# HG changeset patch # User paulson # Date 813851784 -3600 # Node ID 68f6be60ab1c352d6f53aae8666524115021bdf1 # Parent 909079af97b74ec7d059a8b5d31dcd2a9a48a490 The inverse limit construction -- thanks to Sten Agerholm diff -r 909079af97b7 -r 68f6be60ab1c src/ZF/ex/Limit.ML --- /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] + ":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) ==> :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 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(); + diff -r 909079af97b7 -r 68f6be60ab1c src/ZF/ex/Limit.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/ZF/ex/Limit.thy Mon Oct 16 14:56:24 1995 +0100 @@ -0,0 +1,199 @@ +(* Title: ZF/ex/Limit + ID: $Id$ + Author: Sten Agerholm + + A formalization of the inverse limit construction of domain theory. + + The following paper comments on the formalization: + + "A Comparison of HOL-ST and Isabelle/ZF" by Sten Agerholm + In Proceedings of the First Isabelle Users Workshop, Technical + Report No. 379, University of Cambridge Computer Laboratory, 1995. + + This is a condensed version of: + + "A Comparison of HOL-ST and Isabelle/ZF" by Sten Agerholm + Technical Report No. 369, University of Cambridge Computer + Laboratory, 1995. +*) + +Limit = ZF + Perm + Arith + + +consts + + "rel" :: "[i,i,i]=>o" (* rel(D,x,y) *) + "set" :: "i=>i" (* set(D) *) + "po" :: "i=>o" (* po(D) *) + "chain" :: "[i,i]=>o" (* chain(D,X) *) + "isub" :: "[i,i,i]=>o" (* isub(D,X,x) *) + "islub" :: "[i,i,i]=>o" (* islub(D,X,x) *) + "lub" :: "[i,i]=>i" (* lub(D,X) *) + "cpo" :: "i=>o" (* cpo(D) *) + "pcpo" :: "i=>o" (* pcpo(D) *) + "bot" :: "i=>i" (* bot(D) *) + "mono" :: "[i,i]=>i" (* mono(D,E) *) + "cont" :: "[i,i]=>i" (* cont(D,E) *) + "cf" :: "[i,i]=>i" (* cf(D,E) *) + + "suffix" :: "[i,i]=>i" (* suffix(X,n) *) + "subchain" :: "[i,i]=>o" (* subchain(X,Y) *) + "dominate" :: "[i,i,i]=>o" (* dominate(D,X,Y) *) + "matrix" :: "[i,i]=>o" (* matrix(D,M) *) + + "projpair" :: "[i,i,i,i]=>o" (* projpair(D,E,e,p) *) + "emb" :: "[i,i,i]=>o" (* emb(D,E,e) *) + "Rp" :: "[i,i,i]=>i" (* Rp(D,E,e) *) + "iprod" :: "i=>i" (* iprod(DD) *) + "mkcpo" :: "[i,i=>o]=>i" (* mkcpo(D,P) *) + "subcpo" :: "[i,i]=>o" (* subcpo(D,E) *) + "subpcpo" :: "[i,i]=>o" (* subpcpo(D,E) *) + + "emb_chain" :: "[i,i]=>o" (* emb_chain(DD,ee) *) + "Dinf" :: "[i,i]=>i" (* Dinf(DD,ee) *) + + "e_less" :: "[i,i,i,i]=>i" (* e_less(DD,ee,m,n) *) + "e_gr" :: "[i,i,i,i]=>i" (* e_gr(DD,ee,m,n) *) + "eps" :: "[i,i,i,i]=>i" (* eps(DD,ee,m,n) *) + "rho_emb" :: "[i,i,i]=>i" (* rho_emb(DD,ee,n) *) + "rho_proj" :: "[i,i,i]=>i" (* rho_proj(DD,ee,n) *) + "commute" :: "[i,i,i,i=>i]=>o" (* commute(DD,ee,E,r) *) + "mediating" :: "[i,i,i=>i,i=>i,i]=>o" (* mediating(E,G,r,f,t) *) + +rules + + set_def + "set(D) == fst(D)" + + rel_def + "rel(D,x,y) == :snd(D)" + + po_def + "po(D) == \ +\ (ALL x:set(D). rel(D,x,x)) & \ +\ (ALL x:set(D). ALL y:set(D). ALL z:set(D). \ +\ rel(D,x,y) --> rel(D,y,z) --> rel(D,x,z)) & \ +\ (ALL x:set(D). ALL y:set(D). rel(D,x,y) --> rel(D,y,x) --> x = y)" + + (* Chains are object level functions nat->set(D) *) + + chain_def + "chain(D,X) == X:nat->set(D) & (ALL n:nat. rel(D,X`n,X`(succ(n))))" + + isub_def + "isub(D,X,x) == x:set(D) & (ALL n:nat. rel(D,X`n,x))" + + islub_def + "islub(D,X,x) == isub(D,X,x) & (ALL y. isub(D,X,y) --> rel(D,x,y))" + + lub_def + "lub(D,X) == THE x. islub(D,X,x)" + + cpo_def + "cpo(D) == po(D) & (ALL X. chain(D,X) --> (EX x. islub(D,X,x)))" + + pcpo_def + "pcpo(D) == cpo(D) & (EX x:set(D). ALL y:set(D). rel(D,x,y))" + + bot_def + "bot(D) == THE x. x:set(D) & (ALL y:set(D). rel(D,x,y))" + + + mono_def + "mono(D,E) == \ +\ {f:set(D)->set(E). \ +\ ALL x:set(D). ALL y:set(D). rel(D,x,y) --> rel(E,f`x,f`y)}" + + cont_def + "cont(D,E) == \ +\ {f:mono(D,E). \ +\ ALL X. chain(D,X) --> f`(lub(D,X)) = lub(E,lam n:nat. f`(X`n))}" + + cf_def + "cf(D,E) == \ +\ " + + suffix_def + "suffix(X,n) == lam m:nat. X`(n #+ m)" + + subchain_def + "subchain(X,Y) == ALL m:nat. EX n:nat. X`m = Y`(m #+ n)" + + dominate_def + "dominate(D,X,Y) == ALL m:nat. EX n:nat. rel(D,X`m,Y`n)" + + matrix_def + "matrix(D,M) == \ +\ M: nat -> (nat -> set(D)) & \ +\ (ALL n:nat. ALL m:nat. rel(D,M`n`m,M`succ(n)`m)) & \ +\ (ALL n:nat. ALL m:nat. rel(D,M`n`m,M`n`succ(m))) & \ +\ (ALL n:nat. ALL m:nat. rel(D,M`n`m,M`succ(n)`succ(m)))" + + projpair_def + "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)))" + + emb_def + "emb(D,E,e) == EX p. projpair(D,E,e,p)" + + Rp_def + "Rp(D,E,e) == THE p. projpair(D,E,e,p)" + +(* Twice, constructions on cpos are more difficult. *) + + iprod_def + "iprod(DD) == \ +\ <(PROD n:nat. set(DD`n)), \ +\ {x:(PROD n:nat. set(DD`n))*(PROD n:nat. set(DD`n)). \ +\ ALL n:nat. rel(DD`n,fst(x)`n,snd(x)`n)}>" + + mkcpo_def (* Cannot use rel(D), is meta fun, need two more args *) + "mkcpo(D,P) == \ +\ <{x:set(D). P(x)},{x:set(D)*set(D). rel(D,fst(x),snd(x))}>" + + + subcpo_def + "subcpo(D,E) == \ +\ set(D) <= set(E) & \ +\ (ALL x:set(D). ALL y:set(D). rel(D,x,y) <-> rel(E,x,y)) & \ +\ (ALL X. chain(D,X) --> lub(E,X):set(D))" + + subpcpo_def + "subpcpo(D,E) == subcpo(D,E) & bot(E):set(D)" + + emb_chain_def + "emb_chain(DD,ee) == \ +\ (ALL n:nat. cpo(DD`n)) & (ALL n:nat. emb(DD`n,DD`succ(n),ee`n))" + + Dinf_def + "Dinf(DD,ee) == \ +\ mkcpo(iprod(DD)) \ +\ (%x. ALL n:nat. Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n)" + + e_less_def (* Valid for m le n only. *) + "e_less(DD,ee,m,n) == rec(n#-m,id(set(DD`m)),%x y. ee`(m#+x) O y)" + + e_gr_def (* Valid for n le m only. *) + "e_gr(DD,ee,m,n) == \ +\ rec(m#-n,id(set(DD`n)), \ +\ %x y. y O Rp(DD`(n#+x),DD`(succ(n#+x)),ee`(n#+x)))" + + eps_def + "eps(DD,ee,m,n) == if(m le n,e_less(DD,ee,m,n),e_gr(DD,ee,m,n))" + + rho_emb_def + "rho_emb(DD,ee,n) == lam x:set(DD`n). lam m:nat. eps(DD,ee,n,m)`x" + + rho_proj_def + "rho_proj(DD,ee,n) == lam x:set(Dinf(DD,ee)). x`n" + + commute_def + "commute(DD,ee,E,r) == \ +\ (ALL n:nat. emb(DD`n,E,r(n))) & \ +\ (ALL m:nat. ALL n:nat. m le n --> r(n) O eps(DD,ee,m,n) = r(m))" + + mediating_def + "mediating(E,G,r,f,t) == emb(E,G,t) & (ALL n:nat. f(n) = t O r(n))" + +end