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [islub_def] (* islub_isub *)
+ "islub(D,X,x) ==> isub(D,X,x)";
+by(simp_tac (ZF_ss addsimps prems) 1);
+val islub_isub = result();
+
+val prems = goal Limit.thy
+ "islub(D,X,x) ==> x:set(D)";
+br(rewrite_rule[islub_def,isub_def](hd prems) RS conjunct1 RS conjunct1)1;
+val islub_in = result();
+
+val prems = goal Limit.thy
+ "[|islub(D,X,x); n:nat|] ==> rel(D,X`n,x)";
+br(rewrite_rule[islub_def,isub_def](hd prems) RS conjunct1
+ RS conjunct2 RS bspec)1;
+by(resolve_tac prems 1);
+val islub_ub = result();
+
+val prems = goalw Limit.thy [islub_def]
+ "[|islub(D,X,x); isub(D,X,y)|] ==> rel(D,x,y)";
+br(hd prems RS conjunct2 RS spec RS mp)1;
+by(resolve_tac prems 1);
+val islub_least = result();
+
+val prems = goalw Limit.thy [islub_def] (* islubI *)
+ "[|isub(D,X,x); !!y. isub(D,X,y) ==> rel(D,x,y)|] ==> islub(D,X,x)";
+by(safe_tac lemmas_cs);
+by(REPEAT(ares_tac prems 1));
+val islubI = result();
+
+val prems = goalw Limit.thy [isub_def] (* isubI *)
+ "[|x:set(D);!!n. n:nat ==> rel(D,X`n,x)|] ==> isub(D,X,x)";
+by(safe_tac lemmas_cs);
+by(REPEAT(ares_tac prems 1));
+val isubI = result();
+
+val prems = goalw Limit.thy [isub_def] (* isubE *)
+ "!!z.[|isub(D,X,x);[|x:set(D);!!n.n:nat==>rel(D,X`n,x)|] ==> P|] ==> P";
+by(safe_tac lemmas_cs);
+by(asm_simp_tac ZF_ss 1);
+val isubE = result();
+
+val prems = goalw Limit.thy [isub_def] (* isubD1 *)
+ "isub(D,X,x) ==> x:set(D)";
+by(simp_tac (ZF_ss addsimps prems) 1);
+val isubD1 = result();
+
+val prems = goalw Limit.thy [isub_def] (* isubD2 *)
+ "[|isub(D,X,x); n:nat|]==>rel(D,X`n,x)";
+by(simp_tac (ZF_ss addsimps prems) 1);
+val isubD2 = result();
+
+val prems = goal Limit.thy
+ "!!z. [|islub(D,X,x); islub(D,X,y); cpo(D)|] ==> x = y";
+by (etac cpo_antisym 1);
+by (rtac islub_least 2);
+by (rtac islub_least 1);
+brr[islub_isub,islub_in]1;
+val islub_unique = result();
+
+(*----------------------------------------------------------------------*)
+(* lub gives the least upper bound of chains. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [lub_def]
+ "[|chain(D,X); cpo(D)|] ==> islub(D,X,lub(D,X))";
+by (rtac cpo_islub 1);
+brr prems 1;
+by (rtac theI 1); (* loops when repeated *)
+by (rtac ex1I 1);
+by (assume_tac 1);
+by (etac islub_unique 1);
+brr prems 1;
+val cpo_lub = result();
+
+(*----------------------------------------------------------------------*)
+(* Theorems about chains. *)
+(*----------------------------------------------------------------------*)
+
+val chainI = prove_goalw Limit.thy [chain_def]
+ "!!z.[|X:nat->set(D);!!n. n:nat ==> rel(D,X`n,X`succ(n))|] ==> chain(D,X)"
+ (fn prems => [asm_simp_tac ZF_ss 1]);
+
+val prems = goalw Limit.thy [chain_def]
+ "chain(D,X) ==> X : nat -> set(D)";
+by(asm_simp_tac (ZF_ss addsimps prems) 1);
+val chain_fun = result();
+
+val prems = goalw Limit.thy [chain_def]
+ "[|chain(D,X); n:nat|] ==> X`n : set(D)";
+br((hd prems)RS conjunct1 RS apply_type)1;
+br(hd(tl prems))1;
+val chain_in = result();
+
+val prems = goalw Limit.thy [chain_def]
+ "[|chain(D,X); n:nat|] ==> rel(D, X ` n, X ` succ(n))";
+br((hd prems)RS conjunct2 RS bspec)1;
+br(hd(tl prems))1;
+val chain_rel = result();
+
+val prems = goal Limit.thy (* chain_rel_gen_add *)
+ "[|chain(D,X); cpo(D); n:nat; m:nat|] ==> rel(D,X`n,(X`(m #+ n)))";
+by(res_inst_tac [("n","m")] nat_induct 1);
+by(ALLGOALS(simp_tac arith_ss));
+by (rtac cpo_trans 3); (* loops if repeated *)
+brr(cpo_refl::chain_in::chain_rel::nat_succI::add_type::prems)1;
+val chain_rel_gen_add = result();
+
+val prems = goal Limit.thy (* le_succ_eq *)
+ "[| n le succ(x); ~ n le x; x : nat; n:nat |] ==> n = succ(x)";
+by (rtac le_anti_sym 1);
+by(resolve_tac prems 1);
+by(simp_tac arith_ss 1);
+br(not_le_iff_lt RS iffD1)1;
+by(REPEAT(resolve_tac (nat_into_Ord::prems) 1));
+val le_succ_eq = result();
+
+val prems = goal Limit.thy (* chain_rel_gen *)
+ "[|n le m; chain(D,X); cpo(D); n:nat; m:nat|] ==> rel(D,X`n,X`m)";
+by (rtac impE 1); (* The first three steps prepare for the induction proof *)
+by (assume_tac 3);
+br(hd prems)2;
+by(res_inst_tac [("n","m")] nat_induct 1);
+by(safe_tac lemmas_cs);
+by(asm_full_simp_tac (arith_ss addsimps prems) 2);
+by (rtac cpo_trans 4);
+by (rtac (le_succ_eq RS subst) 3);
+brr(cpo_refl::chain_in::chain_rel::nat_0I::nat_succI::prems)1;
+val chain_rel_gen = result();
+
+(*----------------------------------------------------------------------*)
+(* Theorems about pcpos and bottom. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [pcpo_def] (* pcpoI *)
+ "[|!!y.y:set(D)==>rel(D,x,y); x:set(D); cpo(D)|]==>pcpo(D)";
+by (rtac conjI 1);
+by (resolve_tac prems 1);
+by (rtac bexI 1);
+by (rtac ballI 1);
+by (resolve_tac prems 2);
+brr prems 1;
+val pcpoI = result();
+
+val pcpo_cpo = prove_goalw Limit.thy [pcpo_def] "pcpo(D) ==> cpo(D)"
+ (fn [pcpo] => [rtac(pcpo RS conjunct1)1]);
+
+val prems = goalw Limit.thy [pcpo_def] (* pcpo_bot_ex1 *)
+ "pcpo(D) ==> EX! x. x:set(D) & (ALL y:set(D). rel(D,x,y))";
+br(hd prems RS conjunct2 RS bexE)1;
+by (rtac ex1I 1);
+by(safe_tac lemmas_cs);
+by (assume_tac 1);
+by (etac bspec 1);
+by (assume_tac 1);
+by (rtac cpo_antisym 1);
+br(hd prems RS conjunct1)1;
+by (etac bspec 1);
+by (assume_tac 1);
+by (etac bspec 1);
+by(REPEAT(atac 1));
+val pcpo_bot_ex1 = result();
+
+val prems = goalw Limit.thy [bot_def] (* bot_least *)
+ "[| pcpo(D); y:set(D)|] ==> rel(D,bot(D),y)";
+by (rtac theI2 1);
+by (rtac pcpo_bot_ex1 1);
+by (resolve_tac prems 1);
+by (etac conjE 1);
+by (etac bspec 1);
+by (resolve_tac prems 1);
+val bot_least = result();
+
+val prems = goalw Limit.thy [bot_def] (* bot_in *)
+ "pcpo(D) ==> bot(D):set(D)";
+by (rtac theI2 1);
+by (rtac pcpo_bot_ex1 1);
+by (resolve_tac prems 1);
+by (etac conjE 1);
+by (assume_tac 1);
+val bot_in = result();
+
+val prems = goal Limit.thy (* bot_unique *)
+ "[| pcpo(D); x:set(D); !!y. y:set(D) ==> rel(D,x,y)|] ==> x = bot(D)";
+by (rtac cpo_antisym 1);
+brr(pcpo_cpo::bot_in::bot_least::prems)1;
+val bot_unique = result();
+
+(*----------------------------------------------------------------------*)
+(* Constant chains and lubs and cpos. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [chain_def] (* chain_const *)
+ "[|x:set(D); cpo(D)|] ==> chain(D,(lam n:nat. x))";
+by (rtac conjI 1);
+by (rtac lam_type 1);
+by(resolve_tac prems 1);
+by (rtac ballI 1);
+by(asm_simp_tac (ZF_ss addsimps [nat_succI]) 1);
+brr(cpo_refl::prems)1;
+val chain_const = result();
+
+val prems = goalw Limit.thy [islub_def,isub_def] (* islub_const *)
+ "[|x:set(D); cpo(D)|] ==> islub(D,(lam n:nat. x),x)";
+by(simp_tac ZF_ss 1);
+by(safe_tac lemmas_cs);
+by (etac bspec 3);
+brr(cpo_refl::nat_0I::prems)1;
+val islub_const = result();
+
+val prems = goal Limit.thy (* lub_const *)
+ "[|x:set(D); cpo(D)|] ==> lub(D,lam n:nat.x) = x";
+by (rtac islub_unique 1);
+by (rtac cpo_lub 1);
+by (rtac chain_const 1);
+by(REPEAT(resolve_tac prems 1));
+by (rtac islub_const 1);
+by(REPEAT(resolve_tac prems 1));
+val lub_const = result();
+
+(*----------------------------------------------------------------------*)
+(* Taking the suffix of chains has no effect on ub's. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [isub_def,suffix_def] (* isub_suffix *)
+ "[|chain(D,X); cpo(D); n:nat|] ==> isub(D,suffix(X,n),x) <-> isub(D,X,x)";
+by(simp_tac (ZF_ss addsimps prems) 1);
+by(safe_tac lemmas_cs);
+by (dtac bspec 2);
+by (assume_tac 3); (* to instantiate unknowns properly *)
+by (rtac cpo_trans 1);
+by (rtac chain_rel_gen_add 2);
+by (dtac bspec 6);
+by (assume_tac 7); (* to instantiate unknowns properly *)
+brr(chain_in::add_type::prems)1;
+val isub_suffix = result();
+
+val prems = goalw Limit.thy [islub_def] (* islub_suffix *)
+ "[|chain(D,X); cpo(D); n:nat|] ==> islub(D,suffix(X,n),x) <-> islub(D,X,x)";
+by(asm_simp_tac (FOL_ss addsimps isub_suffix::prems) 1);
+val islub_suffix = result();
+
+val prems = goalw Limit.thy [lub_def] (* lub_suffix *)
+ "[|chain(D,X); cpo(D); n:nat|] ==> lub(D,suffix(X,n)) = lub(D,X)";
+by(asm_simp_tac (FOL_ss addsimps islub_suffix::prems) 1);
+val lub_suffix = result();
+
+(*----------------------------------------------------------------------*)
+(* Dominate and subchain. *)
+(*----------------------------------------------------------------------*)
+
+val dominateI = prove_goalw Limit.thy [dominate_def]
+ "[| !!m. m:nat ==> n(m):nat; !!m. m:nat ==> rel(D,X`m,Y`n(m))|] ==> \
+\ dominate(D,X,Y)"
+ (fn prems => [rtac ballI 1,rtac bexI 1,REPEAT(ares_tac prems 1)]);
+
+val [dom,isub,cpo,X,Y] = goal Limit.thy
+ "[|dominate(D,X,Y); isub(D,Y,x); cpo(D); \
+\ X:nat->set(D); Y:nat->set(D)|] ==> isub(D,X,x)";
+by(rewtac isub_def);
+by (rtac conjI 1);
+br(rewrite_rule[isub_def]isub RS conjunct1)1;
+by (rtac ballI 1);
+br(rewrite_rule[dominate_def]dom RS bspec RS bexE)1;
+by (assume_tac 1);
+by (rtac cpo_trans 1);
+by (rtac cpo 1);
+by (assume_tac 1);
+br(rewrite_rule[isub_def]isub RS conjunct2 RS bspec)1;
+by (assume_tac 1);
+be(X RS apply_type)1;
+be(Y RS apply_type)1;
+br(rewrite_rule[isub_def]isub RS conjunct1)1;
+val dominate_isub = result();
+
+val [dom,Xlub,Ylub,cpo,X,Y] = goal Limit.thy
+ "[|dominate(D,X,Y); islub(D,X,x); islub(D,Y,y); cpo(D); \
+\ X:nat->set(D); Y:nat->set(D)|] ==> rel(D,x,y)";
+val Xub = rewrite_rule[islub_def]Xlub RS conjunct1;
+val Yub = rewrite_rule[islub_def]Ylub RS conjunct1;
+val Xub_y = Yub RS (dom RS dominate_isub);
+val lem = Xub_y RS (rewrite_rule[islub_def]Xlub RS conjunct2 RS spec RS mp);
+val thm = Y RS (X RS (cpo RS lem));
+by (rtac thm 1);
+val dominate_islub = result();
+
+val prems = goalw Limit.thy [subchain_def] (* subchainE *)
+ "[|subchain(X,Y); n:nat;!!m. [|m:nat; X`n = Y`(n #+ m)|] ==> Q|] ==> Q";
+br(hd prems RS bspec RS bexE)1;
+by(resolve_tac prems 2);
+by (assume_tac 3);
+by(REPEAT(ares_tac prems 1));
+val subchainE = result();
+
+val prems = goalw Limit.thy [] (* subchain_isub *)
+ "[|subchain(Y,X); isub(D,X,x)|] ==> isub(D,Y,x)";
+by (rtac isubI 1);
+val [subch,ub] = prems;
+br(ub RS isubD1)1;
+br(subch RS subchainE)1;
+by (assume_tac 1);
+by(asm_simp_tac ZF_ss 1);
+by (rtac isubD2 1); (* br with Destruction rule ?? *)
+by(resolve_tac prems 1);
+by(asm_simp_tac arith_ss 1);
+val subchain_isub = result();
+
+val prems = goal Limit.thy (* dominate_islub_eq *)
+ "[|dominate(D,X,Y); subchain(Y,X); islub(D,X,x); islub(D,Y,y); cpo(D); \
+\ X:nat->set(D); Y:nat->set(D)|] ==> x = y";
+by (rtac cpo_antisym 1);
+by(resolve_tac prems 1);
+by (rtac dominate_islub 1);
+by(REPEAT(resolve_tac prems 1));
+by (rtac islub_least 1);
+by(REPEAT(resolve_tac prems 1));
+by (rtac subchain_isub 1);
+by (rtac islub_isub 2);
+by(REPEAT(resolve_tac (islub_in::prems) 1));
+val dominate_islub_eq = result();
+
+(*----------------------------------------------------------------------*)
+(* Matrix. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [matrix_def] (* matrix_fun *)
+ "matrix(D,M) ==> M : nat -> (nat -> set(D))";
+by(simp_tac (ZF_ss addsimps prems) 1);
+val matrix_fun = result();
+
+val prems = goalw Limit.thy [] (* matrix_in_fun *)
+ "[|matrix(D,M); n:nat|] ==> M`n : nat -> set(D)";
+by (rtac apply_type 1);
+by(REPEAT(resolve_tac(matrix_fun::prems)1));
+val matrix_in_fun = result();
+
+val prems = goalw Limit.thy [] (* matrix_in *)
+ "[|matrix(D,M); n:nat; m:nat|] ==> M`n`m : set(D)";
+by (rtac apply_type 1);
+by(REPEAT(resolve_tac(matrix_in_fun::prems)1));
+val matrix_in = result();
+
+val prems = goalw Limit.thy [matrix_def] (* matrix_rel_1_0 *)
+ "[|matrix(D,M); n:nat; m:nat|] ==> rel(D,M`n`m,M`succ(n)`m)";
+by(simp_tac (ZF_ss addsimps prems) 1);
+val matrix_rel_1_0 = result();
+
+val prems = goalw Limit.thy [matrix_def] (* matrix_rel_0_1 *)
+ "[|matrix(D,M); n:nat; m:nat|] ==> rel(D,M`n`m,M`n`succ(m))";
+by(simp_tac (ZF_ss addsimps prems) 1);
+val matrix_rel_0_1 = result();
+
+val prems = goalw Limit.thy [matrix_def] (* matrix_rel_1_1 *)
+ "[|matrix(D,M); n:nat; m:nat|] ==> rel(D,M`n`m,M`succ(n)`succ(m))";
+by(simp_tac (ZF_ss addsimps prems) 1);
+val matrix_rel_1_1 = result();
+
+val prems = goal Limit.thy (* fun_swap *)
+ "f:X->Y->Z ==> (lam y:Y. lam x:X. f`x`y):Y->X->Z";
+by (rtac lam_type 1);
+by (rtac lam_type 1);
+by (rtac apply_type 1);
+by (rtac apply_type 1);
+by(REPEAT(ares_tac prems 1));
+val fun_swap = result();
+
+val prems = goalw Limit.thy [matrix_def] (* matrix_sym_axis *)
+ "!!z. matrix(D,M) ==> matrix(D,lam m:nat. lam n:nat. M`n`m)";
+by(simp_tac arith_ss 1 THEN safe_tac lemmas_cs THEN
+REPEAT(asm_simp_tac (ZF_ss addsimps [fun_swap]) 1));
+val matrix_sym_axis = result();
+
+val prems = goalw Limit.thy [chain_def] (* matrix_chain_diag *)
+ "matrix(D,M) ==> chain(D,lam n:nat. M`n`n)";
+by(safe_tac lemmas_cs);
+by (rtac lam_type 1);
+by (rtac matrix_in 1);
+by(REPEAT(ares_tac prems 1));
+by(asm_simp_tac arith_ss 1);
+by (rtac matrix_rel_1_1 1);
+by(REPEAT(ares_tac prems 1));
+val matrix_chain_diag = result();
+
+val prems = goalw Limit.thy [chain_def] (* matrix_chain_left *)
+ "[|matrix(D,M); n:nat|] ==> chain(D,M`n)";
+by(safe_tac lemmas_cs);
+by (rtac apply_type 1);
+by (rtac matrix_fun 1);
+by(REPEAT(ares_tac prems 1));
+by (rtac matrix_rel_0_1 1);
+by(REPEAT(ares_tac prems 1));
+val matrix_chain_left = result();
+
+val prems = goalw Limit.thy [chain_def] (* matrix_chain_right *)
+ "[|matrix(D,M); m:nat|] ==> chain(D,lam n:nat. M`n`m)";
+by(safe_tac lemmas_cs);
+by(asm_simp_tac(arith_ss addsimps prems)2);
+brr(lam_type::matrix_in::matrix_rel_1_0::prems)1;
+val matrix_chain_right = result();
+
+val prems = goalw Limit.thy [matrix_def] (* matrix_chainI *)
+ "[|!!x.x:nat==>chain(D,M`x);!!y.y:nat==>chain(D,lam x:nat. M`x`y); \
+\ M:nat->nat->set(D); cpo(D)|] ==> matrix(D,M)";
+by(safe_tac (lemmas_cs addSIs [ballI]));
+by(cut_inst_tac[("y1","m"),("n","n")](hd(tl prems) RS chain_rel)2);
+by(asm_full_simp_tac arith_ss 4);
+by (rtac cpo_trans 5);
+by(cut_inst_tac[("y1","m"),("n","n")](hd(tl prems) RS chain_rel)6);
+by(asm_full_simp_tac arith_ss 8);
+by(TRYALL(rtac (chain_fun RS apply_type)));
+brr(chain_rel::nat_succI::prems)1;
+val matrix_chainI = result();
+
+val lemma = prove_goal Limit.thy
+ "!!z.[|m : nat; rel(D, (lam n:nat. M`n`n)`m, y)|] ==> rel(D,M`m`m, y)"
+ (fn prems => [asm_full_simp_tac ZF_ss 1]);
+
+val lemma2 = prove_goal Limit.thy
+ "!!z.[|x:nat; m:nat; rel(D,(lam n:nat.M`n`m1)`x,(lam n:nat.M`n`m1)`m)|] ==> \
+\ rel(D,M`x`m1,M`m`m1)"
+ (fn prems => [asm_full_simp_tac ZF_ss 1]);
+
+val prems = goalw Limit.thy [] (* isub_lemma *)
+ "[|isub(D,(lam n:nat. M`n`n),y); matrix(D,M); cpo(D)|] ==> \
+\ isub(D,(lam n:nat. lub(D,lam m:nat. M`n`m)),y)";
+by(rewtac isub_def);
+by(safe_tac lemmas_cs);
+by (rtac isubD1 1);
+by(resolve_tac prems 1);
+by(asm_simp_tac ZF_ss 1);
+by(cut_inst_tac[("a","n")](hd(tl prems) RS matrix_fun RS apply_type)1);
+by (assume_tac 1);
+by(asm_simp_tac ZF_ss 1);
+by (rtac islub_least 1);
+by (rtac cpo_lub 1);
+by (rtac matrix_chain_left 1);
+by(resolve_tac prems 1);
+by (assume_tac 1);
+by(resolve_tac prems 1);
+by(rewtac isub_def);
+by(safe_tac lemmas_cs);
+by (rtac isubD1 1);
+by(resolve_tac prems 1);
+by(cut_inst_tac[("P","n le na")]excluded_middle 1);
+by(safe_tac lemmas_cs);
+by (rtac cpo_trans 1);
+by(resolve_tac prems 1);
+br(not_le_iff_lt RS iffD1 RS leI RS chain_rel_gen)1;
+by (assume_tac 3);
+by(REPEAT(ares_tac (nat_into_Ord::matrix_chain_left::prems) 1));
+by (rtac lemma 1);
+by (rtac isubD2 2);
+by(REPEAT(ares_tac (matrix_in::isubD1::prems) 1));
+by (rtac cpo_trans 1);
+by(resolve_tac prems 1);
+by (rtac lemma2 1);
+by (rtac lemma 4);
+by (rtac isubD2 5);
+by(REPEAT(ares_tac
+ ([chain_rel_gen,matrix_chain_right,matrix_in,isubD1]@prems)1));
+val isub_lemma = result();
+
+val prems = goalw Limit.thy [chain_def] (* matrix_chain_lub *)
+ "[|matrix(D,M); cpo(D)|] ==> chain(D,lam n:nat.lub(D,lam m:nat.M`n`m))";
+by(safe_tac lemmas_cs);
+by (rtac lam_type 1);
+by (rtac islub_in 1);
+by (rtac cpo_lub 1);
+by(resolve_tac prems 2);
+by(asm_simp_tac arith_ss 2);
+by (rtac chainI 1);
+by (rtac lam_type 1);
+by(REPEAT(ares_tac (matrix_in::prems) 1));
+by(asm_simp_tac arith_ss 1);
+by (rtac matrix_rel_0_1 1);
+by(REPEAT(ares_tac prems 1));
+by(asm_simp_tac (arith_ss addsimps
+ [hd prems RS matrix_chain_left RS chain_fun RS eta]) 1);
+by (rtac dominate_islub 1);
+by (rtac cpo_lub 3);
+by (rtac cpo_lub 2);
+by(rewtac dominate_def);
+by (rtac ballI 1);
+by (rtac bexI 1);
+by (assume_tac 2);
+back(); (* Backtracking...... *)
+by (rtac matrix_rel_1_0 1);
+by(REPEAT(ares_tac (matrix_chain_left::nat_succI::chain_fun::prems) 1));
+val matrix_chain_lub = result();
+
+val prems = goal Limit.thy (* isub_eq *)
+ "[|matrix(D,M); cpo(D)|] ==> \
+\ isub(D,(lam n:nat. lub(D,lam m:nat. M`n`m)),y) <-> \
+\ isub(D,(lam n:nat. M`n`n),y)";
+by (rtac iffI 1);
+by (rtac dominate_isub 1);
+by (assume_tac 2);
+by(rewtac dominate_def);
+by (rtac ballI 1);
+by (rtac bexI 1);
+by (assume_tac 2);
+by(asm_simp_tac ZF_ss 1);
+by(asm_simp_tac (arith_ss addsimps
+ [hd prems RS matrix_chain_left RS chain_fun RS eta]) 1);
+by (rtac islub_ub 1);
+by (rtac cpo_lub 1);
+by(REPEAT(ares_tac
+(matrix_chain_left::matrix_chain_diag::chain_fun::matrix_chain_lub::prems) 1));
+by (rtac isub_lemma 1);
+by(REPEAT(ares_tac prems 1));
+val isub_eq = result();
+
+val lemma1 = prove_goalw Limit.thy [lub_def]
+ "lub(D,(lam n:nat. lub(D,lam m:nat. M`n`m))) = \
+\ (THE x. islub(D, (lam n:nat. lub(D,lam m:nat. M`n`m)), x))"
+ (fn prems => [fast_tac ZF_cs 1]);
+
+val lemma2 = prove_goalw Limit.thy [lub_def]
+ "lub(D,(lam n:nat. M`n`n)) = \
+\ (THE x. islub(D, (lam n:nat. M`n`n), x))"
+ (fn prems => [fast_tac ZF_cs 1]);
+
+val prems = goalw Limit.thy [] (* lub_matrix_diag *)
+ "[|matrix(D,M); cpo(D)|] ==> \
+\ lub(D,(lam n:nat. lub(D,lam m:nat. M`n`m))) = \
+\ lub(D,(lam n:nat. M`n`n))";
+by(simp_tac (arith_ss addsimps [lemma1,lemma2]) 1);
+by(rewtac islub_def);
+by(simp_tac (FOL_ss addsimps [hd(tl prems) RS (hd prems RS isub_eq)]) 1);
+val lub_matrix_diag = result();
+
+val [matrix,cpo] = goalw Limit.thy [] (* lub_matrix_diag_sym *)
+ "[|matrix(D,M); cpo(D)|] ==> \
+\ lub(D,(lam m:nat. lub(D,lam n:nat. M`n`m))) = \
+\ lub(D,(lam n:nat. M`n`n))";
+by(cut_facts_tac[cpo RS (matrix RS matrix_sym_axis RS lub_matrix_diag)]1);
+by(asm_full_simp_tac ZF_ss 1);
+val lub_matrix_diag_sym = result();
+
+(*----------------------------------------------------------------------*)
+(* I/E/D rules for mono and cont. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [mono_def] (* monoI *)
+ "[|f:set(D)->set(E); \
+\ !!x y. [|rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y)|] ==> \
+\ f:mono(D,E)";
+by(fast_tac(ZF_cs addSIs prems)1);
+val monoI = result();
+
+val prems = goal Limit.thy
+ "f:mono(D,E) ==> f:set(D)->set(E)";
+br(rewrite_rule[mono_def](hd prems) RS CollectD1)1;
+val mono_fun = result();
+
+val prems = goal Limit.thy
+ "[|f:mono(D,E); x:set(D)|] ==> f`x:set(E)";
+br(hd prems RS mono_fun RS apply_type)1;
+by(resolve_tac prems 1);
+val mono_map = result();
+
+val prems = goal Limit.thy
+ "[|f:mono(D,E); rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y)";
+br(rewrite_rule[mono_def](hd prems) RS CollectD2 RS bspec RS bspec RS mp)1;
+by(REPEAT(resolve_tac prems 1));
+val mono_mono = result();
+
+val prems = goalw Limit.thy [cont_def,mono_def] (* contI *)
+ "[|f:set(D)->set(E); \
+\ !!x y. [|rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y); \
+\ !!X. chain(D,X) ==> f`lub(D,X) = lub(E,lam n:nat. f`(X`n))|] ==> \
+\ f:cont(D,E)";
+by(fast_tac(ZF_cs addSIs prems)1);
+val contI = result();
+
+val prems = goal Limit.thy
+ "f:cont(D,E) ==> f:mono(D,E)";
+br(rewrite_rule[cont_def](hd prems) RS CollectD1)1;
+val cont2mono = result();
+
+val prems = goal Limit.thy
+ "f:cont(D,E) ==> f:set(D)->set(E)";
+br(rewrite_rule[cont_def](hd prems) RS CollectD1 RS mono_fun)1;
+val cont_fun = result();
+
+val prems = goal Limit.thy
+ "[|f:cont(D,E); x:set(D)|] ==> f`x:set(E)";
+br(hd prems RS cont_fun RS apply_type)1;
+by(resolve_tac prems 1);
+val cont_map = result();
+
+val prems = goal Limit.thy
+ "[|f:cont(D,E); rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y)";
+br(rewrite_rule[cont_def](hd prems) RS CollectD1 RS mono_mono)1;
+by(REPEAT(resolve_tac prems 1));
+val cont_mono = result();
+
+val prems = goal Limit.thy
+ "[|f:cont(D,E); chain(D,X)|] ==> f`(lub(D,X)) = lub(E,lam n:nat. f`(X`n))";
+br(rewrite_rule[cont_def](hd prems) RS CollectD2 RS spec RS mp)1;
+by(REPEAT(resolve_tac prems 1));
+val cont_lub = result();
+
+(*----------------------------------------------------------------------*)
+(* Continuity and chains. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [] (* mono_chain *)
+ "[|f:mono(D,E); chain(D,X)|] ==> chain(E,lam n:nat. f`(X`n))";
+by(rewtac chain_def);
+by(simp_tac arith_ss 1);
+by(safe_tac lemmas_cs);
+by (rtac lam_type 1);
+by (rtac mono_map 1);
+by(resolve_tac prems 1);
+by (rtac chain_in 1);
+by(REPEAT(ares_tac prems 1));
+by (rtac mono_mono 1);
+by(resolve_tac prems 1);
+by (rtac chain_rel 1);
+by(REPEAT(ares_tac prems 1));
+by (rtac chain_in 1);
+by (rtac chain_in 3);
+by(REPEAT(ares_tac (nat_succI::prems) 1));
+val mono_chain = result();
+
+val prems = goalw Limit.thy [] (* cont_chain *)
+ "[|f:cont(D,E); chain(D,X)|] ==> chain(E,lam n:nat. f`(X`n))";
+by (rtac mono_chain 1);
+by(REPEAT(resolve_tac (cont2mono::prems) 1));
+val cont_chain = result();
+
+(*----------------------------------------------------------------------*)
+(* I/E/D rules about (set+rel) cf, the continuous function space. *)
+(*----------------------------------------------------------------------*)
+
+(* The following development more difficult with cpo-as-relation approach. *)
+
+val prems = goalw Limit.thy [set_def,cf_def]
+ "!!z. f:set(cf(D,E)) ==> f:cont(D,E)";
+by(asm_full_simp_tac ZF_ss 1);
+val in_cf = result();
+val cf_cont = result();
+
+val prems = goalw Limit.thy [set_def,cf_def] (* Non-trivial with relation *)
+ "!!z. f:cont(D,E) ==> f:set(cf(D,E))";
+by(asm_full_simp_tac ZF_ss 1);
+val cont_cf = result();
+
+(* rel_cf originally an equality. Now stated as two rules. Seemed easiest.
+ Besides, now complicated by typing assumptions. *)
+
+val prems = goal Limit.thy
+ "[|!!x. x:set(D) ==> rel(E,f`x,g`x); f:cont(D,E); g:cont(D,E)|] ==> \
+\ rel(cf(D,E),f,g)";
+by (rtac rel_I 1);
+by(simp_tac (ZF_ss addsimps [cf_def])1);
+by(safe_tac lemmas_cs);
+brr prems 1;
+val rel_cfI = result();
+
+val prems = goalw Limit.thy [rel_def,cf_def]
+ "!!z. [|rel(cf(D,E),f,g); x:set(D)|] ==> rel(E,f`x,g`x)";
+by(asm_full_simp_tac ZF_ss 1);
+val rel_cf = result();
+
+(*----------------------------------------------------------------------*)
+(* Theorems about the continuous function space. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [] (* chain_cf *)
+ "[| chain(cf(D,E),X); x:set(D)|] ==> chain(E,lam n:nat. X`n`x)";
+by (rtac chainI 1);
+by (rtac lam_type 1);
+by (rtac apply_type 1);
+by (resolve_tac prems 2);
+by(REPEAT(ares_tac([cont_fun,in_cf,chain_in]@prems)1));
+by(asm_simp_tac arith_ss 1);
+by(REPEAT(ares_tac([rel_cf,chain_rel]@prems)1));
+val chain_cf = result();
+
+val prems = goal Limit.thy (* matrix_lemma *)
+ "[|chain(cf(D,E),X); chain(D,Xa); cpo(D); cpo(E) |] ==> \
+\ matrix(E,lam x:nat. lam xa:nat. X`x`(Xa`xa))";
+by (rtac matrix_chainI 1);
+by(asm_simp_tac ZF_ss 1);
+by(asm_simp_tac ZF_ss 2);
+by (rtac chainI 1);
+by (rtac lam_type 1);
+by (rtac apply_type 1);
+by (rtac (chain_in RS cf_cont RS cont_fun) 1);
+by(REPEAT(ares_tac prems 1));
+by (rtac chain_in 1);
+by(REPEAT(ares_tac prems 1));
+by(asm_simp_tac arith_ss 1);
+by (rtac cont_mono 1);
+by (rtac (chain_in RS cf_cont) 1);
+brr prems 1;
+brr (chain_rel::chain_in::nat_succI::prems)1;
+by (rtac chainI 1);
+by (rtac lam_type 1);
+by (rtac apply_type 1);
+by (rtac (chain_in RS cf_cont RS cont_fun) 1);
+by(REPEAT(ares_tac prems 1));
+by (rtac chain_in 1);
+by(REPEAT(ares_tac prems 1));
+by(asm_simp_tac arith_ss 1);
+by (rtac rel_cf 1);
+brr (chain_in::chain_rel::prems)1;
+by (rtac lam_type 1);
+by (rtac lam_type 1);
+by (rtac apply_type 1);
+by (rtac (chain_in RS cf_cont RS cont_fun) 1);
+brr prems 1;
+by (rtac chain_in 1);
+brr prems 1;
+val matrix_lemma = result();
+
+val prems = goal Limit.thy (* chain_cf_lub_cont *)
+ "[|chain(cf(D,E),X); cpo(D); cpo(E) |] ==> \
+\ (lam x:set(D). lub(E, lam n:nat. X ` n ` x)) : cont(D, E)";
+by (rtac contI 1);
+by (rtac lam_type 1);
+by(REPEAT(ares_tac((chain_cf RS cpo_lub RS islub_in)::prems)1));
+by(asm_simp_tac ZF_ss 1);
+by (rtac dominate_islub 1);
+by(REPEAT(ares_tac((chain_cf RS cpo_lub)::prems)2));
+by (rtac dominateI 1);
+by (assume_tac 1);
+by(asm_simp_tac ZF_ss 1);
+by(REPEAT(ares_tac ((chain_in RS cf_cont RS cont_mono)::prems) 1));
+by(REPEAT(ares_tac ((chain_cf RS chain_fun)::prems) 1));
+by (rtac (beta RS ssubst) 1);
+by(REPEAT(ares_tac((cpo_lub RS islub_in)::prems)1));
+by(asm_simp_tac(ZF_ss addsimps[hd prems RS chain_in RS cf_cont RS cont_lub])1);
+by(forward_tac[hd prems RS matrix_lemma RS lub_matrix_diag]1);
+brr prems 1;
+by(asm_full_simp_tac ZF_ss 1);
+by(asm_simp_tac(ZF_ss addsimps[chain_in RS beta])1);
+bd(hd prems RS matrix_lemma RS lub_matrix_diag_sym)1;
+brr prems 1;
+by(asm_full_simp_tac ZF_ss 1);
+val chain_cf_lub_cont = result();
+
+val prems = goal Limit.thy (* islub_cf *)
+ "[| chain(cf(D,E),X); cpo(D); cpo(E)|] ==> \
+\ islub(cf(D,E), X, lam x:set(D). lub(E,lam n:nat. X`n`x))";
+by (rtac islubI 1);
+by (rtac isubI 1);
+by (rtac (chain_cf_lub_cont RS cont_cf) 1);
+brr prems 1;
+by (rtac rel_cfI 1);
+by(asm_simp_tac ZF_ss 1);
+bd(hd(tl(tl prems)) RSN(2,hd prems RS chain_cf RS cpo_lub RS islub_ub))1;
+by (assume_tac 1);
+by(asm_full_simp_tac ZF_ss 1);
+brr(cf_cont::chain_in::prems)1;
+brr(cont_cf::chain_cf_lub_cont::prems)1;
+by (rtac rel_cfI 1);
+by(asm_simp_tac ZF_ss 1);
+by(forward_tac[hd(tl(tl prems)) RSN(2,hd prems RS chain_cf RS cpo_lub RS
+ islub_least)]1);
+by (assume_tac 2);
+brr (chain_cf_lub_cont::isubD1::cf_cont::prems) 2;
+by (rtac isubI 1);
+brr((cf_cont RS cont_fun RS apply_type)::[isubD1])1;
+by(asm_simp_tac ZF_ss 1);
+be(isubD2 RS rel_cf)1;
+brr [] 1;
+val islub_cf = result();
+
+val prems = goal Limit.thy (* cpo_cf *)
+ "[| cpo(D); cpo(E)|] ==> cpo(cf(D,E))";
+by (rtac (poI RS cpoI) 1);
+by (rtac rel_cfI 1);
+brr(cpo_refl::(cf_cont RS cont_fun RS apply_type)::cf_cont::prems)1;
+by (rtac rel_cfI 1);
+by (rtac cpo_trans 1);
+by (resolve_tac prems 1);
+by (etac rel_cf 1);
+by (assume_tac 1);
+by (rtac rel_cf 1);
+by (assume_tac 1);
+brr[cf_cont RS cont_fun RS apply_type,cf_cont]1;
+by (rtac fun_extension 1);
+brr[cf_cont RS cont_fun]1;
+by (rtac cpo_antisym 1);
+br(hd(tl prems))1;
+by (etac rel_cf 1);
+by (assume_tac 1);
+by (rtac rel_cf 1);
+by (assume_tac 1);
+brr[cf_cont RS cont_fun RS apply_type]1;
+by (dtac islub_cf 1);
+brr prems 1;
+val cpo_cf = result();
+
+val prems = goal Limit.thy (* lub_cf *)
+ "[| chain(cf(D,E),X); cpo(D); cpo(E)|] ==> \
+\ lub(cf(D,E), X) = (lam x:set(D). lub(E,lam n:nat. X`n`x))";
+by (rtac islub_unique 1);
+brr (cpo_lub::islub_cf::cpo_cf::prems) 1;
+val lub_cf = result();
+
+val const_fun = prove_goal ZF.thy
+ "y:set(E) ==> (lam x:set(D).y): set(D)->set(E)"
+ (fn prems => [rtac lam_type 1,rtac (hd prems) 1]);
+
+val prems = goal Limit.thy (* const_cont *)
+ "[|y:set(E); cpo(D); cpo(E)|] ==> (lam x:set(D).y) : cont(D,E)";
+by (rtac contI 1);
+by(asm_simp_tac ZF_ss 2);
+brr(const_fun::cpo_refl::prems)1;
+by(asm_simp_tac(ZF_ss addsimps(chain_in::(cpo_lub RS islub_in)::
+ lub_const::prems))1);
+val const_cont = result();
+
+val prems = goal Limit.thy (* cf_least *)
+ "[|cpo(D); pcpo(E); y:cont(D,E)|]==>rel(cf(D,E),(lam x:set(D).bot(E)),y)";
+by (rtac rel_cfI 1);
+by(asm_simp_tac ZF_ss 1);
+brr(bot_least::bot_in::apply_type::cont_fun::const_cont::
+ cpo_cf::(prems@[pcpo_cpo]))1;
+val cf_least = result();
+
+val prems = goal Limit.thy (* pcpo_cf *)
+ "[|cpo(D); pcpo(E)|] ==> pcpo(cf(D,E))";
+by (rtac pcpoI 1);
+brr(cf_least::bot_in::(const_cont RS cont_cf)::cf_cont::
+ cpo_cf::(prems@[pcpo_cpo]))1;
+val pcpo_cf = result();
+
+val prems = goal Limit.thy (* bot_cf *)
+ "[|cpo(D); pcpo(E)|] ==> bot(cf(D,E)) = (lam x:set(D).bot(E))";
+by (rtac (bot_unique RS sym) 1);
+brr(pcpo_cf::cf_least::(bot_in RS const_cont RS cont_cf)::
+ cf_cont::(prems@[pcpo_cpo]))1;
+val bot_cf = result();
+
+(*----------------------------------------------------------------------*)
+(* Identity and composition. *)
+(*----------------------------------------------------------------------*)
+
+val id_thm = prove_goalw Perm.thy [id_def] "x:X ==> (id(X)`x) = x"
+ (fn prems => [simp_tac(ZF_ss addsimps prems)1]);
+
+val prems = goal Limit.thy (* id_cont *)
+ "cpo(D) ==> id(set(D)):cont(D,D)";
+by (rtac contI 1);
+by (rtac id_type 1);
+by(asm_simp_tac (ZF_ss addsimps[id_thm])1);
+by(asm_simp_tac(ZF_ss addsimps(id_thm::(cpo_lub RS islub_in)::
+ chain_in::(chain_fun RS eta)::prems))1);
+val id_cont = result();
+
+val comp_cont_apply = cont_fun RSN(2,cont_fun RS comp_fun_apply);
+
+val prems = goal Limit.thy (* comp_pres_cont *)
+ "[| f:cont(D',E); g:cont(D,D'); cpo(D)|] ==> f O g : cont(D,E)";
+by (rtac contI 1);
+br(comp_cont_apply RS ssubst)2;
+br(comp_cont_apply RS ssubst)5;
+by (rtac cont_mono 8);
+by (rtac cont_mono 9); (* 15 subgoals *)
+brr(comp_fun::cont_fun::cont_map::prems)1; (* proves all but the lub case *)
+br(comp_cont_apply RS ssubst)1;
+br(cont_lub RS ssubst)4;
+br(cont_lub RS ssubst)6;
+by(asm_full_simp_tac(ZF_ss addsimps (* RS: new subgoals contain unknowns *)
+ [hd prems RS (hd(tl prems) RS comp_cont_apply),chain_in])8);
+brr((cpo_lub RS islub_in)::cont_chain::prems)1;
+val comp_pres_cont = result();
+
+val prems = goal Limit.thy (* comp_mono *)
+ "[| f:cont(D',E); g:cont(D,D'); f':cont(D',E); g':cont(D,D'); \
+\ rel(cf(D',E),f,f'); rel(cf(D,D'),g,g'); cpo(D); cpo(E) |] ==> \
+\ rel(cf(D,E),f O g,f' O g')";
+by (rtac rel_cfI 1); (* extra proof obl: f O g and f' O g' cont. Extra asm cpo(D). *)
+br(comp_cont_apply RS ssubst)1;
+br(comp_cont_apply RS ssubst)4;
+by (rtac cpo_trans 7);
+brr(rel_cf::cont_mono::cont_map::comp_pres_cont::prems)1;
+val comp_mono = result();
+
+val prems = goal Limit.thy (* chain_cf_comp *)
+ "[| chain(cf(D',E),X); chain(cf(D,D'),Y); cpo(D); cpo(E)|] ==> \
+\ chain(cf(D,E),lam n:nat. X`n O Y`n)";
+by (rtac chainI 1);
+by(asm_simp_tac arith_ss 2);
+by (rtac rel_cfI 2);
+br(comp_cont_apply RS ssubst)2;
+br(comp_cont_apply RS ssubst)5;
+by (rtac cpo_trans 8);
+by (rtac rel_cf 9);
+by (rtac cont_mono 11);
+brr(lam_type::comp_pres_cont::cont_cf::(chain_in RS cf_cont)::cont_map::
+ chain_rel::rel_cf::nat_succI::prems)1;
+val chain_cf_comp = result();
+
+val prems = goal Limit.thy (* comp_lubs *)
+ "[| chain(cf(D',E),X); chain(cf(D,D'),Y); cpo(D); cpo(D'); cpo(E)|] ==> \
+\ lub(cf(D',E),X) O lub(cf(D,D'),Y) = lub(cf(D,E),lam n:nat. X`n O Y`n)";
+by (rtac fun_extension 1);
+br(lub_cf RS ssubst)3;
+brr(comp_fun::(cf_cont RS cont_fun)::(cpo_lub RS islub_in)::cpo_cf::
+ chain_cf_comp::prems)1;
+by(cut_facts_tac[hd prems,hd(tl prems)]1);
+by(asm_simp_tac(ZF_ss addsimps((chain_in RS cf_cont RSN(3,chain_in RS
+ cf_cont RS comp_cont_apply))::(tl(tl prems))))1);
+br(comp_cont_apply RS ssubst)1;
+brr((cpo_lub RS islub_in RS cf_cont)::cpo_cf::prems)1;
+by(asm_simp_tac(ZF_ss addsimps(lub_cf::
+ (hd(tl prems)RS chain_cf RSN(2,hd prems RS chain_in RS cf_cont RS cont_lub))::
+ (hd(tl prems) RS chain_cf RS cpo_lub RS islub_in)::prems))1);
+by(cut_inst_tac[("M","lam xa:nat. lam xb:nat. X`xa`(Y`xb`x)")]
+ lub_matrix_diag 1);
+by(asm_full_simp_tac ZF_ss 3);
+by (rtac matrix_chainI 1);
+by(asm_simp_tac ZF_ss 1);
+by(asm_simp_tac ZF_ss 2);
+by(forward_tac[hd(tl prems) RSN(2,(hd prems RS chain_in RS cf_cont) RS
+ (chain_cf RSN(2,cont_chain)))]1); (* Here, Isabelle was a bitch! *)
+by(asm_full_simp_tac ZF_ss 2);
+by (assume_tac 1);
+by (rtac chain_cf 1);
+brr((cont_fun RS apply_type)::(chain_in RS cf_cont)::lam_type::prems)1;
+val comp_lubs = result();
+
+(*----------------------------------------------------------------------*)
+(* A couple (out of many) theorems about arithmetic. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goal Arith.thy (* le_cases *)
+ "[|m:nat; n:nat|] ==> m le n | n le m";
+by(safe_tac lemmas_cs);
+brr((not_le_iff_lt RS iffD1 RS leI)::nat_into_Ord::prems)1;
+val le_cases = result();
+
+val prems = goal Arith.thy (* lt_cases *)
+ "[|m:nat; n:nat|] ==> m < n | n le m";
+by(safe_tac lemmas_cs);
+brr((not_le_iff_lt RS iffD1)::nat_into_Ord::prems)1;
+val lt_cases = result();
+
+(*----------------------------------------------------------------------*)
+(* Theorems about projpair. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [projpair_def] (* projpairI *)
+ "!!x. [| e:cont(D,E); p:cont(E,D); p O e = id(set(D)); \
+\ rel(cf(E,E))(e O p)(id(set(E)))|] ==> projpair(D,E,e,p)";
+by(fast_tac FOL_cs 1);
+val projpairI = result();
+
+val prems = goalw Limit.thy [projpair_def] (* projpairE *)
+ "[| projpair(D,E,e,p); \
+\ [| e:cont(D,E); p:cont(E,D); p O e = id(set(D)); \
+\ rel(cf(E,E))(e O p)(id(set(E)))|] ==> Q |] ==> Q";
+br(hd(tl prems))1;
+by(REPEAT(asm_simp_tac(FOL_ss addsimps[hd prems])1));
+val projpairE = result();
+
+val prems = goal Limit.thy (* projpair_e_cont *)
+ "projpair(D,E,e,p) ==> e:cont(D,E)";
+by (rtac projpairE 1);
+by(REPEAT(ares_tac prems 1));
+val projpair_e_cont = result();
+
+val prems = goal Limit.thy (* projpair_p_cont *)
+ "projpair(D,E,e,p) ==> p:cont(E,D)";
+by (rtac projpairE 1);
+by(REPEAT(ares_tac prems 1));
+val projpair_p_cont = result();
+
+val prems = goal Limit.thy (* projpair_eq *)
+ "projpair(D,E,e,p) ==> p O e = id(set(D))";
+by (rtac projpairE 1);
+by(REPEAT(ares_tac prems 1));
+val projpair_eq = result();
+
+val prems = goal Limit.thy (* projpair_rel *)
+ "projpair(D,E,e,p) ==> rel(cf(E,E))(e O p)(id(set(E)))";
+by (rtac projpairE 1);
+by(REPEAT(ares_tac prems 1));
+val projpair_rel = result();
+
+val projpairDs = [projpair_e_cont,projpair_p_cont,projpair_eq,projpair_rel];
+
+(*----------------------------------------------------------------------*)
+(* NB! projpair_e_cont and projpair_p_cont cannot be used repeatedly *)
+(* at the same time since both match a goal of the form f:cont(X,Y).*)
+(*----------------------------------------------------------------------*)
+
+(*----------------------------------------------------------------------*)
+(* Uniqueness of embedding projection pairs. *)
+(*----------------------------------------------------------------------*)
+
+val id_comp = fun_is_rel RS left_comp_id;
+val comp_id = fun_is_rel RS right_comp_id;
+
+val prems = goal Limit.thy (* lemma1 *)
+ "[|cpo(D); cpo(E); projpair(D,E,e,p); projpair(D,E,e',p'); \
+\ rel(cf(D,E),e,e')|] ==> rel(cf(E,D),p',p)";
+val [_,_,p1,p2,_] = prems;
+(* The two theorems proj_e_cont and proj_p_cont are useless unless they
+ are used manually, one at a time. Therefore the following contl. *)
+val contl = [p1 RS projpair_e_cont,p1 RS projpair_p_cont,
+ p2 RS projpair_e_cont,p2 RS projpair_p_cont];
+br(p2 RS projpair_p_cont RS cont_fun RS id_comp RS subst)1;
+br(p1 RS projpair_eq RS subst)1;
+by (rtac cpo_trans 1);
+brr(cpo_cf::prems)1;
+(* The following corresponds to EXISTS_TAC, non-trivial instantiation. *)
+by(res_inst_tac[("f","p O (e' O p')")]cont_cf 4);
+br(comp_assoc RS ssubst)1;
+brr(cpo_refl::cpo_cf::cont_cf::comp_mono::comp_pres_cont::(contl@prems))1;
+by(res_inst_tac[("P","%x. rel(cf(E,D),p O e' O p',x)")]
+ (p1 RS projpair_p_cont RS cont_fun RS comp_id RS subst)1);
+by (rtac comp_mono 1);
+brr(cpo_refl::cpo_cf::cont_cf::comp_mono::comp_pres_cont::id_cont::
+ projpair_rel::(contl@prems))1;
+val lemma1 = result();
+
+val prems = goal Limit.thy (* lemma2 *)
+ "[|cpo(D); cpo(E); projpair(D,E,e,p); projpair(D,E,e',p'); \
+\ rel(cf(E,D),p',p)|] ==> rel(cf(D,E),e,e')";
+val [_,_,p1,p2,_] = prems;
+val contl = [p1 RS projpair_e_cont,p1 RS projpair_p_cont,
+ p2 RS projpair_e_cont,p2 RS projpair_p_cont];
+br(p1 RS projpair_e_cont RS cont_fun RS comp_id RS subst)1;
+br(p2 RS projpair_eq RS subst)1;
+by (rtac cpo_trans 1);
+brr(cpo_cf::prems)1;
+by(res_inst_tac[("f","(e O p) O e'")]cont_cf 4);
+br(comp_assoc RS ssubst)1;
+brr((cpo_cf RS cpo_refl)::cont_cf::comp_mono::comp_pres_cont::(contl@prems))1;
+by(res_inst_tac[("P","%x. rel(cf(D,E),(e O p) O e',x)")]
+ (p2 RS projpair_e_cont RS cont_fun RS id_comp RS subst)1);
+brr((cpo_cf RS cpo_refl)::cont_cf::comp_mono::id_cont::comp_pres_cont::projpair_rel::
+ (contl@prems))1;
+val lemma2 = result();
+
+val prems = goal Limit.thy (* projpair_unique *)
+ "[|cpo(D); cpo(E); projpair(D,E,e,p); projpair(D,E,e',p')|] ==> \
+\ (e=e')<->(p=p')";
+val [_,_,p1,p2] = prems;
+val contl = [p1 RS projpair_e_cont,p1 RS projpair_p_cont,
+ p2 RS projpair_e_cont,p2 RS projpair_p_cont];
+by (rtac iffI 1);
+by (rtac cpo_antisym 1);
+by (rtac lemma1 2);
+(* First some existentials are instantiated. *)
+by (resolve_tac prems 4);
+by (resolve_tac prems 4);
+by(asm_simp_tac FOL_ss 4);
+brr(cpo_cf::cpo_refl::cont_cf::projpair_e_cont::prems)1;
+by (rtac lemma1 1);
+brr prems 1;
+by(asm_simp_tac FOL_ss 1);
+brr(cpo_cf::cpo_refl::cont_cf::(contl @ prems))1;
+by (rtac cpo_antisym 1);
+by (rtac lemma2 2);
+(* First some existentials are instantiated. *)
+by (resolve_tac prems 4);
+by (resolve_tac prems 4);
+by(asm_simp_tac FOL_ss 4);
+brr(cpo_cf::cpo_refl::cont_cf::projpair_p_cont::prems)1;
+by (rtac lemma2 1);
+brr prems 1;
+by(asm_simp_tac FOL_ss 1);
+brr(cpo_cf::cpo_refl::cont_cf::(contl @ prems))1;
+val projpair_unique = result();
+
+(* Slightly different, more asms, since THE chooses the unique element. *)
+
+val prems = goalw Limit.thy [emb_def,Rp_def] (* embRp *)
+ "[|emb(D,E,e); cpo(D); cpo(E)|] ==> projpair(D,E,e,Rp(D,E,e))";
+by (rtac theI2 1);
+by (assume_tac 2);
+by (rtac ((hd prems) RS exE) 1);
+by (rtac ex1I 1);
+by (assume_tac 1);
+br(projpair_unique RS iffD1)1;
+by (assume_tac 3); (* To instantiate variables. *)
+brr (refl::prems) 1;
+val embRp = result();
+
+val embI = prove_goalw Limit.thy [emb_def]
+ "!!x. projpair(D,E,e,p) ==> emb(D,E,e)"
+ (fn prems => [fast_tac FOL_cs 1]);
+
+val prems = goal Limit.thy (* Rp_unique *)
+ "[|projpair(D,E,e,p); cpo(D); cpo(E)|] ==> Rp(D,E,e) = p";
+br(projpair_unique RS iffD1)1;
+by (rtac embRp 3); (* To instantiate variables. *)
+brr (embI::refl::prems) 1;
+val Rp_unique = result();
+
+val emb_cont = prove_goalw Limit.thy [emb_def]
+ "emb(D,E,e) ==> e:cont(D,E)"
+ (fn prems => [rtac(hd prems RS exE)1,rtac projpair_e_cont 1,atac 1]);
+
+(* The following three theorems have cpo asms due to THE (uniqueness). *)
+
+val Rp_cont = embRp RS projpair_p_cont;
+val embRp_eq = embRp RS projpair_eq;
+val embRp_rel = embRp RS projpair_rel;
+
+val id_apply = prove_goalw Perm.thy [id_def]
+ "!!z. x:A ==> id(A)`x = x"
+ (fn prems => [asm_simp_tac ZF_ss 1]);
+
+val prems = goal Limit.thy (* embRp_eq_thm *)
+ "[|emb(D,E,e); x:set(D); cpo(D); cpo(E)|] ==> Rp(D,E,e)`(e`x) = x";
+br(comp_fun_apply RS subst)1;
+brr(Rp_cont::emb_cont::cont_fun::prems)1;
+br(embRp_eq RS ssubst)1;
+brr(id_apply::prems)1;
+val embRp_eq_thm = result();
+
+
+(*----------------------------------------------------------------------*)
+(* The identity embedding. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [projpair_def] (* projpair_id *)
+ "cpo(D) ==> projpair(D,D,id(set(D)),id(set(D)))";
+by(safe_tac lemmas_cs);
+brr(id_cont::id_comp::id_type::prems)1;
+by (rtac (id_comp RS ssubst) 1); (* Matches almost anything *)
+brr(id_cont::id_type::cpo_refl::cpo_cf::cont_cf::prems)1;
+val projpair_id = result();
+
+val prems = goal Limit.thy (* emb_id *)
+ "cpo(D) ==> emb(D,D,id(set(D)))";
+brr(embI::projpair_id::prems)1;
+val emb_id = result();
+
+val prems = goal Limit.thy (* Rp_id *)
+ "cpo(D) ==> Rp(D,D,id(set(D))) = id(set(D))";
+brr(Rp_unique::projpair_id::prems)1;
+val Rp_id = result();
+
+(*----------------------------------------------------------------------*)
+(* Composition preserves embeddings. *)
+(*----------------------------------------------------------------------*)
+
+(* Considerably shorter, only partly due to a simpler comp_assoc. *)
+(* Proof in HOL-ST: 70 lines (minus 14 due to comp_assoc complication). *)
+(* Proof in Isa/ZF: 23 lines (compared to 56: 60% reduction). *)
+
+val prems = goalw Limit.thy [projpair_def] (* lemma *)
+ "[|emb(D,D',e); emb(D',E,e'); cpo(D); cpo(D'); cpo(E)|] ==> \
+\ projpair(D,E,e' O e,(Rp(D,D',e)) O (Rp(D',E,e')))";
+by(safe_tac lemmas_cs);
+brr(comp_pres_cont::Rp_cont::emb_cont::prems)1;
+br(comp_assoc RS subst)1;
+by(res_inst_tac[("t1","e'")](comp_assoc RS ssubst)1);
+br(embRp_eq RS ssubst)1; (* Matches everything due to subst/ssubst. *)
+brr prems 1;
+br(comp_id RS ssubst)1;
+brr(cont_fun::Rp_cont::embRp_eq::prems)1;
+br(comp_assoc RS subst)1;
+by(res_inst_tac[("t1","Rp(D,D',e)")](comp_assoc RS ssubst)1);
+by (rtac cpo_trans 1);
+brr(cpo_cf::prems)1;
+by (rtac comp_mono 1);
+by (rtac cpo_refl 6);
+brr(cont_cf::Rp_cont::prems)7;
+brr(cpo_cf::prems)6;
+by (rtac comp_mono 5);
+brr(embRp_rel::prems)10;
+brr((cpo_cf RS cpo_refl)::cont_cf::Rp_cont::prems)9;
+br(comp_id RS ssubst)10;
+by (rtac embRp_rel 11);
+(* There are 16 subgoals at this point. All are proved immediately by: *)
+brr(comp_pres_cont::Rp_cont::id_cont::emb_cont::cont_fun::cont_cf::prems)1;
+val lemma = result();
+
+(* The use of RS is great in places like the following, both ugly in HOL. *)
+
+val emb_comp = lemma RS embI;
+val Rp_comp = lemma RS Rp_unique;
+
+(*----------------------------------------------------------------------*)
+(* Infinite cartesian product. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [set_def,iprod_def] (* iprodI *)
+ "!!z. x:(PROD n:nat. set(DD`n)) ==> x:set(iprod(DD))";
+by(asm_full_simp_tac ZF_ss 1);
+val iprodI = result();
+
+(* Proof with non-reflexive relation approach:
+by (rtac CollectI 1);
+by (rtac domainI 1);
+by (rtac CollectI 1);
+by(simp_tac(ZF_ss addsimps prems)1);
+by (rtac (hd prems) 1);
+by(simp_tac ZF_ss 1);
+by (rtac ballI 1);
+bd((hd prems) RS apply_type)1;
+by (etac CollectE 1);
+by (assume_tac 1);
+by (rtac rel_I 1);
+by (rtac CollectI 1);
+by(fast_tac(ZF_cs addSIs prems)1);
+by (rtac ballI 1);
+by(simp_tac ZF_ss 1);
+bd((hd prems) RS apply_type)1;
+by (etac CollectE 1);
+by (assume_tac 1);
+*)
+
+val prems = goalw Limit.thy [set_def,iprod_def] (* iprodE *)
+ "!!z. x:set(iprod(DD)) ==> x:(PROD n:nat. set(DD`n))";
+by(asm_full_simp_tac ZF_ss 1);
+val iprodE = result();
+
+(* Contains typing conditions in contrast to HOL-ST *)
+
+val prems = goalw Limit.thy [iprod_def] (* rel_iprodI *)
+ "[|!!n. n:nat ==> rel(DD`n,f`n,g`n); f:(PROD n:nat. set(DD`n)); \
+\ g:(PROD n:nat. set(DD`n))|] ==> rel(iprod(DD),f,g)";
+by (rtac rel_I 1);
+by(simp_tac ZF_ss 1);
+by(safe_tac lemmas_cs);
+brr prems 1;
+val rel_iprodI = result();
+
+val prems = goalw Limit.thy [iprod_def] (* rel_iprodE *)
+ "[|rel(iprod(DD),f,g); n:nat|] ==> rel(DD`n,f`n,g`n)";
+by(cut_facts_tac[hd prems RS rel_E]1);
+by(asm_full_simp_tac ZF_ss 1);
+by(safe_tac lemmas_cs);
+by (etac bspec 1);
+by (resolve_tac prems 1);
+val rel_iprodE = result();
+
+(* Some special theorems like dProdApIn_cpo and other `_cpo'
+ probably not needed in Isabelle, wait and see. *)
+
+val prems = goalw Limit.thy [chain_def] (* chain_iprod *)
+ "[|chain(iprod(DD),X);!!n. n:nat ==> cpo(DD`n); n:nat|] ==> \
+\ chain(DD`n,lam m:nat.X`m`n)";
+by(safe_tac lemmas_cs);
+by (rtac lam_type 1);
+by (rtac apply_type 1);
+by (rtac iprodE 1);
+be(hd prems RS conjunct1 RS apply_type)1;
+by (resolve_tac prems 1);
+by(asm_simp_tac(arith_ss addsimps prems)1);
+by (rtac rel_iprodE 1);
+by(asm_simp_tac (arith_ss addsimps prems) 1);
+by (resolve_tac prems 1);
+val chain_iprod = result();
+
+val prems = goalw Limit.thy [islub_def,isub_def] (* islub_iprod *)
+ "[|chain(iprod(DD),X);!!n. n:nat ==> cpo(DD`n)|] ==> \
+\ islub(iprod(DD),X,lam n:nat. lub(DD`n,lam m:nat.X`m`n))";
+by(safe_tac lemmas_cs);
+by (rtac iprodI 1);
+by (rtac lam_type 1);
+brr((chain_iprod RS cpo_lub RS islub_in)::prems)1;
+by (rtac rel_iprodI 1);
+by(asm_simp_tac ZF_ss 1);
+(* Here, HOL resolution is handy, Isabelle resolution bad. *)
+by(res_inst_tac[("P","%t. rel(DD`na,t,lub(DD`na,lam x:nat. X`x`na))"),
+ ("b1","%n. X`n`na")](beta RS subst)1);
+brr((chain_iprod RS cpo_lub RS islub_ub)::iprodE::chain_in::prems)1;
+brr(iprodI::lam_type::(chain_iprod RS cpo_lub RS islub_in)::prems)1;
+by (rtac rel_iprodI 1);
+by(asm_simp_tac ZF_ss 1);
+brr(islub_least::(chain_iprod RS cpo_lub)::prems)1;
+by(rewtac isub_def);
+by(safe_tac lemmas_cs);
+be(iprodE RS apply_type)1;
+by (assume_tac 1);
+by(asm_simp_tac ZF_ss 1);
+by (dtac bspec 1);
+by (etac rel_iprodE 2);
+brr(lam_type::(chain_iprod RS cpo_lub RS islub_in)::iprodE::prems)1;
+val islub_iprod = result();
+
+val prems = goal Limit.thy (* cpo_iprod *)
+ "(!!n. n:nat ==> cpo(DD`n)) ==> cpo(iprod(DD))";
+brr(cpoI::poI::[])1;
+by (rtac rel_iprodI 1); (* not repeated: want to solve 1 and leave 2 unchanged *)
+brr(cpo_refl::(iprodE RS apply_type)::iprodE::prems)1;
+by (rtac rel_iprodI 1);
+by (dtac rel_iprodE 1);
+by (dtac rel_iprodE 2);
+brr(cpo_trans::(iprodE RS apply_type)::iprodE::prems)1;
+by (rtac fun_extension 1);
+brr(cpo_antisym::rel_iprodE::(iprodE RS apply_type)::iprodE::prems)1;
+brr(islub_iprod::prems)1;
+val cpo_iprod = result();
+
+val prems = goalw Limit.thy [islub_def,isub_def] (* lub_iprod *)
+ "[|chain(iprod(DD),X);!!n. n:nat ==> cpo(DD`n)|] ==> \
+\ lub(iprod(DD),X) = (lam n:nat. lub(DD`n,lam m:nat.X`m`n))";
+brr((cpo_lub RS islub_unique)::islub_iprod::cpo_iprod::prems)1;
+val lub_iprod = result();
+
+(*----------------------------------------------------------------------*)
+(* The notion of subcpo. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [subcpo_def] (* subcpoI *)
+ "[|set(D)<=set(E); \
+\ !!x y. [|x:set(D); y:set(D)|] ==> rel(D,x,y)<->rel(E,x,y); \
+\ !!X. chain(D,X) ==> lub(E,X) : set(D)|] ==> subcpo(D,E)";
+by(safe_tac lemmas_cs);
+by(asm_full_simp_tac(ZF_ss addsimps prems)2);
+by(asm_simp_tac(ZF_ss addsimps prems)2);
+brr(prems@[subsetD])1;
+val subcpoI = result();
+
+val subcpo_subset = prove_goalw Limit.thy [subcpo_def]
+ "!!x. subcpo(D,E) ==> set(D)<=set(E)"
+ (fn prems => [fast_tac FOL_cs 1]);
+
+val subcpo_rel_eq = prove_goalw Limit.thy [subcpo_def]
+ " [|subcpo(D,E); x:set(D); y:set(D)|] ==> rel(D,x,y)<->rel(E,x,y)"
+ (fn prems =>
+ [trr((hd prems RS conjunct2 RS conjunct1 RS bspec RS bspec)::prems)1]);
+
+val subcpo_relD1 = subcpo_rel_eq RS iffD1;
+val subcpo_relD2 = subcpo_rel_eq RS iffD2;
+
+val subcpo_lub = prove_goalw Limit.thy [subcpo_def]
+ "[|subcpo(D,E); chain(D,X)|] ==> lub(E,X) : set(D)"
+ (fn prems =>
+ [rtac(hd prems RS conjunct2 RS conjunct2 RS spec RS impE) 1,trr prems 1]);
+
+val prems = goal Limit.thy (* chain_subcpo *)
+ "[|subcpo(D,E); chain(D,X)|] ==> chain(E,X)";
+by (rtac chainI 1);
+by (rtac Pi_type 1);
+brr(chain_fun::prems)1;
+by (rtac subsetD 1);
+brr(subcpo_subset::chain_in::(hd prems RS subcpo_relD1)::nat_succI::
+ chain_rel::prems)1;
+val chain_subcpo = result();
+
+val prems = goal Limit.thy (* ub_subcpo *)
+ "[|subcpo(D,E); chain(D,X); isub(D,X,x)|] ==> isub(E,X,x)";
+brr(isubI::(hd prems RS subcpo_subset RS subsetD)::
+ (hd prems RS subcpo_relD1)::prems)1;
+brr(isubD1::prems)1;
+brr((hd prems RS subcpo_relD1)::chain_in::isubD1::isubD2::prems)1;
+val ub_subcpo = result();
+
+(* STRIP_TAC and HOL resolution is efficient sometimes. The following
+ theorem is proved easily in HOL without intro and elim rules. *)
+
+val prems = goal Limit.thy (* islub_subcpo *)
+ "[|subcpo(D,E); cpo(E); chain(D,X)|] ==> islub(D,X,lub(E,X))";
+brr[islubI,isubI]1;
+brr(subcpo_lub::(hd prems RS subcpo_relD2)::chain_in::islub_ub::islub_least::
+ cpo_lub::(hd prems RS chain_subcpo)::isubD1::(hd prems RS ub_subcpo)::
+ prems)1;
+val islub_subcpo = result();
+
+val prems = goal Limit.thy (* subcpo_cpo *)
+ "[|subcpo(D,E); cpo(E)|] ==> cpo(D)";
+brr[cpoI,poI]1;
+(* Changing the order of the assumptions, otherwise full_simp doesn't work. *)
+by(asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq])1);
+brr(cpo_refl::(hd prems RS subcpo_subset RS subsetD)::prems)1;
+bd(imp_refl RS mp)1;
+bd(imp_refl RS mp)1;
+by(asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq])1);
+brr(cpo_trans::(hd prems RS subcpo_subset RS subsetD)::prems)1;
+(* Changing the order of the assumptions, otherwise full_simp doesn't work. *)
+bd(imp_refl RS mp)1;
+bd(imp_refl RS mp)1;
+by(asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq])1);
+brr(cpo_antisym::(hd prems RS subcpo_subset RS subsetD)::prems)1;
+brr(islub_subcpo::prems)1;
+val subcpo_cpo = result();
+
+val prems = goal Limit.thy (* lub_subcpo *)
+ "[|subcpo(D,E); cpo(E); chain(D,X)|] ==> lub(D,X) = lub(E,X)";
+brr((cpo_lub RS islub_unique)::islub_subcpo::(hd prems RS subcpo_cpo)::prems)1;
+val lub_subcpo = result();
+
+(*----------------------------------------------------------------------*)
+(* Making subcpos using mkcpo. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [set_def,mkcpo_def] (* mkcpoI *)
+ "!!z. [|x:set(D); P(x)|] ==> x:set(mkcpo(D,P))";
+by(simp_tac ZF_ss 1);
+brr(conjI::prems)1;
+val mkcpoI = result();
+
+(* Old proof where cpos are non-reflexive relations.
+by(rewtac set_def); (* Annoying, cannot just rewrite once. *)
+by (rtac CollectI 1);
+by (rtac domainI 1);
+by (rtac CollectI 1);
+(* Now, work on subgoal 2 (and 3) to instantiate unknown. *)
+by(simp_tac ZF_ss 2);
+by (rtac conjI 2);
+by (rtac conjI 3);
+by (resolve_tac prems 3);
+by(simp_tac(ZF_ss addsimps [rewrite_rule[set_def](hd prems)])1);
+by (resolve_tac prems 1);
+by (rtac cpo_refl 1);
+by (resolve_tac prems 1);
+by (rtac rel_I 1);
+by (rtac CollectI 1);
+by(fast_tac(ZF_cs addSIs [rewrite_rule[set_def](hd prems)])1);
+by(simp_tac ZF_ss 1);
+brr(conjI::cpo_refl::prems)1;
+*)
+
+val prems = goalw Limit.thy [set_def,mkcpo_def] (* mkcpoD1 *)
+ "!!z. x:set(mkcpo(D,P))==> x:set(D)";
+by(asm_full_simp_tac ZF_ss 1);
+val mkcpoD1 = result();
+
+val prems = goalw Limit.thy [set_def,mkcpo_def] (* mkcpoD2 *)
+ "!!z. x:set(mkcpo(D,P))==> P(x)";
+by(asm_full_simp_tac ZF_ss 1);
+val mkcpoD2 = result();
+
+val prems = goalw Limit.thy [rel_def,mkcpo_def] (* rel_mkcpoE *)
+ "!!a. rel(mkcpo(D,P),x,y) ==> rel(D,x,y)";
+by(asm_full_simp_tac ZF_ss 1);
+val rel_mkcpoE = result();
+
+val rel_mkcpo = prove_goalw Limit.thy [mkcpo_def,rel_def,set_def]
+ "!!z. [|x:set(D); y:set(D)|] ==> rel(mkcpo(D,P),x,y) <-> rel(D,x,y)"
+ (fn prems => [asm_simp_tac ZF_ss 1]);
+
+(* The HOL proof is simpler, problems due to cpos as purely in ZF. *)
+(* And chains as set functions. *)
+
+val prems = goal Limit.thy (* chain_mkcpo *)
+ "chain(mkcpo(D,P),X) ==> chain(D,X)";
+by (rtac chainI 1);
+(*---begin additional---*)
+by (rtac Pi_type 1);
+brr(chain_fun::prems)1;
+brr((chain_in RS mkcpoD1)::prems)1;
+(*---end additional---*)
+br(rel_mkcpo RS iffD1)1;
+(*---begin additional---*)
+by (rtac mkcpoD1 1);
+by (rtac mkcpoD1 2);
+brr(chain_in::nat_succI::prems)1;
+(*---end additional---*)
+brr(chain_rel::prems)1;
+val chain_mkcpo = result();
+
+val prems = goal Limit.thy (* subcpo_mkcpo *)
+ "[|!!X. chain(mkcpo(D,P),X) ==> P(lub(D,X)); cpo(D)|] ==> \
+\ subcpo(mkcpo(D,P),D)";
+brr(subcpoI::subsetI::prems)1;
+by (rtac rel_mkcpo 2);
+by(REPEAT(etac mkcpoD1 1));
+brr(mkcpoI::(cpo_lub RS islub_in)::chain_mkcpo::prems)1;
+val subcpo_mkcpo = result();
+
+(*----------------------------------------------------------------------*)
+(* Embedding projection chains of cpos. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [emb_chain_def] (* emb_chainI *)
+ "[|!!n. n:nat ==> cpo(DD`n); \
+\ !!n. n:nat ==> emb(DD`n,DD`succ(n),ee`n)|] ==> emb_chain(DD,ee)";
+by(safe_tac lemmas_cs);
+brr prems 1;
+val emb_chainI = result();
+
+val emb_chain_cpo = prove_goalw Limit.thy [emb_chain_def]
+ "!!x. [|emb_chain(DD,ee); n:nat|] ==> cpo(DD`n)"
+ (fn prems => [fast_tac ZF_cs 1]);
+
+val emb_chain_emb = prove_goalw Limit.thy [emb_chain_def]
+ "!!x. [|emb_chain(DD,ee); n:nat|] ==> emb(DD`n,DD`succ(n),ee`n)"
+ (fn prems => [fast_tac ZF_cs 1]);
+
+(*----------------------------------------------------------------------*)
+(* Dinf, the inverse Limit. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [Dinf_def] (* DinfI *)
+ "[|x:(PROD n:nat. set(DD`n)); \
+\ !!n. n:nat ==> Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n|] ==> \
+\ x:set(Dinf(DD,ee))";
+brr(mkcpoI::iprodI::ballI::prems)1;
+val DinfI = result();
+
+val prems = goalw Limit.thy [Dinf_def] (* DinfD1 *)
+ "x:set(Dinf(DD,ee)) ==> x:(PROD n:nat. set(DD`n))";
+by (rtac iprodE 1);
+by (rtac mkcpoD1 1);
+by (resolve_tac prems 1);
+val DinfD1 = result();
+val Dinf_prod = DinfD1;
+
+val prems = goalw Limit.thy [Dinf_def] (* DinfD2 *)
+ "[|x:set(Dinf(DD,ee)); n:nat|] ==> \
+\ Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n";
+by(asm_simp_tac(ZF_ss addsimps[(hd prems RS mkcpoD2),hd(tl prems)])1);
+val DinfD2 = result();
+val Dinf_eq = DinfD2;
+
+(* At first, rel_DinfI was stated too strongly, because rel_mkcpo was too:
+val prems = goalw Limit.thy [Dinf_def] (* rel_DinfI *)
+ "[|!!n. n:nat ==> rel(DD`n,x`n,y`n); \
+\ x:set(Dinf(DD,ee)); y:set(Dinf(DD,ee))|] ==> rel(Dinf(DD,ee),x,y)";
+br(rel_mkcpo RS iffD2)1;
+brr prems 1;
+brr(rel_iprodI::rewrite_rule[Dinf_def]DinfD1::prems)1;
+val rel_DinfI = result();
+*)
+
+val prems = goalw Limit.thy [Dinf_def] (* rel_DinfI *)
+ "[|!!n. n:nat ==> rel(DD`n,x`n,y`n); \
+\ x:(PROD n:nat. set(DD`n)); y:(PROD n:nat. set(DD`n))|] ==> \
+\ rel(Dinf(DD,ee),x,y)";
+br(rel_mkcpo RS iffD2)1;
+brr(rel_iprodI::iprodI::prems)1;
+val rel_DinfI = result();
+
+val prems = goalw Limit.thy [Dinf_def] (* rel_Dinf *)
+ "[|rel(Dinf(DD,ee),x,y); n:nat|] ==> rel(DD`n,x`n,y`n)";
+br(hd prems RS rel_mkcpoE RS rel_iprodE)1;
+by (resolve_tac prems 1);
+val rel_Dinf = result();
+
+val chain_Dinf = prove_goalw Limit.thy [Dinf_def]
+ "chain(Dinf(DD,ee),X) ==> chain(iprod(DD),X)"
+ (fn prems => [rtac(hd prems RS chain_mkcpo)1]);
+
+val prems = goalw Limit.thy [Dinf_def] (* subcpo_Dinf *)
+ "emb_chain(DD,ee) ==> subcpo(Dinf(DD,ee),iprod(DD))";
+by (rtac subcpo_mkcpo 1);
+by(fold_tac [Dinf_def]);
+by (rtac ballI 1);
+br(lub_iprod RS ssubst)1;
+brr(chain_Dinf::(hd prems RS emb_chain_cpo)::[])1;
+by(asm_simp_tac arith_ss 1);
+br(Rp_cont RS cont_lub RS ssubst)1;
+brr(emb_chain_cpo::emb_chain_emb::nat_succI::chain_iprod::chain_Dinf::prems)1;
+(* Useful simplification, ugly in HOL. *)
+by(asm_simp_tac(arith_ss addsimps(DinfD2::chain_in::[]))1);
+brr(cpo_iprod::emb_chain_cpo::prems)1;
+val subcpo_Dinf = result();
+
+(* Simple example of existential reasoning in Isabelle versus HOL. *)
+
+val prems = goal Limit.thy (* cpo_Dinf *)
+ "emb_chain(DD,ee) ==> cpo(Dinf(DD,ee))";
+by (rtac subcpo_cpo 1);
+brr(subcpo_Dinf::cpo_iprod::emb_chain_cpo::prems)1;;
+val cpo_Dinf = result();
+
+(* Again and again the proofs are much easier to WRITE in Isabelle, but
+ the proof steps are essentially the same (I think). *)
+
+val prems = goal Limit.thy (* lub_Dinf *)
+ "[|chain(Dinf(DD,ee),X); emb_chain(DD,ee)|] ==> \
+\ lub(Dinf(DD,ee),X) = (lam n:nat. lub(DD`n,lam m:nat. X`m`n))";
+br(subcpo_Dinf RS lub_subcpo RS ssubst)1;
+brr(cpo_iprod::emb_chain_cpo::lub_iprod::chain_Dinf::prems)1;
+val lub_Dinf = result();
+
+(*----------------------------------------------------------------------*)
+(* Generalising embedddings D_m -> D_{m+1} to embeddings D_m -> D_n, *)
+(* defined as eps(DD,ee,m,n), via e_less and e_gr. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [e_less_def] (* e_less_eq *)
+ "!!x. m:nat ==> e_less(DD,ee,m,m) = id(set(DD`m))";
+by(asm_simp_tac (arith_ss addsimps[diff_self_eq_0]) 1);
+val e_less_eq = result();
+
+(* ARITH_CONV proves the following in HOL. Would like something similar
+ in Isabelle/ZF. *)
+
+val prems = goalw Arith.thy [] (* lemma_succ_sub *)
+ "!!z. [|n:nat; m:nat|] ==> succ(m#+n)#-m = succ(n)";
+by(asm_simp_tac(arith_ss addsimps [add_succ_right RS sym,diff_add_inverse])1);
+val lemma_succ_sub = result();
+
+val prems = goalw Limit.thy [e_less_def] (* e_less_add *)
+ "!!x. [|m:nat; k:nat|] ==> \
+\ e_less(DD,ee,m,succ(m#+k)) = (ee`(m#+k))O(e_less(DD,ee,m,m#+k))";
+by(asm_simp_tac (arith_ss addsimps [lemma_succ_sub,diff_add_inverse]) 1);
+val e_less_add = result();
+
+(* Again, would like more theorems about arithmetic. *)
+(* Well, HOL has much better support and automation of natural numbers. *)
+
+val add1 = prove_goal Limit.thy
+ "!!x. n:nat ==> succ(n) = n #+ 1"
+ (fn prems =>
+ [asm_simp_tac (arith_ss addsimps[add_succ_right,add_0_right]) 1]);
+
+val prems = goal Limit.thy (* succ_sub1 *)
+ "x:nat ==> 0 < x --> succ(x#-1)=x";
+by(res_inst_tac[("n","x")]nat_induct 1);
+by (resolve_tac prems 1);
+by(fast_tac lt_cs 1);
+by(safe_tac lemmas_cs);
+by(asm_simp_tac arith_ss 1);
+by(asm_simp_tac arith_ss 1);
+val succ_sub1 = result();
+
+val prems = goal Limit.thy (* succ_le_pos *)
+ "[|m:nat; k:nat|] ==> succ(m) le m #+ k --> 0 < k";
+by(res_inst_tac[("n","m")]nat_induct 1);
+by (resolve_tac prems 1);
+by (rtac impI 1);
+by(asm_full_simp_tac(arith_ss addsimps prems)1);
+by(safe_tac lemmas_cs);
+by(asm_full_simp_tac(arith_ss addsimps prems)1); (* Surprise, surprise. *)
+val succ_le_pos = result();
+
+val prems = goal Limit.thy (* lemma_le_exists *)
+ "!!z. [|n:nat; m:nat|] ==> m le n --> (EX k:nat. n = m #+ k)";
+by(res_inst_tac[("n","m")]nat_induct 1);
+by (assume_tac 1);
+by(safe_tac lemmas_cs);
+by (rtac bexI 1);
+br(add_0 RS sym)1;
+by (assume_tac 1);
+by(asm_full_simp_tac arith_ss 1);
+(* Great, by luck I found lt_cs. Such cs's and ss's should be documented. *)
+by(fast_tac lt_cs 1);
+by(asm_simp_tac (arith_ss addsimps[add_succ_right RS sym]) 1);
+by (rtac bexI 1);
+br(succ_sub1 RS mp RS ssubst)1;
+(* Instantiation. *)
+by (rtac refl 3);
+by (assume_tac 1);
+br(succ_le_pos RS mp)1;
+by (assume_tac 3); (* Instantiation *)
+brr[]1;
+by(asm_simp_tac arith_ss 1);
+val lemma_le_exists = result();
+
+val prems = goal Limit.thy (* le_exists *)
+ "[|m le n;!!x. [|n=m#+x; x:nat|] ==> Q; m:nat; n:nat|] ==> Q";
+br(lemma_le_exists RS mp RS bexE)1;
+br(hd(tl prems))4;
+by (assume_tac 4);
+brr prems 1;
+val le_exists = result();
+
+val prems = goal Limit.thy (* e_less_le *)
+ "[|m le n; m:nat; n:nat|] ==> \
+\ e_less(DD,ee,m,succ(n)) = ee`n O e_less(DD,ee,m,n)";
+by (rtac le_exists 1);
+by (resolve_tac prems 1);
+by(asm_simp_tac(ZF_ss addsimps(e_less_add::prems))1);
+brr prems 1;
+val e_less_le = result();
+
+(* All theorems assume variables m and n are natural numbers. *)
+
+val prems = goal Limit.thy (* e_less_succ *)
+ "m:nat ==> e_less(DD,ee,m,succ(m)) = ee`m O id(set(DD`m))";
+by(asm_simp_tac(arith_ss addsimps(e_less_le::e_less_eq::prems))1);
+val e_less_succ = result();
+
+val prems = goal Limit.thy (* e_less_succ_emb *)
+ "[|!!n. n:nat ==> emb(DD`n,DD`succ(n),ee`n); m:nat|] ==> \
+\ e_less(DD,ee,m,succ(m)) = ee`m";
+by(asm_simp_tac(arith_ss addsimps(e_less_succ::prems))1);
+br(comp_id RS ssubst)1;
+brr(emb_cont::cont_fun::refl::prems)1;
+val e_less_succ_emb = result();
+
+(* Compare this proof with the HOL one, here we do type checking. *)
+(* In any case the one below was very easy to write. *)
+
+val prems = goal Limit.thy (* emb_e_less_add *)
+ "[|emb_chain(DD,ee); m:nat; k:nat|] ==> \
+\ emb(DD`m,DD`(m#+k),e_less(DD,ee,m,m#+k))";
+by(res_inst_tac[("n","k")]nat_induct 1);
+by (resolve_tac prems 1);
+by(asm_simp_tac(ZF_ss addsimps(add_0_right::e_less_eq::prems))1);
+brr(emb_id::emb_chain_cpo::prems)1;
+by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_less_add::prems))1);
+brr(emb_comp::emb_chain_emb::emb_chain_cpo::add_type::nat_succI::prems)1;
+val emb_e_less_add = result();
+
+val prems = goal Limit.thy (* emb_e_less *)
+ "[|m le n; emb_chain(DD,ee); m:nat; n:nat|] ==> \
+\ emb(DD`m,DD`n,e_less(DD,ee,m,n))";
+(* same proof as e_less_le *)
+by (rtac le_exists 1);
+by (resolve_tac prems 1);
+by(asm_simp_tac(ZF_ss addsimps(emb_e_less_add::prems))1);
+brr prems 1;
+val emb_e_less = result();
+
+val comp_mono_eq = prove_goal Limit.thy
+ "!!z.[|f=f'; g=g'|] ==> f O g = f' O g'"
+ (fn prems => [asm_simp_tac ZF_ss 1]);
+
+(* Typing, typing, typing, three irritating assumptions. Extra theorems
+ needed in proof, but no real difficulty. *)
+(* Note also the object-level implication for induction on k. This
+ must be removed later to allow the theorems to be used for simp.
+ Therefore this theorem is only a lemma. *)
+
+val prems = goal Limit.thy (* e_less_split_add_lemma *)
+ "[| emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ n le k --> \
+\ e_less(DD,ee,m,m#+k) = e_less(DD,ee,m#+n,m#+k) O e_less(DD,ee,m,m#+n)";
+by(res_inst_tac[("n","k")]nat_induct 1);
+by (resolve_tac prems 1);
+by (rtac impI 1);
+by(asm_full_simp_tac(ZF_ss addsimps
+ (le0_iff::add_0_right::e_less_eq::(id_type RS id_comp)::prems))1);
+by(asm_simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (rtac impI 1);
+by (etac disjE 1);
+by (etac impE 1);
+by (assume_tac 1);
+by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_less_add::
+ add_type::nat_succI::prems))1);
+(* Again and again, simplification is a pain. When does it work, when not? *)
+br(e_less_le RS ssubst)1;
+brr(add_le_mono::nat_le_refl::add_type::nat_succI::prems)1;
+br(comp_assoc RS ssubst)1;
+brr(comp_mono_eq::refl::[])1;
+(* by(asm_simp_tac ZF_ss 1); *)
+by(asm_simp_tac(ZF_ss addsimps(e_less_eq::add_type::nat_succI::prems))1);
+br(id_comp RS ssubst)1; (* simp cannot unify/inst right, use brr below(?). *)
+brr((emb_e_less_add RS emb_cont RS cont_fun)::refl::nat_succI::prems)1;
+val e_less_split_add_lemma = result();
+
+val e_less_split_add = prove_goal Limit.thy
+ "[| n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ e_less(DD,ee,m,m#+k) = e_less(DD,ee,m#+n,m#+k) O e_less(DD,ee,m,m#+n)"
+ (fn prems => [trr((e_less_split_add_lemma RS mp)::prems)1]);
+
+val prems = goalw Limit.thy [e_gr_def] (* e_gr_eq *)
+ "!!x. m:nat ==> e_gr(DD,ee,m,m) = id(set(DD`m))";
+by(asm_simp_tac (arith_ss addsimps[diff_self_eq_0]) 1);
+val e_gr_eq = result();
+
+val prems = goalw Limit.thy [e_gr_def] (* e_gr_add *)
+ "!!x. [|n:nat; k:nat|] ==> \
+\ e_gr(DD,ee,succ(n#+k),n) = \
+\ e_gr(DD,ee,n#+k,n) O Rp(DD`(n#+k),DD`succ(n#+k),ee`(n#+k))";
+by(asm_simp_tac (arith_ss addsimps [lemma_succ_sub,diff_add_inverse]) 1);
+val e_gr_add = result();
+
+val prems = goal Limit.thy (* e_gr_le *)
+ "[|n le m; m:nat; n:nat|] ==> \
+\ e_gr(DD,ee,succ(m),n) = e_gr(DD,ee,m,n) O Rp(DD`m,DD`succ(m),ee`m)";
+by (rtac le_exists 1);
+by (resolve_tac prems 1);
+by(asm_simp_tac(ZF_ss addsimps(e_gr_add::prems))1);
+brr prems 1;
+val e_gr_le = result();
+
+val prems = goal Limit.thy (* e_gr_succ *)
+ "m:nat ==> \
+\ e_gr(DD,ee,succ(m),m) = id(set(DD`m)) O Rp(DD`m,DD`succ(m),ee`m)";
+by(asm_simp_tac(arith_ss addsimps(e_gr_le::e_gr_eq::prems))1);
+val e_gr_succ = result();
+
+(* Cpo asm's due to THE uniqueness. *)
+
+val prems = goal Limit.thy (* e_gr_succ_emb *)
+ "[|emb_chain(DD,ee); m:nat|] ==> \
+\ e_gr(DD,ee,succ(m),m) = Rp(DD`m,DD`succ(m),ee`m)";
+by(asm_simp_tac(arith_ss addsimps(e_gr_succ::prems))1);
+br(id_comp RS ssubst)1;
+brr(Rp_cont::cont_fun::refl::emb_chain_cpo::emb_chain_emb::nat_succI::prems)1;
+val e_gr_succ_emb = result();
+
+val prems = goal Limit.thy (* e_gr_fun_add *)
+ "[|emb_chain(DD,ee); n:nat; k:nat|] ==> \
+\ e_gr(DD,ee,n#+k,n): set(DD`(n#+k))->set(DD`n)";
+by(res_inst_tac[("n","k")]nat_induct 1);
+by (resolve_tac prems 1);
+by(asm_simp_tac(ZF_ss addsimps(add_0_right::e_gr_eq::id_type::prems))1);
+by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_add::prems))1);
+brr(comp_fun::Rp_cont::cont_fun::emb_chain_emb::emb_chain_cpo::add_type::
+ nat_succI::prems)1;
+val e_gr_fun_add = result();
+
+val prems = goal Limit.thy (* e_gr_fun *)
+ "[|n le m; emb_chain(DD,ee); m:nat; n:nat|] ==> \
+\ e_gr(DD,ee,m,n): set(DD`m)->set(DD`n)";
+by (rtac le_exists 1);
+by (resolve_tac prems 1);
+by(asm_simp_tac(ZF_ss addsimps(e_gr_fun_add::prems))1);
+brr prems 1;
+val e_gr_fun = result();
+
+val prems = goal Limit.thy (* e_gr_split_add_lemma *)
+ "[| emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ m le k --> \
+\ e_gr(DD,ee,n#+k,n) = e_gr(DD,ee,n#+m,n) O e_gr(DD,ee,n#+k,n#+m)";
+by(res_inst_tac[("n","k")]nat_induct 1);
+by (resolve_tac prems 1);
+by (rtac impI 1);
+by(asm_full_simp_tac(ZF_ss addsimps
+ (le0_iff::add_0_right::e_gr_eq::(id_type RS comp_id)::prems))1);
+by(asm_simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (rtac impI 1);
+by (etac disjE 1);
+by (etac impE 1);
+by (assume_tac 1);
+by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_add::
+ add_type::nat_succI::prems))1);
+(* Again and again, simplification is a pain. When does it work, when not? *)
+br(e_gr_le RS ssubst)1;
+brr(add_le_mono::nat_le_refl::add_type::nat_succI::prems)1;
+br(comp_assoc RS ssubst)1;
+brr(comp_mono_eq::refl::[])1;
+(* New direct subgoal *)
+by(asm_simp_tac(ZF_ss addsimps(e_gr_eq::add_type::nat_succI::prems))1);
+br(comp_id RS ssubst)1; (* simp cannot unify/inst right, use brr below(?). *)
+brr(e_gr_fun::add_type::refl::add_le_self::nat_succI::prems)1;
+val e_gr_split_add_lemma = result();
+
+val e_gr_split_add = prove_goal Limit.thy
+ "[| m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ e_gr(DD,ee,n#+k,n) = e_gr(DD,ee,n#+m,n) O e_gr(DD,ee,n#+k,n#+m)"
+ (fn prems => [trr((e_gr_split_add_lemma RS mp)::prems)1]);
+
+val e_less_cont = prove_goal Limit.thy
+ "[|m le n; emb_chain(DD,ee); m:nat; n:nat|] ==> \
+\ e_less(DD,ee,m,n):cont(DD`m,DD`n)"
+ (fn prems => [trr(emb_cont::emb_e_less::prems)1]);
+
+val prems = goal Limit.thy (* e_gr_cont_lemma *)
+ "[|emb_chain(DD,ee); m:nat; n:nat|] ==> \
+\ n le m --> e_gr(DD,ee,m,n):cont(DD`m,DD`n)";
+by(res_inst_tac[("n","m")]nat_induct 1);
+by (resolve_tac prems 1);
+by(asm_full_simp_tac(ZF_ss addsimps
+ (le0_iff::e_gr_eq::nat_0I::prems))1);
+brr(impI::id_cont::emb_chain_cpo::nat_0I::prems)1;
+by(asm_full_simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (etac disjE 1);
+by (etac impE 1);
+by (assume_tac 1);
+by(asm_simp_tac(ZF_ss addsimps(e_gr_le::prems))1);
+brr(comp_pres_cont::Rp_cont::emb_chain_cpo::emb_chain_emb::nat_succI::prems)1;
+by(asm_simp_tac(ZF_ss addsimps(e_gr_eq::nat_succI::prems))1);
+brr(id_cont::emb_chain_cpo::nat_succI::prems)1;
+val e_gr_cont_lemma = result();
+
+val prems = goal Limit.thy (* e_gr_cont *)
+ "[|n le m; emb_chain(DD,ee); m:nat; n:nat|] ==> \
+\ e_gr(DD,ee,m,n):cont(DD`m,DD`n)";
+brr((e_gr_cont_lemma RS mp)::prems)1;
+val e_gr_cont = result();
+
+(* Considerably shorter.... 57 against 26 *)
+
+val prems = goal Limit.thy (* e_less_e_gr_split_add *)
+ "[|n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ e_less(DD,ee,m,m#+n) = e_gr(DD,ee,m#+k,m#+n) O e_less(DD,ee,m,m#+k)";
+(* Use mp to prepare for induction. *)
+by (rtac mp 1);
+by (resolve_tac prems 2);
+by(res_inst_tac[("n","k")]nat_induct 1);
+by (resolve_tac prems 1);
+by(asm_full_simp_tac(ZF_ss addsimps
+ (le0_iff::add_0_right::e_gr_eq::e_less_eq::(id_type RS id_comp)::prems))1);
+by(simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (rtac impI 1);
+by (etac disjE 1);
+by (etac impE 1);
+by (assume_tac 1);
+by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_le::e_less_le::
+ add_le_self::nat_le_refl::add_le_mono::add_type::prems))1);
+br(comp_assoc RS ssubst)1;
+by(res_inst_tac[("s1","ee`(m#+x)")](comp_assoc RS subst)1);
+br(embRp_eq RS ssubst)1;
+brr(emb_chain_emb::add_type::emb_chain_cpo::nat_succI::prems)1;
+br(id_comp RS ssubst)1;
+brr((e_less_cont RS cont_fun)::add_type::add_le_self::refl::prems)1;
+by(asm_full_simp_tac(ZF_ss addsimps(e_gr_eq::nat_succI::add_type::prems))1);
+br(id_comp RS ssubst)1;
+brr((e_less_cont RS cont_fun)::add_type::nat_succI::add_le_self::refl::prems)1;
+val e_less_e_gr_split_add = result();
+
+(* Again considerably shorter, and easy to obtain from the previous thm. *)
+
+val prems = goal Limit.thy (* e_gr_e_less_split_add *)
+ "[|m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ e_gr(DD,ee,n#+m,n) = e_gr(DD,ee,n#+k,n) O e_less(DD,ee,n#+m,n#+k)";
+(* Use mp to prepare for induction. *)
+by (rtac mp 1);
+by (resolve_tac prems 2);
+by(res_inst_tac[("n","k")]nat_induct 1);
+by (resolve_tac prems 1);
+by(asm_full_simp_tac(arith_ss addsimps
+ (add_0_right::e_gr_eq::e_less_eq::(id_type RS id_comp)::prems))1);
+by(simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (rtac impI 1);
+by (etac disjE 1);
+by (etac impE 1);
+by (assume_tac 1);
+by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_le::e_less_le::
+ add_le_self::nat_le_refl::add_le_mono::add_type::prems))1);
+br(comp_assoc RS ssubst)1;
+by(res_inst_tac[("s1","ee`(n#+x)")](comp_assoc RS subst)1);
+br(embRp_eq RS ssubst)1;
+brr(emb_chain_emb::add_type::emb_chain_cpo::nat_succI::prems)1;
+br(id_comp RS ssubst)1;
+brr((e_less_cont RS cont_fun)::add_type::add_le_mono::nat_le_refl::refl::
+ prems)1;
+by(asm_full_simp_tac(ZF_ss addsimps(e_less_eq::nat_succI::add_type::prems))1);
+br(comp_id RS ssubst)1;
+brr((e_gr_cont RS cont_fun)::add_type::nat_succI::add_le_self::refl::prems)1;
+val e_gr_e_less_split_add = result();
+
+val prems = goalw Limit.thy [eps_def] (* emb_eps *)
+ "[|m le n; emb_chain(DD,ee); m:nat; n:nat|] ==> \
+\ emb(DD`m,DD`n,eps(DD,ee,m,n))";
+by(asm_simp_tac(ZF_ss addsimps prems)1);
+brr(emb_e_less::prems)1;
+val emb_eps = result();
+
+val prems = goalw Limit.thy [eps_def] (* eps_fun *)
+ "[|emb_chain(DD,ee); m:nat; n:nat|] ==> \
+\ eps(DD,ee,m,n): set(DD`m)->set(DD`n)";
+br(expand_if RS iffD2)1;
+by(safe_tac lemmas_cs);
+brr((e_less_cont RS cont_fun)::prems)1;
+brr((not_le_iff_lt RS iffD1 RS leI)::e_gr_fun::nat_into_Ord::prems)1;
+val eps_fun = result();
+
+val eps_id = prove_goalw Limit.thy [eps_def]
+ "n:nat ==> eps(DD,ee,n,n) = id(set(DD`n))"
+ (fn prems => [simp_tac(ZF_ss addsimps(e_less_eq::nat_le_refl::prems))1]);
+
+val eps_e_less_add = prove_goalw Limit.thy [eps_def]
+ "[|m:nat; n:nat|] ==> eps(DD,ee,m,m#+n) = e_less(DD,ee,m,m#+n)"
+ (fn prems => [simp_tac(ZF_ss addsimps(add_le_self::prems))1]);
+
+val eps_e_less = prove_goalw Limit.thy [eps_def]
+ "[|m le n; m:nat; n:nat|] ==> eps(DD,ee,m,n) = e_less(DD,ee,m,n)"
+ (fn prems => [simp_tac(ZF_ss addsimps prems)1]);
+
+val shift_asm = imp_refl RS mp;
+
+val prems = goalw Limit.thy [eps_def] (* eps_e_gr_add *)
+ "[|n:nat; k:nat|] ==> eps(DD,ee,n#+k,n) = e_gr(DD,ee,n#+k,n)";
+br(expand_if RS iffD2)1;
+by(safe_tac lemmas_cs);
+by (etac leE 1);
+(* Must control rewriting by instantiating a variable. *)
+by(asm_full_simp_tac(arith_ss addsimps
+ ((hd prems RS nat_into_Ord RS not_le_iff_lt RS iff_sym)::nat_into_Ord::
+ add_le_self::prems))1);
+by(asm_simp_tac(ZF_ss addsimps(e_less_eq::e_gr_eq::prems))1);
+val eps_e_gr_add = result();
+
+val prems = goalw Limit.thy [] (* eps_e_gr *)
+ "[|n le m; m:nat; n:nat|] ==> eps(DD,ee,m,n) = e_gr(DD,ee,m,n)";
+by (rtac le_exists 1);
+by (resolve_tac prems 1);
+by(asm_simp_tac(ZF_ss addsimps(eps_e_gr_add::prems))1);
+brr prems 1;
+val eps_e_gr = result();
+
+val prems = goal Limit.thy (* eps_succ_ee *)
+ "[|!!n. n:nat ==> emb(DD`n,DD`succ(n),ee`n); m:nat|] ==> \
+\ eps(DD,ee,m,succ(m)) = ee`m";
+by(asm_simp_tac(arith_ss addsimps(eps_e_less::le_succ_iff::e_less_succ_emb::
+ prems))1);
+val eps_succ_ee = result();
+
+val prems = goal Limit.thy (* eps_succ_Rp *)
+ "[|emb_chain(DD,ee); m:nat|] ==> \
+\ eps(DD,ee,succ(m),m) = Rp(DD`m,DD`succ(m),ee`m)";
+by(asm_simp_tac(arith_ss addsimps(eps_e_gr::le_succ_iff::e_gr_succ_emb::
+ prems))1);
+val eps_succ_Rp = result();
+
+val prems = goal Limit.thy (* eps_cont *)
+ "[|emb_chain(DD,ee); m:nat; n:nat|] ==> eps(DD,ee,m,n): cont(DD`m,DD`n)";
+br(le_cases RS disjE)1;
+by (resolve_tac prems 1);
+br(hd(rev prems))1;
+by(asm_simp_tac(ZF_ss addsimps(eps_e_less::e_less_cont::prems))1);
+by(asm_simp_tac(ZF_ss addsimps(eps_e_gr::e_gr_cont::prems))1);
+val eps_cont = result();
+
+(* Theorems about splitting. *)
+
+val prems = goal Limit.thy (* eps_split_add_left *)
+ "[|n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,m#+k) = eps(DD,ee,m#+n,m#+k) O eps(DD,ee,m,m#+n)";
+by(asm_simp_tac(arith_ss addsimps
+ (eps_e_less::add_le_self::add_le_mono::prems))1);
+brr(e_less_split_add::prems)1;
+val eps_split_add_left = result();
+
+val prems = goal Limit.thy (* eps_split_add_left_rev *)
+ "[|n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,m#+n) = eps(DD,ee,m#+k,m#+n) O eps(DD,ee,m,m#+k)";
+by(asm_simp_tac(arith_ss addsimps
+ (eps_e_less_add::eps_e_gr::add_le_self::add_le_mono::prems))1);
+brr(e_less_e_gr_split_add::prems)1;
+val eps_split_add_left_rev = result();
+
+val prems = goal Limit.thy (* eps_split_add_right *)
+ "[|m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,n#+k,n) = eps(DD,ee,n#+m,n) O eps(DD,ee,n#+k,n#+m)";
+by(asm_simp_tac(arith_ss addsimps
+ (eps_e_gr::add_le_self::add_le_mono::prems))1);
+brr(e_gr_split_add::prems)1;
+val eps_split_add_right = result();
+
+val prems = goal Limit.thy (* eps_split_add_right_rev *)
+ "[|m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,n#+m,n) = eps(DD,ee,n#+k,n) O eps(DD,ee,n#+m,n#+k)";
+by(asm_simp_tac(arith_ss addsimps
+ (eps_e_gr_add::eps_e_less::add_le_self::add_le_mono::prems))1);
+brr(e_gr_e_less_split_add::prems)1;
+val eps_split_add_right_rev = result();
+
+(* Arithmetic, little support in Isabelle/ZF. *)
+
+val prems = goal Arith.thy (* add_le_elim1 *)
+ "[|m#+n le m#+k; m:nat; n:nat; k:nat|] ==> n le k";
+by (rtac mp 1);
+by (resolve_tac prems 2);
+by(res_inst_tac[("n","n")]nat_induct 1);
+by (resolve_tac prems 1);
+by(simp_tac (arith_ss addsimps prems) 1);
+by(safe_tac lemmas_cs);
+by(asm_full_simp_tac (ZF_ss addsimps
+ (not_le_iff_lt::succ_le_iff::add_succ::add_succ_right::
+ add_type::nat_into_Ord::prems)) 1);
+by (etac lt_asym 1);
+by (assume_tac 1);
+by(asm_full_simp_tac (ZF_ss addsimps
+ (succ_le_iff::add_succ::add_succ_right::le_iff::
+ add_type::nat_into_Ord::prems)) 1);
+by(safe_tac lemmas_cs);
+by (etac lt_irrefl 1);
+val add_le_elim1 = result();
+
+val prems = goal Limit.thy (* le_exists_lemma *)
+ "[|n le k; k le m; \
+\ !!p q. [|p le q; k=n#+p; m=n#+q; p:nat; q:nat|] ==> R; \
+\ m:nat; n:nat; k:nat|]==>R";
+br(hd prems RS le_exists)1;
+br(le_exists)1;
+by (rtac le_trans 1);
+(* Careful *)
+by (resolve_tac prems 1);
+by (resolve_tac prems 1);
+by (resolve_tac prems 1);
+by (assume_tac 2);
+by (assume_tac 2);
+by(cut_facts_tac[hd prems,hd(tl prems)]1);
+by(asm_full_simp_tac arith_ss 1);
+by (etac add_le_elim1 1);
+brr prems 1;
+val le_exists_lemma = result();
+
+val prems = goal Limit.thy (* eps_split_left_le *)
+ "[|m le k; k le n; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
+by (rtac le_exists_lemma 1);
+brr prems 1;
+by(asm_simp_tac ZF_ss 1);
+brr(eps_split_add_left::prems)1;
+val eps_split_left_le = result();
+
+val prems = goal Limit.thy (* eps_split_left_le_rev *)
+ "[|m le n; n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
+by (rtac le_exists_lemma 1);
+brr prems 1;
+by(asm_simp_tac ZF_ss 1);
+brr(eps_split_add_left_rev::prems)1;
+val eps_split_left_le_rev = result();
+
+val prems = goal Limit.thy (* eps_split_right_le *)
+ "[|n le k; k le m; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
+by (rtac le_exists_lemma 1);
+brr prems 1;
+by(asm_simp_tac ZF_ss 1);
+brr(eps_split_add_right::prems)1;
+val eps_split_right_le = result();
+
+val prems = goal Limit.thy (* eps_split_right_le_rev *)
+ "[|n le m; m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
+by (rtac le_exists_lemma 1);
+brr prems 1;
+by(asm_simp_tac ZF_ss 1);
+brr(eps_split_add_right_rev::prems)1;
+val eps_split_right_le_rev = result();
+
+(* The desired two theorems about `splitting'. *)
+
+val prems = goal Limit.thy (* eps_split_left *)
+ "[|m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
+br(le_cases RS disjE)1;
+by (rtac eps_split_right_le_rev 4);
+by (assume_tac 4);
+br(le_cases RS disjE)3;
+by (rtac eps_split_left_le 5);
+by (assume_tac 6);
+by (rtac eps_split_left_le_rev 10);
+brr prems 1; (* 20 trivial subgoals *)
+val eps_split_left = result();
+
+val prems = goal Limit.thy (* eps_split_right *)
+ "[|n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==> \
+\ eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
+br(le_cases RS disjE)1;
+by (rtac eps_split_left_le_rev 3);
+by (assume_tac 3);
+br(le_cases RS disjE)8;
+by (rtac eps_split_right_le 10);
+by (assume_tac 11);
+by (rtac eps_split_right_le_rev 15);
+brr prems 1; (* 20 trivial subgoals *)
+val eps_split_right = result();
+
+(*----------------------------------------------------------------------*)
+(* That was eps: D_m -> D_n, NEXT rho_emb: D_n -> Dinf. *)
+(*----------------------------------------------------------------------*)
+
+(* Considerably shorter. *)
+
+val prems = goalw Limit.thy [rho_emb_def] (* rho_emb_fun *)
+ "[|emb_chain(DD,ee); n:nat|] ==> \
+\ rho_emb(DD,ee,n): set(DD`n) -> set(Dinf(DD,ee))";
+brr(lam_type::DinfI::(eps_cont RS cont_fun RS apply_type)::prems)1;
+by(asm_simp_tac arith_ss 1);
+br(le_cases RS disjE)1;
+by (rtac nat_succI 1);
+by (assume_tac 1);
+by (resolve_tac prems 1);
+(* The easiest would be to apply add1 everywhere also in the assumptions,
+ but since x le y is x<succ(y) simplification does too much with this thm. *)
+br(eps_split_right_le RS ssubst)1;
+by (assume_tac 2);
+by(asm_simp_tac(ZF_ss addsimps (add1::[])) 1);
+brr(add_le_self::nat_0I::nat_succI::prems)1;
+by(asm_simp_tac(ZF_ss addsimps(eps_succ_Rp::prems))1);
+br(comp_fun_apply RS ssubst)1;
+brr(eps_fun::nat_succI::(Rp_cont RS cont_fun)::emb_chain_emb::
+ emb_chain_cpo::refl::prems)1;
+(* Now the second part of the proof. Slightly different than HOL. *)
+by(asm_simp_tac(ZF_ss addsimps(eps_e_less::nat_succI::prems))1);
+be(le_iff RS iffD1 RS disjE)1;
+by(asm_simp_tac(ZF_ss addsimps(e_less_le::prems))1);
+br(comp_fun_apply RS ssubst)1;
+brr(e_less_cont::cont_fun::emb_chain_emb::emb_cont::prems)1;
+br(embRp_eq_thm RS ssubst)1;
+brr(emb_chain_emb::(e_less_cont RS cont_fun RS apply_type)::emb_chain_cpo::
+ nat_succI::prems)1;
+by(asm_simp_tac(ZF_ss addsimps(eps_e_less::prems))1);
+by (dtac shift_asm 1);
+by(asm_full_simp_tac(ZF_ss addsimps(eps_succ_Rp::e_less_eq::id_apply::
+ nat_succI::prems))1);
+val rho_emb_fun = result();
+
+val rho_emb_apply1 = prove_goalw Limit.thy [rho_emb_def]
+ "!!z. x:set(DD`n) ==> rho_emb(DD,ee,n)`x = (lam m:nat. eps(DD,ee,n,m)`x)"
+ (fn prems => [asm_simp_tac ZF_ss 1]);
+
+val rho_emb_apply2 = prove_goalw Limit.thy [rho_emb_def]
+ "!!z. [|x:set(DD`n); m:nat|] ==> rho_emb(DD,ee,n)`x`m = eps(DD,ee,n,m)`x"
+ (fn prems => [asm_simp_tac ZF_ss 1]);
+
+val rho_emb_id = prove_goal Limit.thy
+ "!!z. [| x:set(DD`n); n:nat|] ==> rho_emb(DD,ee,n)`x`n = x"
+ (fn prems => [asm_simp_tac(ZF_ss addsimps[rho_emb_apply2,eps_id,id_thm])1]);
+
+(* Shorter proof, 23 against 62. *)
+
+val prems = goalw Limit.thy [] (* rho_emb_cont *)
+ "[|emb_chain(DD,ee); n:nat|] ==> \
+\ rho_emb(DD,ee,n): cont(DD`n,Dinf(DD,ee))";
+by (rtac contI 1);
+brr(rho_emb_fun::prems)1;
+by (rtac rel_DinfI 1);
+by(SELECT_GOAL(rewtac rho_emb_def)1);
+by(asm_simp_tac ZF_ss 1);
+brr((eps_cont RS cont_mono)::Dinf_prod::apply_type::rho_emb_fun::prems)1;
+(* Continuity, different order, slightly different proofs. *)
+br(lub_Dinf RS ssubst)1;
+by (rtac chainI 1);
+brr(lam_type::(rho_emb_fun RS apply_type)::chain_in::prems)1;
+by(asm_simp_tac arith_ss 1);
+by (rtac rel_DinfI 1);
+by(asm_simp_tac(arith_ss addsimps (rho_emb_apply2::chain_in::[]))1);
+brr((eps_cont RS cont_mono)::chain_rel::Dinf_prod::
+ (rho_emb_fun RS apply_type)::chain_in::nat_succI::prems)1;
+(* Now, back to the result of applying lub_Dinf *)
+by(asm_simp_tac(arith_ss addsimps (rho_emb_apply2::chain_in::[]))1);
+by (rtac (rho_emb_apply1 RS ssubst) 1);
+brr((cpo_lub RS islub_in)::emb_chain_cpo::prems)1;
+by (rtac fun_extension 1);
+brr(lam_type::(eps_cont RS cont_fun RS apply_type)::(cpo_lub RS islub_in)::
+ emb_chain_cpo::prems)1;
+brr(cont_chain::eps_cont::emb_chain_cpo::prems)1;
+by(asm_simp_tac arith_ss 1);
+by(asm_simp_tac(ZF_ss addsimps((eps_cont RS cont_lub)::prems))1);
+val rho_emb_cont = result();
+
+(* 32 vs 61, using safe_tac with imp in asm would be unfortunate (5steps) *)
+
+val prems = goalw Limit.thy [] (* lemma1 *)
+ "[|m le n; emb_chain(DD,ee); x:set(Dinf(DD,ee)); m:nat; n:nat|] ==> \
+\ rel(DD`n,e_less(DD,ee,m,n)`(x`m),x`n)";
+by(rtac impE 1 THEN atac 3 THEN rtac(hd prems)2); (* For induction proof *)
+by(res_inst_tac[("n","n")]nat_induct 1);
+by (rtac impI 2);
+by(asm_full_simp_tac (arith_ss addsimps (e_less_eq::prems)) 2);
+br(id_thm RS ssubst)2;
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_0I::prems)1;
+by(asm_full_simp_tac (arith_ss addsimps [le_succ_iff]) 1);
+by (rtac impI 1);
+by (etac disjE 1);
+by(dtac mp 1 THEN atac 1);
+by (rtac cpo_trans 1);
+br(e_less_le RS ssubst)2;
+brr(emb_chain_cpo::nat_succI::prems)1;
+br(comp_fun_apply RS ssubst)1;
+brr((emb_chain_emb RS emb_cont)::e_less_cont::cont_fun::apply_type::
+ Dinf_prod::prems)1;
+by(res_inst_tac[("y","x`xa")](emb_chain_emb RS emb_cont RS cont_mono)1);
+brr((e_less_cont RS cont_fun)::apply_type::Dinf_prod::prems)1;
+by(res_inst_tac[("x1","x"),("n1","xa")](Dinf_eq RS subst)1);
+br(comp_fun_apply RS subst)3;
+by(res_inst_tac
+ [("P",
+ "%z. rel(DD ` succ(xa), \
+\ (ee ` xa O Rp(?DD46(xa) ` xa,?DD46(xa) ` succ(xa),?ee46(xa) ` xa)) ` \
+\ (x ` succ(xa)),z)")](id_thm RS subst)6);
+by (rtac rel_cf 7);
+(* Dinf and cont_fun doesn't go well together, both Pi(_,%x._). *)
+(* brr solves 11 of 12 subgoals *)
+brr((hd(tl(tl prems)) RS Dinf_prod RS apply_type)::cont_fun::Rp_cont::
+ e_less_cont::emb_cont::emb_chain_emb::emb_chain_cpo::apply_type::
+ embRp_rel::(disjI1 RS (le_succ_iff RS iffD2))::nat_succI::prems)1;
+by(asm_full_simp_tac (arith_ss addsimps (e_less_eq::prems)) 1);
+br(id_thm RS ssubst)1;
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_succI::prems)1;
+val lemma1 = result();
+
+(* 18 vs 40 *)
+
+val prems = goalw Limit.thy [] (* lemma2 *)
+ "[|n le m; emb_chain(DD,ee); x:set(Dinf(DD,ee)); m:nat; n:nat|] ==> \
+\ rel(DD`n,e_gr(DD,ee,m,n)`(x`m),x`n)";
+by(rtac impE 1 THEN atac 3 THEN rtac(hd prems)2); (* For induction proof *)
+by(res_inst_tac[("n","m")]nat_induct 1);
+by (rtac impI 2);
+by(asm_full_simp_tac (arith_ss addsimps (e_gr_eq::prems)) 2);
+br(id_thm RS ssubst)2;
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_0I::prems)1;
+by(asm_full_simp_tac (arith_ss addsimps [le_succ_iff]) 1);
+by (rtac impI 1);
+by (etac disjE 1);
+by(dtac mp 1 THEN atac 1);
+br(e_gr_le RS ssubst)1;
+br(comp_fun_apply RS ssubst)4;
+br(Dinf_eq RS ssubst)7;
+brr(emb_chain_emb::emb_chain_cpo::Rp_cont::e_gr_cont::cont_fun::emb_cont::
+ apply_type::Dinf_prod::nat_succI::prems)1;
+by(asm_full_simp_tac (arith_ss addsimps (e_gr_eq::prems)) 1);
+br(id_thm RS ssubst)1;
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_succI::prems)1;
+val lemma2 = result();
+
+val prems = goalw ZF.thy [if_def]
+ "[| P==>R(a); ~P==>R(b) |] ==> R(if(P,a,b))";
+by(excluded_middle_tac"P"1);
+by(ALLGOALS(asm_simp_tac ZF_ss THEN' rtac theI2));
+by(ALLGOALS(asm_full_simp_tac FOL_ss));
+brr(ex1I::refl::prems)1;
+val if_case = result();
+
+val prems = goalw Limit.thy [eps_def] (* eps1 *)
+ "[|emb_chain(DD,ee); x:set(Dinf(DD,ee)); m:nat; n:nat|] ==> \
+\ rel(DD`n,eps(DD,ee,m,n)`(x`m),x`n)";
+by (rtac if_case 1);
+brr(lemma1::(not_le_iff_lt RS iffD1 RS leI RS lemma2)::nat_into_Ord::prems)1;
+val eps1 = result();
+
+(* The following theorem is needed/useful due to type check for rel_cfI,
+ but also elsewhere.
+ Look for occurences of rel_cfI, rel_DinfI, etc to evaluate the problem. *)
+
+val prems = goal Limit.thy (* lam_Dinf_cont *)
+ "[| emb_chain(DD,ee); n:nat |] ==> \
+\ (lam x:set(Dinf(DD,ee)). x`n) : cont(Dinf(DD,ee),DD`n)";
+by (rtac contI 1);
+brr(lam_type::apply_type::Dinf_prod::prems)1;
+by(asm_simp_tac ZF_ss 1);
+brr(rel_Dinf::prems)1;
+br(beta RS ssubst)1;
+brr(cpo_Dinf::islub_in::cpo_lub::prems)1;
+by(asm_simp_tac(ZF_ss addsimps(chain_in::lub_Dinf::prems))1);
+val lam_Dinf_cont = result();
+
+val prems = goalw Limit.thy [rho_proj_def] (* rho_projpair *)
+ "[| emb_chain(DD,ee); n:nat |] ==> \
+\ projpair(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n),rho_proj(DD,ee,n))";
+by (rtac projpairI 1);
+brr(rho_emb_cont::prems)1;
+(* lemma used, introduced because same fact needed below due to rel_cfI. *)
+brr(lam_Dinf_cont::prems)1;
+(*-----------------------------------------------*)
+(* This part is 7 lines, but 30 in HOL (75% reduction!) *)
+by (rtac fun_extension 1);
+br(id_thm RS ssubst)3;
+br(comp_fun_apply RS ssubst)4;
+br(beta RS ssubst)7;
+br(rho_emb_id RS ssubst)8;
+brr(comp_fun::id_type::lam_type::rho_emb_fun::(Dinf_prod RS apply_type)::
+ apply_type::refl::prems)1;
+(*^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
+by (rtac rel_cfI 1); (* ------------------>>>Yields type cond, not in HOL *)
+br(id_thm RS ssubst)1;
+br(comp_fun_apply RS ssubst)2;
+br(beta RS ssubst)5;
+br(rho_emb_apply1 RS ssubst)6;
+by (rtac rel_DinfI 7); (* ------------------>>>Yields type cond, not in HOL *)
+br(beta RS ssubst)7;
+brr(eps1::lam_type::rho_emb_fun::eps_fun:: (* Dinf_prod bad with lam_type *)
+ (Dinf_prod RS apply_type)::refl::prems)1;
+brr(apply_type::eps_fun::Dinf_prod::comp_pres_cont::rho_emb_cont::
+ lam_Dinf_cont::id_cont::cpo_Dinf::emb_chain_cpo::prems)1;
+val rho_projpair = result();
+
+val prems = goalw Limit.thy [emb_def]
+ "[| emb_chain(DD,ee); n:nat |] ==> emb(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))";
+brr(exI::rho_projpair::prems)1;
+val emb_rho_emb = result();
+
+val prems = goal Limit.thy
+ "[| emb_chain(DD,ee); n:nat |] ==> \
+\ rho_proj(DD,ee,n) : cont(Dinf(DD,ee),DD`n)";
+brr(rho_projpair::projpair_p_cont::prems)1;
+val rho_proj_cont = result();
+
+(*----------------------------------------------------------------------*)
+(* Commutivity and universality. *)
+(*----------------------------------------------------------------------*)
+
+val prems = goalw Limit.thy [commute_def] (* commuteI *)
+ "[| !!n. n:nat ==> emb(DD`n,E,r(n)); \
+\ !!m n. [|m le n; m:nat; n:nat|] ==> r(n) O eps(DD,ee,m,n) = r(m) |] ==> \
+\ commute(DD,ee,E,r)";
+by(safe_tac lemmas_cs);
+brr prems 1;
+val commuteI = result();
+
+val prems = goalw Limit.thy [commute_def] (* commute_emb *)
+ "!!z. [| commute(DD,ee,E,r); n:nat |] ==> emb(DD`n,E,r(n))";
+by(fast_tac ZF_cs 1);
+val commute_emb = result();
+
+val prems = goalw Limit.thy [commute_def] (* commute_eq *)
+ "!!z. [| commute(DD,ee,E,r); m le n; m:nat; n:nat |] ==> \
+\ r(n) O eps(DD,ee,m,n) = r(m) ";
+by(fast_tac ZF_cs 1);
+val commute_eq = result();
+
+(* Shorter proof: 11 vs 46 lines. *)
+
+val prems = goal Limit.thy (* rho_emb_commute *)
+ "emb_chain(DD,ee) ==> commute(DD,ee,Dinf(DD,ee),rho_emb(DD,ee))";
+by (rtac commuteI 1);
+brr(emb_rho_emb::prems)1;
+by (rtac fun_extension 1); (* Manual instantiation in HOL. *)
+br(comp_fun_apply RS ssubst)3;
+by (rtac fun_extension 6); (* Next, clean up and instantiate unknowns *)
+brr(comp_fun::rho_emb_fun::eps_fun::Dinf_prod::apply_type::prems)1;
+by(asm_simp_tac
+ (ZF_ss addsimps(rho_emb_apply2::(eps_fun RS apply_type)::prems))1);
+br(comp_fun_apply RS subst)1;
+br(eps_split_left RS subst)4;
+brr(eps_fun::refl::prems)1;
+val rho_emb_commute = result();
+
+val le_succ = prove_goal Arith.thy "n:nat ==> n le succ(n)"
+ (fn prems =>
+ [REPEAT (ares_tac
+ ((disjI1 RS(le_succ_iff RS iffD2))::le_refl::nat_into_Ord::prems)1)]);
+
+(* Shorter proof: 21 vs 83 (106 - 23, due to OAssoc complication) *)
+
+val prems = goal Limit.thy (* commute_chain *)
+ "[| commute(DD,ee,E,r); emb_chain(DD,ee); cpo(E) |] ==> \
+\ chain(cf(E,E),lam n:nat. r(n) O Rp(DD`n,E,r(n)))";
+val emb_r = hd prems RS commute_emb; (* To avoid BACKTRACKING !! *)
+by (rtac chainI 1);
+brr(lam_type::cont_cf::comp_pres_cont::emb_r::Rp_cont::emb_cont::
+ emb_chain_cpo::prems)1;
+by(asm_simp_tac arith_ss 1);
+by(res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst)1);
+brr(le_succ::nat_succI::prems)1;
+br(Rp_comp RS ssubst)1;
+brr(emb_eps::emb_r::emb_chain_cpo::le_succ::nat_succI::prems)1;
+br(comp_assoc RS subst)1; (* Remember that comp_assoc is simpler in Isa *)
+by(res_inst_tac[("r1","r(succ(n))")](comp_assoc RS ssubst)1);
+by (rtac comp_mono 1);
+brr(comp_pres_cont::eps_cont::emb_eps::emb_r::Rp_cont::emb_cont::
+ emb_chain_cpo::le_succ::nat_succI::prems)1;
+by(res_inst_tac[("b","r(succ(n))")](comp_id RS subst)1); (* 1 subst too much *)
+by (rtac comp_mono 2);
+brr(comp_pres_cont::eps_cont::emb_eps::emb_id::emb_r::Rp_cont::emb_cont::
+ cont_fun::emb_chain_cpo::le_succ::nat_succI::prems)1;
+br(comp_id RS ssubst)1; (* Undo's "1 subst too much", typing next anyway *)
+brr(cont_fun::Rp_cont::emb_cont::emb_r::cpo_refl::cont_cf::cpo_cf::
+ emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems)1;
+val commute_chain = result();
+
+val prems = goal Limit.thy (* rho_emb_chain *)
+ "emb_chain(DD,ee) ==> \
+\ chain(cf(Dinf(DD,ee),Dinf(DD,ee)), \
+\ lam n:nat. rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))";
+brr(commute_chain::rho_emb_commute::cpo_Dinf::prems)1;
+val rho_emb_chain = result();
+
+val prems = goal Limit.thy (* rho_emb_chain_apply1 *)
+ "[| emb_chain(DD,ee); x:set(Dinf(DD,ee)) |] ==> \
+\ chain(Dinf(DD,ee), \
+\ lam n:nat. \
+\ (rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))`x)";
+by(cut_facts_tac[hd(tl prems) RS (hd prems RS (rho_emb_chain RS chain_cf))]1);
+by(asm_full_simp_tac ZF_ss 1);
+val rho_emb_chain_apply1 = result();
+
+val prems = goal Limit.thy
+ "[| chain(iprod(DD),X); emb_chain(DD,ee); n:nat |] ==> \
+\ chain(DD`n,lam m:nat. X `m `n)";
+brr(chain_iprod::emb_chain_cpo::prems)1;
+val chain_iprod_emb_chain = result();
+
+val prems = goal Limit.thy (* rho_emb_chain_apply2 *)
+ "[| emb_chain(DD,ee); x:set(Dinf(DD,ee)); n:nat |] ==> \
+\ chain \
+\ (DD`n, \
+\ lam xa:nat. \
+\ (rho_emb(DD, ee, xa) O Rp(DD ` xa, Dinf(DD, ee),rho_emb(DD, ee, xa))) ` \
+\ x ` n)";
+by(cut_facts_tac
+ [hd(tl(tl prems)) RS (hd prems RS (hd(tl prems) RS (hd prems RS
+ (rho_emb_chain_apply1 RS chain_Dinf RS chain_iprod_emb_chain))))]1);
+by(asm_full_simp_tac ZF_ss 1);
+val rho_emb_chain_apply2 = result();
+
+(* Shorter proof: 32 vs 72 (roughly), Isabelle proof has lemmas. *)
+
+val prems = goal Limit.thy (* rho_emb_lub *)
+ "emb_chain(DD,ee) ==> \
+\ lub(cf(Dinf(DD,ee),Dinf(DD,ee)), \
+\ lam n:nat. rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))) = \
+\ id(set(Dinf(DD,ee)))";
+by (rtac cpo_antisym 1);
+by (rtac cpo_cf 1); (* Instantiate variable, continued below (would loop otherwise) *)
+brr(cpo_Dinf::prems)1;
+by (rtac islub_least 1);
+brr(cpo_lub::rho_emb_chain::cpo_cf::cpo_Dinf::isubI::cont_cf::id_cont::prems)1;
+by(asm_simp_tac ZF_ss 1);
+brr(embRp_rel::emb_rho_emb::emb_chain_cpo::cpo_Dinf::prems)1;
+by (rtac rel_cfI 1);
+by(asm_simp_tac
+ (ZF_ss addsimps(id_thm::lub_cf::rho_emb_chain::cpo_Dinf::prems))1);
+by (rtac rel_DinfI 1); (* Addtional assumptions *)
+br(lub_Dinf RS ssubst)1;
+brr(rho_emb_chain_apply1::prems)1;
+brr(Dinf_prod::(cpo_lub RS islub_in)::id_cont::cpo_Dinf::cpo_cf::cf_cont::
+ rho_emb_chain::rho_emb_chain_apply1::(id_cont RS cont_cf)::prems)2;
+by(asm_simp_tac ZF_ss 1);
+by (rtac dominate_islub 1);
+by (rtac cpo_lub 3);
+brr(rho_emb_chain_apply2::emb_chain_cpo::prems)3;
+by(res_inst_tac[("x1","x`n")](chain_const RS chain_fun)3);
+brr(islub_const::apply_type::Dinf_prod::emb_chain_cpo::chain_fun::
+ rho_emb_chain_apply2::prems)2;
+by (rtac dominateI 1);
+by (assume_tac 1);
+by(asm_simp_tac ZF_ss 1);
+br(comp_fun_apply RS ssubst)1;
+brr(cont_fun::Rp_cont::emb_cont::emb_rho_emb::cpo_Dinf::emb_chain_cpo::prems)1;
+br((rho_projpair RS Rp_unique) RS ssubst)1;
+by(SELECT_GOAL(rewtac rho_proj_def)5);
+by(asm_simp_tac ZF_ss 5);
+br(rho_emb_id RS ssubst)5;
+brr(cpo_refl::cpo_Dinf::apply_type::Dinf_prod::emb_chain_cpo::prems)1;
+val rho_emb_lub = result();
+
+val prems = goal Limit.thy (* theta_chain, almost same prf as commute_chain *)
+ "[| commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G) |] ==> \
+\ chain(cf(E,G),lam n:nat. f(n) O Rp(DD`n,E,r(n)))";
+val emb_r = hd prems RS commute_emb; (* Used in the rest of the FILE *)
+val emb_f = hd(tl prems) RS commute_emb; (* Used in the rest of the FILE *)
+by (rtac chainI 1);
+brr(lam_type::cont_cf::comp_pres_cont::emb_r::emb_f::
+ Rp_cont::emb_cont::emb_chain_cpo::prems)1;
+by(asm_simp_tac arith_ss 1);
+by(res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst)1);
+by(res_inst_tac[("r1","f"),("m1","n")](commute_eq RS subst)5);
+brr(le_succ::nat_succI::prems)1;
+br(Rp_comp RS ssubst)1;
+brr(emb_eps::emb_r::emb_chain_cpo::le_succ::nat_succI::prems)1;
+br(comp_assoc RS subst)1; (* Remember that comp_assoc is simpler in Isa *)
+by(res_inst_tac[("r1","f(succ(n))")](comp_assoc RS ssubst)1);
+by (rtac comp_mono 1);
+brr(comp_pres_cont::eps_cont::emb_eps::emb_r::emb_f::Rp_cont::
+ emb_cont::emb_chain_cpo::le_succ::nat_succI::prems)1;
+by(res_inst_tac[("b","f(succ(n))")](comp_id RS subst)1); (* 1 subst too much *)
+by (rtac comp_mono 2);
+brr(comp_pres_cont::eps_cont::emb_eps::emb_id::emb_r::emb_f::Rp_cont::
+ emb_cont::cont_fun::emb_chain_cpo::le_succ::nat_succI::prems)1;
+br(comp_id RS ssubst)1; (* Undo's "1 subst too much", typing next anyway *)
+brr(cont_fun::Rp_cont::emb_cont::emb_r::emb_f::cpo_refl::cont_cf::
+ cpo_cf::emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems)1;
+val theta_chain = result();
+
+val prems = goal Limit.thy (* theta_proj_chain, same prf as theta_chain *)
+ "[| commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G) |] ==> \
+\ chain(cf(G,E),lam n:nat. r(n) O Rp(DD`n,G,f(n)))";
+by (rtac chainI 1);
+brr(lam_type::cont_cf::comp_pres_cont::emb_r::emb_f::
+ Rp_cont::emb_cont::emb_chain_cpo::prems)1;
+by(asm_simp_tac arith_ss 1);
+by(res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst)1);
+by(res_inst_tac[("r1","f"),("m1","n")](commute_eq RS subst)5);
+brr(le_succ::nat_succI::prems)1;
+br(Rp_comp RS ssubst)1;
+brr(emb_eps::emb_f::emb_chain_cpo::le_succ::nat_succI::prems)1;
+br(comp_assoc RS subst)1; (* Remember that comp_assoc is simpler in Isa *)
+by(res_inst_tac[("r1","r(succ(n))")](comp_assoc RS ssubst)1);
+by (rtac comp_mono 1);
+brr(comp_pres_cont::eps_cont::emb_eps::emb_r::emb_f::Rp_cont::
+ emb_cont::emb_chain_cpo::le_succ::nat_succI::prems)1;
+by(res_inst_tac[("b","r(succ(n))")](comp_id RS subst)1); (* 1 subst too much *)
+by (rtac comp_mono 2);
+brr(comp_pres_cont::eps_cont::emb_eps::emb_id::emb_r::emb_f::Rp_cont::
+ emb_cont::cont_fun::emb_chain_cpo::le_succ::nat_succI::prems)1;
+br(comp_id RS ssubst)1; (* Undo's "1 subst too much", typing next anyway *)
+brr(cont_fun::Rp_cont::emb_cont::emb_r::emb_f::cpo_refl::cont_cf::
+ cpo_cf::emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems)1;
+val theta_proj_chain = result();
+
+(* Simplification with comp_assoc is possible inside a lam-abstraction,
+ because it does not have assumptions. If it had, as the HOL-ST theorem
+ too strongly has, we would be in deep trouble due to the lack of proper
+ conditional rewriting (a HOL contrib provides something that works). *)
+
+(* Controlled simplification inside lambda: introduce lemmas *)
+
+val prems = goal Limit.thy
+ "[| commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G); x:nat |] ==> \
+\ r(x) O Rp(DD ` x, G, f(x)) O f(x) O Rp(DD ` x, E, r(x)) = \
+\ r(x) O Rp(DD ` x, E, r(x))";
+by(res_inst_tac[("s1","f(x)")](comp_assoc RS subst)1);
+br(embRp_eq RS ssubst)1;
+br(id_comp RS ssubst)4;
+brr(cont_fun::Rp_cont::emb_r::emb_f::emb_chain_cpo::refl::prems)1;
+val lemma = result();
+
+val lemma_assoc = prove_goal Limit.thy "a O b O c O d = a O (b O c) O d"
+ (fn prems => [simp_tac (ZF_ss addsimps[comp_assoc]) 1]);
+
+fun elem n l = if n = 1 then hd l else elem(n-1)(tl l);
+
+(* Shorter proof (but lemmas): 19 vs 79 (103 - 24, due to OAssoc) *)
+
+val prems = goalw Limit.thy [projpair_def,rho_proj_def] (* theta_projpair *)
+ "[| lub(cf(E,E), lam n:nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E)); \
+\ commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G) |] ==> \
+\ projpair \
+\ (E,G, \
+\ lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n))), \
+\ lub(cf(G,E), lam n:nat. r(n) O Rp(DD`n,G,f(n))))";
+by(safe_tac lemmas_cs);
+br(comp_lubs RS ssubst)3;
+(* The following one line is 15 lines in HOL, and includes existentials. *)
+brr(cf_cont::islub_in::cpo_lub::cpo_cf::theta_chain::theta_proj_chain::prems)1;
+by(simp_tac (ZF_ss addsimps[comp_assoc]) 1);
+by(simp_tac (ZF_ss addsimps[(tl prems) MRS lemma]) 1);
+br(comp_lubs RS ssubst)2;
+brr(cf_cont::islub_in::cpo_lub::cpo_cf::theta_chain::theta_proj_chain::prems)1;
+by(simp_tac (ZF_ss addsimps[comp_assoc]) 1);
+by(simp_tac (ZF_ss addsimps[
+ [elem 3 prems,elem 2 prems,elem 4 prems,elem 6 prems, elem 5 prems]
+ MRS lemma]) 1);
+by (rtac dominate_islub 1);
+by (rtac cpo_lub 2);
+brr(commute_chain::emb_f::islub_const::cont_cf::id_cont::cpo_cf::
+ chain_fun::chain_const::prems)2;
+by (rtac dominateI 1);
+by (assume_tac 1);
+by(asm_simp_tac ZF_ss 1);
+brr(embRp_rel::emb_f::emb_chain_cpo::prems)1;
+val theta_projpair = result();
+
+val prems = goalw Limit.thy [emb_def] (* emb_theta *)
+ "[| lub(cf(E,E), lam n:nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E)); \
+\ commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G) |] ==> \
+\ emb(E,G,lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n))))";
+by (rtac exI 1);
+br(prems MRS theta_projpair)1;
+val emb_theta = result();
+
+val prems = goal Limit.thy (* mono_lemma *)
+ "[| g:cont(D,D'); cpo(D); cpo(D'); cpo(E) |] ==> \
+\ (lam f : cont(D',E). f O g) : mono(cf(D',E),cf(D,E))";
+by (rtac monoI 1);
+by(REPEAT(dtac cf_cont 2));
+by(asm_simp_tac ZF_ss 2);
+by (rtac comp_mono 2);
+by(SELECT_GOAL(rewrite_goals_tac[set_def,cf_def])1);
+by(asm_simp_tac ZF_ss 1);
+brr(lam_type::comp_pres_cont::cpo_cf::cpo_refl::cont_cf::prems)1;
+val mono_lemma = result();
+
+(* PAINFUL: wish condrew with difficult conds on term bound in lam-abs. *)
+(* Introduces need for lemmas. *)
+
+val prems = goal Limit.thy
+ "[| commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G); n:nat |] ==> \
+\ (lam na:nat. (lam f:cont(E, G). f O r(n)) ` \
+\ ((lam n:nat. f(n) O Rp(DD ` n, E, r(n))) ` na)) = \
+\ (lam na:nat. (f(na) O Rp(DD ` na, E, r(na))) O r(n))";
+by (rtac fun_extension 1);
+br(beta RS ssubst)3;
+br(beta RS ssubst)4;
+br(beta RS ssubst)5;
+by (rtac lam_type 1);
+br(beta RS ssubst)1;
+by(ALLGOALS(asm_simp_tac (ZF_ss addsimps prems)));
+brr(lam_type::comp_pres_cont::Rp_cont::emb_cont::emb_r::emb_f::
+ emb_chain_cpo::prems)1;
+val lemma = result();
+
+val prems = goal Limit.thy (* chain_lemma *)
+ "[| commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G); n:nat |] ==> \
+\ chain(cf(DD`n,G),lam x:nat. (f(x) O Rp(DD ` x, E, r(x))) O r(n))";
+by(cut_facts_tac[(rev(tl(rev prems)) MRS theta_chain) RS
+ (elem 5 prems RS (elem 4 prems RS ((elem 6 prems RS
+ (elem 3 prems RS emb_chain_cpo)) RS (elem 6 prems RS
+ (emb_r RS emb_cont RS mono_lemma RS mono_chain)))))]1);
+br((prems MRS lemma) RS subst)1;
+by (assume_tac 1);
+val chain_lemma = result();
+
+val prems = goalw Limit.thy [suffix_def] (* suffix_lemma *)
+ "[| commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G); cpo(DD`x); x:nat |] ==> \
+\ suffix(lam n:nat. (f(n) O Rp(DD`n,E,r(n))) O r(x),x) = (lam n:nat. f(x))";
+by(simp_tac (arith_ss addsimps prems) 1);
+by (rtac fun_extension 1);
+brr(lam_type::comp_fun::cont_fun::Rp_cont::emb_cont::emb_r::emb_f::
+ add_type::emb_chain_cpo::prems)1;
+by(asm_simp_tac ZF_ss 1);
+by(res_inst_tac[("r1","r"),("m1","x")](commute_eq RS subst)1);
+brr(emb_r::add_le_self::add_type::prems)1;
+br(comp_assoc RS ssubst)1;
+br(lemma_assoc RS ssubst)1;
+br(embRp_eq RS ssubst)1;
+br(id_comp RS ssubst)4;
+br((hd(tl prems) RS commute_eq) RS ssubst)5; (* avoid eta_contraction:=true. *)
+brr(emb_r::add_type::eps_fun::add_le_self::refl::emb_chain_cpo::prems)1;
+val suffix_lemma = result();
+
+val mediatingI = prove_goalw Limit.thy [mediating_def]
+ "[|emb(E,G,t);!!n.n:nat ==> f(n) = t O r(n) |]==>mediating(E,G,r,f,t)"
+ (fn prems => [safe_tac lemmas_cs,trr prems 1]);
+
+val mediating_emb = prove_goalw Limit.thy [mediating_def]
+ "!!z. mediating(E,G,r,f,t) ==> emb(E,G,t)"
+ (fn prems => [fast_tac ZF_cs 1]);
+
+val mediating_eq = prove_goalw Limit.thy [mediating_def]
+ "!!z. [| mediating(E,G,r,f,t); n:nat |] ==> f(n) = t O r(n)"
+ (fn prems => [fast_tac ZF_cs 1]);
+
+val prems = goal Limit.thy (* lub_universal_mediating *)
+ "[| lub(cf(E,E), lam n:nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E)); \
+\ commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G) |] ==> \
+\ mediating(E,G,r,f,lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n))))";
+brr(mediatingI::emb_theta::prems)1;
+by(res_inst_tac[("b","r(n)")](lub_const RS subst)1);
+br(comp_lubs RS ssubst)3;
+brr(cont_cf::emb_cont::emb_r::cpo_cf::theta_chain::chain_const::
+ emb_chain_cpo::prems)1;
+by(simp_tac ZF_ss 1);
+br(lub_suffix RS subst)1;
+brr(chain_lemma::cpo_cf::emb_chain_cpo::prems)1;
+br((tl prems MRS suffix_lemma) RS ssubst)1;
+br(lub_const RS ssubst)3;
+brr(cont_cf::emb_cont::emb_f::cpo_cf::emb_chain_cpo::refl::prems)1;
+val lub_universal_mediating = result();
+
+val prems = goal Limit.thy (* lub_universal_unique *)
+ "[| mediating(E,G,r,f,t); \
+\ lub(cf(E,E), lam n:nat. r(n) O Rp(DD`n,E,r(n))) = id(set(E)); \
+\ commute(DD,ee,E,r); commute(DD,ee,G,f); \
+\ emb_chain(DD,ee); cpo(E); cpo(G) |] ==> \
+\ t = lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n)))";
+by(res_inst_tac[("b","t")](comp_id RS subst)1);
+br(hd(tl prems) RS subst)2;
+by(res_inst_tac[("b","t")](lub_const RS subst)2);
+br(comp_lubs RS ssubst)4;
+by(simp_tac (ZF_ss addsimps(comp_assoc::(hd prems RS mediating_eq)::prems))9);
+brr(cont_fun::emb_cont::mediating_emb::cont_cf::cpo_cf::chain_const::
+ commute_chain::emb_chain_cpo::prems)1;
+val lub_universal_unique = result();
+
+(*---------------------------------------------------------------------*)
+(* Dinf yields the inverse_limit, stated as rho_emb_commute and *)
+(* Dinf_universal. *)
+(*---------------------------------------------------------------------*)
+
+val prems = goal Limit.thy (* Dinf_universal *)
+ "[| commute(DD,ee,G,f); emb_chain(DD,ee); cpo(G) |] ==> \
+\ mediating \
+\ (Dinf(DD,ee),G,rho_emb(DD,ee),f, \
+\ lub(cf(Dinf(DD,ee),G), \
+\ lam n:nat. f(n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))) & \
+\ (ALL t. mediating(Dinf(DD,ee),G,rho_emb(DD,ee),f,t) --> \
+\ t = lub(cf(Dinf(DD,ee),G), \
+\ lam n:nat. f(n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))))";
+by(safe_tac lemmas_cs);
+brr(lub_universal_mediating::rho_emb_commute::rho_emb_lub::cpo_Dinf::prems)1;
+brr(lub_universal_unique::rho_emb_commute::rho_emb_lub::cpo_Dinf::prems)1;
+val Dinf_universal = result();
+