# HG changeset patch # User kleing # Date 1049556306 -7200 # Node ID f62f9a75f68575e4cbdc240078e33d018107c151 # Parent 717bd79b976f8a760526c0b66e974095ed46f029 cleanup, mark old (<1994) deleted files as dead diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cfun3.ML --- a/src/HOLCF/cfun3.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,403 +0,0 @@ -(* Title: HOLCF/cfun3.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen -*) - -open Cfun3; - -(* ------------------------------------------------------------------------ *) -(* the contlub property for fapp its 'first' argument *) -(* ------------------------------------------------------------------------ *) - -val contlub_fapp1 = prove_goal Cfun3.thy "contlub(fapp)" -(fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (expand_fun_eq RS iffD2) 1), - (strip_tac 1), - (rtac (lub_cfun RS thelubI RS ssubst) 1), - (atac 1), - (rtac (Cfunapp2 RS ssubst) 1), - (etac contX_lubcfun 1), - (rtac (lub_fun RS thelubI RS ssubst) 1), - (etac (monofun_fapp1 RS ch2ch_monofun) 1), - (rtac refl 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* the contX property for fapp in its first argument *) -(* ------------------------------------------------------------------------ *) - -val contX_fapp1 = prove_goal Cfun3.thy "contX(fapp)" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_fapp1 1), - (rtac contlub_fapp1 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* contlub, contX properties of fapp in its first argument in mixfix _[_] *) -(* ------------------------------------------------------------------------ *) - -val contlub_cfun_fun = prove_goal Cfun3.thy -"is_chain(FY) ==>\ -\ lub(range(FY))[x] = lub(range(%i.FY(i)[x]))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans 1), - (etac (contlub_fapp1 RS contlubE RS spec RS mp RS fun_cong) 1), - (rtac (thelub_fun RS ssubst) 1), - (etac (monofun_fapp1 RS ch2ch_monofun) 1), - (rtac refl 1) - ]); - - -val contX_cfun_fun = prove_goal Cfun3.thy -"is_chain(FY) ==>\ -\ range(%i.FY(i)[x]) <<| lub(range(FY))[x]" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac thelubE 1), - (etac ch2ch_fappL 1), - (etac (contlub_cfun_fun RS sym) 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* contlub, contX properties of fapp in both argument in mixfix _[_] *) -(* ------------------------------------------------------------------------ *) - -val contlub_cfun = prove_goal Cfun3.thy -"[|is_chain(FY);is_chain(TY)|] ==>\ -\ lub(range(FY))[lub(range(TY))] = lub(range(%i.FY(i)[TY(i)]))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac contlub_CF2 1), - (rtac contX_fapp1 1), - (rtac allI 1), - (rtac contX_fapp2 1), - (atac 1), - (atac 1) - ]); - -val contX_cfun = prove_goal Cfun3.thy -"[|is_chain(FY);is_chain(TY)|] ==>\ -\ range(%i.FY(i)[TY(i)]) <<| lub(range(FY))[lub(range(TY))]" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac thelubE 1), - (rtac (monofun_fapp1 RS ch2ch_MF2LR) 1), - (rtac allI 1), - (rtac monofun_fapp2 1), - (atac 1), - (atac 1), - (etac (contlub_cfun RS sym) 1), - (atac 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* contX2contX lemma for fapp *) -(* ------------------------------------------------------------------------ *) - -val contX2contX_fapp = prove_goal Cfun3.thy - "[|contX(%x.ft(x));contX(%x.tt(x))|] ==> contX(%x.(ft(x))[tt(x)])" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac contX2contX_app2 1), - (rtac contX2contX_app2 1), - (rtac contX_const 1), - (rtac contX_fapp1 1), - (atac 1), - (rtac contX_fapp2 1), - (atac 1) - ]); - - - -(* ------------------------------------------------------------------------ *) -(* contX2mono Lemma for %x. LAM y. c1(x,y) *) -(* ------------------------------------------------------------------------ *) - -val contX2mono_LAM = prove_goal Cfun3.thy - "[|!x.contX(c1(x)); !y.monofun(%x.c1(x,y))|] ==>\ -\ monofun(%x. LAM y. c1(x,y))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac monofunI 1), - (strip_tac 1), - (rtac (less_cfun RS ssubst) 1), - (rtac (less_fun RS ssubst) 1), - (rtac allI 1), - (rtac (beta_cfun RS ssubst) 1), - (etac spec 1), - (rtac (beta_cfun RS ssubst) 1), - (etac spec 1), - (etac ((hd (tl prems)) RS spec RS monofunE RS spec RS spec RS mp) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* contX2contX Lemma for %x. LAM y. c1(x,y) *) -(* ------------------------------------------------------------------------ *) - -val contX2contX_LAM = prove_goal Cfun3.thy - "[| !x.contX(c1(x)); !y.contX(%x.c1(x,y)) |] ==> contX(%x. LAM y. c1(x,y))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac monocontlub2contX 1), - (etac contX2mono_LAM 1), - (rtac (contX2mono RS allI) 1), - (etac spec 1), - (rtac contlubI 1), - (strip_tac 1), - (rtac (thelub_cfun RS ssubst) 1), - (rtac (contX2mono_LAM RS ch2ch_monofun) 1), - (atac 1), - (rtac (contX2mono RS allI) 1), - (etac spec 1), - (atac 1), - (res_inst_tac [("f","fabs")] arg_cong 1), - (rtac ext 1), - (rtac (beta_cfun RS ext RS ssubst) 1), - (etac spec 1), - (rtac (contX2contlub RS contlubE - RS spec RS mp ) 1), - (etac spec 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* elimination of quantifier in premisses of contX2contX_LAM yields good *) -(* lemma for the contX tactic *) -(* ------------------------------------------------------------------------ *) - -val contX2contX_LAM2 = (allI RSN (2,(allI RS contX2contX_LAM))); -(* - [| !!x. contX(?c1.0(x)); !!y. contX(%x. ?c1.0(x,y)) |] ==> - contX(%x. LAM y. ?c1.0(x,y)) -*) - -(* ------------------------------------------------------------------------ *) -(* contX2contX tactic *) -(* ------------------------------------------------------------------------ *) - -val contX_lemmas = [contX_const, contX_id, contX_fapp2, - contX2contX_fapp,contX2contX_LAM2]; - - -val contX_tac = (fn i => (resolve_tac contX_lemmas i)); - -val contX_tacR = (fn i => (REPEAT (contX_tac i))); - -(* ------------------------------------------------------------------------ *) -(* function application _[_] is strict in its first arguments *) -(* ------------------------------------------------------------------------ *) - -val strict_fapp1 = prove_goal Cfun3.thy "UU[x] = UU" - (fn prems => - [ - (rtac (inst_cfun_pcpo RS ssubst) 1), - (rewrite_goals_tac [UU_cfun_def]), - (rtac (beta_cfun RS ssubst) 1), - (contX_tac 1), - (rtac refl 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* results about strictify *) -(* ------------------------------------------------------------------------ *) - -val Istrictify1 = prove_goalw Cfun3.thy [Istrictify_def] - "Istrictify(f)(UU)=UU" - (fn prems => - [ - (rtac select_equality 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val Istrictify2 = prove_goalw Cfun3.thy [Istrictify_def] - "~x=UU ==> Istrictify(f)(x)=f[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val monofun_Istrictify1 = prove_goal Cfun3.thy "monofun(Istrictify)" - (fn prems => - [ - (rtac monofunI 1), - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (res_inst_tac [("Q","xa=UU")] (excluded_middle RS disjE) 1), - (rtac (Istrictify2 RS ssubst) 1), - (atac 1), - (rtac (Istrictify2 RS ssubst) 1), - (atac 1), - (rtac monofun_cfun_fun 1), - (atac 1), - (hyp_subst_tac 1), - (rtac (Istrictify1 RS ssubst) 1), - (rtac (Istrictify1 RS ssubst) 1), - (rtac refl_less 1) - ]); - -val monofun_Istrictify2 = prove_goal Cfun3.thy "monofun(Istrictify(f))" - (fn prems => - [ - (rtac monofunI 1), - (strip_tac 1), - (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1), - (rtac (Istrictify2 RS ssubst) 1), - (etac notUU_I 1), - (atac 1), - (rtac (Istrictify2 RS ssubst) 1), - (atac 1), - (rtac monofun_cfun_arg 1), - (atac 1), - (hyp_subst_tac 1), - (rtac (Istrictify1 RS ssubst) 1), - (rtac minimal 1) - ]); - - -val contlub_Istrictify1 = prove_goal Cfun3.thy "contlub(Istrictify)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (expand_fun_eq RS iffD2) 1), - (strip_tac 1), - (rtac (thelub_fun RS ssubst) 1), - (etac (monofun_Istrictify1 RS ch2ch_monofun) 1), - (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1), - (rtac (Istrictify2 RS ssubst) 1), - (atac 1), - (rtac (Istrictify2 RS ext RS ssubst) 1), - (atac 1), - (rtac (thelub_cfun RS ssubst) 1), - (atac 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_lubcfun 1), - (atac 1), - (rtac refl 1), - (hyp_subst_tac 1), - (rtac (Istrictify1 RS ssubst) 1), - (rtac (Istrictify1 RS ext RS ssubst) 1), - (rtac (chain_UU_I_inverse RS sym) 1), - (rtac (refl RS allI) 1) - ]); - -val contlub_Istrictify2 = prove_goal Cfun3.thy "contlub(Istrictify(f))" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (res_inst_tac [("Q","lub(range(Y))=UU")] classical2 1), - (res_inst_tac [("t","lub(range(Y))")] subst 1), - (rtac sym 1), - (atac 1), - (rtac (Istrictify1 RS ssubst) 1), - (rtac sym 1), - (rtac chain_UU_I_inverse 1), - (strip_tac 1), - (res_inst_tac [("t","Y(i)"),("s","UU")] subst 1), - (rtac sym 1), - (rtac (chain_UU_I RS spec) 1), - (atac 1), - (atac 1), - (rtac Istrictify1 1), - (rtac (Istrictify2 RS ssubst) 1), - (atac 1), - (res_inst_tac [("s","lub(range(%i. f[Y(i)]))")] trans 1), - (rtac contlub_cfun_arg 1), - (atac 1), - (rtac lub_equal2 1), - (rtac (chain_mono2 RS exE) 1), - (atac 2), - (rtac chain_UU_I_inverse2 1), - (atac 1), - (rtac exI 1), - (strip_tac 1), - (rtac (Istrictify2 RS sym) 1), - (fast_tac HOL_cs 1), - (rtac ch2ch_monofun 1), - (rtac monofun_fapp2 1), - (atac 1), - (rtac ch2ch_monofun 1), - (rtac monofun_Istrictify2 1), - (atac 1) - ]); - - -val contX_Istrictify1 = (contlub_Istrictify1 RS - (monofun_Istrictify1 RS monocontlub2contX)); - -val contX_Istrictify2 = (contlub_Istrictify2 RS - (monofun_Istrictify2 RS monocontlub2contX)); - - -val strictify1 = prove_goalw Cfun3.thy [strictify_def] - "strictify[f][UU]=UU" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tac 1), - (rtac contX_Istrictify2 1), - (rtac contX2contX_CF1L 1), - (rtac contX_Istrictify1 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Istrictify2 1), - (rtac Istrictify1 1) - ]); - -val strictify2 = prove_goalw Cfun3.thy [strictify_def] - "~x=UU ==> strictify[f][x]=f[x]" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tac 1), - (rtac contX_Istrictify2 1), - (rtac contX2contX_CF1L 1), - (rtac contX_Istrictify1 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Istrictify2 1), - (rtac Istrictify2 1), - (resolve_tac prems 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Instantiate the simplifier *) -(* ------------------------------------------------------------------------ *) - -val Cfun_rews = [minimal,refl_less,beta_cfun,strict_fapp1,strictify1, - strictify2]; - -(* ------------------------------------------------------------------------ *) -(* use contX_tac as autotac. *) -(* ------------------------------------------------------------------------ *) - -val Cfun_ss = HOL_ss - addsimps Cfun_rews - setsolver - (fn thms => (resolve_tac (TrueI::refl::thms)) ORELSE' atac ORELSE' - (fn i => DEPTH_SOLVE_1 (contX_tac i)) - ); diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cfun3.thy --- a/src/HOLCF/cfun3.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,31 +0,0 @@ -(* Title: HOLCF/cfun3.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class instance of -> for class pcpo - -*) - -Cfun3 = Cfun2 + - -arities "->" :: (pcpo,pcpo)pcpo (* Witness cfun2.ML *) - -consts - Istrictify :: "('a->'b)=>'a=>'b" - strictify :: "('a->'b)->'a->'b" - -rules - -inst_cfun_pcpo "UU::'a->'b = UU_cfun" - -Istrictify_def "Istrictify(f,x) == (@z.\ -\ ( x=UU --> z = UU)\ -\ & (~x=UU --> z = f[x]))" - -strictify_def "strictify == (LAM f x.Istrictify(f,x))" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cinfix.ML --- a/src/HOLCF/cinfix.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,73 +0,0 @@ -(* Title: HOLCF/cinfix.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -Some functions for the print and parse translation of continuous functions - -Suppose the user introduces the following notation for the continuous -infixl . and the cont. infixr # with binding power 100 - -consts - "." :: "'a => 'b => ('a**'b)" ("_._" [100,101] 100) - "cop ." :: "'a -> 'b -> ('a**'b)" ("spair") - - "#" :: "'a => 'b => ('a**'b)" ("_#_" [101,100] 100) - "cop #" :: "'a -> 'b -> ('a**'b)" ("spair2") - -the following functions are needed to set up proper translations -*) - -(* ----------------------------------------------------------------------- - a general purpose parse translation for continuous infix operators - this functions must be used for every cont. infix - ----------------------------------------------------------------------- *) - -fun mk_cinfixtr id = - (fn ts => - let val Cfapp = Const("fapp",dummyT) in - Cfapp $ (Cfapp$Const("cop "^id,dummyT)$(nth_elem (0,ts)))$ - (nth_elem (1,ts)) - end); - - - -(* ----------------------------------------------------------------------- - make a print translation for a cont. infix operator "cop ???" - this is a print translation for fapp and is installed only once - special translations for other mixfixes (e.g. If_then_else_fi) are also - defined. - ----------------------------------------------------------------------- *) - -fun fapptr' ts = - case ts of - [Const("fapp",T1)$Const(s,T2)$t1,t2] => - if ["c","o","p"," "] = take(4, explode s) - then Const(implode(drop(4, explode s)),dummyT)$t1$t2 - else raise Match - | [Const("fapp",dummyT)$ - (Const("fapp",T1)$Const("Icifte",T2)$t)$e1,e2] - => Const("@cifte",dummyT)$t$e1$e2 - | _ => raise Match; - - -(* ----------------------------------------------------------------------- - -for the example above, the following must be setup in the ML section - -val parse_translation = [(".",mk_cinfixtr "."), - ("#",mk_cinfixtr "#")]; - - -the print translation for fapp is setup only once in the system - -val print_translation = [("fapp",fapptr')]; - - ----------------------------------------------------------------------- *) - - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cont.ML --- a/src/HOLCF/cont.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,670 +0,0 @@ -(* Title: HOLCF/cont.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for cont.thy -*) - -open Cont; - -(* ------------------------------------------------------------------------ *) -(* access to definition *) -(* ------------------------------------------------------------------------ *) - -val contlubI = prove_goalw Cont.thy [contlub] - "! Y. is_chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))==>\ -\ contlub(f)" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - -val contlubE = prove_goalw Cont.thy [contlub] - " contlub(f)==>\ -\ ! Y. is_chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - - -val contXI = prove_goalw Cont.thy [contX] - "! Y. is_chain(Y) --> range(% i.f(Y(i))) <<| f(lub(range(Y))) ==> contX(f)" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - -val contXE = prove_goalw Cont.thy [contX] - "contX(f) ==> ! Y. is_chain(Y) --> range(% i.f(Y(i))) <<| f(lub(range(Y)))" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - - -val monofunI = prove_goalw Cont.thy [monofun] - "! x y. x << y --> f(x) << f(y) ==> monofun(f)" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - -val monofunE = prove_goalw Cont.thy [monofun] - "monofun(f) ==> ! x y. x << y --> f(x) << f(y)" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the main purpose of cont.thy is to show: *) -(* monofun(f) & contlub(f) <==> contX(f) *) -(* ------------------------------------------------------------------------ *) - -(* ------------------------------------------------------------------------ *) -(* monotone functions map chains to chains *) -(* ------------------------------------------------------------------------ *) - -val ch2ch_monofun= prove_goal Cont.thy - "[| monofun(f); is_chain(Y) |] ==> is_chain(%i. f(Y(i)))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_chainI 1), - (rtac allI 1), - (etac (monofunE RS spec RS spec RS mp) 1), - (etac (is_chainE RS spec) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* monotone functions map upper bound to upper bounds *) -(* ------------------------------------------------------------------------ *) - -val ub2ub_monofun = prove_goal Cont.thy - "[| monofun(f); range(Y) <| u|] ==> range(%i.f(Y(i))) <| f(u)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (etac (monofunE RS spec RS spec RS mp) 1), - (etac (ub_rangeE RS spec) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* left to right: monofun(f) & contlub(f) ==> contX(f) *) -(* ------------------------------------------------------------------------ *) - -val monocontlub2contX = prove_goalw Cont.thy [contX] - "[|monofun(f);contlub(f)|] ==> contX(f)" -(fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac thelubE 1), - (etac ch2ch_monofun 1), - (atac 1), - (etac (contlubE RS spec RS mp RS sym) 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* first a lemma about binary chains *) -(* ------------------------------------------------------------------------ *) - -val binchain_contX = prove_goal Cont.thy -"[| contX(f); x << y |] ==> range(%i. f(if(i = 0,x,y))) <<| f(y)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac subst 1), - (etac (contXE RS spec RS mp) 2), - (etac bin_chain 2), - (res_inst_tac [("y","y")] arg_cong 1), - (etac (lub_bin_chain RS thelubI) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* right to left: contX(f) ==> monofun(f) & contlub(f) *) -(* part1: contX(f) ==> monofun(f *) -(* ------------------------------------------------------------------------ *) - -val contX2mono = prove_goalw Cont.thy [monofun] - "contX(f) ==> monofun(f)" -(fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (res_inst_tac [("s","if(0 = 0,x,y)")] subst 1), - (rtac (binchain_contX RS is_ub_lub) 2), - (atac 2), - (atac 2), - (simp_tac nat_ss 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* right to left: contX(f) ==> monofun(f) & contlub(f) *) -(* part2: contX(f) ==> contlub(f) *) -(* ------------------------------------------------------------------------ *) - -val contX2contlub = prove_goalw Cont.thy [contlub] - "contX(f) ==> contlub(f)" -(fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac (thelubI RS sym) 1), - (etac (contXE RS spec RS mp) 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* The following results are about a curried function that is monotone *) -(* in both arguments *) -(* ------------------------------------------------------------------------ *) - -val ch2ch_MF2L = prove_goal Cont.thy -"[|monofun(MF2::('a::po=>'b::po=>'c::po));\ -\ is_chain(F)|] ==> is_chain(%i. MF2(F(i),x))" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac (ch2ch_monofun RS ch2ch_fun) 1), - (atac 1) - ]); - - -val ch2ch_MF2R = prove_goal Cont.thy "[|monofun(MF2(f)::('b::po=>'c::po));\ -\ is_chain(Y)|] ==> is_chain(%i. MF2(f,Y(i)))" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac ch2ch_monofun 1), - (atac 1) - ]); - -val ch2ch_MF2LR = prove_goal Cont.thy -"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\ -\ !f.monofun(MF2(f)::('b::po=>'c::pcpo));\ -\ is_chain(F); is_chain(Y)|] ==> \ -\ is_chain(%i. MF2(F(i))(Y(i)))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_chainI 1), - (strip_tac 1 ), - (rtac trans_less 1), - (etac (ch2ch_MF2L RS is_chainE RS spec) 1), - (atac 1), - ((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1)), - (etac (is_chainE RS spec) 1) - ]); - -val ch2ch_lubMF2R = prove_goal Cont.thy -"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\ -\ !f.monofun(MF2(f)::('b::po=>'c::pcpo));\ -\ is_chain(F);is_chain(Y)|] ==> \ -\ is_chain(%j. lub(range(%i. MF2(F(j),Y(i)))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (lub_mono RS allI RS is_chainI) 1), - ((rtac ch2ch_MF2R 1) THEN (etac spec 1)), - (atac 1), - ((rtac ch2ch_MF2R 1) THEN (etac spec 1)), - (atac 1), - (strip_tac 1), - (rtac (is_chainE RS spec) 1), - (etac ch2ch_MF2L 1), - (atac 1) - ]); - - -val ch2ch_lubMF2L = prove_goal Cont.thy -"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\ -\ !f.monofun(MF2(f)::('b::po=>'c::pcpo));\ -\ is_chain(F);is_chain(Y)|] ==> \ -\ is_chain(%i. lub(range(%j. MF2(F(j),Y(i)))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (lub_mono RS allI RS is_chainI) 1), - (etac ch2ch_MF2L 1), - (atac 1), - (etac ch2ch_MF2L 1), - (atac 1), - (strip_tac 1), - (rtac (is_chainE RS spec) 1), - ((rtac ch2ch_MF2R 1) THEN (etac spec 1)), - (atac 1) - ]); - - -val lub_MF2_mono = prove_goal Cont.thy -"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\ -\ !f.monofun(MF2(f)::('b::po=>'c::pcpo));\ -\ is_chain(F)|] ==> \ -\ monofun(% x.lub(range(% j.MF2(F(j),x))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac monofunI 1), - (strip_tac 1), - (rtac lub_mono 1), - (etac ch2ch_MF2L 1), - (atac 1), - (etac ch2ch_MF2L 1), - (atac 1), - (strip_tac 1), - ((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1)), - (atac 1) - ]); - - -val ex_lubMF2 = prove_goal Cont.thy -"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\ -\ !f.monofun(MF2(f)::('b::po=>'c::pcpo));\ -\ is_chain(F); is_chain(Y)|] ==> \ -\ lub(range(%j. lub(range(%i. MF2(F(j),Y(i)))))) =\ -\ lub(range(%i. lub(range(%j. MF2(F(j),Y(i))))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac antisym_less 1), - (rtac is_lub_thelub 1), - (etac ch2ch_lubMF2R 1), - (atac 1),(atac 1),(atac 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (rtac lub_mono 1), - ((rtac ch2ch_MF2R 1) THEN (etac spec 1)), - (atac 1), - (etac ch2ch_lubMF2L 1), - (atac 1),(atac 1),(atac 1), - (strip_tac 1), - (rtac is_ub_thelub 1), - (etac ch2ch_MF2L 1),(atac 1), - (rtac is_lub_thelub 1), - (etac ch2ch_lubMF2L 1), - (atac 1),(atac 1),(atac 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (rtac lub_mono 1), - (etac ch2ch_MF2L 1),(atac 1), - (etac ch2ch_lubMF2R 1), - (atac 1),(atac 1),(atac 1), - (strip_tac 1), - (rtac is_ub_thelub 1), - ((rtac ch2ch_MF2R 1) THEN (etac spec 1)), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* The following results are about a curried function that is continuous *) -(* in both arguments *) -(* ------------------------------------------------------------------------ *) - -val diag_lubCF2_1 = prove_goal Cont.thy -"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\ -\ lub(range(%i. lub(range(%j. CF2(FY(j))(TY(i)))))) =\ -\ lub(range(%i. CF2(FY(i))(TY(i))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac antisym_less 1), - (rtac is_lub_thelub 1), - (rtac ch2ch_lubMF2L 1), - (rtac contX2mono 1), - (atac 1), - (rtac allI 1), - (rtac contX2mono 1), - (etac spec 1), - (atac 1), - (atac 1), - (rtac ub_rangeI 1), - (strip_tac 1 ), - (rtac is_lub_thelub 1), - ((rtac ch2ch_MF2L 1) THEN (rtac contX2mono 1) THEN (atac 1)), - (atac 1), - (rtac ub_rangeI 1), - (strip_tac 1 ), - (res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1), - (rtac trans_less 1), - (rtac is_ub_thelub 2), - (rtac (chain_mono RS mp) 1), - ((rtac ch2ch_MF2R 1) THEN (rtac contX2mono 1) THEN (etac spec 1)), - (atac 1), - (atac 1), - ((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)), - (rtac allI 1), - ((rtac contX2mono 1) THEN (etac spec 1)), - (atac 1), - (atac 1), - (hyp_subst_tac 1), - (rtac is_ub_thelub 1), - ((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)), - (rtac allI 1), - ((rtac contX2mono 1) THEN (etac spec 1)), - (atac 1), - (atac 1), - (rtac trans_less 1), - (rtac is_ub_thelub 2), - (res_inst_tac [("x1","ia")] (chain_mono RS mp) 1), - ((rtac ch2ch_MF2L 1) THEN (etac contX2mono 1)), - (atac 1), - (atac 1), - ((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)), - (rtac allI 1), - ((rtac contX2mono 1) THEN (etac spec 1)), - (atac 1), - (atac 1), - (rtac lub_mono 1), - ((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)), - (rtac allI 1), - ((rtac contX2mono 1) THEN (etac spec 1)), - (atac 1), - (atac 1), - (rtac ch2ch_lubMF2L 1), - (rtac contX2mono 1), - (atac 1), - (rtac allI 1), - ((rtac contX2mono 1) THEN (etac spec 1)), - (atac 1), - (atac 1), - (strip_tac 1 ), - (rtac is_ub_thelub 1), - ((rtac ch2ch_MF2L 1) THEN (etac contX2mono 1)), - (atac 1) - ]); - - -val diag_lubCF2_2 = prove_goal Cont.thy -"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\ -\ lub(range(%j. lub(range(%i. CF2(FY(j))(TY(i)))))) =\ -\ lub(range(%i. CF2(FY(i))(TY(i))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans 1), - (rtac ex_lubMF2 1), - (rtac ((hd prems) RS contX2mono) 1), - (rtac allI 1), - (rtac (((hd (tl prems)) RS spec RS contX2mono)) 1), - (atac 1), - (atac 1), - (rtac diag_lubCF2_1 1), - (atac 1), - (atac 1), - (atac 1), - (atac 1) - ]); - - -val contlub_CF2 = prove_goal Cont.thy -"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\ -\ CF2(lub(range(FY)))(lub(range(TY))) = lub(range(%i.CF2(FY(i))(TY(i))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac ((hd prems) RS contX2contlub RS contlubE RS - spec RS mp RS ssubst) 1), - (atac 1), - (rtac (thelub_fun RS ssubst) 1), - (rtac ((hd prems) RS contX2mono RS ch2ch_monofun) 1), - (atac 1), - (rtac trans 1), - (rtac (((hd (tl prems)) RS spec RS contX2contlub) RS - contlubE RS spec RS mp RS ext RS arg_cong RS arg_cong) 1), - (atac 1), - (rtac diag_lubCF2_2 1), - (atac 1), - (atac 1), - (atac 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* The following results are about application for functions in 'a=>'b *) -(* ------------------------------------------------------------------------ *) - -val monofun_fun_fun = prove_goal Cont.thy - "f1 << f2 ==> f1(x) << f2(x)" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac (less_fun RS iffD1 RS spec) 1) - ]); - -val monofun_fun_arg = prove_goal Cont.thy - "[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac (monofunE RS spec RS spec RS mp) 1), - (atac 1) - ]); - -val monofun_fun = prove_goal Cont.thy -"[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans_less 1), - (etac monofun_fun_arg 1), - (atac 1), - (etac monofun_fun_fun 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* The following results are about the propagation of monotonicity and *) -(* continuity *) -(* ------------------------------------------------------------------------ *) - -val mono2mono_MF1L = prove_goal Cont.thy - "[|monofun(c1)|] ==> monofun(%x. c1(x,y))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac monofunI 1), - (strip_tac 1), - (etac (monofun_fun_arg RS monofun_fun_fun) 1), - (atac 1) - ]); - -val contX2contX_CF1L = prove_goal Cont.thy - "[|contX(c1)|] ==> contX(%x. c1(x,y))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac monocontlub2contX 1), - (etac (contX2mono RS mono2mono_MF1L) 1), - (rtac contlubI 1), - (strip_tac 1), - (rtac ((hd prems) RS contX2contlub RS - contlubE RS spec RS mp RS ssubst) 1), - (atac 1), - (rtac (thelub_fun RS ssubst) 1), - (rtac ch2ch_monofun 1), - (etac contX2mono 1), - (atac 1), - (rtac refl 1) - ]); - -(********* Note "(%x.%y.c1(x,y)) = c1" ***********) - -val mono2mono_MF1L_rev = prove_goal Cont.thy - "!y.monofun(%x.c1(x,y)) ==> monofun(c1)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac monofunI 1), - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (rtac ((hd prems) RS spec RS monofunE RS spec RS spec RS mp) 1), - (atac 1) - ]); - -val contX2contX_CF1L_rev = prove_goal Cont.thy - "!y.contX(%x.c1(x,y)) ==> contX(c1)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac monocontlub2contX 1), - (rtac (contX2mono RS allI RS mono2mono_MF1L_rev ) 1), - (etac spec 1), - (rtac contlubI 1), - (strip_tac 1), - (rtac ext 1), - (rtac (thelub_fun RS ssubst) 1), - (rtac (contX2mono RS allI RS mono2mono_MF1L_rev RS ch2ch_monofun) 1), - (etac spec 1), - (atac 1), - (rtac - ((hd prems) RS spec RS contX2contlub RS contlubE RS spec RS mp) 1), - (atac 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* What D.A.Schmidt calls continuity of abstraction *) -(* never used here *) -(* ------------------------------------------------------------------------ *) - -val contlub_abstraction = prove_goal Cont.thy -"[|is_chain(Y::nat=>'a);!y.contX(%x.(c::'a=>'b=>'c)(x,y))|] ==>\ -\ (%y.lub(range(%i.c(Y(i),y)))) = (lub(range(%i.%y.c(Y(i),y))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans 1), - (rtac (contX2contlub RS contlubE RS spec RS mp) 2), - (atac 3), - (etac contX2contX_CF1L_rev 2), - (rtac ext 1), - (rtac (contX2contlub RS contlubE RS spec RS mp RS sym) 1), - (etac spec 1), - (atac 1) - ]); - - -val mono2mono_app = prove_goal Cont.thy -"[|monofun(ft);!x.monofun(ft(x));monofun(tt)|] ==>\ -\ monofun(%x.(ft(x))(tt(x)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac monofunI 1), - (strip_tac 1), - (res_inst_tac [("f1.0","ft(x)"),("f2.0","ft(y)")] monofun_fun 1), - (etac spec 1), - (etac spec 1), - (etac (monofunE RS spec RS spec RS mp) 1), - (atac 1), - (etac (monofunE RS spec RS spec RS mp) 1), - (atac 1) - ]); - -val contX2contlub_app = prove_goal Cont.thy -"[|contX(ft);!x.contX(ft(x));contX(tt)|] ==>\ -\ contlub(%x.(ft(x))(tt(x)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac contlubI 1), - (strip_tac 1), - (res_inst_tac [("f3","tt")] (contlubE RS spec RS mp RS ssubst) 1), - (rtac contX2contlub 1), - (resolve_tac prems 1), - (atac 1), - (rtac contlub_CF2 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (atac 1), - (rtac (contX2mono RS ch2ch_monofun) 1), - (resolve_tac prems 1), - (atac 1) - ]); - - -val contX2contX_app = prove_goal Cont.thy -"[|contX(ft);!x.contX(ft(x));contX(tt)|] ==>\ -\ contX(%x.(ft(x))(tt(x)))" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac mono2mono_app 1), - (rtac contX2mono 1), - (resolve_tac prems 1), - (strip_tac 1), - (rtac contX2mono 1), - (cut_facts_tac prems 1), - (etac spec 1), - (rtac contX2mono 1), - (resolve_tac prems 1), - (rtac contX2contlub_app 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); - - -val contX2contX_app2 = (allI RSN (2,contX2contX_app)); -(* [| contX(?ft); !!x. contX(?ft(x)); contX(?tt) |] ==> *) -(* contX(%x. ?ft(x,?tt(x))) *) - - -(* ------------------------------------------------------------------------ *) -(* The identity function is continuous *) -(* ------------------------------------------------------------------------ *) - -val contX_id = prove_goal Cont.thy "contX(% x.x)" - (fn prems => - [ - (rtac contXI 1), - (strip_tac 1), - (etac thelubE 1), - (rtac refl 1) - ]); - - - -(* ------------------------------------------------------------------------ *) -(* constant functions are continuous *) -(* ------------------------------------------------------------------------ *) - -val contX_const = prove_goalw Cont.thy [contX] "contX(%x.c)" - (fn prems => - [ - (strip_tac 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (rtac refl_less 1), - (strip_tac 1), - (dtac ub_rangeE 1), - (etac spec 1) - ]); - - -val contX2contX_app3 = prove_goal Cont.thy - "[|contX(f);contX(t) |] ==> contX(%x. f(t(x)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac contX2contX_app2 1), - (rtac contX_const 1), - (atac 1), - (atac 1) - ]); - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cont.thy --- a/src/HOLCF/cont.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,41 +0,0 @@ -(* Title: HOLCF/cont.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - Results about continuity and monotonicity -*) - -Cont = Fun3 + - -(* - - Now we change the default class! Form now on all untyped typevariables are - of default class pcpo - -*) - - -default pcpo - -consts - monofun :: "('a::po => 'b::po) => bool" (* monotonicity *) - contlub :: "('a => 'b) => bool" (* first cont. def *) - contX :: "('a => 'b) => bool" (* secnd cont. def *) - -rules - -monofun "monofun(f) == ! x y. x << y --> f(x) << f(y)" - -contlub "contlub(f) == ! Y. is_chain(Y) --> \ -\ f(lub(range(Y))) = lub(range(% i.f(Y(i))))" - -contX "contX(f) == ! Y. is_chain(Y) --> \ -\ range(% i.f(Y(i))) <<| f(lub(range(Y)))" - -(* ------------------------------------------------------------------------ *) -(* the main purpose of cont.thy is to show: *) -(* monofun(f) & contlub(f) <==> contX(f) *) -(* ------------------------------------------------------------------------ *) - -end diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cprod1.ML --- a/src/HOLCF/cprod1.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,117 +0,0 @@ -(* Title: HOLCF/cprod1.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory cprod1.thy -*) - -open Cprod1; - -val less_cprod1b = prove_goalw Cprod1.thy [less_cprod_def] - "less_cprod(p1,p2) = ( fst(p1) << fst(p2) & snd(p1) << snd(p2))" - (fn prems => - [ - (rtac refl 1) - ]); - -val less_cprod2a = prove_goalw Cprod1.thy [less_cprod_def] - "less_cprod(,) ==> x = UU & y = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac conjE 1), - (dtac (fst_conv RS subst) 1), - (dtac (fst_conv RS subst) 1), - (dtac (fst_conv RS subst) 1), - (dtac (snd_conv RS subst) 1), - (dtac (snd_conv RS subst) 1), - (dtac (snd_conv RS subst) 1), - (rtac conjI 1), - (etac UU_I 1), - (etac UU_I 1) - ]); - -val less_cprod2b = prove_goal Cprod1.thy - "less_cprod(p,) ==> p=" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p")] PairE 1), - (hyp_subst_tac 1), - (dtac less_cprod2a 1), - (asm_simp_tac HOL_ss 1) - ]); - -val less_cprod2c = prove_goalw Cprod1.thy [less_cprod_def] - "less_cprod(,) ==> x1 << x2 & y1 << y2" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac conjE 1), - (dtac (fst_conv RS subst) 1), - (dtac (fst_conv RS subst) 1), - (dtac (fst_conv RS subst) 1), - (dtac (snd_conv RS subst) 1), - (dtac (snd_conv RS subst) 1), - (dtac (snd_conv RS subst) 1), - (rtac conjI 1), - (atac 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* less_cprod is a partial order on 'a * 'b *) -(* ------------------------------------------------------------------------ *) - -val refl_less_cprod = prove_goalw Cprod1.thy [less_cprod_def] "less_cprod(p,p)" - (fn prems => - [ - (res_inst_tac [("p","p")] PairE 1), - (hyp_subst_tac 1), - (simp_tac pair_ss 1), - (simp_tac Cfun_ss 1) - ]); - -val antisym_less_cprod = prove_goal Cprod1.thy - "[|less_cprod(p1,p2);less_cprod(p2,p1)|] ==> p1=p2" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] PairE 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] PairE 1), - (hyp_subst_tac 1), - (dtac less_cprod2c 1), - (dtac less_cprod2c 1), - (etac conjE 1), - (etac conjE 1), - (rtac (Pair_eq RS ssubst) 1), - (fast_tac (HOL_cs addSIs [antisym_less]) 1) - ]); - - -val trans_less_cprod = prove_goal Cprod1.thy - "[|less_cprod(p1,p2);less_cprod(p2,p3)|] ==> less_cprod(p1,p3)" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] PairE 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p3")] PairE 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] PairE 1), - (hyp_subst_tac 1), - (dtac less_cprod2c 1), - (dtac less_cprod2c 1), - (rtac (less_cprod1b RS ssubst) 1), - (simp_tac pair_ss 1), - (etac conjE 1), - (etac conjE 1), - (rtac conjI 1), - (etac trans_less 1), - (atac 1), - (etac trans_less 1), - (atac 1) - ]); - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cprod1.thy --- a/src/HOLCF/cprod1.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,23 +0,0 @@ -(* Title: HOLCF/cprod1.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -Partial ordering for cartesian product of HOL theory prod.thy - -*) - -Cprod1 = Cfun3 + - - -consts - less_cprod :: "[('a::pcpo * 'b::pcpo),('a * 'b)] => bool" - -rules - - less_cprod_def "less_cprod(p1,p2) == ( fst(p1) << fst(p2) &\ -\ snd(p1) << snd(p2))" - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cprod2.ML --- a/src/HOLCF/cprod2.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,181 +0,0 @@ -(* Title: HOLCF/cprod2.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for cprod2.thy -*) - -open Cprod2; - -val less_cprod3a = prove_goal Cprod2.thy - "p1= ==> p1 << p2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (inst_cprod_po RS ssubst) 1), - (rtac (less_cprod1b RS ssubst) 1), - (hyp_subst_tac 1), - (asm_simp_tac pair_ss 1), - (rtac conjI 1), - (rtac minimal 1), - (rtac minimal 1) - ]); - -val less_cprod3b = prove_goal Cprod2.thy - "(p1 << p2) = (fst(p1)< - [ - (rtac (inst_cprod_po RS ssubst) 1), - (rtac less_cprod1b 1) - ]); - -val less_cprod4a = prove_goal Cprod2.thy - " << ==> x1=UU & x2=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac less_cprod2a 1), - (etac (inst_cprod_po RS subst) 1) - ]); - -val less_cprod4b = prove_goal Cprod2.thy - "p << ==> p = " -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac less_cprod2b 1), - (etac (inst_cprod_po RS subst) 1) - ]); - -val less_cprod4c = prove_goal Cprod2.thy - " << ==> xa< - [ - (cut_facts_tac prems 1), - (rtac less_cprod2c 1), - (etac (inst_cprod_po RS subst) 1), - (REPEAT (atac 1)) - ]); - -(* ------------------------------------------------------------------------ *) -(* type cprod is pointed *) -(* ------------------------------------------------------------------------ *) - -val minimal_cprod = prove_goal Cprod2.thy "< - [ - (rtac less_cprod3a 1), - (rtac refl 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Pair <_,_> is monotone in both arguments *) -(* ------------------------------------------------------------------------ *) - -val monofun_pair1 = prove_goalw Cprod2.thy [monofun] "monofun(Pair)" - (fn prems => - [ - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (rtac (less_cprod3b RS iffD2) 1), - (simp_tac pair_ss 1), - (asm_simp_tac Cfun_ss 1) - ]); - -val monofun_pair2 = prove_goalw Cprod2.thy [monofun] "monofun(Pair(x))" - (fn prems => - [ - (strip_tac 1), - (rtac (less_cprod3b RS iffD2) 1), - (simp_tac pair_ss 1), - (asm_simp_tac Cfun_ss 1) - ]); - -val monofun_pair = prove_goal Cprod2.thy - "[|x1< << " - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans_less 1), - (rtac (monofun_pair1 RS monofunE RS spec RS spec RS mp RS - (less_fun RS iffD1 RS spec)) 1), - (rtac (monofun_pair2 RS monofunE RS spec RS spec RS mp) 2), - (atac 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* fst and snd are monotone *) -(* ------------------------------------------------------------------------ *) - -val monofun_fst = prove_goalw Cprod2.thy [monofun] "monofun(fst)" - (fn prems => - [ - (strip_tac 1), - (res_inst_tac [("p","x")] PairE 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","y")] PairE 1), - (hyp_subst_tac 1), - (asm_simp_tac pair_ss 1), - (etac (less_cprod4c RS conjunct1) 1) - ]); - -val monofun_snd = prove_goalw Cprod2.thy [monofun] "monofun(snd)" - (fn prems => - [ - (strip_tac 1), - (res_inst_tac [("p","x")] PairE 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","y")] PairE 1), - (hyp_subst_tac 1), - (asm_simp_tac pair_ss 1), - (etac (less_cprod4c RS conjunct2) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the type 'a * 'b is a cpo *) -(* ------------------------------------------------------------------------ *) - -val lub_cprod = prove_goal Cprod2.thy -" is_chain(S) ==> range(S) <<| \ -\ < lub(range(%i.fst(S(i)))),lub(range(%i.snd(S(i))))> " - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (res_inst_tac [("t","S(i)")] (surjective_pairing RS ssubst) 1), - (rtac monofun_pair 1), - (rtac is_ub_thelub 1), - (etac (monofun_fst RS ch2ch_monofun) 1), - (rtac is_ub_thelub 1), - (etac (monofun_snd RS ch2ch_monofun) 1), - (strip_tac 1), - (res_inst_tac [("t","u")] (surjective_pairing RS ssubst) 1), - (rtac monofun_pair 1), - (rtac is_lub_thelub 1), - (etac (monofun_fst RS ch2ch_monofun) 1), - (etac (monofun_fst RS ub2ub_monofun) 1), - (rtac is_lub_thelub 1), - (etac (monofun_snd RS ch2ch_monofun) 1), - (etac (monofun_snd RS ub2ub_monofun) 1) - ]); - -val thelub_cprod = (lub_cprod RS thelubI); -(* "is_chain(?S1) ==> lub(range(?S1)) = *) -(* " *) - - -val cpo_cprod = prove_goal Cprod2.thy - "is_chain(S::nat=>'a*'b)==>? x.range(S)<<| x" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac exI 1), - (etac lub_cprod 1) - ]); - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cprod2.thy --- a/src/HOLCF/cprod2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,25 +0,0 @@ -(* Title: HOLCF/cprod2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class Instance *::(pcpo,pcpo)po - -*) - -Cprod2 = Cprod1 + - -(* Witness for the above arity axiom is cprod1.ML *) - -arities "*" :: (pcpo,pcpo)po - -rules - -(* instance of << for type ['a * 'b] *) - -inst_cprod_po "(op <<)::['a * 'b,'a * 'b]=>bool = less_cprod" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cprod3.ML --- a/src/HOLCF/cprod3.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,315 +0,0 @@ -(* Title: HOLCF/cprod3.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for Cprod3.thy -*) - -open Cprod3; - -(* ------------------------------------------------------------------------ *) -(* continuity of <_,_> , fst, snd *) -(* ------------------------------------------------------------------------ *) - -val Cprod3_lemma1 = prove_goal Cprod3.thy -"is_chain(Y::(nat=>'a)) ==>\ -\ =\ -\ ))),lub(range(%i. snd()))>" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("f1","Pair")] (arg_cong RS cong) 1), - (rtac lub_equal 1), - (atac 1), - (rtac (monofun_fst RS ch2ch_monofun) 1), - (rtac ch2ch_fun 1), - (rtac (monofun_pair1 RS ch2ch_monofun) 1), - (atac 1), - (rtac allI 1), - (simp_tac pair_ss 1), - (rtac sym 1), - (simp_tac pair_ss 1), - (rtac (lub_const RS thelubI) 1) - ]); - -val contlub_pair1 = prove_goal Cprod3.thy "contlub(Pair)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (expand_fun_eq RS iffD2) 1), - (strip_tac 1), - (rtac (lub_fun RS thelubI RS ssubst) 1), - (etac (monofun_pair1 RS ch2ch_monofun) 1), - (rtac trans 1), - (rtac (thelub_cprod RS sym) 2), - (rtac ch2ch_fun 2), - (etac (monofun_pair1 RS ch2ch_monofun) 2), - (etac Cprod3_lemma1 1) - ]); - -val Cprod3_lemma2 = prove_goal Cprod3.thy -"is_chain(Y::(nat=>'a)) ==>\ -\ <(x::'b),lub(range(Y))> =\ -\ ))),lub(range(%i. snd()))>" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("f1","Pair")] (arg_cong RS cong) 1), - (rtac sym 1), - (simp_tac pair_ss 1), - (rtac (lub_const RS thelubI) 1), - (rtac lub_equal 1), - (atac 1), - (rtac (monofun_snd RS ch2ch_monofun) 1), - (rtac (monofun_pair2 RS ch2ch_monofun) 1), - (atac 1), - (rtac allI 1), - (simp_tac pair_ss 1) - ]); - -val contlub_pair2 = prove_goal Cprod3.thy "contlub(Pair(x))" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_cprod RS sym) 2), - (etac (monofun_pair2 RS ch2ch_monofun) 2), - (etac Cprod3_lemma2 1) - ]); - -val contX_pair1 = prove_goal Cprod3.thy "contX(Pair)" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_pair1 1), - (rtac contlub_pair1 1) - ]); - -val contX_pair2 = prove_goal Cprod3.thy "contX(Pair(x))" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_pair2 1), - (rtac contlub_pair2 1) - ]); - -val contlub_fst = prove_goal Cprod3.thy "contlub(fst)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (lub_cprod RS thelubI RS ssubst) 1), - (atac 1), - (simp_tac pair_ss 1) - ]); - -val contlub_snd = prove_goal Cprod3.thy "contlub(snd)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (lub_cprod RS thelubI RS ssubst) 1), - (atac 1), - (simp_tac pair_ss 1) - ]); - -val contX_fst = prove_goal Cprod3.thy "contX(fst)" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_fst 1), - (rtac contlub_fst 1) - ]); - -val contX_snd = prove_goal Cprod3.thy "contX(snd)" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_snd 1), - (rtac contlub_snd 1) - ]); - -(* - -------------------------------------------------------------------------- - more lemmas for Cprod3.thy - - -------------------------------------------------------------------------- -*) - -(* ------------------------------------------------------------------------ *) -(* convert all lemmas to the continuous versions *) -(* ------------------------------------------------------------------------ *) - -val beta_cfun_cprod = prove_goalw Cprod3.thy [cpair_def] - "(LAM x y.)[a][b] = " - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tac 1), - (rtac contX_pair2 1), - (rtac contX2contX_CF1L 1), - (rtac contX_pair1 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_pair2 1), - (rtac refl 1) - ]); - -val inject_cpair = prove_goalw Cprod3.thy [cpair_def] - " (a#b)=(aa#ba) ==> a=aa & b=ba" - (fn prems => - [ - (cut_facts_tac prems 1), - (dtac (beta_cfun_cprod RS subst) 1), - (dtac (beta_cfun_cprod RS subst) 1), - (etac Pair_inject 1), - (fast_tac HOL_cs 1) - ]); - -val inst_cprod_pcpo2 = prove_goalw Cprod3.thy [cpair_def] "UU = (UU#UU)" - (fn prems => - [ - (rtac sym 1), - (rtac trans 1), - (rtac beta_cfun_cprod 1), - (rtac sym 1), - (rtac inst_cprod_pcpo 1) - ]); - -val defined_cpair_rev = prove_goal Cprod3.thy - "(a#b) = UU ==> a = UU & b = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (dtac (inst_cprod_pcpo2 RS subst) 1), - (etac inject_cpair 1) - ]); - -val Exh_Cprod2 = prove_goalw Cprod3.thy [cpair_def] - "? a b. z=(a#b) " - (fn prems => - [ - (rtac PairE 1), - (rtac exI 1), - (rtac exI 1), - (etac (beta_cfun_cprod RS ssubst) 1) - ]); - -val cprodE = prove_goalw Cprod3.thy [cpair_def] -"[|!!x y. [|p=(x#y) |] ==> Q|] ==> Q" - (fn prems => - [ - (rtac PairE 1), - (resolve_tac prems 1), - (etac (beta_cfun_cprod RS ssubst) 1) - ]); - -val cfst2 = prove_goalw Cprod3.thy [cfst_def,cpair_def] - "cfst[x#y]=x" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun_cprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_fst 1), - (simp_tac pair_ss 1) - ]); - -val csnd2 = prove_goalw Cprod3.thy [csnd_def,cpair_def] - "csnd[x#y]=y" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun_cprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_snd 1), - (simp_tac pair_ss 1) - ]); - -val surjective_pairing_Cprod2 = prove_goalw Cprod3.thy - [cfst_def,csnd_def,cpair_def] "(cfst[p] # csnd[p]) = p" - (fn prems => - [ - (rtac (beta_cfun_cprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_snd 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_fst 1), - (rtac (surjective_pairing RS sym) 1) - ]); - - -val less_cprod5b = prove_goalw Cprod3.thy [cfst_def,csnd_def,cpair_def] - " (p1 << p2) = (cfst[p1]< - [ - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_snd 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_snd 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_fst 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_fst 1), - (rtac less_cprod3b 1) - ]); - -val less_cprod5c = prove_goalw Cprod3.thy [cfst_def,csnd_def,cpair_def] - "xa#ya << x#y ==>xa< - [ - (cut_facts_tac prems 1), - (rtac less_cprod4c 1), - (dtac (beta_cfun_cprod RS subst) 1), - (dtac (beta_cfun_cprod RS subst) 1), - (atac 1) - ]); - - -val lub_cprod2 = prove_goalw Cprod3.thy [cfst_def,csnd_def,cpair_def] -"[|is_chain(S)|] ==> range(S) <<| \ -\ (lub(range(%i.cfst[S(i)])) # lub(range(%i.csnd[S(i)])))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun_cprod RS ssubst) 1), - (rtac (beta_cfun RS ext RS ssubst) 1), - (rtac contX_snd 1), - (rtac (beta_cfun RS ext RS ssubst) 1), - (rtac contX_fst 1), - (rtac lub_cprod 1), - (atac 1) - ]); - -val thelub_cprod2 = (lub_cprod2 RS thelubI); -(* "is_chain(?S1) ==> lub(range(?S1)) = *) -(* lub(range(%i. cfst[?S1(i)]))#lub(range(%i. csnd[?S1(i)]))" *) - -val csplit2 = prove_goalw Cprod3.thy [csplit_def] - "csplit[f][x#y]=f[x][y]" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tacR 1), - (simp_tac Cfun_ss 1), - (simp_tac (Cfun_ss addsimps [cfst2,csnd2]) 1) - ]); - -val csplit3 = prove_goalw Cprod3.thy [csplit_def] - "csplit[cpair][z]=z" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tacR 1), - (simp_tac (Cfun_ss addsimps [surjective_pairing_Cprod2]) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* install simplifier for Cprod *) -(* ------------------------------------------------------------------------ *) - -val Cprod_rews = [cfst2,csnd2,csplit2]; - -val Cprod_ss = Cfun_ss addsimps Cprod_rews; diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/cprod3.thy --- a/src/HOLCF/cprod3.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,43 +0,0 @@ -(* Title: HOLCF/cprod3.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -Class instance of * for class pcpo - -*) - -Cprod3 = Cprod2 + - -arities "*" :: (pcpo,pcpo)pcpo (* Witness cprod2.ML *) - -consts - "@cpair" :: "'a => 'b => ('a*'b)" ("_#_" [101,100] 100) - "cop @cpair" :: "'a -> 'b -> ('a*'b)" ("cpair") - (* continuous pairing *) - cfst :: "('a*'b)->'a" - csnd :: "('a*'b)->'b" - csplit :: "('a->'b->'c)->('a*'b)->'c" - -rules - -inst_cprod_pcpo "UU::'a*'b = " - -cpair_def "cpair == (LAM x y.)" -cfst_def "cfst == (LAM p.fst(p))" -csnd_def "csnd == (LAM p.snd(p))" -csplit_def "csplit == (LAM f p.f[cfst[p]][csnd[p]])" - -end - -ML - -(* ----------------------------------------------------------------------*) -(* parse translations for the above mixfix *) -(* ----------------------------------------------------------------------*) - -val parse_translation = [("@cpair",mk_cinfixtr "@cpair")]; - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/dlist.ML --- a/src/HOLCF/dlist.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,563 +0,0 @@ -(* Title: HOLCF/dlist.ML - Author: Franz Regensburger - ID: $ $ - Copyright 1994 Technische Universitaet Muenchen - -Lemmas for dlist.thy -*) - -open Dlist; - -(* ------------------------------------------------------------------------*) -(* The isomorphisms dlist_rep_iso and dlist_abs_iso are strict *) -(* ------------------------------------------------------------------------*) - -val dlist_iso_strict= dlist_rep_iso RS (dlist_abs_iso RS - (allI RSN (2,allI RS iso_strict))); - -val dlist_rews = [dlist_iso_strict RS conjunct1, - dlist_iso_strict RS conjunct2]; - -(* ------------------------------------------------------------------------*) -(* Properties of dlist_copy *) -(* ------------------------------------------------------------------------*) - -val temp = prove_goalw Dlist.thy [dlist_copy_def] "dlist_copy[f][UU]=UU" - (fn prems => - [ - (asm_simp_tac (HOLCF_ss addsimps - (dlist_rews @ [dlist_abs_iso,dlist_rep_iso])) 1) - ]); - -val dlist_copy = [temp]; - - -val temp = prove_goalw Dlist.thy [dlist_copy_def,dnil_def] - "dlist_copy[f][dnil]=dnil" - (fn prems => - [ - (asm_simp_tac (HOLCF_ss addsimps - (dlist_rews @ [dlist_abs_iso,dlist_rep_iso])) 1) - ]); - -val dlist_copy = temp :: dlist_copy; - - -val temp = prove_goalw Dlist.thy [dlist_copy_def,dcons_def] - "xl~=UU ==> dlist_copy[f][dcons[x][xl]]= dcons[x][f[xl]]" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps - (dlist_rews @ [dlist_abs_iso,dlist_rep_iso])) 1), - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac HOLCF_ss 1), - (asm_simp_tac (HOLCF_ss addsimps [defined_spair]) 1) - ]); - -val dlist_copy = temp :: dlist_copy; - -val dlist_rews = dlist_copy @ dlist_rews; - -(* ------------------------------------------------------------------------*) -(* Exhaustion and elimination for dlists *) -(* ------------------------------------------------------------------------*) - -val Exh_dlist = prove_goalw Dlist.thy [dcons_def,dnil_def] - "l = UU | l = dnil | (? x xl. x~=UU &xl~=UU & l = dcons[x][xl])" - (fn prems => - [ - (simp_tac HOLCF_ss 1), - (rtac (dlist_rep_iso RS subst) 1), - (res_inst_tac [("p","dlist_rep[l]")] ssumE 1), - (rtac disjI1 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (rtac disjI2 1), - (rtac disjI1 1), - (res_inst_tac [("p","x")] oneE 1), - (contr_tac 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (rtac disjI2 1), - (rtac disjI2 1), - (res_inst_tac [("p","y")] sprodE 1), - (contr_tac 1), - (rtac exI 1), - (rtac exI 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (fast_tac HOL_cs 1) - ]); - - -val dlistE = prove_goal Dlist.thy -"[| l=UU ==> Q; l=dnil ==> Q;!!x xl.[|l=dcons[x][xl];x~=UU;xl~=UU|]==>Q|]==>Q" - (fn prems => - [ - (rtac (Exh_dlist RS disjE) 1), - (eresolve_tac prems 1), - (etac disjE 1), - (eresolve_tac prems 1), - (etac exE 1), - (etac exE 1), - (resolve_tac prems 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -(* ------------------------------------------------------------------------*) -(* Properties of dlist_when *) -(* ------------------------------------------------------------------------*) - -val temp = prove_goalw Dlist.thy [dlist_when_def] "dlist_when[f1][f2][UU]=UU" - (fn prems => - [ - (asm_simp_tac (HOLCF_ss addsimps [dlist_iso_strict]) 1) - ]); - -val dlist_when = [temp]; - -val temp = prove_goalw Dlist.thy [dlist_when_def,dnil_def] - "dlist_when[f1][f2][dnil]= f1" - (fn prems => - [ - (asm_simp_tac (HOLCF_ss addsimps [dlist_abs_iso]) 1) - ]); - -val dlist_when = temp::dlist_when; - -val temp = prove_goalw Dlist.thy [dlist_when_def,dcons_def] - "[|x~=UU;xl~=UU|] ==> dlist_when[f1][f2][dcons[x][xl]]= f2[x][xl]" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps [dlist_abs_iso,defined_spair]) 1) - ]); - -val dlist_when = temp::dlist_when; - -val dlist_rews = dlist_when @ dlist_rews; - -(* ------------------------------------------------------------------------*) -(* Rewrites for discriminators and selectors *) -(* ------------------------------------------------------------------------*) - -fun prover defs thm = prove_goalw Dlist.thy defs thm - (fn prems => - [ - (simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_discsel = [ - prover [is_dnil_def] "is_dnil[UU]=UU", - prover [is_dcons_def] "is_dcons[UU]=UU", - prover [dhd_def] "dhd[UU]=UU", - prover [dtl_def] "dtl[UU]=UU" - ]; - -fun prover defs thm = prove_goalw Dlist.thy defs thm - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_discsel = [ -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "is_dnil[dnil]=TT", -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "[|x~=UU;xl~=UU|] ==> is_dnil[dcons[x][xl]]=FF", -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "is_dcons[dnil]=FF", -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "[|x~=UU;xl~=UU|] ==> is_dcons[dcons[x][xl]]=TT", -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "dhd[dnil]=UU", -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "[|x~=UU;xl~=UU|] ==> dhd[dcons[x][xl]]=x", -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "dtl[dnil]=UU", -prover [is_dnil_def,is_dcons_def,dhd_def,dtl_def] - "[|x~=UU;xl~=UU|] ==> dtl[dcons[x][xl]]=xl"] @ dlist_discsel; - -val dlist_rews = dlist_discsel @ dlist_rews; - -(* ------------------------------------------------------------------------*) -(* Definedness and strictness *) -(* ------------------------------------------------------------------------*) - -fun prover contr thm = prove_goal Dlist.thy thm - (fn prems => - [ - (res_inst_tac [("P1",contr)] classical3 1), - (simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (dtac sym 1), - (asm_simp_tac HOLCF_ss 1), - (simp_tac (HOLCF_ss addsimps (prems @ dlist_rews)) 1) - ]); - - -val dlist_constrdef = [ -prover "is_dnil[UU] ~= UU" "dnil~=UU", -prover "is_dcons[UU] ~= UU" "[|x~=UU;xl~=UU|] ==> dcons[x][xl]~=UU" - ]; - - -fun prover defs thm = prove_goalw Dlist.thy defs thm - (fn prems => - [ - (simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_constrdef = [ - prover [dcons_def] "dcons[UU][xl]=UU", - prover [dcons_def] "dcons[x][UU]=UU" - ] @ dlist_constrdef; - -val dlist_rews = dlist_constrdef @ dlist_rews; - - -(* ------------------------------------------------------------------------*) -(* Distinctness wrt. << and = *) -(* ------------------------------------------------------------------------*) - -val temp = prove_goal Dlist.thy "~dnil << dcons[x][xl]" - (fn prems => - [ - (res_inst_tac [("P1","TT << FF")] classical3 1), - (resolve_tac dist_less_tr 1), - (dres_inst_tac [("fo5","is_dnil")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (res_inst_tac [("Q","xl=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_dist_less = [temp]; - -val temp = prove_goal Dlist.thy "[|x~=UU;xl~=UU|]==>~dcons[x][xl] << dnil" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("P1","TT << FF")] classical3 1), - (resolve_tac dist_less_tr 1), - (dres_inst_tac [("fo5","is_dcons")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_dist_less = temp::dlist_dist_less; - -val temp = prove_goal Dlist.thy "dnil ~= dcons[x][xl]" - (fn prems => - [ - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (res_inst_tac [("Q","xl=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (res_inst_tac [("P1","TT = FF")] classical3 1), - (resolve_tac dist_eq_tr 1), - (dres_inst_tac [("f","is_dnil")] cfun_arg_cong 1), - (etac box_equals 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_dist_eq = [temp,temp RS not_sym]; - -val dlist_rews = dlist_dist_less @ dlist_dist_eq @ dlist_rews; - -(* ------------------------------------------------------------------------*) -(* Invertibility *) -(* ------------------------------------------------------------------------*) - -val temp = prove_goal Dlist.thy "[|x1~=UU; y1~=UU;x2~=UU; y2~=UU;\ -\ dcons[x1][x2] << dcons[y1][y2]|] ==> x1<< y1 & x2 << y2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac conjI 1), - (dres_inst_tac [("fo5","dlist_when[UU][LAM x l.x]")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1), - (dres_inst_tac [("fo5","dlist_when[UU][LAM x l.l]")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1) - ]); - -val dlist_invert =[temp]; - -(* ------------------------------------------------------------------------*) -(* Injectivity *) -(* ------------------------------------------------------------------------*) - -val temp = prove_goal Dlist.thy "[|x1~=UU; y1~=UU;x2~=UU; y2~=UU;\ -\ dcons[x1][x2] = dcons[y1][y2]|] ==> x1= y1 & x2 = y2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac conjI 1), - (dres_inst_tac [("f","dlist_when[UU][LAM x l.x]")] cfun_arg_cong 1), - (etac box_equals 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1), - (dres_inst_tac [("f","dlist_when[UU][LAM x l.l]")] cfun_arg_cong 1), - (etac box_equals 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_when) 1) - ]); - -val dlist_inject = [temp]; - - -(* ------------------------------------------------------------------------*) -(* definedness for discriminators and selectors *) -(* ------------------------------------------------------------------------*) - -fun prover thm = prove_goal Dlist.thy thm - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac dlistE 1), - (contr_tac 1), - (REPEAT (asm_simp_tac (HOLCF_ss addsimps dlist_discsel) 1)) - ]); - -val dlist_discsel_def = - [ - prover "l~=UU ==> is_dnil[l]~=UU", - prover "l~=UU ==> is_dcons[l]~=UU" - ]; - -val dlist_rews = dlist_discsel_def @ dlist_rews; - -(* ------------------------------------------------------------------------*) -(* enhance the simplifier *) -(* ------------------------------------------------------------------------*) - -val dhd2 = prove_goal Dlist.thy "xl~=UU ==> dhd[dcons[x][xl]]=x" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dtl2 = prove_goal Dlist.thy "x~=UU ==> dtl[dcons[x][xl]]=xl" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("Q","xl=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_rews = dhd2 :: dtl2 :: dlist_rews; - -(* ------------------------------------------------------------------------*) -(* Properties dlist_take *) -(* ------------------------------------------------------------------------*) - -val temp = prove_goalw Dlist.thy [dlist_take_def] "dlist_take(n)[UU]=UU" - (fn prems => - [ - (res_inst_tac [("n","n")] natE 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1), - (simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_take = [temp]; - -val temp = prove_goalw Dlist.thy [dlist_take_def] "dlist_take(0)[xs]=UU" - (fn prems => - [ - (asm_simp_tac iterate_ss 1) - ]); - -val dlist_take = temp::dlist_take; - -val temp = prove_goalw Dlist.thy [dlist_take_def] - "dlist_take(Suc(n))[dnil]=dnil" - (fn prems => - [ - (asm_simp_tac iterate_ss 1), - (simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_take = temp::dlist_take; - -val temp = prove_goalw Dlist.thy [dlist_take_def] - "dlist_take(Suc(n))[dcons[x][xl]]=dcons[x][dlist_take(n)[xl]]" - (fn prems => - [ - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (res_inst_tac [("Q","xl=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (res_inst_tac [("n","n")] natE 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_take = temp::dlist_take; - -val dlist_rews = dlist_take @ dlist_rews; - -(* ------------------------------------------------------------------------*) -(* take lemma for dlists *) -(* ------------------------------------------------------------------------*) - -fun prover reach defs thm = prove_goalw Dlist.thy defs thm - (fn prems => - [ - (res_inst_tac [("t","l1")] (reach RS subst) 1), - (res_inst_tac [("t","l2")] (reach RS subst) 1), - (rtac (fix_def2 RS ssubst) 1), - (rtac (contlub_cfun_fun RS ssubst) 1), - (rtac is_chain_iterate 1), - (rtac (contlub_cfun_fun RS ssubst) 1), - (rtac is_chain_iterate 1), - (rtac lub_equal 1), - (rtac (is_chain_iterate RS ch2ch_fappL) 1), - (rtac (is_chain_iterate RS ch2ch_fappL) 1), - (rtac allI 1), - (resolve_tac prems 1) - ]); - -val dlist_take_lemma = prover dlist_reach [dlist_take_def] - "(!!n.dlist_take(n)[l1]=dlist_take(n)[l2]) ==> l1=l2"; - - -(* ------------------------------------------------------------------------*) -(* Co -induction for dlists *) -(* ------------------------------------------------------------------------*) - -val dlist_coind_lemma = prove_goalw Dlist.thy [dlist_bisim_def] -"dlist_bisim(R) ==> ! p q.R(p,q) --> dlist_take(n)[p]=dlist_take(n)[q]" - (fn prems => - [ - (cut_facts_tac prems 1), - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (strip_tac 1), - ((etac allE 1) THEN (etac allE 1) THEN (etac (mp RS disjE) 1)), - (atac 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (etac disjE 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (etac exE 1), - (etac exE 1), - (etac exE 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (REPEAT (etac conjE 1)), - (rtac cfun_arg_cong 1), - (fast_tac HOL_cs 1) - ]); - -val dlist_coind = prove_goal Dlist.thy "[|dlist_bisim(R);R(p,q)|] ==> p = q" - (fn prems => - [ - (rtac dlist_take_lemma 1), - (rtac (dlist_coind_lemma RS spec RS spec RS mp) 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); - -(* ------------------------------------------------------------------------*) -(* structural induction *) -(* ------------------------------------------------------------------------*) - -val dlist_finite_ind = prove_goal Dlist.thy -"[|P(UU);P(dnil);\ -\ !! x l1.[|x~=UU;l1~=UU;P(l1)|] ==> P(dcons[x][l1])\ -\ |] ==> !l.P(dlist_take(n)[l])" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (resolve_tac prems 1), - (rtac allI 1), - (res_inst_tac [("l","l")] dlistE 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (resolve_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (resolve_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (res_inst_tac [("Q","dlist_take(n1)[xl]=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (atac 1), - (atac 1), - (etac spec 1) - ]); - -val dlist_all_finite_lemma1 = prove_goal Dlist.thy -"!l.dlist_take(n)[l]=UU |dlist_take(n)[l]=l" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (rtac allI 1), - (res_inst_tac [("l","l")] dlistE 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (eres_inst_tac [("x","xl")] allE 1), - (etac disjE 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1) - ]); - -val dlist_all_finite_lemma2 = prove_goal Dlist.thy "? n.dlist_take(n)[l]=l" - (fn prems => - [ - (res_inst_tac [("Q","l=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (subgoal_tac "(!n.dlist_take(n)[l]=UU) |(? n.dlist_take(n)[l]=l)" 1), - (etac disjE 1), - (eres_inst_tac [("P","l=UU")] notE 1), - (rtac dlist_take_lemma 1), - (asm_simp_tac (HOLCF_ss addsimps dlist_rews) 1), - (atac 1), - (subgoal_tac "!n.!l.dlist_take(n)[l]=UU |dlist_take(n)[l]=l" 1), - (fast_tac HOL_cs 1), - (rtac allI 1), - (rtac dlist_all_finite_lemma1 1) - ]); - -val dlist_all_finite= prove_goalw Dlist.thy [dlist_finite_def] "dlist_finite(l)" - (fn prems => - [ - (rtac dlist_all_finite_lemma2 1) - ]); - -val dlist_ind = prove_goal Dlist.thy -"[|P(UU);P(dnil);\ -\ !! x l1.[|x~=UU;l1~=UU;P(l1)|] ==> P(dcons[x][l1])|] ==> P(l)" - (fn prems => - [ - (rtac (dlist_all_finite_lemma2 RS exE) 1), - (etac subst 1), - (rtac (dlist_finite_ind RS spec) 1), - (REPEAT (resolve_tac prems 1)), - (REPEAT (atac 1)) - ]); - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/dlist.thy --- a/src/HOLCF/dlist.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,111 +0,0 @@ -(* Title: HOLCF/dlist.thy - - Author: Franz Regensburger - ID: $ $ - Copyright 1994 Technische Universitaet Muenchen - -Theory for lists -*) - -Dlist = Stream2 + - -types dlist 1 - -(* ----------------------------------------------------------------------- *) -(* arity axiom is validated by semantic reasoning *) -(* partial ordering is implicit in the isomorphism axioms and their cont. *) - -arities dlist::(pcpo)pcpo - -consts - -(* ----------------------------------------------------------------------- *) -(* essential constants *) - -dlist_rep :: "('a dlist) -> (one ++ 'a ** 'a dlist)" -dlist_abs :: "(one ++ 'a ** 'a dlist) -> ('a dlist)" - -(* ----------------------------------------------------------------------- *) -(* abstract constants and auxiliary constants *) - -dlist_copy :: "('a dlist -> 'a dlist) ->'a dlist -> 'a dlist" - -dnil :: "'a dlist" -dcons :: "'a -> 'a dlist -> 'a dlist" -dlist_when :: " 'b -> ('a -> 'a dlist -> 'b) -> 'a dlist -> 'b" -is_dnil :: "'a dlist -> tr" -is_dcons :: "'a dlist -> tr" -dhd :: "'a dlist -> 'a" -dtl :: "'a dlist -> 'a dlist" -dlist_take :: "nat => 'a dlist -> 'a dlist" -dlist_finite :: "'a dlist => bool" -dlist_bisim :: "('a dlist => 'a dlist => bool) => bool" - -rules - -(* ----------------------------------------------------------------------- *) -(* axiomatization of recursive type 'a dlist *) -(* ----------------------------------------------------------------------- *) -(* ('a dlist,dlist_abs) is the initial F-algebra where *) -(* F is the locally continuous functor determined by domain equation *) -(* X = one ++ 'a ** X *) -(* ----------------------------------------------------------------------- *) -(* dlist_abs is an isomorphism with inverse dlist_rep *) -(* identity is the least endomorphism on 'a dlist *) - -dlist_abs_iso "dlist_rep[dlist_abs[x]] = x" -dlist_rep_iso "dlist_abs[dlist_rep[x]] = x" -dlist_copy_def "dlist_copy == (LAM f. dlist_abs oo \ -\ (when[sinl][sinr oo (ssplit[LAM x y. x ## f[y]])])\ -\ oo dlist_rep)" -dlist_reach "(fix[dlist_copy])[x]=x" - -(* ----------------------------------------------------------------------- *) -(* properties of additional constants *) -(* ----------------------------------------------------------------------- *) -(* constructors *) - -dnil_def "dnil == dlist_abs[sinl[one]]" -dcons_def "dcons == (LAM x l. dlist_abs[sinr[x##l]])" - -(* ----------------------------------------------------------------------- *) -(* discriminator functional *) - -dlist_when_def -"dlist_when == (LAM f1 f2 l.\ -\ when[LAM x.f1][ssplit[LAM x l.f2[x][l]]][dlist_rep[l]])" - -(* ----------------------------------------------------------------------- *) -(* discriminators and selectors *) - -is_dnil_def "is_dnil == dlist_when[TT][LAM x l.FF]" -is_dcons_def "is_dcons == dlist_when[FF][LAM x l.TT]" -dhd_def "dhd == dlist_when[UU][LAM x l.x]" -dtl_def "dtl == dlist_when[UU][LAM x l.l]" - -(* ----------------------------------------------------------------------- *) -(* the taker for dlists *) - -dlist_take_def "dlist_take == (%n.iterate(n,dlist_copy,UU))" - -(* ----------------------------------------------------------------------- *) - -dlist_finite_def "dlist_finite == (%s.? n.dlist_take(n)[s]=s)" - -(* ----------------------------------------------------------------------- *) -(* definition of bisimulation is determined by domain equation *) -(* simplification and rewriting for abstract constants yields def below *) - -dlist_bisim_def "dlist_bisim ==\ -\ ( %R.!l1 l2.\ -\ R(l1,l2) -->\ -\ ((l1=UU & l2=UU) |\ -\ (l1=dnil & l2=dnil) |\ -\ (? x l11 l21. x~=UU & l11~=UU & l21~=UU & \ -\ l1=dcons[x][l11] & l2 = dcons[x][l21] & R(l11,l21))))" - -end - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/dnat.ML --- a/src/HOLCF/dnat.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,532 +0,0 @@ -(* Title: HOLCF/dnat.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for dnat.thy -*) - -open Dnat; - -(* ------------------------------------------------------------------------*) -(* The isomorphisms dnat_rep_iso and dnat_abs_iso are strict *) -(* ------------------------------------------------------------------------*) - -val dnat_iso_strict= dnat_rep_iso RS (dnat_abs_iso RS - (allI RSN (2,allI RS iso_strict))); - -val dnat_rews = [dnat_iso_strict RS conjunct1, - dnat_iso_strict RS conjunct2]; - -(* ------------------------------------------------------------------------*) -(* Properties of dnat_copy *) -(* ------------------------------------------------------------------------*) - -fun prover defs thm = prove_goalw Dnat.thy defs thm - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps - (dnat_rews @ [dnat_abs_iso,dnat_rep_iso])) 1) - ]); - -val dnat_copy = - [ - prover [dnat_copy_def] "dnat_copy[f][UU]=UU", - prover [dnat_copy_def,dzero_def] "dnat_copy[f][dzero]= dzero", - prover [dnat_copy_def,dsucc_def] - "n~=UU ==> dnat_copy[f][dsucc[n]] = dsucc[f[n]]" - ]; - -val dnat_rews = dnat_copy @ dnat_rews; - -(* ------------------------------------------------------------------------*) -(* Exhaustion and elimination for dnat *) -(* ------------------------------------------------------------------------*) - -val Exh_dnat = prove_goalw Dnat.thy [dsucc_def,dzero_def] - "n = UU | n = dzero | (? x . x~=UU & n = dsucc[x])" - (fn prems => - [ - (simp_tac HOLCF_ss 1), - (rtac (dnat_rep_iso RS subst) 1), - (res_inst_tac [("p","dnat_rep[n]")] ssumE 1), - (rtac disjI1 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (rtac (disjI1 RS disjI2) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (res_inst_tac [("p","x")] oneE 1), - (contr_tac 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (rtac (disjI2 RS disjI2) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (fast_tac HOL_cs 1) - ]); - -val dnatE = prove_goal Dnat.thy - "[| n=UU ==> Q; n=dzero ==> Q; !!x.[|n=dsucc[x];x~=UU|]==>Q|]==>Q" - (fn prems => - [ - (rtac (Exh_dnat RS disjE) 1), - (eresolve_tac prems 1), - (etac disjE 1), - (eresolve_tac prems 1), - (REPEAT (etac exE 1)), - (resolve_tac prems 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -(* ------------------------------------------------------------------------*) -(* Properties of dnat_when *) -(* ------------------------------------------------------------------------*) - -fun prover defs thm = prove_goalw Dnat.thy defs thm - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps - (dnat_rews @ [dnat_abs_iso,dnat_rep_iso])) 1) - ]); - - -val dnat_when = [ - prover [dnat_when_def] "dnat_when[c][f][UU]=UU", - prover [dnat_when_def,dzero_def] "dnat_when[c][f][dzero]=c", - prover [dnat_when_def,dsucc_def] - "n~=UU ==> dnat_when[c][f][dsucc[n]]=f[n]" - ]; - -val dnat_rews = dnat_when @ dnat_rews; - -(* ------------------------------------------------------------------------*) -(* Rewrites for discriminators and selectors *) -(* ------------------------------------------------------------------------*) - -fun prover defs thm = prove_goalw Dnat.thy defs thm - (fn prems => - [ - (simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_discsel = [ - prover [is_dzero_def] "is_dzero[UU]=UU", - prover [is_dsucc_def] "is_dsucc[UU]=UU", - prover [dpred_def] "dpred[UU]=UU" - ]; - - -fun prover defs thm = prove_goalw Dnat.thy defs thm - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_discsel = [ - prover [is_dzero_def] "is_dzero[dzero]=TT", - prover [is_dzero_def] "n~=UU ==>is_dzero[dsucc[n]]=FF", - prover [is_dsucc_def] "is_dsucc[dzero]=FF", - prover [is_dsucc_def] "n~=UU ==> is_dsucc[dsucc[n]]=TT", - prover [dpred_def] "dpred[dzero]=UU", - prover [dpred_def] "n~=UU ==> dpred[dsucc[n]]=n" - ] @ dnat_discsel; - -val dnat_rews = dnat_discsel @ dnat_rews; - -(* ------------------------------------------------------------------------*) -(* Definedness and strictness *) -(* ------------------------------------------------------------------------*) - -fun prover contr thm = prove_goal Dnat.thy thm - (fn prems => - [ - (res_inst_tac [("P1",contr)] classical3 1), - (simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (dtac sym 1), - (asm_simp_tac HOLCF_ss 1), - (simp_tac (HOLCF_ss addsimps (prems @ dnat_rews)) 1) - ]); - -val dnat_constrdef = [ - prover "is_dzero[UU] ~= UU" "dzero~=UU", - prover "is_dsucc[UU] ~= UU" "n~=UU ==> dsucc[n]~=UU" - ]; - - -fun prover defs thm = prove_goalw Dnat.thy defs thm - (fn prems => - [ - (simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_constrdef = [ - prover [dsucc_def] "dsucc[UU]=UU" - ] @ dnat_constrdef; - -val dnat_rews = dnat_constrdef @ dnat_rews; - - -(* ------------------------------------------------------------------------*) -(* Distinctness wrt. << and = *) -(* ------------------------------------------------------------------------*) - -val temp = prove_goal Dnat.thy "~dzero << dsucc[n]" - (fn prems => - [ - (res_inst_tac [("P1","TT << FF")] classical3 1), - (resolve_tac dist_less_tr 1), - (dres_inst_tac [("fo5","is_dzero")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (res_inst_tac [("Q","n=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_dist_less = [temp]; - -val temp = prove_goal Dnat.thy "n~=UU ==> ~dsucc[n] << dzero" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("P1","TT << FF")] classical3 1), - (resolve_tac dist_less_tr 1), - (dres_inst_tac [("fo5","is_dsucc")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_dist_less = temp::dnat_dist_less; - -val temp = prove_goal Dnat.thy "dzero ~= dsucc[n]" - (fn prems => - [ - (res_inst_tac [("Q","n=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (res_inst_tac [("P1","TT = FF")] classical3 1), - (resolve_tac dist_eq_tr 1), - (dres_inst_tac [("f","is_dzero")] cfun_arg_cong 1), - (etac box_equals 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_dist_eq = [temp, temp RS not_sym]; - -val dnat_rews = dnat_dist_less @ dnat_dist_eq @ dnat_rews; - -(* ------------------------------------------------------------------------*) -(* Invertibility *) -(* ------------------------------------------------------------------------*) - -val dnat_invert = - [ -prove_goal Dnat.thy -"[|x1~=UU; y1~=UU; dsucc[x1] << dsucc[y1] |] ==> x1<< y1" - (fn prems => - [ - (cut_facts_tac prems 1), - (dres_inst_tac [("fo5","dnat_when[c][LAM x.x]")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]) - ]; - -(* ------------------------------------------------------------------------*) -(* Injectivity *) -(* ------------------------------------------------------------------------*) - -val dnat_inject = - [ -prove_goal Dnat.thy -"[|x1~=UU; y1~=UU; dsucc[x1] = dsucc[y1] |] ==> x1= y1" - (fn prems => - [ - (cut_facts_tac prems 1), - (dres_inst_tac [("f","dnat_when[c][LAM x.x]")] cfun_arg_cong 1), - (etac box_equals 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]) - ]; - -(* ------------------------------------------------------------------------*) -(* definedness for discriminators and selectors *) -(* ------------------------------------------------------------------------*) - - -fun prover thm = prove_goal Dnat.thy thm - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac dnatE 1), - (contr_tac 1), - (REPEAT (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1)) - ]); - -val dnat_discsel_def = - [ - prover "n~=UU ==> is_dzero[n]~=UU", - prover "n~=UU ==> is_dsucc[n]~=UU" - ]; - -val dnat_rews = dnat_discsel_def @ dnat_rews; - - -(* ------------------------------------------------------------------------*) -(* Properties dnat_take *) -(* ------------------------------------------------------------------------*) -val temp = prove_goalw Dnat.thy [dnat_take_def] "dnat_take(n)[UU]=UU" - (fn prems => - [ - (res_inst_tac [("n","n")] natE 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1), - (simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_take = [temp]; - -val temp = prove_goalw Dnat.thy [dnat_take_def] "dnat_take(0)[xs]=UU" - (fn prems => - [ - (asm_simp_tac iterate_ss 1) - ]); - -val dnat_take = temp::dnat_take; - -val temp = prove_goalw Dnat.thy [dnat_take_def] - "dnat_take(Suc(n))[dzero]=dzero" - (fn prems => - [ - (asm_simp_tac iterate_ss 1), - (simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_take = temp::dnat_take; - -val temp = prove_goalw Dnat.thy [dnat_take_def] - "dnat_take(Suc(n))[dsucc[xs]]=dsucc[dnat_take(n)[xs]]" - (fn prems => - [ - (res_inst_tac [("Q","xs=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (res_inst_tac [("n","n")] natE 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_take = temp::dnat_take; - -val dnat_rews = dnat_take @ dnat_rews; - - -(* ------------------------------------------------------------------------*) -(* take lemma for dnats *) -(* ------------------------------------------------------------------------*) - -fun prover reach defs thm = prove_goalw Dnat.thy defs thm - (fn prems => - [ - (res_inst_tac [("t","s1")] (reach RS subst) 1), - (res_inst_tac [("t","s2")] (reach RS subst) 1), - (rtac (fix_def2 RS ssubst) 1), - (rtac (contlub_cfun_fun RS ssubst) 1), - (rtac is_chain_iterate 1), - (rtac (contlub_cfun_fun RS ssubst) 1), - (rtac is_chain_iterate 1), - (rtac lub_equal 1), - (rtac (is_chain_iterate RS ch2ch_fappL) 1), - (rtac (is_chain_iterate RS ch2ch_fappL) 1), - (rtac allI 1), - (resolve_tac prems 1) - ]); - -val dnat_take_lemma = prover dnat_reach [dnat_take_def] - "(!!n.dnat_take(n)[s1]=dnat_take(n)[s2]) ==> s1=s2"; - - -(* ------------------------------------------------------------------------*) -(* Co -induction for dnats *) -(* ------------------------------------------------------------------------*) - -val dnat_coind_lemma = prove_goalw Dnat.thy [dnat_bisim_def] -"dnat_bisim(R) ==> ! p q.R(p,q) --> dnat_take(n)[p]=dnat_take(n)[q]" - (fn prems => - [ - (cut_facts_tac prems 1), - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps dnat_take) 1), - (strip_tac 1), - ((etac allE 1) THEN (etac allE 1) THEN (etac (mp RS disjE) 1)), - (atac 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_take) 1), - (etac disjE 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_take) 1), - (etac exE 1), - (etac exE 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_take) 1), - (REPEAT (etac conjE 1)), - (rtac cfun_arg_cong 1), - (fast_tac HOL_cs 1) - ]); - -val dnat_coind = prove_goal Dnat.thy "[|dnat_bisim(R);R(p,q)|] ==> p = q" - (fn prems => - [ - (rtac dnat_take_lemma 1), - (rtac (dnat_coind_lemma RS spec RS spec RS mp) 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); - - -(* ------------------------------------------------------------------------*) -(* structural induction for admissible predicates *) -(* ------------------------------------------------------------------------*) - -(* not needed any longer -val dnat_ind = prove_goal Dnat.thy -"[| adm(P);\ -\ P(UU);\ -\ P(dzero);\ -\ !! s1.[|s1~=UU ; P(s1)|] ==> P(dsucc[s1])|] ==> P(s)" - (fn prems => - [ - (rtac (dnat_reach RS subst) 1), - (res_inst_tac [("x","s")] spec 1), - (rtac fix_ind 1), - (rtac adm_all2 1), - (rtac adm_subst 1), - (contX_tacR 1), - (resolve_tac prems 1), - (simp_tac HOLCF_ss 1), - (resolve_tac prems 1), - (strip_tac 1), - (res_inst_tac [("n","xa")] dnatE 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1), - (resolve_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1), - (resolve_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1), - (res_inst_tac [("Q","x[xb]=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (resolve_tac prems 1), - (eresolve_tac prems 1), - (etac spec 1) - ]); -*) - -val dnat_finite_ind = prove_goal Dnat.thy -"[|P(UU);P(dzero);\ -\ !! s1.[|s1~=UU;P(s1)|] ==> P(dsucc[s1])\ -\ |] ==> !s.P(dnat_take(n)[s])" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (resolve_tac prems 1), - (rtac allI 1), - (res_inst_tac [("n","s")] dnatE 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (resolve_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (resolve_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (res_inst_tac [("Q","dnat_take(n1)[x]=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (atac 1), - (etac spec 1) - ]); - -val dnat_all_finite_lemma1 = prove_goal Dnat.thy -"!s.dnat_take(n)[s]=UU |dnat_take(n)[s]=s" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (rtac allI 1), - (res_inst_tac [("n","s")] dnatE 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (eres_inst_tac [("x","x")] allE 1), - (etac disjE 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1) - ]); - -val dnat_all_finite_lemma2 = prove_goal Dnat.thy "? n.dnat_take(n)[s]=s" - (fn prems => - [ - (res_inst_tac [("Q","s=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (subgoal_tac "(!n.dnat_take(n)[s]=UU) |(? n.dnat_take(n)[s]=s)" 1), - (etac disjE 1), - (eres_inst_tac [("P","s=UU")] notE 1), - (rtac dnat_take_lemma 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (atac 1), - (subgoal_tac "!n.!s.dnat_take(n)[s]=UU |dnat_take(n)[s]=s" 1), - (fast_tac HOL_cs 1), - (rtac allI 1), - (rtac dnat_all_finite_lemma1 1) - ]); - - -val dnat_ind = prove_goal Dnat.thy -"[|P(UU);P(dzero);\ -\ !! s1.[|s1~=UU;P(s1)|] ==> P(dsucc[s1])\ -\ |] ==> P(s)" - (fn prems => - [ - (rtac (dnat_all_finite_lemma2 RS exE) 1), - (etac subst 1), - (rtac (dnat_finite_ind RS spec) 1), - (REPEAT (resolve_tac prems 1)), - (REPEAT (atac 1)) - ]); - - -val dnat_flat = prove_goalw Dnat.thy [flat_def] "flat(dzero)" - (fn prems => - [ - (rtac allI 1), - (res_inst_tac [("s","x")] dnat_ind 1), - (fast_tac HOL_cs 1), - (rtac allI 1), - (res_inst_tac [("n","y")] dnatE 1), - (fast_tac (HOL_cs addSIs [UU_I]) 1), - (asm_simp_tac HOLCF_ss 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_dist_less) 1), - (rtac allI 1), - (res_inst_tac [("n","y")] dnatE 1), - (fast_tac (HOL_cs addSIs [UU_I]) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_dist_less) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (strip_tac 1), - (subgoal_tac "s1< (one ++ dnat)" -dnat_abs :: "(one ++ dnat) -> dnat" - -(* ----------------------------------------------------------------------- *) -(* abstract constants and auxiliary constants *) - -dnat_copy :: "(dnat -> dnat) -> dnat -> dnat" - -dzero :: "dnat" -dsucc :: "dnat -> dnat" -dnat_when :: "'b -> (dnat -> 'b) -> dnat -> 'b" -is_dzero :: "dnat -> tr" -is_dsucc :: "dnat -> tr" -dpred :: "dnat -> dnat" -dnat_take :: "nat => dnat -> dnat" -dnat_bisim :: "(dnat => dnat => bool) => bool" - -rules - -(* ----------------------------------------------------------------------- *) -(* axiomatization of recursive type dnat *) -(* ----------------------------------------------------------------------- *) -(* (dnat,dnat_abs) is the initial F-algebra where *) -(* F is the locally continuous functor determined by domain equation *) -(* X = one ++ X *) -(* ----------------------------------------------------------------------- *) -(* dnat_abs is an isomorphism with inverse dnat_rep *) -(* identity is the least endomorphism on dnat *) - -dnat_abs_iso "dnat_rep[dnat_abs[x]] = x" -dnat_rep_iso "dnat_abs[dnat_rep[x]] = x" -dnat_copy_def "dnat_copy == (LAM f. dnat_abs oo \ -\ (when[sinl][sinr oo f]) oo dnat_rep )" -dnat_reach "(fix[dnat_copy])[x]=x" - -(* ----------------------------------------------------------------------- *) -(* properties of additional constants *) -(* ----------------------------------------------------------------------- *) -(* constructors *) - -dzero_def "dzero == dnat_abs[sinl[one]]" -dsucc_def "dsucc == (LAM n. dnat_abs[sinr[n]])" - -(* ----------------------------------------------------------------------- *) -(* discriminator functional *) - -dnat_when_def "dnat_when == (LAM f1 f2 n.when[LAM x.f1][f2][dnat_rep[n]])" - - -(* ----------------------------------------------------------------------- *) -(* discriminators and selectors *) - -is_dzero_def "is_dzero == dnat_when[TT][LAM x.FF]" -is_dsucc_def "is_dsucc == dnat_when[FF][LAM x.TT]" -dpred_def "dpred == dnat_when[UU][LAM x.x]" - - -(* ----------------------------------------------------------------------- *) -(* the taker for dnats *) - -dnat_take_def "dnat_take == (%n.iterate(n,dnat_copy,UU))" - -(* ----------------------------------------------------------------------- *) -(* definition of bisimulation is determined by domain equation *) -(* simplification and rewriting for abstract constants yields def below *) - -dnat_bisim_def "dnat_bisim ==\ -\(%R.!s1 s2.\ -\ R(s1,s2) -->\ -\ ((s1=UU & s2=UU) |(s1=dzero & s2=dzero) |\ -\ (? s11 s21. s11~=UU & s21~=UU & s1=dsucc[s11] &\ -\ s2 = dsucc[s21] & R(s11,s21))))" - -end - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/dnat2.ML --- a/src/HOLCF/dnat2.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,52 +0,0 @@ -(* Title: HOLCF/dnat2.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory Dnat2.thy -*) - -open Dnat2; - - -(* ------------------------------------------------------------------------- *) -(* expand fixed point properties *) -(* ------------------------------------------------------------------------- *) - -val iterator_def2 = fix_prover Dnat2.thy iterator_def - "iterator = (LAM n f x. dnat_when[x][LAM m.f[iterator[m][f][x]]][n])"; - -(* ------------------------------------------------------------------------- *) -(* recursive properties *) -(* ------------------------------------------------------------------------- *) - -val iterator1 = prove_goal Dnat2.thy "iterator[UU][f][x] = UU" - (fn prems => - [ - (rtac (iterator_def2 RS ssubst) 1), - (simp_tac (HOLCF_ss addsimps dnat_when) 1) - ]); - -val iterator2 = prove_goal Dnat2.thy "iterator[dzero][f][x] = x" - (fn prems => - [ - (rtac (iterator_def2 RS ssubst) 1), - (simp_tac (HOLCF_ss addsimps dnat_when) 1) - ]); - -val iterator3 = prove_goal Dnat2.thy -"n~=UU ==> iterator[dsucc[n]][f][x] = f[iterator[n][f][x]]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans 1), - (rtac (iterator_def2 RS ssubst) 1), - (asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1), - (rtac refl 1) - ]); - -val dnat2_rews = - [iterator1, iterator2, iterator3]; - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/dnat2.thy --- a/src/HOLCF/dnat2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,32 +0,0 @@ -(* Title: HOLCF/dnat2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Additional constants for dnat - -*) - -Dnat2 = Dnat + - -consts - -iterator :: "dnat -> ('a -> 'a) -> 'a -> 'a" - - -rules - -iterator_def "iterator = fix[LAM h n f x.\ -\ dnat_when[x][LAM m.f[h[m][f][x]]][n]]" - - -end - - -(* - - iterator[UU][f][x] = UU - iterator[dzero][f][x] = x - n~=UU --> iterator[dsucc[n]][f][x] = f[iterator[n][f][x]] -*) - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ex/coind.thy --- a/src/HOLCF/ex/coind.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,37 +0,0 @@ -(* Title: HOLCF/coind.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Example for co-induction on streams -*) - -Coind = Stream2 + - - -consts - -nats :: "dnat stream" -from :: "dnat -> dnat stream" - -rules - -nats_def "nats = fix[LAM h.scons[dzero][smap[dsucc][h]]]" - -from_def "from = fix[LAM h n.scons[n][h[dsucc[n]]]]" - -end - -(* - - smap[f][UU] = UU - x~=UU --> smap[f][scons[x][xs]] = scons[f[x]][smap[f][xs]] - - nats = scons[dzero][smap[dsucc][nats]] - - from[n] = scons[n][from[dsucc[n]]] - - -*) - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ex/dagstuhl.ML --- a/src/HOLCF/ex/dagstuhl.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,149 +0,0 @@ -(* - ID: $ $ -*) - - -open Dagstuhl; - -val YS_def2 = fix_prover Dagstuhl.thy YS_def "YS = scons[y][YS]"; -val YYS_def2 = fix_prover Dagstuhl.thy YYS_def "YYS = scons[y][scons[y][YYS]]"; - - -val prems = goal Dagstuhl.thy "YYS << scons[y][YYS]"; -by (rtac (YYS_def RS ssubst) 1); -by (rtac fix_ind 1); -by (resolve_tac adm_thms 1); -by (contX_tacR 1); -by (rtac minimal 1); -by (rtac (beta_cfun RS ssubst) 1); -by (contX_tacR 1); -by (rtac monofun_cfun_arg 1); -by (rtac monofun_cfun_arg 1); -by (atac 1); -val lemma3 = result(); - -val prems = goal Dagstuhl.thy "scons[y][YYS] << YYS"; -by (rtac (YYS_def2 RS ssubst) 1); -back(); -by (rtac monofun_cfun_arg 1); -by (rtac lemma3 1); -val lemma4 = result(); - -(* val lemma5 = lemma3 RS (lemma4 RS antisym_less) *) - -val prems = goal Dagstuhl.thy "scons[y][YYS] = YYS"; -by (rtac antisym_less 1); -by (rtac lemma4 1); -by (rtac lemma3 1); -val lemma5 = result(); - -val prems = goal Dagstuhl.thy "YS = YYS"; -by (rtac stream_take_lemma 1); -by (nat_ind_tac "n" 1); -by (simp_tac (HOLCF_ss addsimps stream_rews) 1); -by (res_inst_tac [("y1","y")] (YS_def2 RS ssubst) 1); -by (res_inst_tac [("y1","y")] (YYS_def2 RS ssubst) 1); -by (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1); -by (rtac (lemma5 RS sym RS subst) 1); -by (rtac refl 1); -val wir_moel = result(); - -(* ------------------------------------------------------------------------ *) -(* Zweite L"osung: Bernhard M"oller *) -(* statt Beweis von wir_moel "uber take_lemma beidseitige Inclusion *) -(* verwendet lemma5 *) -(* ------------------------------------------------------------------------ *) - -val prems = goal Dagstuhl.thy "YYS << YS"; -by (rtac (YYS_def RS ssubst) 1); -by (rtac fix_least 1); -by (rtac (beta_cfun RS ssubst) 1); -by (contX_tacR 1); -by (simp_tac (HOLCF_ss addsimps [YS_def2 RS sym]) 1); -val lemma6 = result(); - -val prems = goal Dagstuhl.thy "YS << YYS"; -by (rtac (YS_def RS ssubst) 1); -by (rtac fix_ind 1); -by (resolve_tac adm_thms 1); -by (contX_tacR 1); -by (rtac minimal 1); -by (rtac (beta_cfun RS ssubst) 1); -by (contX_tacR 1); -by (res_inst_tac [("y2","y10")] (lemma5 RS sym RS ssubst) 1); -by (etac monofun_cfun_arg 1); -val lemma7 = result(); - -val wir_moel = lemma6 RS (lemma7 RS antisym_less); - - -(* ------------------------------------------------------------------------ *) -(* L"osung aus Dagstuhl (F.R.) *) -(* Wie oben, jedoch ohne Konstantendefinition f"ur YS, YYS *) -(* ------------------------------------------------------------------------ *) - -val prems = goal Stream2.thy - "(fix[LAM x. scons[y][x]]) = scons[y][fix[LAM x. scons[y][x]]]"; -by (rtac (fix_eq RS ssubst) 1); -back(); -back(); -by (rtac (beta_cfun RS ssubst) 1); -by (contX_tacR 1); -by (rtac refl 1); -val lemma1 = result(); - -val prems = goal Stream2.thy - "(fix[ LAM z. scons[y][scons[y][z]]]) = \ -\ scons[y][scons[y][(fix[ LAM z. scons[y][scons[y][z]]])]]"; -by (rtac (fix_eq RS ssubst) 1); -back(); -back(); -by (rtac (beta_cfun RS ssubst) 1); -by (contX_tacR 1); -by (rtac refl 1); -val lemma2 = result(); - -val prems = goal Stream2.thy -"fix[LAM z. scons[y][scons[y][z]]] << \ -\ scons[y][fix[LAM z. scons[y][scons[y][z]]]]"; -by (rtac fix_ind 1); -by (resolve_tac adm_thms 1); -by (contX_tacR 1); -by (rtac minimal 1); -by (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1); -by (rtac monofun_cfun_arg 1); -by (rtac monofun_cfun_arg 1); -by (atac 1); -val lemma3 = result(); - -val prems = goal Stream2.thy -" scons[y][fix[LAM z. scons[y][scons[y][z]]]] <<\ -\ fix[LAM z. scons[y][scons[y][z]]]"; -by (rtac (lemma2 RS ssubst) 1); -back(); -by (rtac monofun_cfun_arg 1); -by (rtac lemma3 1); -val lemma4 = result(); - -val prems = goal Stream2.thy -" scons[y][fix[LAM z. scons[y][scons[y][z]]]] =\ -\ fix[LAM z. scons[y][scons[y][z]]]"; -by (rtac antisym_less 1); -by (rtac lemma4 1); -by (rtac lemma3 1); -val lemma5 = result(); - -val prems = goal Stream2.thy - "(fix[LAM x. scons[y][x]]) = (fix[ LAM z. scons[y][scons[y][z]]])"; -by (rtac stream_take_lemma 1); -by (nat_ind_tac "n" 1); -by (simp_tac (HOLCF_ss addsimps stream_rews) 1); -by (rtac (lemma1 RS ssubst) 1); -by (rtac (lemma2 RS ssubst) 1); -by (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1); -by (rtac (lemma5 RS sym RS subst) 1); -by (rtac refl 1); -val wir_moel = result(); - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ex/dagstuhl.thy --- a/src/HOLCF/ex/dagstuhl.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,17 +0,0 @@ -(* - ID: $ $ -*) - -Dagstuhl = Stream2 + - -consts - YS :: "'a stream" - YYS :: "'a stream" - -rules - -YS_def "YS = fix[LAM x. scons[y][x]]" -YYS_def "YYS = fix[LAM z. scons[y][scons[y][z]]]" - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ex/hoare.ML --- a/src/HOLCF/ex/hoare.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,540 +0,0 @@ -(* Title: HOLCF/ex/hoare.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen -*) - -open Hoare; - -(* --------- pure HOLCF logic, some little lemmas ------ *) - -val hoare_lemma2 = prove_goal HOLCF.thy "~b=TT ==> b=FF | b=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (Exh_tr RS disjE) 1), - (fast_tac HOL_cs 1), - (etac disjE 1), - (contr_tac 1), - (fast_tac HOL_cs 1) - ]); - -val hoare_lemma3 = prove_goal HOLCF.thy -" (!k.b1[iterate(k,g,x)]=TT) | (? k.~ b1[iterate(k,g,x)]=TT)" - (fn prems => - [ - (fast_tac HOL_cs 1) - ]); - -val hoare_lemma4 = prove_goal HOLCF.thy -"(? k.~ b1[iterate(k,g,x)]=TT) ==> \ -\ ? k.b1[iterate(k,g,x)]=FF | b1[iterate(k,g,x)]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (rtac exI 1), - (rtac hoare_lemma2 1), - (atac 1) - ]); - -val hoare_lemma5 = prove_goal HOLCF.thy -"[|(? k.~ b1[iterate(k,g,x)]=TT);\ -\ k=theleast(%n.~ b1[iterate(n,g,x)]=TT)|] ==> \ -\ b1[iterate(k,g,x)]=FF | b1[iterate(k,g,x)]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (rtac hoare_lemma2 1), - (etac exE 1), - (etac theleast1 1) - ]); - -val hoare_lemma6 = prove_goal HOLCF.thy "b=UU ==> ~b=TT" - (fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (resolve_tac dist_eq_tr 1) - ]); - -val hoare_lemma7 = prove_goal HOLCF.thy "b=FF ==> ~b=TT" - (fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (resolve_tac dist_eq_tr 1) - ]); - -val hoare_lemma8 = prove_goal HOLCF.thy -"[|(? k.~ b1[iterate(k,g,x)]=TT);\ -\ k=theleast(%n.~ b1[iterate(n,g,x)]=TT)|] ==> \ -\ !m. m b1[iterate(m,g,x)]=TT" - (fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (etac exE 1), - (strip_tac 1), - (res_inst_tac [("p","b1[iterate(m,g,x)]")] trE 1), - (atac 2), - (rtac (le_less_trans RS less_anti_refl) 1), - (atac 2), - (rtac theleast2 1), - (etac hoare_lemma6 1), - (rtac (le_less_trans RS less_anti_refl) 1), - (atac 2), - (rtac theleast2 1), - (etac hoare_lemma7 1) - ]); - -val hoare_lemma28 = prove_goal HOLCF.thy -"b1[y::'a]=UU::tr ==> b1[UU] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac (flat_tr RS flat_codom RS disjE) 1), - (atac 1), - (etac spec 1) - ]); - - -(* ----- access to definitions ----- *) - -val p_def2 = prove_goalw Hoare.thy [p_def] -"p = fix[LAM f x. If b1[x] then f[g[x]] else x fi]" - (fn prems => - [ - (rtac refl 1) - ]); - -val q_def2 = prove_goalw Hoare.thy [q_def] -"q = fix[LAM f x. If b1[x] orelse b2[x] then \ -\ f[g[x]] else x fi]" - (fn prems => - [ - (rtac refl 1) - ]); - - -val p_def3 = prove_goal Hoare.thy -"p[x] = If b1[x] then p[g[x]] else x fi" - (fn prems => - [ - (fix_tac3 p_def 1), - (simp_tac HOLCF_ss 1) - ]); - -val q_def3 = prove_goal Hoare.thy -"q[x] = If b1[x] orelse b2[x] then q[g[x]] else x fi" - (fn prems => - [ - (fix_tac3 q_def 1), - (simp_tac HOLCF_ss 1) - ]); - -(** --------- proves about iterations of p and q ---------- **) - -val hoare_lemma9 = prove_goal Hoare.thy -"(! m. m b1[iterate(m,g,x)]=TT) -->\ -\ p[iterate(k,g,x)]=p[x]" - (fn prems => - [ - (nat_ind_tac "k" 1), - (simp_tac iterate_ss 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (res_inst_tac [("s","p[iterate(k1,g,x)]")] trans 1), - (rtac trans 1), - (rtac (p_def3 RS sym) 2), - (res_inst_tac [("s","TT"),("t","b1[iterate(k1,g,x)]")] ssubst 1), - (rtac mp 1), - (etac spec 1), - (simp_tac nat_ss 1), - (simp_tac HOLCF_ss 1), - (etac mp 1), - (strip_tac 1), - (rtac mp 1), - (etac spec 1), - (etac less_trans 1), - (simp_tac nat_ss 1) - ]); - -val hoare_lemma24 = prove_goal Hoare.thy -"(! m. m b1[iterate(m,g,x)]=TT) --> \ -\ q[iterate(k,g,x)]=q[x]" - (fn prems => - [ - (nat_ind_tac "k" 1), - (simp_tac iterate_ss 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (res_inst_tac [("s","q[iterate(k1,g,x)]")] trans 1), - (rtac trans 1), - (rtac (q_def3 RS sym) 2), - (res_inst_tac [("s","TT"),("t","b1[iterate(k1,g,x)]")] ssubst 1), - (rtac mp 1), - (etac spec 1), - (simp_tac nat_ss 1), - (simp_tac HOLCF_ss 1), - (etac mp 1), - (strip_tac 1), - (rtac mp 1), - (etac spec 1), - (etac less_trans 1), - (simp_tac nat_ss 1) - ]); - -(* -------- results about p for case (? k.~ b1[iterate(k,g,x)]=TT) ------- *) - - -val hoare_lemma10 = (hoare_lemma8 RS (hoare_lemma9 RS mp)); -(* -[| ? k. ~ b1[iterate(k,g,?x1)] = TT; - Suc(?k3) = theleast(%n. ~ b1[iterate(n,g,?x1)] = TT) |] ==> -p[iterate(?k3,g,?x1)] = p[?x1] -*) - -val hoare_lemma11 = prove_goal Hoare.thy -"(? n.b1[iterate(n,g,x)]~=TT) ==>\ -\ k=theleast(%n.b1[iterate(n,g,x)]~=TT) & b1[iterate(k,g,x)]=FF \ -\ --> p[x] = iterate(k,g,x)" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("n","k")] natE 1), - (hyp_subst_tac 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (etac conjE 1), - (rtac trans 1), - (rtac p_def3 1), - (asm_simp_tac HOLCF_ss 1), - (eres_inst_tac [("s","0"),("t","theleast(%n. b1[iterate(n, g, x)] ~= TT)")] - subst 1), - (simp_tac iterate_ss 1), - (hyp_subst_tac 1), - (strip_tac 1), - (etac conjE 1), - (rtac trans 1), - (etac (hoare_lemma10 RS sym) 1), - (atac 1), - (rtac trans 1), - (rtac p_def3 1), - (res_inst_tac [("s","TT"),("t","b1[iterate(xa,g,x)]")] ssubst 1), - (rtac (hoare_lemma8 RS spec RS mp) 1), - (atac 1), - (atac 1), - (simp_tac nat_ss 1), - (simp_tac HOLCF_ss 1), - (rtac trans 1), - (rtac p_def3 1), - (simp_tac (HOLCF_ss addsimps [iterate_Suc RS sym]) 1), - (eres_inst_tac [("s","FF")] ssubst 1), - (simp_tac HOLCF_ss 1) - ]); - -val hoare_lemma12 = prove_goal Hoare.thy -"(? n.~ b1[iterate(n,g,x)]=TT) ==>\ -\ k=theleast(%n.~ b1[iterate(n,g,x)]=TT) & b1[iterate(k,g,x)]=UU \ -\ --> p[x] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("n","k")] natE 1), - (hyp_subst_tac 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (etac conjE 1), - (rtac trans 1), - (rtac p_def3 1), - (asm_simp_tac HOLCF_ss 1), - (hyp_subst_tac 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (etac conjE 1), - (rtac trans 1), - (rtac (hoare_lemma10 RS sym) 1), - (atac 1), - (atac 1), - (rtac trans 1), - (rtac p_def3 1), - (res_inst_tac [("s","TT"),("t","b1[iterate(xa,g,x)]")] ssubst 1), - (rtac (hoare_lemma8 RS spec RS mp) 1), - (atac 1), - (atac 1), - (simp_tac nat_ss 1), - (asm_simp_tac HOLCF_ss 1), - (rtac trans 1), - (rtac p_def3 1), - (asm_simp_tac HOLCF_ss 1) - ]); - -(* -------- results about p for case (! k. b1[iterate(k,g,x)]=TT) ------- *) - -val fernpass_lemma = prove_goal Hoare.thy -"(! k. b1[iterate(k,g,x)]=TT) ==> !k.p[iterate(k,g,x)] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (p_def2 RS ssubst) 1), - (rtac fix_ind 1), - (rtac adm_all 1), - (rtac allI 1), - (rtac adm_eq 1), - (contX_tacR 1), - (rtac allI 1), - (rtac (strict_fapp1 RS ssubst) 1), - (rtac refl 1), - (simp_tac iterate_ss 1), - (rtac allI 1), - (res_inst_tac [("s","TT"),("t","b1[iterate(k,g,x)]")] ssubst 1), - (etac spec 1), - (asm_simp_tac HOLCF_ss 1), - (rtac (iterate_Suc RS subst) 1), - (etac spec 1) - ]); - -val hoare_lemma16 = prove_goal Hoare.thy -"(! k. b1[iterate(k,g,x)]=TT) ==> p[x] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("F1","g"),("t","x")] (iterate_0 RS subst) 1), - (etac (fernpass_lemma RS spec) 1) - ]); - -(* -------- results about q for case (! k. b1[iterate(k,g,x)]=TT) ------- *) - -val hoare_lemma17 = prove_goal Hoare.thy -"(! k. b1[iterate(k,g,x)]=TT) ==> !k.q[iterate(k,g,x)] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (q_def2 RS ssubst) 1), - (rtac fix_ind 1), - (rtac adm_all 1), - (rtac allI 1), - (rtac adm_eq 1), - (contX_tacR 1), - (rtac allI 1), - (rtac (strict_fapp1 RS ssubst) 1), - (rtac refl 1), - (rtac allI 1), - (simp_tac iterate_ss 1), - (res_inst_tac [("s","TT"),("t","b1[iterate(k,g,x)]")] ssubst 1), - (etac spec 1), - (asm_simp_tac HOLCF_ss 1), - (rtac (iterate_Suc RS subst) 1), - (etac spec 1) - ]); - -val hoare_lemma18 = prove_goal Hoare.thy -"(! k. b1[iterate(k,g,x)]=TT) ==> q[x] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("F1","g"),("t","x")] (iterate_0 RS subst) 1), - (etac (hoare_lemma17 RS spec) 1) - ]); - -val hoare_lemma19 = prove_goal Hoare.thy -"(!k. (b1::'a->tr)[iterate(k,g,x)]=TT) ==> b1[UU::'a] = UU | (!y.b1[y::'a]=TT)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (flat_tr RS flat_codom) 1), - (res_inst_tac [("t","x1")] (iterate_0 RS subst) 1), - (etac spec 1) - ]); - -val hoare_lemma20 = prove_goal Hoare.thy -"(! y. b1[y::'a]=TT) ==> !k.q[iterate(k,g,x::'a)] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (q_def2 RS ssubst) 1), - (rtac fix_ind 1), - (rtac adm_all 1), - (rtac allI 1), - (rtac adm_eq 1), - (contX_tacR 1), - (rtac allI 1), - (rtac (strict_fapp1 RS ssubst) 1), - (rtac refl 1), - (rtac allI 1), - (simp_tac iterate_ss 1), - (res_inst_tac [("s","TT"),("t","b1[iterate(k,g,x::'a)]")] ssubst 1), - (etac spec 1), - (asm_simp_tac HOLCF_ss 1), - (rtac (iterate_Suc RS subst) 1), - (etac spec 1) - ]); - -val hoare_lemma21 = prove_goal Hoare.thy -"(! y. b1[y::'a]=TT) ==> q[x::'a] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("F1","g"),("t","x")] (iterate_0 RS subst) 1), - (etac (hoare_lemma20 RS spec) 1) - ]); - -val hoare_lemma22 = prove_goal Hoare.thy -"b1[UU::'a]=UU ==> q[UU::'a] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (q_def3 RS ssubst) 1), - (asm_simp_tac HOLCF_ss 1) - ]); - -(* -------- results about q for case (? k.~ b1[iterate(k,g,x)]=TT) ------- *) - -val hoare_lemma25 = (hoare_lemma8 RS (hoare_lemma24 RS mp) ); -(* -[| ? k. ~ ?b1.1[iterate(k,?g1,?x1)] = TT; - Suc(?k3) = theleast(%n. ~ ?b1.1[iterate(n,?g1,?x1)] = TT) |] ==> -q[iterate(?k3,?g1,?x1)] = q[?x1] -*) - -val hoare_lemma26 = prove_goal Hoare.thy -"(? n.~ b1[iterate(n,g,x)]=TT) ==>\ -\ k=theleast(%n.~ b1[iterate(n,g,x)]=TT) & b1[iterate(k,g,x)]=FF \ -\ --> q[x] = q[iterate(k,g,x)]" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("n","k")] natE 1), - (hyp_subst_tac 1), - (strip_tac 1), - (simp_tac iterate_ss 1), - (hyp_subst_tac 1), - (strip_tac 1), - (etac conjE 1), - (rtac trans 1), - (rtac (hoare_lemma25 RS sym) 1), - (atac 1), - (atac 1), - (rtac trans 1), - (rtac q_def3 1), - (res_inst_tac [("s","TT"),("t","b1[iterate(xa,g,x)]")] ssubst 1), - (rtac (hoare_lemma8 RS spec RS mp) 1), - (atac 1), - (atac 1), - (simp_tac nat_ss 1), - (simp_tac (HOLCF_ss addsimps [iterate_Suc]) 1) - ]); - - -val hoare_lemma27 = prove_goal Hoare.thy -"(? n.~ b1[iterate(n,g,x)]=TT) ==>\ -\ k=theleast(%n.~ b1[iterate(n,g,x)]=TT) & b1[iterate(k,g,x)]=UU \ -\ --> q[x] = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("n","k")] natE 1), - (hyp_subst_tac 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (etac conjE 1), - (rtac (q_def3 RS ssubst) 1), - (asm_simp_tac HOLCF_ss 1), - (hyp_subst_tac 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (etac conjE 1), - (rtac trans 1), - (rtac (hoare_lemma25 RS sym) 1), - (atac 1), - (atac 1), - (rtac trans 1), - (rtac q_def3 1), - (res_inst_tac [("s","TT"),("t","b1[iterate(xa,g,x)]")] ssubst 1), - (rtac (hoare_lemma8 RS spec RS mp) 1), - (atac 1), - (atac 1), - (simp_tac nat_ss 1), - (asm_simp_tac HOLCF_ss 1), - (rtac trans 1), - (rtac q_def3 1), - (asm_simp_tac HOLCF_ss 1) - ]); - -(* ------- (! k. b1[iterate(k,g,x)]=TT) ==> q o p = q ----- *) - -val hoare_lemma23 = prove_goal Hoare.thy -"(! k. b1[iterate(k,g,x)]=TT) ==> q[p[x]] = q[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (hoare_lemma16 RS ssubst) 1), - (atac 1), - (rtac (hoare_lemma19 RS disjE) 1), - (atac 1), - (rtac (hoare_lemma18 RS ssubst) 1), - (atac 1), - (rtac (hoare_lemma22 RS ssubst) 1), - (atac 1), - (rtac refl 1), - (rtac (hoare_lemma21 RS ssubst) 1), - (atac 1), - (rtac (hoare_lemma21 RS ssubst) 1), - (atac 1), - (rtac refl 1) - ]); - -(* ------------ ? k. ~ b1[iterate(k,g,x)] = TT ==> q o p = q ----- *) - -val hoare_lemma29 = prove_goal Hoare.thy -"? k. ~ b1[iterate(k,g,x)] = TT ==> q[p[x]] = q[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (hoare_lemma5 RS disjE) 1), - (atac 1), - (rtac refl 1), - (rtac (hoare_lemma11 RS mp RS ssubst) 1), - (atac 1), - (rtac conjI 1), - (rtac refl 1), - (atac 1), - (rtac (hoare_lemma26 RS mp RS subst) 1), - (atac 1), - (rtac conjI 1), - (rtac refl 1), - (atac 1), - (rtac refl 1), - (rtac (hoare_lemma12 RS mp RS ssubst) 1), - (atac 1), - (rtac conjI 1), - (rtac refl 1), - (atac 1), - (rtac (hoare_lemma22 RS ssubst) 1), - (rtac (hoare_lemma28 RS ssubst) 1), - (atac 1), - (rtac refl 1), - (rtac sym 1), - (rtac (hoare_lemma27 RS mp RS ssubst) 1), - (atac 1), - (rtac conjI 1), - (rtac refl 1), - (atac 1), - (rtac refl 1) - ]); - -(* ------ the main prove q o p = q ------ *) - -val hoare_main = prove_goal Hoare.thy "q oo p = q" - (fn prems => - [ - (rtac ext_cfun 1), - (rtac (cfcomp2 RS ssubst) 1), - (rtac (hoare_lemma3 RS disjE) 1), - (etac hoare_lemma23 1), - (etac hoare_lemma29 1) - ]); - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ex/hoare.thy --- a/src/HOLCF/ex/hoare.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,43 +0,0 @@ -(* Title: HOLCF/ex/hoare.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Theory for an example by C.A.R. Hoare - -p x = if b1(x) - then p(g(x)) - else x fi - -q x = if b1(x) orelse b2(x) - then q (g(x)) - else x fi - -Prove: for all b1 b2 g . - q o p = q - -In order to get a nice notation we fix the functions b1,b2 and g in the -signature of this example - -*) - -Hoare = Tr2 + - -consts - b1:: "'a -> tr" - b2:: "'a -> tr" - g:: "'a -> 'a" - p :: "'a -> 'a" - q :: "'a -> 'a" - -rules - - p_def "p == fix[LAM f. LAM x.\ -\ If b1[x] then f[g[x]] else x fi]" - - q_def "q == fix[LAM f. LAM x.\ -\ If b1[x] orelse b2[x] then f[g[x]] else x fi]" - - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ex/loop.ML --- a/src/HOLCF/ex/loop.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,282 +0,0 @@ -(* Title: HOLCF/ex/loop.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory loop.thy -*) - -open Loop; - -(* --------------------------------------------------------------------------- *) -(* access to definitions *) -(* --------------------------------------------------------------------------- *) - -val step_def2 = prove_goalw Loop.thy [step_def] -"step[b][g][x] = If b[x] then g[x] else x fi" - (fn prems => - [ - (simp_tac Cfun_ss 1) - ]); - -val while_def2 = prove_goalw Loop.thy [while_def] -"while[b][g] = fix[LAM f x. If b[x] then f[g[x]] else x fi]" - (fn prems => - [ - (simp_tac Cfun_ss 1) - ]); - - -(* ------------------------------------------------------------------------- *) -(* rekursive properties of while *) -(* ------------------------------------------------------------------------- *) - -val while_unfold = prove_goal Loop.thy -"while[b][g][x] = If b[x] then while[b][g][g[x]] else x fi" - (fn prems => - [ - (fix_tac5 while_def2 1), - (simp_tac Cfun_ss 1) - ]); - -val while_unfold2 = prove_goal Loop.thy - "!x.while[b][g][x] = while[b][g][iterate(k,step[b][g],x)]" - (fn prems => - [ - (nat_ind_tac "k" 1), - (simp_tac (HOLCF_ss addsimps [iterate_0,iterate_Suc]) 1), - (rtac allI 1), - (rtac trans 1), - (rtac (while_unfold RS ssubst) 1), - (rtac refl 2), - (rtac (iterate_Suc2 RS ssubst) 1), - (rtac trans 1), - (etac spec 2), - (rtac (step_def2 RS ssubst) 1), - (res_inst_tac [("p","b[x]")] trE 1), - (asm_simp_tac HOLCF_ss 1), - (rtac (while_unfold RS ssubst) 1), - (res_inst_tac [("s","UU"),("t","b[UU]")] ssubst 1), - (etac (flat_tr RS flat_codom RS disjE) 1), - (atac 1), - (etac spec 1), - (simp_tac HOLCF_ss 1), - (asm_simp_tac HOLCF_ss 1), - (asm_simp_tac HOLCF_ss 1), - (rtac (while_unfold RS ssubst) 1), - (asm_simp_tac HOLCF_ss 1) - ]); - -val while_unfold3 = prove_goal Loop.thy - "while[b][g][x] = while[b][g][step[b][g][x]]" - (fn prems => - [ - (res_inst_tac [("s","while[b][g][iterate(Suc(0),step[b][g],x)]")] trans 1), - (rtac (while_unfold2 RS spec) 1), - (simp_tac iterate_ss 1) - ]); - - -(* --------------------------------------------------------------------------- *) -(* properties of while and iterations *) -(* --------------------------------------------------------------------------- *) - -val loop_lemma1 = prove_goal Loop.thy -"[|? y.b[y]=FF; iterate(k,step[b][g],x)=UU|]==>iterate(Suc(k),step[b][g],x)=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (simp_tac iterate_ss 1), - (rtac trans 1), - (rtac step_def2 1), - (asm_simp_tac HOLCF_ss 1), - (etac exE 1), - (etac (flat_tr RS flat_codom RS disjE) 1), - (asm_simp_tac HOLCF_ss 1), - (asm_simp_tac HOLCF_ss 1) - ]); - -val loop_lemma2 = prove_goal Loop.thy -"[|? y.b[y]=FF;~iterate(Suc(k),step[b][g],x)=UU |]==>\ -\~iterate(k,step[b][g],x)=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac contrapos 1), - (etac loop_lemma1 2), - (atac 1), - (atac 1) - ]); - -val loop_lemma3 = prove_goal Loop.thy -"[|!x. INV(x) & b[x]=TT & ~g[x]=UU --> INV(g[x]);\ -\? y.b[y]=FF; INV(x)|] ==>\ -\~iterate(k,step[b][g],x)=UU --> INV(iterate(k,step[b][g],x))" - (fn prems => - [ - (cut_facts_tac prems 1), - (nat_ind_tac "k" 1), - (asm_simp_tac iterate_ss 1), - (strip_tac 1), - (simp_tac (iterate_ss addsimps [step_def2]) 1), - (res_inst_tac [("p","b[iterate(k1, step[b][g], x)]")] trE 1), - (etac notE 1), - (asm_simp_tac (HOLCF_ss addsimps [step_def2,iterate_Suc] ) 1), - (asm_simp_tac HOLCF_ss 1), - (rtac mp 1), - (etac spec 1), - (asm_simp_tac (HOLCF_ss addsimps [loop_lemma2] ) 1), - (res_inst_tac [("s","iterate(Suc(k1), step[b][g], x)"), - ("t","g[iterate(k1, step[b][g], x)]")] ssubst 1), - (atac 2), - (asm_simp_tac (HOLCF_ss addsimps [iterate_Suc,step_def2] ) 1), - (asm_simp_tac (HOLCF_ss addsimps [loop_lemma2] ) 1) - ]); - - -val loop_lemma4 = prove_goal Loop.thy -"!x. b[iterate(k,step[b][g],x)]=FF --> while[b][g][x]=iterate(k,step[b][g],x)" - (fn prems => - [ - (nat_ind_tac "k" 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (rtac (while_unfold RS ssubst) 1), - (asm_simp_tac HOLCF_ss 1), - (rtac allI 1), - (rtac (iterate_Suc2 RS ssubst) 1), - (strip_tac 1), - (rtac trans 1), - (rtac while_unfold3 1), - (asm_simp_tac HOLCF_ss 1) - ]); - -val loop_lemma5 = prove_goal Loop.thy -"!k. ~b[iterate(k,step[b][g],x)]=FF ==>\ -\ !m. while[b][g][iterate(m,step[b][g],x)]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (while_def2 RS ssubst) 1), - (rtac fix_ind 1), - (rtac (allI RS adm_all) 1), - (rtac adm_eq 1), - (contX_tacR 1), - (simp_tac HOLCF_ss 1), - (rtac allI 1), - (simp_tac HOLCF_ss 1), - (res_inst_tac [("p","b[iterate(m, step[b][g],x)]")] trE 1), - (asm_simp_tac HOLCF_ss 1), - (asm_simp_tac HOLCF_ss 1), - (res_inst_tac [("s","xa[iterate(Suc(m), step[b][g], x)]")] trans 1), - (etac spec 2), - (rtac cfun_arg_cong 1), - (rtac trans 1), - (rtac (iterate_Suc RS sym) 2), - (asm_simp_tac (HOLCF_ss addsimps [step_def2]) 1), - (dtac spec 1), - (contr_tac 1) - ]); - - -val loop_lemma6 = prove_goal Loop.thy -"!k. ~b[iterate(k,step[b][g],x)]=FF ==> while[b][g][x]=UU" - (fn prems => - [ - (res_inst_tac [("t","x")] (iterate_0 RS subst) 1), - (rtac (loop_lemma5 RS spec) 1), - (resolve_tac prems 1) - ]); - -val loop_lemma7 = prove_goal Loop.thy -"~while[b][g][x]=UU ==> ? k. b[iterate(k,step[b][g],x)]=FF" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac swap 1), - (rtac loop_lemma6 1), - (fast_tac HOL_cs 1) - ]); - -val loop_lemma8 = prove_goal Loop.thy -"~while[b][g][x]=UU ==> ? y. b[y]=FF" - (fn prems => - [ - (cut_facts_tac prems 1), - (dtac loop_lemma7 1), - (fast_tac HOL_cs 1) - ]); - - -(* --------------------------------------------------------------------------- *) -(* an invariant rule for loops *) -(* --------------------------------------------------------------------------- *) - -val loop_inv2 = prove_goal Loop.thy -"[| (!y. INV(y) & b[y]=TT & ~g[y]=UU --> INV(g[y]));\ -\ (!y. INV(y) & b[y]=FF --> Q(y));\ -\ INV(x); ~while[b][g][x]=UU |] ==> Q(while[b][g][x])" - (fn prems => - [ - (res_inst_tac [("P","%k.b[iterate(k,step[b][g],x)]=FF")] exE 1), - (rtac loop_lemma7 1), - (resolve_tac prems 1), - (rtac ((loop_lemma4 RS spec RS mp) RS ssubst) 1), - (atac 1), - (rtac (nth_elem (1,prems) RS spec RS mp) 1), - (rtac conjI 1), - (atac 2), - (rtac (loop_lemma3 RS mp) 1), - (resolve_tac prems 1), - (rtac loop_lemma8 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (rtac classical3 1), - (resolve_tac prems 1), - (etac box_equals 1), - (rtac (loop_lemma4 RS spec RS mp RS sym) 1), - (atac 1), - (rtac refl 1) - ]); - -val loop_inv3 = prove_goal Loop.thy -"[| !!y.[| INV(y); b[y]=TT; ~g[y]=UU|] ==> INV(g[y]);\ -\ !!y.[| INV(y); b[y]=FF|]==> Q(y);\ -\ INV(x); ~while[b][g][x]=UU |] ==> Q(while[b][g][x])" - (fn prems => - [ - (rtac loop_inv2 1), - (rtac allI 1), - (rtac impI 1), - (resolve_tac prems 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1), - (rtac allI 1), - (rtac impI 1), - (resolve_tac prems 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); - -val loop_inv = prove_goal Loop.thy -"[| P(x);\ -\ !!y.P(y) ==> INV(y);\ -\ !!y.[| INV(y); b[y]=TT; ~g[y]=UU|] ==> INV(g[y]);\ -\ !!y.[| INV(y); b[y]=FF|]==> Q(y);\ -\ ~while[b][g][x]=UU |] ==> Q(while[b][g][x])" - (fn prems => - [ - (rtac loop_inv3 1), - (eresolve_tac prems 1), - (atac 1), - (atac 1), - (resolve_tac prems 1), - (atac 1), - (atac 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ex/loop.thy --- a/src/HOLCF/ex/loop.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,24 +0,0 @@ -(* Title: HOLCF/ex/loop.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Theory for a loop primitive like while -*) - -Loop = Tr2 + - -consts - - step :: "('a -> tr)->('a -> 'a)->'a->'a" - while :: "('a -> tr)->('a -> 'a)->'a->'a" - -rules - - step_def "step == (LAM b g x. If b[x] then g[x] else x fi)" - while_def "while == (LAM b g. fix[LAM f x.\ -\ If b[x] then f[g[x]] else x fi])" - - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fix.ML --- a/src/HOLCF/fix.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1153 +0,0 @@ -(* Title: HOLCF/fix.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for fix.thy -*) - -open Fix; - -(* ------------------------------------------------------------------------ *) -(* derive inductive properties of iterate from primitive recursion *) -(* ------------------------------------------------------------------------ *) - -val iterate_0 = prove_goal Fix.thy "iterate(0,F,x) = x" - (fn prems => - [ - (resolve_tac (nat_recs iterate_def) 1) - ]); - -val iterate_Suc = prove_goal Fix.thy "iterate(Suc(n),F,x) = F[iterate(n,F,x)]" - (fn prems => - [ - (resolve_tac (nat_recs iterate_def) 1) - ]); - -val iterate_ss = Cfun_ss addsimps [iterate_0,iterate_Suc]; - -val iterate_Suc2 = prove_goal Fix.thy "iterate(Suc(n),F,x) = iterate(n,F,F[x])" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the sequence of function itertaions is a chain *) -(* This property is essential since monotonicity of iterate makes no sense *) -(* ------------------------------------------------------------------------ *) - -val is_chain_iterate2 = prove_goalw Fix.thy [is_chain] - " x << F[x] ==> is_chain(%i.iterate(i,F,x))" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (simp_tac iterate_ss 1), - (nat_ind_tac "i" 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1), - (etac monofun_cfun_arg 1) - ]); - - -val is_chain_iterate = prove_goal Fix.thy - "is_chain(%i.iterate(i,F,UU))" - (fn prems => - [ - (rtac is_chain_iterate2 1), - (rtac minimal 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Kleene's fixed point theorems for continuous functions in pointed *) -(* omega cpo's *) -(* ------------------------------------------------------------------------ *) - - -val Ifix_eq = prove_goalw Fix.thy [Ifix_def] "Ifix(F)=F[Ifix(F)]" - (fn prems => - [ - (rtac (contlub_cfun_arg RS ssubst) 1), - (rtac is_chain_iterate 1), - (rtac antisym_less 1), - (rtac lub_mono 1), - (rtac is_chain_iterate 1), - (rtac ch2ch_fappR 1), - (rtac is_chain_iterate 1), - (rtac allI 1), - (rtac (iterate_Suc RS subst) 1), - (rtac (is_chain_iterate RS is_chainE RS spec) 1), - (rtac is_lub_thelub 1), - (rtac ch2ch_fappR 1), - (rtac is_chain_iterate 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (rtac (iterate_Suc RS subst) 1), - (rtac is_ub_thelub 1), - (rtac is_chain_iterate 1) - ]); - - -val Ifix_least = prove_goalw Fix.thy [Ifix_def] "F[x]=x ==> Ifix(F) << x" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lub_thelub 1), - (rtac is_chain_iterate 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (nat_ind_tac "i" 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1), - (res_inst_tac [("t","x")] subst 1), - (atac 1), - (etac monofun_cfun_arg 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* monotonicity and continuity of iterate *) -(* ------------------------------------------------------------------------ *) - -val monofun_iterate = prove_goalw Fix.thy [monofun] "monofun(iterate(i))" - (fn prems => - [ - (strip_tac 1), - (nat_ind_tac "i" 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1), - (rtac (less_fun RS iffD2) 1), - (rtac allI 1), - (rtac monofun_cfun 1), - (atac 1), - (rtac (less_fun RS iffD1 RS spec) 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the following lemma uses contlub_cfun which itself is based on a *) -(* diagonalisation lemma for continuous functions with two arguments. *) -(* In this special case it is the application function fapp *) -(* ------------------------------------------------------------------------ *) - -val contlub_iterate = prove_goalw Fix.thy [contlub] "contlub(iterate(i))" - (fn prems => - [ - (strip_tac 1), - (nat_ind_tac "i" 1), - (asm_simp_tac iterate_ss 1), - (rtac (lub_const RS thelubI RS sym) 1), - (asm_simp_tac iterate_ss 1), - (rtac ext 1), - (rtac (thelub_fun RS ssubst) 1), - (rtac is_chainI 1), - (rtac allI 1), - (rtac (less_fun RS iffD2) 1), - (rtac allI 1), - (rtac (is_chainE RS spec) 1), - (rtac (monofun_fapp1 RS ch2ch_MF2LR) 1), - (rtac allI 1), - (rtac monofun_fapp2 1), - (atac 1), - (rtac ch2ch_fun 1), - (rtac (monofun_iterate RS ch2ch_monofun) 1), - (atac 1), - (rtac (thelub_fun RS ssubst) 1), - (rtac (monofun_iterate RS ch2ch_monofun) 1), - (atac 1), - (rtac contlub_cfun 1), - (atac 1), - (etac (monofun_iterate RS ch2ch_monofun RS ch2ch_fun) 1) - ]); - - -val contX_iterate = prove_goal Fix.thy "contX(iterate(i))" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_iterate 1), - (rtac contlub_iterate 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* a lemma about continuity of iterate in its third argument *) -(* ------------------------------------------------------------------------ *) - -val monofun_iterate2 = prove_goal Fix.thy "monofun(iterate(n,F))" - (fn prems => - [ - (rtac monofunI 1), - (strip_tac 1), - (nat_ind_tac "n" 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1), - (etac monofun_cfun_arg 1) - ]); - -val contlub_iterate2 = prove_goal Fix.thy "contlub(iterate(n,F))" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (nat_ind_tac "n" 1), - (simp_tac iterate_ss 1), - (simp_tac iterate_ss 1), - (res_inst_tac [("t","iterate(n1, F, lub(range(%u. Y(u))))"), - ("s","lub(range(%i. iterate(n1, F, Y(i))))")] ssubst 1), - (atac 1), - (rtac contlub_cfun_arg 1), - (etac (monofun_iterate2 RS ch2ch_monofun) 1) - ]); - -val contX_iterate2 = prove_goal Fix.thy "contX(iterate(n,F))" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_iterate2 1), - (rtac contlub_iterate2 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* monotonicity and continuity of Ifix *) -(* ------------------------------------------------------------------------ *) - -val monofun_Ifix = prove_goalw Fix.thy [monofun,Ifix_def] "monofun(Ifix)" - (fn prems => - [ - (strip_tac 1), - (rtac lub_mono 1), - (rtac is_chain_iterate 1), - (rtac is_chain_iterate 1), - (rtac allI 1), - (rtac (less_fun RS iffD1 RS spec) 1), - (etac (monofun_iterate RS monofunE RS spec RS spec RS mp) 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* since iterate is not monotone in its first argument, special lemmas must *) -(* be derived for lubs in this argument *) -(* ------------------------------------------------------------------------ *) - -val is_chain_iterate_lub = prove_goal Fix.thy -"is_chain(Y) ==> is_chain(%i. lub(range(%ia. iterate(ia,Y(i),UU))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_chainI 1), - (strip_tac 1), - (rtac lub_mono 1), - (rtac is_chain_iterate 1), - (rtac is_chain_iterate 1), - (strip_tac 1), - (etac (monofun_iterate RS ch2ch_monofun RS ch2ch_fun RS is_chainE - RS spec) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* this exchange lemma is analog to the one for monotone functions *) -(* observe that monotonicity is not really needed. The propagation of *) -(* chains is the essential argument which is usually derived from monot. *) -(* ------------------------------------------------------------------------ *) - -val contlub_Ifix_lemma1 = prove_goal Fix.thy -"is_chain(Y) ==> iterate(n,lub(range(Y)),y) = lub(range(%i. iterate(n,Y(i),y)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (thelub_fun RS subst) 1), - (rtac (monofun_iterate RS ch2ch_monofun) 1), - (atac 1), - (rtac fun_cong 1), - (rtac (contlub_iterate RS contlubE RS spec RS mp RS ssubst) 1), - (atac 1), - (rtac refl 1) - ]); - - -val ex_lub_iterate = prove_goal Fix.thy "is_chain(Y) ==>\ -\ lub(range(%i. lub(range(%ia. iterate(i,Y(ia),UU))))) =\ -\ lub(range(%i. lub(range(%ia. iterate(ia,Y(i),UU)))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac antisym_less 1), - (rtac is_lub_thelub 1), - (rtac (contlub_Ifix_lemma1 RS ext RS subst) 1), - (atac 1), - (rtac is_chain_iterate 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (rtac lub_mono 1), - (etac (monofun_iterate RS ch2ch_monofun RS ch2ch_fun) 1), - (etac is_chain_iterate_lub 1), - (strip_tac 1), - (rtac is_ub_thelub 1), - (rtac is_chain_iterate 1), - (rtac is_lub_thelub 1), - (etac is_chain_iterate_lub 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (rtac lub_mono 1), - (rtac is_chain_iterate 1), - (rtac (contlub_Ifix_lemma1 RS ext RS subst) 1), - (atac 1), - (rtac is_chain_iterate 1), - (strip_tac 1), - (rtac is_ub_thelub 1), - (etac (monofun_iterate RS ch2ch_monofun RS ch2ch_fun) 1) - ]); - - -val contlub_Ifix = prove_goalw Fix.thy [contlub,Ifix_def] "contlub(Ifix)" - (fn prems => - [ - (strip_tac 1), - (rtac (contlub_Ifix_lemma1 RS ext RS ssubst) 1), - (atac 1), - (etac ex_lub_iterate 1) - ]); - - -val contX_Ifix = prove_goal Fix.thy "contX(Ifix)" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Ifix 1), - (rtac contlub_Ifix 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* propagate properties of Ifix to its continuous counterpart *) -(* ------------------------------------------------------------------------ *) - -val fix_eq = prove_goalw Fix.thy [fix_def] "fix[F]=F[fix[F]]" - (fn prems => - [ - (asm_simp_tac (Cfun_ss addsimps [contX_Ifix]) 1), - (rtac Ifix_eq 1) - ]); - -val fix_least = prove_goalw Fix.thy [fix_def] "F[x]=x ==> fix[F] << x" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (Cfun_ss addsimps [contX_Ifix]) 1), - (etac Ifix_least 1) - ]); - - -val fix_eq2 = prove_goal Fix.thy "f == fix[F] ==> f = F[f]" - (fn prems => - [ - (rewrite_goals_tac prems), - (rtac fix_eq 1) - ]); - -val fix_eq3 = prove_goal Fix.thy "f == fix[F] ==> f[x] = F[f][x]" - (fn prems => - [ - (rtac trans 1), - (rtac ((hd prems) RS fix_eq2 RS cfun_fun_cong) 1), - (rtac refl 1) - ]); - -fun fix_tac3 thm i = ((rtac trans i) THEN (rtac (thm RS fix_eq3) i)); - -val fix_eq4 = prove_goal Fix.thy "f = fix[F] ==> f = F[f]" - (fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (rtac fix_eq 1) - ]); - -val fix_eq5 = prove_goal Fix.thy "f = fix[F] ==> f[x] = F[f][x]" - (fn prems => - [ - (rtac trans 1), - (rtac ((hd prems) RS fix_eq4 RS cfun_fun_cong) 1), - (rtac refl 1) - ]); - -fun fix_tac5 thm i = ((rtac trans i) THEN (rtac (thm RS fix_eq5) i)); - -fun fix_prover thy fixdef thm = prove_goal thy thm - (fn prems => - [ - (rtac trans 1), - (rtac (fixdef RS fix_eq4) 1), - (rtac trans 1), - (rtac beta_cfun 1), - (contX_tacR 1), - (rtac refl 1) - ]); - -(* ------------------------------------------------------------------------ - -given the definition - -smap_def - "smap = fix[LAM h f s. stream_when[LAM x l.scons[f[x]][h[f][l]]][s]]" - -use fix_prover for - -val smap_def2 = fix_prover Stream2.thy smap_def - "smap = (LAM f s. stream_when[LAM x l.scons[f[x]][smap[f][l]]][s])"; - - ------------------------------------------------------------------------ *) - -(* ------------------------------------------------------------------------ *) -(* better access to definitions *) -(* ------------------------------------------------------------------------ *) - - -val Ifix_def2 = prove_goal Fix.thy "Ifix=(%x. lub(range(%i. iterate(i,x,UU))))" - (fn prems => - [ - (rtac ext 1), - (rewrite_goals_tac [Ifix_def]), - (rtac refl 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* direct connection between fix and iteration without Ifix *) -(* ------------------------------------------------------------------------ *) - -val fix_def2 = prove_goalw Fix.thy [fix_def] - "fix[F] = lub(range(%i. iterate(i,F,UU)))" - (fn prems => - [ - (fold_goals_tac [Ifix_def]), - (asm_simp_tac (Cfun_ss addsimps [contX_Ifix]) 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Lemmas about admissibility and fixed point induction *) -(* ------------------------------------------------------------------------ *) - -(* ------------------------------------------------------------------------ *) -(* access to definitions *) -(* ------------------------------------------------------------------------ *) - -val adm_def2 = prove_goalw Fix.thy [adm_def] - "adm(P) = (!Y. is_chain(Y) --> (!i.P(Y(i))) --> P(lub(range(Y))))" - (fn prems => - [ - (rtac refl 1) - ]); - -val admw_def2 = prove_goalw Fix.thy [admw_def] - "admw(P) = (!F.((!n.P(iterate(n,F,UU)))-->\ -\ P(lub(range(%i.iterate(i,F,UU))))))" - (fn prems => - [ - (rtac refl 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* an admissible formula is also weak admissible *) -(* ------------------------------------------------------------------------ *) - -val adm_impl_admw = prove_goalw Fix.thy [admw_def] "adm(P)==>admw(P)" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac (adm_def2 RS iffD1 RS spec RS mp RS mp) 1), - (atac 1), - (rtac is_chain_iterate 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* fixed point induction *) -(* ------------------------------------------------------------------------ *) - -val fix_ind = prove_goal Fix.thy -"[| adm(P);P(UU);!!x. P(x) ==> P(F[x])|] ==> P(fix[F])" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (fix_def2 RS ssubst) 1), - (rtac (adm_def2 RS iffD1 RS spec RS mp RS mp) 1), - (atac 1), - (rtac is_chain_iterate 1), - (rtac allI 1), - (nat_ind_tac "i" 1), - (rtac (iterate_0 RS ssubst) 1), - (atac 1), - (rtac (iterate_Suc RS ssubst) 1), - (resolve_tac prems 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* computational induction for weak admissible formulae *) -(* ------------------------------------------------------------------------ *) - -val wfix_ind = prove_goal Fix.thy -"[| admw(P); !n. P(iterate(n,F,UU))|] ==> P(fix[F])" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (fix_def2 RS ssubst) 1), - (rtac (admw_def2 RS iffD1 RS spec RS mp) 1), - (atac 1), - (rtac allI 1), - (etac spec 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* for chain-finite (easy) types every formula is admissible *) -(* ------------------------------------------------------------------------ *) - -val adm_max_in_chain = prove_goalw Fix.thy [adm_def] -"!Y. is_chain(Y::nat=>'a) --> (? n.max_in_chain(n,Y)) ==> adm(P::'a=>bool)" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac exE 1), - (rtac mp 1), - (etac spec 1), - (atac 1), - (rtac (lub_finch1 RS thelubI RS ssubst) 1), - (atac 1), - (atac 1), - (etac spec 1) - ]); - - -val adm_chain_finite = prove_goalw Fix.thy [chain_finite_def] - "chain_finite(x::'a) ==> adm(P::'a=>bool)" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac adm_max_in_chain 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* flat types are chain_finite *) -(* ------------------------------------------------------------------------ *) - -val flat_imp_chain_finite = prove_goalw Fix.thy [flat_def,chain_finite_def] - "flat(x::'a)==>chain_finite(x::'a)" - (fn prems => - [ - (rewrite_goals_tac [max_in_chain_def]), - (cut_facts_tac prems 1), - (strip_tac 1), - (res_inst_tac [("Q","!i.Y(i)=UU")] classical2 1), - (res_inst_tac [("x","0")] exI 1), - (strip_tac 1), - (rtac trans 1), - (etac spec 1), - (rtac sym 1), - (etac spec 1), - (rtac (chain_mono2 RS exE) 1), - (fast_tac HOL_cs 1), - (atac 1), - (res_inst_tac [("x","Suc(x)")] exI 1), - (strip_tac 1), - (rtac disjE 1), - (atac 3), - (rtac mp 1), - (dtac spec 1), - (etac spec 1), - (etac (le_imp_less_or_eq RS disjE) 1), - (etac (chain_mono RS mp) 1), - (atac 1), - (hyp_subst_tac 1), - (rtac refl_less 1), - (res_inst_tac [("P","Y(Suc(x)) = UU")] notE 1), - (atac 2), - (rtac mp 1), - (etac spec 1), - (asm_simp_tac nat_ss 1) - ]); - - -val adm_flat = flat_imp_chain_finite RS adm_chain_finite; -(* flat(?x::?'a) ==> adm(?P::?'a => bool) *) - -val flat_void = prove_goalw Fix.thy [flat_def] "flat(UU::void)" - (fn prems => - [ - (strip_tac 1), - (rtac disjI1 1), - (rtac unique_void2 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* continuous isomorphisms are strict *) -(* a prove for embedding projection pairs is similar *) -(* ------------------------------------------------------------------------ *) - -val iso_strict = prove_goal Fix.thy -"!!f g.[|!y.f[g[y]]=(y::'b) ; !x.g[f[x]]=(x::'a) |] \ -\ ==> f[UU]=UU & g[UU]=UU" - (fn prems => - [ - (rtac conjI 1), - (rtac UU_I 1), - (res_inst_tac [("s","f[g[UU::'b]]"),("t","UU::'b")] subst 1), - (etac spec 1), - (rtac (minimal RS monofun_cfun_arg) 1), - (rtac UU_I 1), - (res_inst_tac [("s","g[f[UU::'a]]"),("t","UU::'a")] subst 1), - (etac spec 1), - (rtac (minimal RS monofun_cfun_arg) 1) - ]); - - -val isorep_defined = prove_goal Fix.thy - "[|!x.rep[abs[x]]=x;!y.abs[rep[y]]=y;z~=UU|] ==> rep[z]~=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac swap 1), - (dtac notnotD 1), - (dres_inst_tac [("f","abs")] cfun_arg_cong 1), - (etac box_equals 1), - (fast_tac HOL_cs 1), - (etac (iso_strict RS conjunct1) 1), - (atac 1) - ]); - -val isoabs_defined = prove_goal Fix.thy - "[|!x.rep[abs[x]]=x;!y.abs[rep[y]]=y;z~=UU|] ==> abs[z]~=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac swap 1), - (dtac notnotD 1), - (dres_inst_tac [("f","rep")] cfun_arg_cong 1), - (etac box_equals 1), - (fast_tac HOL_cs 1), - (etac (iso_strict RS conjunct2) 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* propagation of flatness and chainfiniteness by continuous isomorphisms *) -(* ------------------------------------------------------------------------ *) - -val chfin2chfin = prove_goalw Fix.thy [chain_finite_def] -"!!f g.[|chain_finite(x::'a); !y.f[g[y]]=(y::'b) ; !x.g[f[x]]=(x::'a) |] \ -\ ==> chain_finite(y::'b)" - (fn prems => - [ - (rewrite_goals_tac [max_in_chain_def]), - (strip_tac 1), - (rtac exE 1), - (res_inst_tac [("P","is_chain(%i.g[Y(i)])")] mp 1), - (etac spec 1), - (etac ch2ch_fappR 1), - (rtac exI 1), - (strip_tac 1), - (res_inst_tac [("s","f[g[Y(x)]]"),("t","Y(x)")] subst 1), - (etac spec 1), - (res_inst_tac [("s","f[g[Y(j)]]"),("t","Y(j)")] subst 1), - (etac spec 1), - (rtac cfun_arg_cong 1), - (rtac mp 1), - (etac spec 1), - (atac 1) - ]); - -val flat2flat = prove_goalw Fix.thy [flat_def] -"!!f g.[|flat(x::'a); !y.f[g[y]]=(y::'b) ; !x.g[f[x]]=(x::'a) |] \ -\ ==> flat(y::'b)" - (fn prems => - [ - (strip_tac 1), - (rtac disjE 1), - (res_inst_tac [("P","g[x]< adm(%x.u(x)< - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (etac (contX2contlub RS contlubE RS spec RS mp RS ssubst) 1), - (atac 1), - (etac (contX2contlub RS contlubE RS spec RS mp RS ssubst) 1), - (atac 1), - (rtac lub_mono 1), - (cut_facts_tac prems 1), - (etac (contX2mono RS ch2ch_monofun) 1), - (atac 1), - (cut_facts_tac prems 1), - (etac (contX2mono RS ch2ch_monofun) 1), - (atac 1), - (atac 1) - ]); - -val adm_conj = prove_goal Fix.thy - "[| adm(P); adm(Q) |] ==> adm(%x.P(x)&Q(x))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (adm_def2 RS iffD2) 1), - (strip_tac 1), - (rtac conjI 1), - (rtac (adm_def2 RS iffD1 RS spec RS mp RS mp) 1), - (atac 1), - (atac 1), - (fast_tac HOL_cs 1), - (rtac (adm_def2 RS iffD1 RS spec RS mp RS mp) 1), - (atac 1), - (atac 1), - (fast_tac HOL_cs 1) - ]); - -val adm_cong = prove_goal Fix.thy - "(!x. P(x) = Q(x)) ==> adm(P)=adm(Q)" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("s","P"),("t","Q")] subst 1), - (rtac refl 2), - (rtac ext 1), - (etac spec 1) - ]); - -val adm_not_free = prove_goalw Fix.thy [adm_def] "adm(%x.t)" - (fn prems => - [ - (fast_tac HOL_cs 1) - ]); - -val adm_not_less = prove_goalw Fix.thy [adm_def] - "contX(t) ==> adm(%x.~ t(x) << u)" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac contrapos 1), - (etac spec 1), - (rtac trans_less 1), - (atac 2), - (etac (contX2mono RS monofun_fun_arg) 1), - (rtac is_ub_thelub 1), - (atac 1) - ]); - -val adm_all = prove_goal Fix.thy - " !y.adm(P(y)) ==> adm(%x.!y.P(y,x))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (adm_def2 RS iffD2) 1), - (strip_tac 1), - (rtac (adm_def2 RS iffD1 RS spec RS mp RS mp) 1), - (etac spec 1), - (atac 1), - (rtac allI 1), - (dtac spec 1), - (etac spec 1) - ]); - -val adm_subst = prove_goal Fix.thy - "[|contX(t); adm(P)|] ==> adm(%x.P(t(x)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (adm_def2 RS iffD2) 1), - (strip_tac 1), - (rtac (contX2contlub RS contlubE RS spec RS mp RS ssubst) 1), - (atac 1), - (atac 1), - (rtac (adm_def2 RS iffD1 RS spec RS mp RS mp) 1), - (atac 1), - (rtac (contX2mono RS ch2ch_monofun) 1), - (atac 1), - (atac 1), - (atac 1) - ]); - -val adm_UU_not_less = prove_goal Fix.thy "adm(%x.~ UU << t(x))" - (fn prems => - [ - (res_inst_tac [("P2","%x.False")] (adm_cong RS iffD1) 1), - (asm_simp_tac Cfun_ss 1), - (rtac adm_not_free 1) - ]); - -val adm_not_UU = prove_goalw Fix.thy [adm_def] - "contX(t)==> adm(%x.~ t(x) = UU)" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac contrapos 1), - (etac spec 1), - (rtac (chain_UU_I RS spec) 1), - (rtac (contX2mono RS ch2ch_monofun) 1), - (atac 1), - (atac 1), - (rtac (contX2contlub RS contlubE RS spec RS mp RS subst) 1), - (atac 1), - (atac 1), - (atac 1) - ]); - -val adm_eq = prove_goal Fix.thy - "[|contX(u);contX(v)|]==> adm(%x.u(x)= v(x))" - (fn prems => - [ - (rtac (adm_cong RS iffD1) 1), - (rtac allI 1), - (rtac iffI 1), - (rtac antisym_less 1), - (rtac antisym_less_inverse 3), - (atac 3), - (etac conjunct1 1), - (etac conjunct2 1), - (rtac adm_conj 1), - (rtac adm_less 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (rtac adm_less 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* admissibility for disjunction is hard to prove. It takes 10 Lemmas *) -(* ------------------------------------------------------------------------ *) - -val adm_disj_lemma1 = prove_goal Pcpo.thy -"[| is_chain(Y); !n.P(Y(n))|Q(Y(n))|]\ -\ ==> (? i.!j. i Q(Y(j))) | (!i.? j.i - [ - (cut_facts_tac prems 1), - (fast_tac HOL_cs 1) - ]); - -val adm_disj_lemma2 = prove_goal Fix.thy -"[| adm(Q); ? X.is_chain(X) & (!n.Q(X(n))) &\ -\ lub(range(Y))=lub(range(X))|] ==> Q(lub(range(Y)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (etac conjE 1), - (etac conjE 1), - (res_inst_tac [("s","lub(range(X))"),("t","lub(range(Y))")] ssubst 1), - (atac 1), - (rtac (adm_def2 RS iffD1 RS spec RS mp RS mp) 1), - (atac 1), - (atac 1), - (atac 1) - ]); - -val adm_disj_lemma3 = prove_goal Fix.thy -"[| is_chain(Y); ! j. i < j --> Q(Y(j)) |] ==>\ -\ is_chain(%m. if(m < Suc(i),Y(Suc(i)),Y(m)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_chainI 1), - (rtac allI 1), - (res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1), - (res_inst_tac [("s","False"),("t","ia < Suc(i)")] ssubst 1), - (rtac iffI 1), - (etac FalseE 2), - (rtac notE 1), - (rtac (not_less_eq RS iffD2) 1), - (atac 1), - (atac 1), - (res_inst_tac [("s","False"),("t","Suc(ia) < Suc(i)")] ssubst 1), - (asm_simp_tac nat_ss 1), - (rtac iffI 1), - (etac FalseE 2), - (rtac notE 1), - (etac less_not_sym 1), - (atac 1), - (asm_simp_tac Cfun_ss 1), - (etac (is_chainE RS spec) 1), - (hyp_subst_tac 1), - (asm_simp_tac nat_ss 1), - (rtac refl_less 1), - (asm_simp_tac nat_ss 1), - (rtac refl_less 1) - ]); - -val adm_disj_lemma4 = prove_goal Fix.thy -"[| ! j. i < j --> Q(Y(j)) |] ==>\ -\ ! n. Q(if(n < Suc(i),Y(Suc(i)),Y(n)))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac allI 1), - (res_inst_tac [("m","n"),("n","Suc(i)")] nat_less_cases 1), - (res_inst_tac[("s","Y(Suc(i))"),("t","if(n'a); ! j. i < j --> Q(Y(j)) |] ==>\ -\ lub(range(Y)) = lub(range(%m. if(m < Suc(i),Y(Suc(i)),Y(m))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac lub_equal2 1), - (atac 2), - (rtac adm_disj_lemma3 2), - (atac 2), - (atac 2), - (res_inst_tac [("x","i")] exI 1), - (strip_tac 1), - (res_inst_tac [("s","False"),("t","ia < Suc(i)")] ssubst 1), - (rtac iffI 1), - (etac FalseE 2), - (rtac notE 1), - (rtac (not_less_eq RS iffD2) 1), - (atac 1), - (atac 1), - (rtac (if_False RS ssubst) 1), - (rtac refl 1) - ]); - -val adm_disj_lemma6 = prove_goal Fix.thy -"[| is_chain(Y::nat=>'a); ? i. ! j. i < j --> Q(Y(j)) |] ==>\ -\ ? X. is_chain(X) & (! n. Q(X(n))) & lub(range(Y)) = lub(range(X))" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (res_inst_tac [("x","%m.if(m< Suc(i),Y(Suc(i)),Y(m))")] exI 1), - (rtac conjI 1), - (rtac adm_disj_lemma3 1), - (atac 1), - (atac 1), - (rtac conjI 1), - (rtac adm_disj_lemma4 1), - (atac 1), - (rtac adm_disj_lemma5 1), - (atac 1), - (atac 1) - ]); - - -val adm_disj_lemma7 = prove_goal Fix.thy -"[| is_chain(Y::nat=>'a); ! i. ? j. i < j & P(Y(j)) |] ==>\ -\ is_chain(%m. Y(theleast(%j. m - [ - (cut_facts_tac prems 1), - (rtac is_chainI 1), - (rtac allI 1), - (rtac chain_mono3 1), - (atac 1), - (rtac theleast2 1), - (rtac conjI 1), - (rtac Suc_lessD 1), - (etac allE 1), - (etac exE 1), - (rtac (theleast1 RS conjunct1) 1), - (atac 1), - (etac allE 1), - (etac exE 1), - (rtac (theleast1 RS conjunct2) 1), - (atac 1) - ]); - -val adm_disj_lemma8 = prove_goal Fix.thy -"[| ! i. ? j. i < j & P(Y(j)) |] ==> ! m. P(Y(theleast(%j. m - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (etac allE 1), - (etac exE 1), - (etac (theleast1 RS conjunct2) 1) - ]); - -val adm_disj_lemma9 = prove_goal Fix.thy -"[| is_chain(Y::nat=>'a); ! i. ? j. i < j & P(Y(j)) |] ==>\ -\ lub(range(Y)) = lub(range(%m. Y(theleast(%j. m - [ - (cut_facts_tac prems 1), - (rtac antisym_less 1), - (rtac lub_mono 1), - (atac 1), - (rtac adm_disj_lemma7 1), - (atac 1), - (atac 1), - (strip_tac 1), - (rtac (chain_mono RS mp) 1), - (atac 1), - (etac allE 1), - (etac exE 1), - (rtac (theleast1 RS conjunct1) 1), - (atac 1), - (rtac lub_mono3 1), - (rtac adm_disj_lemma7 1), - (atac 1), - (atac 1), - (atac 1), - (strip_tac 1), - (rtac exI 1), - (rtac (chain_mono RS mp) 1), - (atac 1), - (rtac lessI 1) - ]); - -val adm_disj_lemma10 = prove_goal Fix.thy -"[| is_chain(Y::nat=>'a); ! i. ? j. i < j & P(Y(j)) |] ==>\ -\ ? X. is_chain(X) & (! n. P(X(n))) & lub(range(Y)) = lub(range(X))" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("x","%m. Y(theleast(%j. m adm(%x.P(x)|Q(x))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (adm_def2 RS iffD2) 1), - (strip_tac 1), - (rtac (adm_disj_lemma1 RS disjE) 1), - (atac 1), - (atac 1), - (rtac disjI2 1), - (rtac adm_disj_lemma2 1), - (atac 1), - (rtac adm_disj_lemma6 1), - (atac 1), - (atac 1), - (rtac disjI1 1), - (rtac adm_disj_lemma2 1), - (atac 1), - (rtac adm_disj_lemma10 1), - (atac 1), - (atac 1) - ]); - -val adm_impl = prove_goal Fix.thy - "[| adm(%x.~P(x)); adm(Q) |] ==> adm(%x.P(x)-->Q(x))" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("P2","%x.~P(x)|Q(x)")] (adm_cong RS iffD1) 1), - (fast_tac HOL_cs 1), - (rtac adm_disj 1), - (atac 1), - (atac 1) - ]); - - -val adm_all2 = (allI RS adm_all); - -val adm_thms = [adm_impl,adm_disj,adm_eq,adm_not_UU,adm_UU_not_less, - adm_all2,adm_not_less,adm_not_free,adm_conj,adm_less - ]; - -(* ------------------------------------------------------------------------- *) -(* a result about functions with flat codomain *) -(* ------------------------------------------------------------------------- *) - -val flat_codom = prove_goalw Fix.thy [flat_def] -"[|flat(y::'b);f[x::'a]=(c::'b)|] ==> f[UU::'a]=UU::'b | (!z.f[z::'a]=c)" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("Q","f[x::'a]=UU::'b")] classical2 1), - (rtac disjI1 1), - (rtac UU_I 1), - (res_inst_tac [("s","f[x]"),("t","UU::'b")] subst 1), - (atac 1), - (rtac (minimal RS monofun_cfun_arg) 1), - (res_inst_tac [("Q","f[UU::'a]=UU::'b")] classical2 1), - (etac disjI1 1), - (rtac disjI2 1), - (rtac allI 1), - (res_inst_tac [("s","f[x]"),("t","c")] subst 1), - (atac 1), - (res_inst_tac [("a","f[UU::'a]")] (refl RS box_equals) 1), - (etac allE 1),(etac allE 1), - (dtac mp 1), - (res_inst_tac [("fo5","f")] (minimal RS monofun_cfun_arg) 1), - (etac disjE 1), - (contr_tac 1), - (atac 1), - (etac allE 1), - (etac allE 1), - (dtac mp 1), - (res_inst_tac [("fo5","f")] (minimal RS monofun_cfun_arg) 1), - (etac disjE 1), - (contr_tac 1), - (atac 1) - ]); diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fix.thy --- a/src/HOLCF/fix.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,42 +0,0 @@ -(* Title: HOLCF/fix.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -definitions for fixed point operator and admissibility - -*) - -Fix = Cfun3 + - -consts - -iterate :: "nat=>('a->'a)=>'a=>'a" -Ifix :: "('a->'a)=>'a" -fix :: "('a->'a)->'a" -adm :: "('a=>bool)=>bool" -admw :: "('a=>bool)=>bool" -chain_finite :: "'a=>bool" -flat :: "'a=>bool" - -rules - -iterate_def "iterate(n,F,c) == nat_rec(n,c,%n x.F[x])" -Ifix_def "Ifix(F) == lub(range(%i.iterate(i,F,UU)))" -fix_def "fix == (LAM f. Ifix(f))" - -adm_def "adm(P) == !Y. is_chain(Y) --> \ -\ (!i.P(Y(i))) --> P(lub(range(Y)))" - -admw_def "admw(P)== (!F.((!n.P(iterate(n,F,UU)))-->\ -\ P(lub(range(%i.iterate(i,F,UU))))))" - -chain_finite_def "chain_finite(x::'a)==\ -\ !Y. is_chain(Y::nat=>'a) --> (? n.max_in_chain(n,Y))" - -flat_def "flat(x::'a) ==\ -\ ! x y. x::'a << y --> (x = UU) | (x=y)" - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fun1.ML --- a/src/HOLCF/fun1.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,49 +0,0 @@ -(* Title: HOLCF/fun1.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for fun1.thy -*) - -open Fun1; - -(* ------------------------------------------------------------------------ *) -(* less_fun is a partial order on 'a => 'b *) -(* ------------------------------------------------------------------------ *) - -val refl_less_fun = prove_goalw Fun1.thy [less_fun_def] "less_fun(f,f)" -(fn prems => - [ - (fast_tac (HOL_cs addSIs [refl_less]) 1) - ]); - -val antisym_less_fun = prove_goalw Fun1.thy [less_fun_def] - "[|less_fun(f1,f2); less_fun(f2,f1)|] ==> f1 = f2" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (expand_fun_eq RS ssubst) 1), - (fast_tac (HOL_cs addSIs [antisym_less]) 1) - ]); - -val trans_less_fun = prove_goalw Fun1.thy [less_fun_def] - "[|less_fun(f1,f2); less_fun(f2,f3)|] ==> less_fun(f1,f3)" -(fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac trans_less 1), - (etac allE 1), - (atac 1), - ((etac allE 1) THEN (atac 1)) - ]); - -(* - -------------------------------------------------------------------------- - Since less_fun :: "['a::term=>'b::po,'a::term=>'b::po] => bool" the - lemmas refl_less_fun, antisym_less_fun, trans_less_fun justify - the class arity fun::(term,po)po !! - -------------------------------------------------------------------------- -*) - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fun1.thy --- a/src/HOLCF/fun1.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,30 +0,0 @@ -(* Title: HOLCF/fun1.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -Definition of the partial ordering for the type of all functions => (fun) - -REMARK: The ordering on 'a => 'b is only defined if 'b is in class po !! - -*) - -Fun1 = Pcpo + - -(* default class is still term *) - -consts - less_fun :: "['a=>'b::po,'a=>'b] => bool" - -rules - (* definition of the ordering less_fun *) - (* in fun1.ML it is proved that less_fun is a po *) - - less_fun_def "less_fun(f1,f2) == ! x. f1(x) << f2(x)" - -end - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fun2.ML --- a/src/HOLCF/fun2.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,106 +0,0 @@ -(* Title: HOLCF/fun2.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for fun2.thy -*) - -open Fun2; - -(* ------------------------------------------------------------------------ *) -(* Type 'a::term => 'b::pcpo is pointed *) -(* ------------------------------------------------------------------------ *) - -val minimal_fun = prove_goalw Fun2.thy [UU_fun_def] "UU_fun << f" -(fn prems => - [ - (rtac (inst_fun_po RS ssubst) 1), - (rewrite_goals_tac [less_fun_def]), - (fast_tac (HOL_cs addSIs [minimal]) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* make the symbol << accessible for type fun *) -(* ------------------------------------------------------------------------ *) - -val less_fun = prove_goal Fun2.thy "(f1 << f2) = (! x. f1(x) << f2(x))" -(fn prems => - [ - (rtac (inst_fun_po RS ssubst) 1), - (fold_goals_tac [less_fun_def]), - (rtac refl 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* chains of functions yield chains in the po range *) -(* ------------------------------------------------------------------------ *) - -val ch2ch_fun = prove_goal Fun2.thy - "is_chain(S::nat=>('a::term => 'b::po)) ==> is_chain(% i.S(i)(x))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rewrite_goals_tac [is_chain]), - (rtac allI 1), - (rtac spec 1), - (rtac (less_fun RS subst) 1), - (etac allE 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* upper bounds of function chains yield upper bound in the po range *) -(* ------------------------------------------------------------------------ *) - -val ub2ub_fun = prove_goal Fun2.thy - " range(S::nat=>('a::term => 'b::po)) <| u ==> range(%i. S(i,x)) <| u(x)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (rtac allE 1), - (rtac (less_fun RS subst) 1), - (etac (ub_rangeE RS spec) 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Type 'a::term => 'b::pcpo is chain complete *) -(* ------------------------------------------------------------------------ *) - -val lub_fun = prove_goal Fun2.thy - "is_chain(S::nat=>('a::term => 'b::pcpo)) ==> \ -\ range(S) <<| (% x.lub(range(% i.S(i)(x))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (rtac (less_fun RS ssubst) 1), - (rtac allI 1), - (rtac is_ub_thelub 1), - (etac ch2ch_fun 1), - (strip_tac 1), - (rtac (less_fun RS ssubst) 1), - (rtac allI 1), - (rtac is_lub_thelub 1), - (etac ch2ch_fun 1), - (etac ub2ub_fun 1) - ]); - -val thelub_fun = (lub_fun RS thelubI); -(* is_chain(?S1) ==> lub(range(?S1)) = (%x. lub(range(%i. ?S1(i,x)))) *) - -val cpo_fun = prove_goal Fun2.thy - "is_chain(S::nat=>('a::term => 'b::pcpo)) ==> ? x. range(S) <<| x" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac exI 1), - (etac lub_fun 1) - ]); - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fun2.thy --- a/src/HOLCF/fun2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,40 +0,0 @@ -(* Title: HOLCF/fun2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class Instance =>::(term,po)po -Definiton of least element -*) - -Fun2 = Fun1 + - -(* default class is still term !*) - -(* Witness for the above arity axiom is fun1.ML *) - -arities fun :: (term,po)po - -consts - UU_fun :: "'a::term => 'b::pcpo" - -rules - -(* instance of << for type ['a::term => 'b::po] *) - -inst_fun_po "(op <<)::['a=>'b::po,'a=>'b::po ]=>bool = less_fun" - -(* definitions *) -(* The least element in type 'a::term => 'b::pcpo *) - -UU_fun_def "UU_fun == (% x.UU)" - -end - - - - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fun3.ML --- a/src/HOLCF/fun3.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,7 +0,0 @@ -(* Title: HOLCF/fun3.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen -*) - -open Fun3; diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/fun3.thy --- a/src/HOLCF/fun3.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,23 +0,0 @@ -(* Title: HOLCF/fun3.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class instance of => (fun) for class pcpo - -*) - -Fun3 = Fun2 + - -(* default class is still term *) - -arities fun :: (term,pcpo)pcpo (* Witness fun2.ML *) - -rules - -inst_fun_pcpo "UU::'a=>'b::pcpo = UU_fun" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/holcf.ML --- a/src/HOLCF/holcf.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,20 +0,0 @@ -(* Title: HOLCF/HOLCF.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen -*) - -open HOLCF; - -val HOLCF_ss = ccc1_ss - addsimps one_when - addsimps dist_less_one - addsimps dist_eq_one - addsimps dist_less_tr - addsimps dist_eq_tr - addsimps tr_when - addsimps andalso_thms - addsimps orelse_thms - addsimps ifte_thms; - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/holcf.thy --- a/src/HOLCF/holcf.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,13 +0,0 @@ -(* Title: HOLCF/HOLCF.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -Top theory for HOLCF system - -*) - -HOLCF = Tr2 - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/holcfb.ML --- a/src/HOLCF/holcfb.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,152 +0,0 @@ -(* Title: HOLCF/holcfb.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for Holcfb.thy -*) - -open Holcfb; - -(* ------------------------------------------------------------------------ *) -(* <::nat=>nat=>bool is well-founded *) -(* ------------------------------------------------------------------------ *) - -val well_founded_nat = prove_goal Nat.thy - "!P. P(x::nat) --> (? y. P(y) & (! x. P(x) --> y <= x))" - (fn prems => - [ - (res_inst_tac [("n","x")] less_induct 1), - (strip_tac 1), - (res_inst_tac [("Q","? k.k P(theleast(P)) & (!x.P(x)--> theleast(P) <= x)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (well_founded_nat RS spec RS mp RS exE) 1), - (atac 1), - (rtac selectI 1), - (atac 1) - ]); - -val theleast1= theleast RS conjunct1; -(* ?P1(?x1) ==> ?P1(theleast(?P1)) *) - -val theleast2 = prove_goal Holcfb.thy "P(x) ==> theleast(P) <= x" - (fn prems => - [ - (rtac (theleast RS conjunct2 RS spec RS mp) 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Some lemmas in HOL.thy *) -(* ------------------------------------------------------------------------ *) - - -val de_morgan1 = prove_goal HOL.thy "(~a & ~b)=(~(a | b))" -(fn prems => - [ - (rtac iffI 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val de_morgan2 = prove_goal HOL.thy "(~a | ~b)=(~(a & b))" -(fn prems => - [ - (rtac iffI 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - - -val notall2ex = prove_goal HOL.thy "(~ (!x.P(x))) = (? x.~P(x))" -(fn prems => - [ - (rtac iffI 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val notex2all = prove_goal HOL.thy "(~ (? x.P(x))) = (!x.~P(x))" -(fn prems => - [ - (rtac iffI 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - - -val selectI2 = prove_goal HOL.thy "(? x. P(x)) ==> P(@ x.P(x))" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (etac selectI 1) - ]); - -val selectE = prove_goal HOL.thy "P(@ x.P(x)) ==> (? x. P(x))" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac exI 1) - ]); - -val select_eq_Ex = prove_goal HOL.thy "(P(@ x.P(x))) = (? x. P(x))" -(fn prems => - [ - (rtac (iff RS mp RS mp) 1), - (strip_tac 1), - (etac selectE 1), - (strip_tac 1), - (etac selectI2 1) - ]); - - -val notnotI = prove_goal HOL.thy "P ==> ~~P" -(fn prems => - [ - (cut_facts_tac prems 1), - (fast_tac HOL_cs 1) - ]); - - -val classical2 = prove_goal HOL.thy "[|Q ==> R; ~Q ==> R|]==>R" - (fn prems => - [ - (rtac disjE 1), - (rtac excluded_middle 1), - (eresolve_tac prems 1), - (eresolve_tac prems 1) - ]); - - - -val classical3 = (notE RS notI); -(* [| ?P ==> ~ ?P1; ?P ==> ?P1 |] ==> ~ ?P *) - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/holcfb.thy --- a/src/HOLCF/holcfb.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,25 +0,0 @@ -(* Title: HOLCF/holcfb.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Basic definitions for the embedding of LCF into HOL. - -*) - -Holcfb = Nat + - -consts - -theleast :: "(nat=>bool)=>nat" - -rules - -theleast_def "theleast(P) == (@z.(P(z) & (!n.P(n)-->z<=n)))" - -end - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/lift1.ML --- a/src/HOLCF/lift1.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,188 +0,0 @@ -(* Title: HOLCF/lift1.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen -*) - -open Lift1; - -val Exh_Lift = prove_goalw Lift1.thy [UU_lift_def,Iup_def ] - "z = UU_lift | (? x. z = Iup(x))" - (fn prems => - [ - (rtac (Rep_Lift_inverse RS subst) 1), - (res_inst_tac [("s","Rep_Lift(z)")] sumE 1), - (rtac disjI1 1), - (res_inst_tac [("f","Abs_Lift")] arg_cong 1), - (rtac (unique_void2 RS subst) 1), - (atac 1), - (rtac disjI2 1), - (rtac exI 1), - (res_inst_tac [("f","Abs_Lift")] arg_cong 1), - (atac 1) - ]); - -val inj_Abs_Lift = prove_goal Lift1.thy "inj(Abs_Lift)" - (fn prems => - [ - (rtac inj_inverseI 1), - (rtac Abs_Lift_inverse 1) - ]); - -val inj_Rep_Lift = prove_goal Lift1.thy "inj(Rep_Lift)" - (fn prems => - [ - (rtac inj_inverseI 1), - (rtac Rep_Lift_inverse 1) - ]); - -val inject_Iup = prove_goalw Lift1.thy [Iup_def] "Iup(x)=Iup(y) ==> x=y" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (inj_Inr RS injD) 1), - (rtac (inj_Abs_Lift RS injD) 1), - (atac 1) - ]); - -val defined_Iup=prove_goalw Lift1.thy [Iup_def,UU_lift_def] "~ Iup(x)=UU_lift" - (fn prems => - [ - (rtac notI 1), - (rtac notE 1), - (rtac Inl_not_Inr 1), - (rtac sym 1), - (etac (inj_Abs_Lift RS injD) 1) - ]); - - -val liftE = prove_goal Lift1.thy - "[| p=UU_lift ==> Q; !!x. p=Iup(x)==>Q|] ==>Q" - (fn prems => - [ - (rtac (Exh_Lift RS disjE) 1), - (eresolve_tac prems 1), - (etac exE 1), - (eresolve_tac prems 1) - ]); - -val Ilift1 = prove_goalw Lift1.thy [Ilift_def,UU_lift_def] - "Ilift(f)(UU_lift)=UU" - (fn prems => - [ - (rtac (Abs_Lift_inverse RS ssubst) 1), - (rtac (sum_case_Inl RS ssubst) 1), - (rtac refl 1) - ]); - -val Ilift2 = prove_goalw Lift1.thy [Ilift_def,Iup_def] - "Ilift(f)(Iup(x))=f[x]" - (fn prems => - [ - (rtac (Abs_Lift_inverse RS ssubst) 1), - (rtac (sum_case_Inr RS ssubst) 1), - (rtac refl 1) - ]); - -val Lift_ss = Cfun_ss addsimps [Ilift1,Ilift2]; - -val less_lift1a = prove_goalw Lift1.thy [less_lift_def,UU_lift_def] - "less_lift(UU_lift)(z)" - (fn prems => - [ - (rtac (Abs_Lift_inverse RS ssubst) 1), - (rtac (sum_case_Inl RS ssubst) 1), - (rtac TrueI 1) - ]); - -val less_lift1b = prove_goalw Lift1.thy [Iup_def,less_lift_def,UU_lift_def] - "~less_lift(Iup(x),UU_lift)" - (fn prems => - [ - (rtac notI 1), - (rtac iffD1 1), - (atac 2), - (rtac (Abs_Lift_inverse RS ssubst) 1), - (rtac (Abs_Lift_inverse RS ssubst) 1), - (rtac (sum_case_Inr RS ssubst) 1), - (rtac (sum_case_Inl RS ssubst) 1), - (rtac refl 1) - ]); - -val less_lift1c = prove_goalw Lift1.thy [Iup_def,less_lift_def,UU_lift_def] - "less_lift(Iup(x),Iup(y))=(x< - [ - (rtac (Abs_Lift_inverse RS ssubst) 1), - (rtac (Abs_Lift_inverse RS ssubst) 1), - (rtac (sum_case_Inr RS ssubst) 1), - (rtac (sum_case_Inr RS ssubst) 1), - (rtac refl 1) - ]); - - -val refl_less_lift = prove_goal Lift1.thy "less_lift(p,p)" - (fn prems => - [ - (res_inst_tac [("p","p")] liftE 1), - (hyp_subst_tac 1), - (rtac less_lift1a 1), - (hyp_subst_tac 1), - (rtac (less_lift1c RS iffD2) 1), - (rtac refl_less 1) - ]); - -val antisym_less_lift = prove_goal Lift1.thy - "[|less_lift(p1,p2);less_lift(p2,p1)|] ==> p1=p2" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] liftE 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] liftE 1), - (hyp_subst_tac 1), - (rtac refl 1), - (hyp_subst_tac 1), - (res_inst_tac [("P","less_lift(Iup(x),UU_lift)")] notE 1), - (rtac less_lift1b 1), - (atac 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] liftE 1), - (hyp_subst_tac 1), - (res_inst_tac [("P","less_lift(Iup(x),UU_lift)")] notE 1), - (rtac less_lift1b 1), - (atac 1), - (hyp_subst_tac 1), - (rtac arg_cong 1), - (rtac antisym_less 1), - (etac (less_lift1c RS iffD1) 1), - (etac (less_lift1c RS iffD1) 1) - ]); - -val trans_less_lift = prove_goal Lift1.thy - "[|less_lift(p1,p2);less_lift(p2,p3)|] ==> less_lift(p1,p3)" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] liftE 1), - (hyp_subst_tac 1), - (rtac less_lift1a 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] liftE 1), - (hyp_subst_tac 1), - (rtac notE 1), - (rtac less_lift1b 1), - (atac 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p3")] liftE 1), - (hyp_subst_tac 1), - (rtac notE 1), - (rtac less_lift1b 1), - (atac 1), - (hyp_subst_tac 1), - (rtac (less_lift1c RS iffD2) 1), - (rtac trans_less 1), - (etac (less_lift1c RS iffD1) 1), - (etac (less_lift1c RS iffD1) 1) - ]); - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/lift1.thy --- a/src/HOLCF/lift1.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,55 +0,0 @@ -(* Title: HOLCF/lift1.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -Lifting - -*) - -Lift1 = Cfun3 + - -(* new type for lifting *) - -types "u" 1 - -arities "u" :: (pcpo)term - -consts - - Rep_Lift :: "('a)u => (void + 'a)" - Abs_Lift :: "(void + 'a) => ('a)u" - - Iup :: "'a => ('a)u" - UU_lift :: "('a)u" - Ilift :: "('a->'b)=>('a)u => 'b" - less_lift :: "('a)u => ('a)u => bool" - -rules - - (*faking a type definition... *) - (* ('a)u is isomorphic to void + 'a *) - - Rep_Lift_inverse "Abs_Lift(Rep_Lift(p)) = p" - Abs_Lift_inverse "Rep_Lift(Abs_Lift(p)) = p" - - (*defining the abstract constants*) - - UU_lift_def "UU_lift == Abs_Lift(Inl(UU))" - Iup_def "Iup(x) == Abs_Lift(Inr(x))" - - Ilift_def "Ilift(f)(x)==\ -\ sum_case (Rep_Lift(x)) (%y.UU) (%z.f[z])" - - less_lift_def "less_lift(x1)(x2) == \ -\ (sum_case (Rep_Lift(x1))\ -\ (% y1.True)\ -\ (% y2.sum_case (Rep_Lift(x2))\ -\ (% z1.False)\ -\ (% z2.y2< - [ - (rtac (inst_lift_po RS ssubst) 1), - (rtac less_lift1a 1) - ]); - -(* -------------------------------------------------------------------------*) -(* access to less_lift in class po *) -(* ------------------------------------------------------------------------ *) - -val less_lift2b = prove_goal Lift2.thy "~ Iup(x) << UU_lift" - (fn prems => - [ - (rtac (inst_lift_po RS ssubst) 1), - (rtac less_lift1b 1) - ]); - -val less_lift2c = prove_goal Lift2.thy "(Iup(x)< - [ - (rtac (inst_lift_po RS ssubst) 1), - (rtac less_lift1c 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Iup and Ilift are monotone *) -(* ------------------------------------------------------------------------ *) - -val monofun_Iup = prove_goalw Lift2.thy [monofun] "monofun(Iup)" - (fn prems => - [ - (strip_tac 1), - (etac (less_lift2c RS iffD2) 1) - ]); - -val monofun_Ilift1 = prove_goalw Lift2.thy [monofun] "monofun(Ilift)" - (fn prems => - [ - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (res_inst_tac [("p","xa")] liftE 1), - (asm_simp_tac Lift_ss 1), - (asm_simp_tac Lift_ss 1), - (etac monofun_cfun_fun 1) - ]); - -val monofun_Ilift2 = prove_goalw Lift2.thy [monofun] "monofun(Ilift(f))" - (fn prems => - [ - (strip_tac 1), - (res_inst_tac [("p","x")] liftE 1), - (asm_simp_tac Lift_ss 1), - (asm_simp_tac Lift_ss 1), - (res_inst_tac [("p","y")] liftE 1), - (hyp_subst_tac 1), - (hyp_subst_tac 1), - (rtac notE 1), - (rtac less_lift2b 1), - (atac 1), - (asm_simp_tac Lift_ss 1), - (rtac monofun_cfun_arg 1), - (hyp_subst_tac 1), - (hyp_subst_tac 1), - (etac (less_lift2c RS iffD1) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Some kind of surjectivity lemma *) -(* ------------------------------------------------------------------------ *) - - -val lift_lemma1 = prove_goal Lift2.thy "z=Iup(x) ==> Iup(Ilift(LAM x.x)(z)) = z" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac Lift_ss 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* ('a)u is a cpo *) -(* ------------------------------------------------------------------------ *) - -val lub_lift1a = prove_goal Lift2.thy -"[|is_chain(Y);? i x.Y(i)=Iup(x)|] ==>\ -\ range(Y) <<| Iup(lub(range(%i.(Ilift (LAM x.x) (Y(i))))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (res_inst_tac [("p","Y(i)")] liftE 1), - (res_inst_tac [("s","UU_lift"),("t","Y(i)")] subst 1), - (etac sym 1), - (rtac minimal_lift 1), - (res_inst_tac [("t","Y(i)")] (lift_lemma1 RS subst) 1), - (atac 1), - (rtac (less_lift2c RS iffD2) 1), - (rtac is_ub_thelub 1), - (etac (monofun_Ilift2 RS ch2ch_monofun) 1), - (strip_tac 1), - (res_inst_tac [("p","u")] liftE 1), - (etac exE 1), - (etac exE 1), - (res_inst_tac [("P","Y(i)<\ -\ range(Y) <<| UU_lift" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (res_inst_tac [("p","Y(i)")] liftE 1), - (res_inst_tac [("s","UU_lift"),("t","Y(i)")] ssubst 1), - (atac 1), - (rtac refl_less 1), - (rtac notE 1), - (dtac spec 1), - (dtac spec 1), - (atac 1), - (atac 1), - (strip_tac 1), - (rtac minimal_lift 1) - ]); - -val thelub_lift1a = lub_lift1a RS thelubI; -(* [| is_chain(?Y1); ? i x. ?Y1(i) = Iup(x) |] ==> *) -(* lub(range(?Y1)) = Iup(lub(range(%i. Ilift(LAM x. x,?Y1(i))))) *) - -val thelub_lift1b = lub_lift1b RS thelubI; -(* [| is_chain(?Y1); ! i x. ~ ?Y1(i) = Iup(x) |] ==> *) -(* lub(range(?Y1)) = UU_lift *) - - -val cpo_lift = prove_goal Lift2.thy - "is_chain(Y::nat=>('a)u) ==> ? x.range(Y) <<|x" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac disjE 1), - (rtac exI 2), - (etac lub_lift1a 2), - (atac 2), - (rtac exI 2), - (etac lub_lift1b 2), - (atac 2), - (fast_tac HOL_cs 1) - ]); - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/lift2.thy --- a/src/HOLCF/lift2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,25 +0,0 @@ -(* Title: HOLCF/lift2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class Instance u::(pcpo)po - -*) - -Lift2 = Lift1 + - -(* Witness for the above arity axiom is lift1.ML *) - -arities "u" :: (pcpo)po - -rules - -(* instance of << for type ('a)u *) - -inst_lift_po "(op <<)::[('a)u,('a)u]=>bool = less_lift" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/lift3.ML --- a/src/HOLCF/lift3.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,349 +0,0 @@ -(* Title: HOLCF/lift3.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for lift3.thy -*) - -open Lift3; - -(* -------------------------------------------------------------------------*) -(* some lemmas restated for class pcpo *) -(* ------------------------------------------------------------------------ *) - -val less_lift3b = prove_goal Lift3.thy "~ Iup(x) << UU" - (fn prems => - [ - (rtac (inst_lift_pcpo RS ssubst) 1), - (rtac less_lift2b 1) - ]); - -val defined_Iup2 = prove_goal Lift3.thy "~ Iup(x) = UU" - (fn prems => - [ - (rtac (inst_lift_pcpo RS ssubst) 1), - (rtac defined_Iup 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* continuity for Iup *) -(* ------------------------------------------------------------------------ *) - -val contlub_Iup = prove_goal Lift3.thy "contlub(Iup)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_lift1a RS sym) 2), - (fast_tac HOL_cs 3), - (etac (monofun_Iup RS ch2ch_monofun) 2), - (res_inst_tac [("f","Iup")] arg_cong 1), - (rtac lub_equal 1), - (atac 1), - (rtac (monofun_Ilift2 RS ch2ch_monofun) 1), - (etac (monofun_Iup RS ch2ch_monofun) 1), - (asm_simp_tac Lift_ss 1) - ]); - -val contX_Iup = prove_goal Lift3.thy "contX(Iup)" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Iup 1), - (rtac contlub_Iup 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* continuity for Ilift *) -(* ------------------------------------------------------------------------ *) - -val contlub_Ilift1 = prove_goal Lift3.thy "contlub(Ilift)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_fun RS sym) 2), - (etac (monofun_Ilift1 RS ch2ch_monofun) 2), - (rtac ext 1), - (res_inst_tac [("p","x")] liftE 1), - (asm_simp_tac Lift_ss 1), - (rtac (lub_const RS thelubI RS sym) 1), - (asm_simp_tac Lift_ss 1), - (etac contlub_cfun_fun 1) - ]); - - -val contlub_Ilift2 = prove_goal Lift3.thy "contlub(Ilift(f))" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac disjE 1), - (rtac (thelub_lift1a RS ssubst) 2), - (atac 2), - (atac 2), - (asm_simp_tac Lift_ss 2), - (rtac (thelub_lift1b RS ssubst) 3), - (atac 3), - (atac 3), - (fast_tac HOL_cs 1), - (asm_simp_tac Lift_ss 2), - (rtac (chain_UU_I_inverse RS sym) 2), - (rtac allI 2), - (res_inst_tac [("p","Y(i)")] liftE 2), - (asm_simp_tac Lift_ss 2), - (rtac notE 2), - (dtac spec 2), - (etac spec 2), - (atac 2), - (rtac (contlub_cfun_arg RS ssubst) 1), - (etac (monofun_Ilift2 RS ch2ch_monofun) 1), - (rtac lub_equal2 1), - (rtac (monofun_fapp2 RS ch2ch_monofun) 2), - (etac (monofun_Ilift2 RS ch2ch_monofun) 2), - (etac (monofun_Ilift2 RS ch2ch_monofun) 2), - (rtac (chain_mono2 RS exE) 1), - (atac 2), - (etac exE 1), - (etac exE 1), - (rtac exI 1), - (res_inst_tac [("s","Iup(x)"),("t","Y(i)")] ssubst 1), - (atac 1), - (rtac defined_Iup2 1), - (rtac exI 1), - (strip_tac 1), - (res_inst_tac [("p","Y(i)")] liftE 1), - (asm_simp_tac Lift_ss 2), - (res_inst_tac [("P","Y(i) = UU")] notE 1), - (fast_tac HOL_cs 1), - (rtac (inst_lift_pcpo RS ssubst) 1), - (atac 1) - ]); - -val contX_Ilift1 = prove_goal Lift3.thy "contX(Ilift)" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Ilift1 1), - (rtac contlub_Ilift1 1) - ]); - -val contX_Ilift2 = prove_goal Lift3.thy "contX(Ilift(f))" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Ilift2 1), - (rtac contlub_Ilift2 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* continuous versions of lemmas for ('a)u *) -(* ------------------------------------------------------------------------ *) - -val Exh_Lift1 = prove_goalw Lift3.thy [up_def] "z = UU | (? x. z = up[x])" - (fn prems => - [ - (simp_tac (Lift_ss addsimps [contX_Iup]) 1), - (rtac (inst_lift_pcpo RS ssubst) 1), - (rtac Exh_Lift 1) - ]); - -val inject_up = prove_goalw Lift3.thy [up_def] "up[x]=up[y] ==> x=y" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac inject_Iup 1), - (etac box_equals 1), - (simp_tac (Lift_ss addsimps [contX_Iup]) 1), - (simp_tac (Lift_ss addsimps [contX_Iup]) 1) - ]); - -val defined_up = prove_goalw Lift3.thy [up_def] "~ up[x]=UU" - (fn prems => - [ - (simp_tac (Lift_ss addsimps [contX_Iup]) 1), - (rtac defined_Iup2 1) - ]); - -val liftE1 = prove_goalw Lift3.thy [up_def] - "[| p=UU ==> Q; !!x. p=up[x]==>Q|] ==>Q" - (fn prems => - [ - (rtac liftE 1), - (resolve_tac prems 1), - (etac (inst_lift_pcpo RS ssubst) 1), - (resolve_tac (tl prems) 1), - (asm_simp_tac (Lift_ss addsimps [contX_Iup]) 1) - ]); - - -val lift1 = prove_goalw Lift3.thy [up_def,lift_def] "lift[f][UU]=UU" - (fn prems => - [ - (rtac (inst_lift_pcpo RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iup,contX_Ilift1, - contX_Ilift2,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iup,contX_Ilift1, - contX_Ilift2,contX2contX_CF1L]) 1)), - (simp_tac (Lift_ss addsimps [contX_Iup,contX_Ilift1,contX_Ilift2]) 1) - ]); - -val lift2 = prove_goalw Lift3.thy [up_def,lift_def] "lift[f][up[x]]=f[x]" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Iup 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iup,contX_Ilift1, - contX_Ilift2,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Ilift2 1), - (simp_tac (Lift_ss addsimps [contX_Iup,contX_Ilift1,contX_Ilift2]) 1) - ]); - -val less_lift4b = prove_goalw Lift3.thy [up_def,lift_def] "~ up[x] << UU" - (fn prems => - [ - (simp_tac (Lift_ss addsimps [contX_Iup]) 1), - (rtac less_lift3b 1) - ]); - -val less_lift4c = prove_goalw Lift3.thy [up_def,lift_def] - "(up[x]< - [ - (simp_tac (Lift_ss addsimps [contX_Iup]) 1), - (rtac less_lift2c 1) - ]); - -val thelub_lift2a = prove_goalw Lift3.thy [up_def,lift_def] -"[| is_chain(Y); ? i x. Y(i) = up[x] |] ==>\ -\ lub(range(Y)) = up[lub(range(%i. lift[LAM x. x][Y(i)]))]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iup,contX_Ilift1, - contX_Ilift2,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iup,contX_Ilift1, - contX_Ilift2,contX2contX_CF1L]) 1)), - - (rtac (beta_cfun RS ext RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iup,contX_Ilift1, - contX_Ilift2,contX2contX_CF1L]) 1)), - (rtac thelub_lift1a 1), - (atac 1), - (etac exE 1), - (etac exE 1), - (rtac exI 1), - (rtac exI 1), - (etac box_equals 1), - (rtac refl 1), - (simp_tac (Lift_ss addsimps [contX_Iup]) 1) - ]); - - - -val thelub_lift2b = prove_goalw Lift3.thy [up_def,lift_def] -"[| is_chain(Y); ! i x. ~ Y(i) = up[x] |] ==> lub(range(Y)) = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (inst_lift_pcpo RS ssubst) 1), - (rtac thelub_lift1b 1), - (atac 1), - (strip_tac 1), - (dtac spec 1), - (dtac spec 1), - (rtac swap 1), - (atac 1), - (dtac notnotD 1), - (etac box_equals 1), - (rtac refl 1), - (simp_tac (Lift_ss addsimps [contX_Iup]) 1) - ]); - - -val lift_lemma2 = prove_goal Lift3.thy " (? x.z = up[x]) = (~z=UU)" - (fn prems => - [ - (rtac iffI 1), - (etac exE 1), - (hyp_subst_tac 1), - (rtac defined_up 1), - (res_inst_tac [("p","z")] liftE1 1), - (etac notE 1), - (atac 1), - (etac exI 1) - ]); - - -val thelub_lift2a_rev = prove_goal Lift3.thy -"[| is_chain(Y); lub(range(Y)) = up[x] |] ==> ? i x. Y(i) = up[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac exE 1), - (rtac chain_UU_I_inverse2 1), - (rtac (lift_lemma2 RS iffD1) 1), - (etac exI 1), - (rtac exI 1), - (rtac (lift_lemma2 RS iffD2) 1), - (atac 1) - ]); - -val thelub_lift2b_rev = prove_goal Lift3.thy -"[| is_chain(Y); lub(range(Y)) = UU |] ==> ! i x. ~ Y(i) = up[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac allI 1), - (rtac (notex2all RS iffD1) 1), - (rtac contrapos 1), - (etac (lift_lemma2 RS iffD1) 2), - (rtac notnotI 1), - (rtac (chain_UU_I RS spec) 1), - (atac 1), - (atac 1) - ]); - - -val thelub_lift3 = prove_goal Lift3.thy -"is_chain(Y) ==> lub(range(Y)) = UU |\ -\ lub(range(Y)) = up[lub(range(%i. lift[LAM x. x][Y(i)]))]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac disjE 1), - (rtac disjI1 2), - (rtac thelub_lift2b 2), - (atac 2), - (atac 2), - (rtac disjI2 2), - (rtac thelub_lift2a 2), - (atac 2), - (atac 2), - (fast_tac HOL_cs 1) - ]); - -val lift3 = prove_goal Lift3.thy "lift[up][x]=x" - (fn prems => - [ - (res_inst_tac [("p","x")] liftE1 1), - (asm_simp_tac (Cfun_ss addsimps [lift1,lift2]) 1), - (asm_simp_tac (Cfun_ss addsimps [lift1,lift2]) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* install simplifier for ('a)u *) -(* ------------------------------------------------------------------------ *) - -val lift_rews = [lift1,lift2,defined_up]; diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/lift3.thy --- a/src/HOLCF/lift3.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,29 +0,0 @@ -(* Title: HOLCF/lift3.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - - -Class instance of ('a)u for class pcpo - -*) - -Lift3 = Lift2 + - -arities "u" :: (pcpo)pcpo (* Witness lift2.ML *) - -consts - up :: "'a -> ('a)u" - lift :: "('a->'c)-> ('a)u -> 'c" - -rules - -inst_lift_pcpo "UU::('a)u = UU_lift" - -up_def "up == (LAM x.Iup(x))" -lift_def "lift == (LAM f p.Ilift(f)(p))" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/one.ML --- a/src/HOLCF/one.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,96 +0,0 @@ -(* Title: HOLCF/one.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for one.thy -*) - -open One; - -(* ------------------------------------------------------------------------ *) -(* Exhaustion and Elimination for type one *) -(* ------------------------------------------------------------------------ *) - -val Exh_one = prove_goalw One.thy [one_def] "z=UU | z = one" - (fn prems => - [ - (res_inst_tac [("p","rep_one[z]")] liftE1 1), - (rtac disjI1 1), - (rtac ((abs_one_iso RS allI) RS ((rep_one_iso RS allI) RS iso_strict ) - RS conjunct2 RS subst) 1), - (rtac (abs_one_iso RS subst) 1), - (etac cfun_arg_cong 1), - (rtac disjI2 1), - (rtac (abs_one_iso RS subst) 1), - (rtac cfun_arg_cong 1), - (rtac (unique_void2 RS subst) 1), - (atac 1) - ]); - -val oneE = prove_goal One.thy - "[| p=UU ==> Q; p = one ==>Q|] ==>Q" - (fn prems => - [ - (rtac (Exh_one RS disjE) 1), - (eresolve_tac prems 1), - (eresolve_tac prems 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* distinctness for type one : stored in a list *) -(* ------------------------------------------------------------------------ *) - -val dist_less_one = [ -prove_goalw One.thy [one_def] "~one << UU" - (fn prems => - [ - (rtac classical3 1), - (rtac less_lift4b 1), - (rtac (rep_one_iso RS subst) 1), - (rtac (rep_one_iso RS subst) 1), - (rtac monofun_cfun_arg 1), - (etac ((abs_one_iso RS allI) RS ((rep_one_iso RS allI) RS iso_strict ) - RS conjunct2 RS ssubst) 1) - ]) -]; - -val dist_eq_one = [prove_goal One.thy "~one=UU" - (fn prems => - [ - (rtac not_less2not_eq 1), - (resolve_tac dist_less_one 1) - ])]; - -val dist_eq_one = dist_eq_one @ (map (fn thm => (thm RS not_sym)) dist_eq_one); - -(* ------------------------------------------------------------------------ *) -(* one is flat *) -(* ------------------------------------------------------------------------ *) - -val prems = goalw One.thy [flat_def] "flat(one)"; -by (rtac allI 1); -by (rtac allI 1); -by (res_inst_tac [("p","x")] oneE 1); -by (asm_simp_tac ccc1_ss 1); -by (res_inst_tac [("p","y")] oneE 1); -by (asm_simp_tac (ccc1_ss addsimps dist_less_one) 1); -by (asm_simp_tac ccc1_ss 1); -val flat_one = result(); - - -(* ------------------------------------------------------------------------ *) -(* properties of one_when *) -(* here I tried a generic prove procedure *) -(* ------------------------------------------------------------------------ *) - -fun prover s = prove_goalw One.thy [one_when_def,one_def] s - (fn prems => - [ - (simp_tac (ccc1_ss addsimps [(rep_one_iso ), - (abs_one_iso RS allI) RS ((rep_one_iso RS allI) - RS iso_strict) RS conjunct1] )1) - ]); - -val one_when = map prover ["one_when[x][UU] = UU","one_when[x][one] = x"]; - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/one.thy --- a/src/HOLCF/one.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,53 +0,0 @@ -(* Title: HOLCF/one.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Introduve atomic type one = (void)u - -This is the first type that is introduced using a domain isomorphism. -The type axiom - - arities one :: pcpo - -and the continuity of the Isomorphisms are taken for granted. Since the -type is not recursive, it could be easily introduced in a purely conservative -style as it was used for the types sprod, ssum, lift. The definition of the -ordering is canonical using abstraction and representation, but this would take -again a lot of internal constants. It can easily be seen that the axioms below -are consistent. - -The partial ordering on type one is implicitly defined via the -isomorphism axioms and the continuity of abs_one and rep_one. - -We could also introduce the function less_one with definition - -less_one(x,y) = rep_one[x] << rep_one[y] - - -*) - -One = ccc1+ - -types one 0 -arities one :: pcpo - -consts - abs_one :: "(void)u -> one" - rep_one :: "one -> (void)u" - one :: "one" - one_when :: "'c -> one -> 'c" - -rules - abs_one_iso "abs_one[rep_one[u]] = u" - rep_one_iso "rep_one[abs_one[x]] = x" - - one_def "one == abs_one[up[UU]]" - one_when_def "one_when == (LAM c u.lift[LAM x.c][rep_one[u]])" - -end - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/pcpo.ML --- a/src/HOLCF/pcpo.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,272 +0,0 @@ -(* Title: HOLCF/pcpo.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for pcpo.thy -*) - -open Pcpo; - -(* ------------------------------------------------------------------------ *) -(* in pcpo's everthing equal to THE lub has lub properties for every chain *) -(* ------------------------------------------------------------------------ *) - -val thelubE = prove_goal Pcpo.thy - "[| is_chain(S);lub(range(S)) = l::'a::pcpo|] ==> range(S) <<| l " -(fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (rtac lubI 1), - (etac cpo 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Properties of the lub *) -(* ------------------------------------------------------------------------ *) - - -val is_ub_thelub = (cpo RS lubI RS is_ub_lub); -(* is_chain(?S1) ==> ?S1(?x) << lub(range(?S1)) *) - -val is_lub_thelub = (cpo RS lubI RS is_lub_lub); -(* [| is_chain(?S5); range(?S5) <| ?x1 |] ==> lub(range(?S5)) << ?x1 *) - - -(* ------------------------------------------------------------------------ *) -(* the << relation between two chains is preserved by their lubs *) -(* ------------------------------------------------------------------------ *) - -val lub_mono = prove_goal Pcpo.thy - "[|is_chain(C1::(nat=>'a::pcpo));is_chain(C2); ! k. C1(k) << C2(k)|]\ -\ ==> lub(range(C1)) << lub(range(C2))" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac is_lub_thelub 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (rtac trans_less 1), - (etac spec 1), - (etac is_ub_thelub 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the = relation between two chains is preserved by their lubs *) -(* ------------------------------------------------------------------------ *) - -val lub_equal = prove_goal Pcpo.thy -"[| is_chain(C1::(nat=>'a::pcpo));is_chain(C2);!k.C1(k)=C2(k)|]\ -\ ==> lub(range(C1))=lub(range(C2))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac antisym_less 1), - (rtac lub_mono 1), - (atac 1), - (atac 1), - (strip_tac 1), - (rtac (antisym_less_inverse RS conjunct1) 1), - (etac spec 1), - (rtac lub_mono 1), - (atac 1), - (atac 1), - (strip_tac 1), - (rtac (antisym_less_inverse RS conjunct2) 1), - (etac spec 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* more results about mono and = of lubs of chains *) -(* ------------------------------------------------------------------------ *) - -val lub_mono2 = prove_goal Pcpo.thy -"[|? j.!i. j X(i::nat)=Y(i);is_chain(X::nat=>'a::pcpo);is_chain(Y)|]\ -\ ==> lub(range(X))< - [ - (rtac exE 1), - (resolve_tac prems 1), - (rtac is_lub_thelub 1), - (resolve_tac prems 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (res_inst_tac [("Q","x X(i)=Y(i);is_chain(X::nat=>'a::pcpo);is_chain(Y)|]\ -\ ==> lub(range(X))=lub(range(Y))" - (fn prems => - [ - (rtac antisym_less 1), - (rtac lub_mono2 1), - (REPEAT (resolve_tac prems 1)), - (cut_facts_tac prems 1), - (rtac lub_mono2 1), - (safe_tac HOL_cs), - (step_tac HOL_cs 1), - (safe_tac HOL_cs), - (rtac sym 1), - (fast_tac HOL_cs 1) - ]); - -val lub_mono3 = prove_goal Pcpo.thy "[|is_chain(Y::nat=>'a::pcpo);is_chain(X);\ -\! i. ? j. Y(i)<< X(j)|]==> lub(range(Y))< - [ - (cut_facts_tac prems 1), - (rtac is_lub_thelub 1), - (atac 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (etac allE 1), - (etac exE 1), - (rtac trans_less 1), - (rtac is_ub_thelub 2), - (atac 2), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* usefull lemmas about UU *) -(* ------------------------------------------------------------------------ *) - -val eq_UU_iff = prove_goal Pcpo.thy "(x=UU)=(x< - [ - (rtac iffI 1), - (hyp_subst_tac 1), - (rtac refl_less 1), - (rtac antisym_less 1), - (atac 1), - (rtac minimal 1) - ]); - -val UU_I = prove_goal Pcpo.thy "x << UU ==> x = UU" - (fn prems => - [ - (rtac (eq_UU_iff RS ssubst) 1), - (resolve_tac prems 1) - ]); - -val not_less2not_eq = prove_goal Pcpo.thy "~x< ~x=y" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac classical3 1), - (atac 1), - (hyp_subst_tac 1), - (rtac refl_less 1) - ]); - - -val chain_UU_I = prove_goal Pcpo.thy - "[|is_chain(Y);lub(range(Y))=UU|] ==> ! i.Y(i)=UU" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac allI 1), - (rtac antisym_less 1), - (rtac minimal 2), - (res_inst_tac [("t","UU")] subst 1), - (atac 1), - (etac is_ub_thelub 1) - ]); - - -val chain_UU_I_inverse = prove_goal Pcpo.thy - "!i.Y(i::nat)=UU ==> lub(range(Y::(nat=>'a::pcpo)))=UU" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac lub_chain_maxelem 1), - (rtac is_chainI 1), - (rtac allI 1), - (res_inst_tac [("s","UU"),("t","Y(i)")] subst 1), - (rtac sym 1), - (etac spec 1), - (rtac minimal 1), - (rtac exI 1), - (etac spec 1), - (rtac allI 1), - (rtac (antisym_less_inverse RS conjunct1) 1), - (etac spec 1) - ]); - -val chain_UU_I_inverse2 = prove_goal Pcpo.thy - "~lub(range(Y::(nat=>'a::pcpo)))=UU ==> ? i.~ Y(i)=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (notall2ex RS iffD1) 1), - (rtac swap 1), - (rtac chain_UU_I_inverse 2), - (etac notnotD 2), - (atac 1) - ]); - - -val notUU_I = prove_goal Pcpo.thy "[| x< ~y=UU" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac contrapos 1), - (rtac UU_I 1), - (hyp_subst_tac 1), - (atac 1) - ]); - - -val chain_mono2 = prove_goal Pcpo.thy -"[|? j.~Y(j)=UU;is_chain(Y::nat=>'a::pcpo)|]\ -\ ==> ? j.!i.j~Y(i)=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (safe_tac HOL_cs), - (step_tac HOL_cs 1), - (strip_tac 1), - (rtac notUU_I 1), - (atac 2), - (etac (chain_mono RS mp) 1), - (atac 1) - ]); - - - - -(* ------------------------------------------------------------------------ *) -(* uniqueness in void *) -(* ------------------------------------------------------------------------ *) - -val unique_void2 = prove_goal Pcpo.thy "x::void=UU" - (fn prems => - [ - (rtac (inst_void_pcpo RS ssubst) 1), - (rtac (Rep_Void_inverse RS subst) 1), - (rtac (Rep_Void_inverse RS subst) 1), - (rtac arg_cong 1), - (rtac box_equals 1), - (rtac refl 1), - (rtac (unique_void RS sym) 1), - (rtac (unique_void RS sym) 1) - ]); - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/pcpo.thy --- a/src/HOLCF/pcpo.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,39 +0,0 @@ -(* Title: HOLCF/pcpo.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Definition of class pcpo (pointed complete partial order) - -The prototype theory for this class is porder.thy - -*) - -Pcpo = Porder + - -(* Introduction of new class. The witness is type void. *) - -classes pcpo < po - -(* default class is still term *) -(* void is the prototype in pcpo *) - -arities void :: pcpo - -consts - UU :: "'a::pcpo" (* UU is the least element *) -rules - -(* class axioms: justification is theory Porder *) - -minimal "UU << x" (* witness is minimal_void *) - -cpo "is_chain(S) ==> ? x. range(S) <<| x::('a::pcpo)" - (* witness is cpo_void *) - (* needs explicit type *) - -(* instance of UU for the prototype void *) - -inst_void_pcpo "UU::void = UU_void" - -end diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/porder.ML --- a/src/HOLCF/porder.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,427 +0,0 @@ -(* Title: HOLCF/porder.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory porder.thy -*) - -open Porder0; -open Porder; - -val box_less = prove_goal Porder.thy -"[| a << b; c << a; b << d|] ==> c << d" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac trans_less 1), - (etac trans_less 1), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* lubs are unique *) -(* ------------------------------------------------------------------------ *) - -val unique_lub = prove_goalw Porder.thy [is_lub, is_ub] - "[| S <<| x ; S <<| y |] ==> x=y" -( fn prems => - [ - (cut_facts_tac prems 1), - (etac conjE 1), - (etac conjE 1), - (rtac antisym_less 1), - (rtac mp 1),((etac allE 1) THEN (atac 1) THEN (atac 1)), - (rtac mp 1),((etac allE 1) THEN (atac 1) THEN (atac 1)) - ]); - -(* ------------------------------------------------------------------------ *) -(* chains are monotone functions *) -(* ------------------------------------------------------------------------ *) - -val chain_mono = prove_goalw Porder.thy [is_chain] - " is_chain(F) ==> x F(x)< - [ - (cut_facts_tac prems 1), - (nat_ind_tac "y" 1), - (rtac impI 1), - (etac less_zeroE 1), - (rtac (less_Suc_eq RS ssubst) 1), - (strip_tac 1), - (etac disjE 1), - (rtac trans_less 1), - (etac allE 2), - (atac 2), - (fast_tac HOL_cs 1), - (hyp_subst_tac 1), - (etac allE 1), - (atac 1) - ]); - -val chain_mono3 = prove_goal Porder.thy - "[| is_chain(F); x <= y |] ==> F(x) << F(y)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (le_imp_less_or_eq RS disjE) 1), - (atac 1), - (etac (chain_mono RS mp) 1), - (atac 1), - (hyp_subst_tac 1), - (rtac refl_less 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Lemma for reasoning by cases on the natural numbers *) -(* ------------------------------------------------------------------------ *) - -val nat_less_cases = prove_goal Porder.thy - "[| m::nat < n ==> P(n,m); m=n ==> P(n,m);n < m ==> P(n,m)|]==>P(n,m)" -( fn prems => - [ - (res_inst_tac [("m1","n"),("n1","m")] (less_linear RS disjE) 1), - (etac disjE 2), - (etac (hd (tl (tl prems))) 1), - (etac (sym RS hd (tl prems)) 1), - (etac (hd prems) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* The range of a chain is a totaly ordered << *) -(* ------------------------------------------------------------------------ *) - -val chain_is_tord = prove_goalw Porder.thy [is_tord] - "is_chain(F) ==> is_tord(range(F))" -( fn prems => - [ - (cut_facts_tac prems 1), - (rewrite_goals_tac [range_def]), - (rtac allI 1 ), - (rtac allI 1 ), - (rtac (mem_Collect_eq RS ssubst) 1), - (rtac (mem_Collect_eq RS ssubst) 1), - (strip_tac 1), - (etac conjE 1), - (etac exE 1), - (etac exE 1), - (hyp_subst_tac 1), - (hyp_subst_tac 1), - (res_inst_tac [("m","xa"),("n","xb")] (nat_less_cases) 1), - (rtac disjI1 1), - (rtac (chain_mono RS mp) 1), - (atac 1), - (atac 1), - (rtac disjI1 1), - (hyp_subst_tac 1), - (rtac refl_less 1), - (rtac disjI2 1), - (rtac (chain_mono RS mp) 1), - (atac 1), - (atac 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* technical lemmas about lub and is_lub, use above results about @ *) -(* ------------------------------------------------------------------------ *) - -val lubI = prove_goal Porder.thy "(? x. M <<| x) ==> M <<| lub(M)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (lub RS ssubst) 1), - (etac selectI2 1) - ]); - -val lubE = prove_goal Porder.thy " M <<| lub(M) ==> ? x. M <<| x" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac exI 1) - ]); - -val lub_eq = prove_goal Porder.thy - "(? x. M <<| x) = M <<| lub(M)" -(fn prems => - [ - (rtac (lub RS ssubst) 1), - (rtac (select_eq_Ex RS subst) 1), - (rtac refl 1) - ]); - - -val thelubI = prove_goal Porder.thy " M <<| l ==> lub(M) = l" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac unique_lub 1), - (rtac (lub RS ssubst) 1), - (etac selectI 1), - (atac 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* access to some definition as inference rule *) -(* ------------------------------------------------------------------------ *) - -val is_lubE = prove_goalw Porder.thy [is_lub] - "S <<| x ==> S <| x & (! u. S <| u --> x << u)" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - -val is_lubI = prove_goalw Porder.thy [is_lub] - "S <| x & (! u. S <| u --> x << u) ==> S <<| x" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1) - ]); - -val is_chainE = prove_goalw Porder.thy [is_chain] - "is_chain(F) ==> ! i. F(i) << F(Suc(i))" -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1)]); - -val is_chainI = prove_goalw Porder.thy [is_chain] - "! i. F(i) << F(Suc(i)) ==> is_chain(F) " -(fn prems => - [ - (cut_facts_tac prems 1), - (atac 1)]); - -(* ------------------------------------------------------------------------ *) -(* technical lemmas about (least) upper bounds of chains *) -(* ------------------------------------------------------------------------ *) - -val ub_rangeE = prove_goalw Porder.thy [is_ub] - "range(S) <| x ==> ! i. S(i) << x" -(fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (rtac mp 1), - (etac spec 1), - (rtac rangeI 1) - ]); - -val ub_rangeI = prove_goalw Porder.thy [is_ub] - "! i. S(i) << x ==> range(S) <| x" -(fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (etac rangeE 1), - (hyp_subst_tac 1), - (etac spec 1) - ]); - -val is_ub_lub = (is_lubE RS conjunct1 RS ub_rangeE RS spec); -(* range(?S1) <<| ?x1 ==> ?S1(?x) << ?x1 *) - -val is_lub_lub = (is_lubE RS conjunct2 RS spec RS mp); -(* [| ?S3 <<| ?x3; ?S3 <| ?x1 |] ==> ?x3 << ?x1 *) - -(* ------------------------------------------------------------------------ *) -(* Prototype lemmas for class pcpo *) -(* ------------------------------------------------------------------------ *) - -(* ------------------------------------------------------------------------ *) -(* a technical argument about << on void *) -(* ------------------------------------------------------------------------ *) - -val less_void = prove_goal Porder.thy "(u1::void << u2) = (u1 = u2)" -(fn prems => - [ - (rtac (inst_void_po RS ssubst) 1), - (rewrite_goals_tac [less_void_def]), - (rtac iffI 1), - (rtac injD 1), - (atac 2), - (rtac inj_inverseI 1), - (rtac Rep_Void_inverse 1), - (etac arg_cong 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* void is pointed. The least element is UU_void *) -(* ------------------------------------------------------------------------ *) - -val minimal_void = prove_goal Porder.thy "UU_void << x" -(fn prems => - [ - (rtac (inst_void_po RS ssubst) 1), - (rewrite_goals_tac [less_void_def]), - (simp_tac (HOL_ss addsimps [unique_void]) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* UU_void is the trivial lub of all chains in void *) -(* ------------------------------------------------------------------------ *) - -val lub_void = prove_goalw Porder.thy [is_lub] "M <<| UU_void" -(fn prems => - [ - (rtac conjI 1), - (rewrite_goals_tac [is_ub]), - (strip_tac 1), - (rtac (inst_void_po RS ssubst) 1), - (rewrite_goals_tac [less_void_def]), - (simp_tac (HOL_ss addsimps [unique_void]) 1), - (strip_tac 1), - (rtac minimal_void 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* lub(?M) = UU_void *) -(* ------------------------------------------------------------------------ *) - -val thelub_void = (lub_void RS thelubI); - -(* ------------------------------------------------------------------------ *) -(* void is a cpo wrt. countable chains *) -(* ------------------------------------------------------------------------ *) - -val cpo_void = prove_goal Porder.thy - "is_chain(S::nat=>void) ==> ? x. range(S) <<| x " -(fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("x","UU_void")] exI 1), - (rtac lub_void 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* end of prototype lemmas for class pcpo *) -(* ------------------------------------------------------------------------ *) - - -(* ------------------------------------------------------------------------ *) -(* the reverse law of anti--symmetrie of << *) -(* ------------------------------------------------------------------------ *) - -val antisym_less_inverse = prove_goal Porder.thy "x=y ==> x << y & y << x" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac conjI 1), - ((rtac subst 1) THEN (rtac refl_less 2) THEN (atac 1)), - ((rtac subst 1) THEN (rtac refl_less 2) THEN (etac sym 1)) - ]); - -(* ------------------------------------------------------------------------ *) -(* results about finite chains *) -(* ------------------------------------------------------------------------ *) - -val lub_finch1 = prove_goalw Porder.thy [max_in_chain_def] - "[| is_chain(C) ; max_in_chain(i,C)|] ==> range(C) <<| C(i)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (res_inst_tac [("m","i")] nat_less_cases 1), - (rtac (antisym_less_inverse RS conjunct2) 1), - (etac (disjI1 RS less_or_eq_imp_le RS rev_mp) 1), - (etac spec 1), - (rtac (antisym_less_inverse RS conjunct2) 1), - (etac (disjI2 RS less_or_eq_imp_le RS rev_mp) 1), - (etac spec 1), - (etac (chain_mono RS mp) 1), - (atac 1), - (strip_tac 1), - (etac (ub_rangeE RS spec) 1) - ]); - -val lub_finch2 = prove_goalw Porder.thy [finite_chain_def] - "finite_chain(C) ==> range(C) <<| C(@ i. max_in_chain(i,C))" - (fn prems=> - [ - (cut_facts_tac prems 1), - (rtac lub_finch1 1), - (etac conjunct1 1), - (rtac selectI2 1), - (etac conjunct2 1) - ]); - - -val bin_chain = prove_goal Porder.thy "x< is_chain(%i. if(i=0,x,y))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_chainI 1), - (rtac allI 1), - (nat_ind_tac "i" 1), - (asm_simp_tac nat_ss 1), - (asm_simp_tac nat_ss 1), - (rtac refl_less 1) - ]); - -val bin_chainmax = prove_goalw Porder.thy [max_in_chain_def,le_def] - "x< max_in_chain(Suc(0),%i. if(i=0,x,y))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac allI 1), - (nat_ind_tac "j" 1), - (asm_simp_tac nat_ss 1), - (asm_simp_tac nat_ss 1) - ]); - -val lub_bin_chain = prove_goal Porder.thy - "x << y ==> range(%i. if(i = 0,x,y)) <<| y" -(fn prems=> - [ (cut_facts_tac prems 1), - (res_inst_tac [("s","if(Suc(0) = 0,x,y)")] subst 1), - (rtac lub_finch1 2), - (etac bin_chain 2), - (etac bin_chainmax 2), - (simp_tac nat_ss 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the maximal element in a chain is its lub *) -(* ------------------------------------------------------------------------ *) - -val lub_chain_maxelem = prove_goal Porder.thy -"[|is_chain(Y);? i.Y(i)=c;!i.Y(i)< lub(range(Y)) = c" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac thelubI 1), - (rtac is_lubI 1), - (rtac conjI 1), - (etac ub_rangeI 1), - (strip_tac 1), - (res_inst_tac [("P","%i.Y(i)=c")] exE 1), - (atac 1), - (hyp_subst_tac 1), - (etac (ub_rangeE RS spec) 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the lub of a constant chain is the constant *) -(* ------------------------------------------------------------------------ *) - -val lub_const = prove_goal Porder.thy "range(%x.c) <<| c" - (fn prems => - [ - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (strip_tac 1), - (rtac refl_less 1), - (strip_tac 1), - (etac (ub_rangeE RS spec) 1) - ]); - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/porder.thy --- a/src/HOLCF/porder.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,42 +0,0 @@ -(* Title: HOLCF/porder.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Conservative extension of theory Porder0 by constant definitions - -*) - -Porder = Porder0 + - -consts - "<|" :: "['a set,'a::po] => bool" (infixl 55) - "<<|" :: "['a set,'a::po] => bool" (infixl 55) - lub :: "'a set => 'a::po" - is_tord :: "'a::po set => bool" - is_chain :: "(nat=>'a::po) => bool" - max_in_chain :: "[nat,nat=>'a::po]=>bool" - finite_chain :: "(nat=>'a::po)=>bool" - -rules - -(* class definitions *) - -is_ub "S <| x == ! y.y:S --> y< x << u)" - -lub "lub(S) = (@x. S <<| x)" - -(* Arbitrary chains are total orders *) -is_tord "is_tord(S) == ! x y. x:S & y:S --> (x< C(i) = C(j)" - -finite_chain_def "finite_chain(C) == is_chain(C) & (? i. max_in_chain(i,C))" - -end diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/porder0.thy --- a/src/HOLCF/porder0.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,42 +0,0 @@ -(* Title: HOLCF/porder0.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Definition of class porder (partial order) - -The prototype theory for this class is void.thy - -*) - -Porder0 = Void + - -(* Introduction of new class. The witness is type void. *) - -classes po < term - -(* default type is still term ! *) -(* void is the prototype in po *) - -arities void :: po - -consts "<<" :: "['a,'a::po] => bool" (infixl 55) - -rules - -(* class axioms: justification is theory Void *) - -refl_less "x << x" - (* witness refl_less_void *) - -antisym_less "[|x< x = y" - (* witness antisym_less_void *) - -trans_less "[|x< x<bool = less_void" - -end diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod0.ML --- a/src/HOLCF/sprod0.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,389 +0,0 @@ -(* Title: HOLCF/sprod0.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory sprod0.thy -*) - -open Sprod0; - -(* ------------------------------------------------------------------------ *) -(* A non-emptyness result for Sprod *) -(* ------------------------------------------------------------------------ *) - -val SprodI = prove_goalw Sprod0.thy [Sprod_def] - "Spair_Rep(a,b):Sprod" -(fn prems => - [ - (EVERY1 [rtac CollectI, rtac exI,rtac exI, rtac refl]) - ]); - - -val inj_onto_Abs_Sprod = prove_goal Sprod0.thy - "inj_onto(Abs_Sprod,Sprod)" -(fn prems => - [ - (rtac inj_onto_inverseI 1), - (etac Abs_Sprod_inverse 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Strictness and definedness of Spair_Rep *) -(* ------------------------------------------------------------------------ *) - - -val strict_Spair_Rep = prove_goalw Sprod0.thy [Spair_Rep_def] - "(a=UU | b=UU) ==> (Spair_Rep(a,b) = Spair_Rep(UU,UU))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac ext 1), - (rtac ext 1), - (rtac iffI 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val defined_Spair_Rep_rev = prove_goalw Sprod0.thy [Spair_Rep_def] - "(Spair_Rep(a,b) = Spair_Rep(UU,UU)) ==> (a=UU | b=UU)" - (fn prems => - [ - (res_inst_tac [("Q","a=UU|b=UU")] classical2 1), - (atac 1), - (rtac disjI1 1), - (rtac ((hd prems) RS fun_cong RS fun_cong RS iffD2 RS mp RS - conjunct1 RS sym) 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* injectivity of Spair_Rep and Ispair *) -(* ------------------------------------------------------------------------ *) - -val inject_Spair_Rep = prove_goalw Sprod0.thy [Spair_Rep_def] -"[|~aa=UU ; ~ba=UU ; Spair_Rep(a,b)=Spair_Rep(aa,ba) |] ==> a=aa & b=ba" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac ((nth_elem (2,prems)) RS fun_cong RS fun_cong - RS iffD1 RS mp) 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - - -val inject_Ispair = prove_goalw Sprod0.thy [Ispair_def] - "[|~aa=UU ; ~ba=UU ; Ispair(a,b)=Ispair(aa,ba) |] ==> a=aa & b=ba" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac inject_Spair_Rep 1), - (atac 1), - (etac (inj_onto_Abs_Sprod RS inj_ontoD) 1), - (rtac SprodI 1), - (rtac SprodI 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* strictness and definedness of Ispair *) -(* ------------------------------------------------------------------------ *) - -val strict_Ispair = prove_goalw Sprod0.thy [Ispair_def] - "(a=UU | b=UU) ==> Ispair(a,b)=Ispair(UU,UU)" -(fn prems => - [ - (cut_facts_tac prems 1), - (etac (strict_Spair_Rep RS arg_cong) 1) - ]); - -val strict_Ispair1 = prove_goalw Sprod0.thy [Ispair_def] - "Ispair(UU,b) = Ispair(UU,UU)" -(fn prems => - [ - (rtac (strict_Spair_Rep RS arg_cong) 1), - (rtac disjI1 1), - (rtac refl 1) - ]); - -val strict_Ispair2 = prove_goalw Sprod0.thy [Ispair_def] - "Ispair(a,UU) = Ispair(UU,UU)" -(fn prems => - [ - (rtac (strict_Spair_Rep RS arg_cong) 1), - (rtac disjI2 1), - (rtac refl 1) - ]); - -val strict_Ispair_rev = prove_goal Sprod0.thy - "~Ispair(x,y)=Ispair(UU,UU) ==> ~x=UU & ~y=UU" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (de_morgan1 RS ssubst) 1), - (etac contrapos 1), - (etac strict_Ispair 1) - ]); - -val defined_Ispair_rev = prove_goalw Sprod0.thy [Ispair_def] - "Ispair(a,b) = Ispair(UU,UU) ==> (a = UU | b = UU)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac defined_Spair_Rep_rev 1), - (rtac (inj_onto_Abs_Sprod RS inj_ontoD) 1), - (atac 1), - (rtac SprodI 1), - (rtac SprodI 1) - ]); - -val defined_Ispair = prove_goal Sprod0.thy -"[|~a=UU; ~b=UU|] ==> ~(Ispair(a,b) = Ispair(UU,UU))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac contrapos 1), - (etac defined_Ispair_rev 2), - (rtac (de_morgan1 RS iffD1) 1), - (etac conjI 1), - (atac 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Exhaustion of the strict product ** *) -(* ------------------------------------------------------------------------ *) - -val Exh_Sprod = prove_goalw Sprod0.thy [Ispair_def] - "z=Ispair(UU,UU) | (? a b. z=Ispair(a,b) & ~a=UU & ~b=UU)" -(fn prems => - [ - (rtac (rewrite_rule [Sprod_def] Rep_Sprod RS CollectE) 1), - (etac exE 1), - (etac exE 1), - (rtac (excluded_middle RS disjE) 1), - (rtac disjI2 1), - (rtac exI 1), - (rtac exI 1), - (rtac conjI 1), - (rtac (Rep_Sprod_inverse RS sym RS trans) 1), - (etac arg_cong 1), - (rtac (de_morgan1 RS ssubst) 1), - (atac 1), - (rtac disjI1 1), - (rtac (Rep_Sprod_inverse RS sym RS trans) 1), - (res_inst_tac [("f","Abs_Sprod")] arg_cong 1), - (etac trans 1), - (etac strict_Spair_Rep 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* general elimination rule for strict product *) -(* ------------------------------------------------------------------------ *) - -val IsprodE = prove_goal Sprod0.thy -"[|p=Ispair(UU,UU) ==> Q ;!!x y. [|p=Ispair(x,y); ~x=UU ; ~y=UU|] ==> Q|] ==> Q" -(fn prems => - [ - (rtac (Exh_Sprod RS disjE) 1), - (etac (hd prems) 1), - (etac exE 1), - (etac exE 1), - (etac conjE 1), - (etac conjE 1), - (etac (hd (tl prems)) 1), - (atac 1), - (atac 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* some results about the selectors Isfst, Issnd *) -(* ------------------------------------------------------------------------ *) - -val strict_Isfst = prove_goalw Sprod0.thy [Isfst_def] - "p=Ispair(UU,UU)==>Isfst(p)=UU" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (rtac conjI 1), - (fast_tac HOL_cs 1), - (strip_tac 1), - (res_inst_tac [("P","Ispair(UU,UU) = Ispair(a,b)")] notE 1), - (rtac not_sym 1), - (rtac defined_Ispair 1), - (REPEAT (fast_tac HOL_cs 1)) - ]); - - -val strict_Isfst1 = prove_goal Sprod0.thy - "Isfst(Ispair(UU,y)) = UU" -(fn prems => - [ - (rtac (strict_Ispair1 RS ssubst) 1), - (rtac strict_Isfst 1), - (rtac refl 1) - ]); - -val strict_Isfst2 = prove_goal Sprod0.thy - "Isfst(Ispair(x,UU)) = UU" -(fn prems => - [ - (rtac (strict_Ispair2 RS ssubst) 1), - (rtac strict_Isfst 1), - (rtac refl 1) - ]); - - -val strict_Issnd = prove_goalw Sprod0.thy [Issnd_def] - "p=Ispair(UU,UU)==>Issnd(p)=UU" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (rtac conjI 1), - (fast_tac HOL_cs 1), - (strip_tac 1), - (res_inst_tac [("P","Ispair(UU,UU) = Ispair(a,b)")] notE 1), - (rtac not_sym 1), - (rtac defined_Ispair 1), - (REPEAT (fast_tac HOL_cs 1)) - ]); - -val strict_Issnd1 = prove_goal Sprod0.thy - "Issnd(Ispair(UU,y)) = UU" -(fn prems => - [ - (rtac (strict_Ispair1 RS ssubst) 1), - (rtac strict_Issnd 1), - (rtac refl 1) - ]); - -val strict_Issnd2 = prove_goal Sprod0.thy - "Issnd(Ispair(x,UU)) = UU" -(fn prems => - [ - (rtac (strict_Ispair2 RS ssubst) 1), - (rtac strict_Issnd 1), - (rtac refl 1) - ]); - -val Isfst = prove_goalw Sprod0.thy [Isfst_def] - "[|~x=UU ;~y=UU |] ==> Isfst(Ispair(x,y)) = x" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (rtac conjI 1), - (strip_tac 1), - (res_inst_tac [("P","Ispair(x,y) = Ispair(UU,UU)")] notE 1), - (etac defined_Ispair 1), - (atac 1), - (atac 1), - (strip_tac 1), - (rtac (inject_Ispair RS conjunct1) 1), - (fast_tac HOL_cs 3), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val Issnd = prove_goalw Sprod0.thy [Issnd_def] - "[|~x=UU ;~y=UU |] ==> Issnd(Ispair(x,y)) = y" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (rtac conjI 1), - (strip_tac 1), - (res_inst_tac [("P","Ispair(x,y) = Ispair(UU,UU)")] notE 1), - (etac defined_Ispair 1), - (atac 1), - (atac 1), - (strip_tac 1), - (rtac (inject_Ispair RS conjunct2) 1), - (fast_tac HOL_cs 3), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val Isfst2 = prove_goal Sprod0.thy "~y=UU ==>Isfst(Ispair(x,y))=x" -(fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1), - (etac Isfst 1), - (atac 1), - (hyp_subst_tac 1), - (rtac strict_Isfst1 1) - ]); - -val Issnd2 = prove_goal Sprod0.thy "~x=UU ==>Issnd(Ispair(x,y))=y" -(fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("Q","y=UU")] (excluded_middle RS disjE) 1), - (etac Issnd 1), - (atac 1), - (hyp_subst_tac 1), - (rtac strict_Issnd2 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* instantiate the simplifier *) -(* ------------------------------------------------------------------------ *) - -val Sprod_ss = - HOL_ss - addsimps [strict_Isfst1,strict_Isfst2,strict_Issnd1,strict_Issnd2, - Isfst2,Issnd2]; - - -val defined_IsfstIssnd = prove_goal Sprod0.thy - "~p=Ispair(UU,UU) ==> ~Isfst(p)=UU & ~Issnd(p)=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p")] IsprodE 1), - (contr_tac 1), - (hyp_subst_tac 1), - (rtac conjI 1), - (asm_simp_tac Sprod_ss 1), - (asm_simp_tac Sprod_ss 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Surjective pairing: equivalent to Exh_Sprod *) -(* ------------------------------------------------------------------------ *) - -val surjective_pairing_Sprod = prove_goal Sprod0.thy - "z = Ispair(Isfst(z))(Issnd(z))" -(fn prems => - [ - (res_inst_tac [("z1","z")] (Exh_Sprod RS disjE) 1), - (asm_simp_tac Sprod_ss 1), - (etac exE 1), - (etac exE 1), - (asm_simp_tac Sprod_ss 1) - ]); - - - - - - - - - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod0.thy --- a/src/HOLCF/sprod0.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,53 +0,0 @@ -(* Title: HOLCF/sprod0.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Strict product -*) - -Sprod0 = Cfun3 + - -(* new type for strict product *) - -types "**" 2 (infixr 20) - -arities "**" :: (pcpo,pcpo)term - -consts - Sprod :: "('a => 'b => bool)set" - Spair_Rep :: "['a,'b] => ['a,'b] => bool" - Rep_Sprod :: "('a ** 'b) => ('a => 'b => bool)" - Abs_Sprod :: "('a => 'b => bool) => ('a ** 'b)" - Ispair :: "['a,'b] => ('a ** 'b)" - Isfst :: "('a ** 'b) => 'a" - Issnd :: "('a ** 'b) => 'b" - -rules - - Spair_Rep_def "Spair_Rep == (%a b. %x y.\ -\ (~a=UU & ~b=UU --> x=a & y=b ))" - - Sprod_def "Sprod == {f. ? a b. f = Spair_Rep(a,b)}" - - (*faking a type definition... *) - (* "**" is isomorphic to Sprod *) - - Rep_Sprod "Rep_Sprod(p):Sprod" - Rep_Sprod_inverse "Abs_Sprod(Rep_Sprod(p)) = p" - Abs_Sprod_inverse "f:Sprod ==> Rep_Sprod(Abs_Sprod(f)) = f" - - (*defining the abstract constants*) - - Ispair_def "Ispair(a,b) == Abs_Sprod(Spair_Rep(a,b))" - - Isfst_def "Isfst(p) == @z.\ -\ (p=Ispair(UU,UU) --> z=UU)\ -\ &(! a b. ~a=UU & ~b=UU & p=Ispair(a,b) --> z=a)" - - Issnd_def "Issnd(p) == @z.\ -\ (p=Ispair(UU,UU) --> z=UU)\ -\ &(! a b. ~a=UU & ~b=UU & p=Ispair(a,b) --> z=b)" - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod1.ML --- a/src/HOLCF/sprod1.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,204 +0,0 @@ -(* Title: HOLCF/sprod1.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory sprod1.thy -*) - -open Sprod1; - -(* ------------------------------------------------------------------------ *) -(* reduction properties for less_sprod *) -(* ------------------------------------------------------------------------ *) - - -val less_sprod1a = prove_goalw Sprod1.thy [less_sprod_def] - "p1=Ispair(UU,UU) ==> less_sprod(p1,p2)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac eqTrueE 1), - (rtac select_equality 1), - (rtac conjI 1), - (fast_tac HOL_cs 1), - (strip_tac 1), - (contr_tac 1), - (dtac conjunct1 1), - (etac rev_mp 1), - (atac 1) - ]); - -val less_sprod1b = prove_goalw Sprod1.thy [less_sprod_def] - "~p1=Ispair(UU,UU) ==> \ -\ less_sprod(p1,p2) = ( Isfst(p1) << Isfst(p2) & Issnd(p1) << Issnd(p2))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (rtac conjI 1), - (strip_tac 1), - (contr_tac 1), - (fast_tac HOL_cs 1), - (dtac conjunct2 1), - (etac rev_mp 1), - (atac 1) - ]); - -val less_sprod2a = prove_goal Sprod1.thy - "less_sprod(Ispair(x,y),Ispair(UU,UU)) ==> x = UU | y = UU" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (excluded_middle RS disjE) 1), - (atac 2), - (rtac disjI1 1), - (rtac antisym_less 1), - (rtac minimal 2), - (res_inst_tac [("s","Isfst(Ispair(x,y))"),("t","x")] subst 1), - (rtac Isfst 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1), - (res_inst_tac [("s","Isfst(Ispair(UU,UU))"),("t","UU")] subst 1), - (simp_tac Sprod_ss 1), - (rtac (defined_Ispair RS less_sprod1b RS iffD1 RS conjunct1) 1), - (REPEAT (fast_tac HOL_cs 1)) - ]); - -val less_sprod2b = prove_goal Sprod1.thy - "less_sprod(p,Ispair(UU,UU)) ==> p = Ispair(UU,UU)" -(fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p")] IsprodE 1), - (atac 1), - (hyp_subst_tac 1), - (rtac strict_Ispair 1), - (etac less_sprod2a 1) - ]); - -val less_sprod2c = prove_goal Sprod1.thy - "[|less_sprod(Ispair(xa,ya),Ispair(x,y));\ -\~ xa = UU ; ~ ya = UU;~ x = UU ; ~ y = UU |] ==> xa << x & ya << y" -(fn prems => - [ - (rtac conjI 1), - (res_inst_tac [("s","Isfst(Ispair(xa,ya))"),("t","xa")] subst 1), - (simp_tac (Sprod_ss addsimps prems)1), - (res_inst_tac [("s","Isfst(Ispair(x,y))"),("t","x")] subst 1), - (simp_tac (Sprod_ss addsimps prems)1), - (rtac (defined_Ispair RS less_sprod1b RS iffD1 RS conjunct1) 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (simp_tac (Sprod_ss addsimps prems)1), - (res_inst_tac [("s","Issnd(Ispair(xa,ya))"),("t","ya")] subst 1), - (simp_tac (Sprod_ss addsimps prems)1), - (res_inst_tac [("s","Issnd(Ispair(x,y))"),("t","y")] subst 1), - (simp_tac (Sprod_ss addsimps prems)1), - (rtac (defined_Ispair RS less_sprod1b RS iffD1 RS conjunct2) 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (simp_tac (Sprod_ss addsimps prems)1) - ]); - -(* ------------------------------------------------------------------------ *) -(* less_sprod is a partial order on Sprod *) -(* ------------------------------------------------------------------------ *) - -val refl_less_sprod = prove_goal Sprod1.thy "less_sprod(p,p)" -(fn prems => - [ - (res_inst_tac [("p","p")] IsprodE 1), - (etac less_sprod1a 1), - (hyp_subst_tac 1), - (rtac (less_sprod1b RS ssubst) 1), - (rtac defined_Ispair 1), - (REPEAT (fast_tac (HOL_cs addIs [refl_less]) 1)) - ]); - - -val antisym_less_sprod = prove_goal Sprod1.thy - "[|less_sprod(p1,p2);less_sprod(p2,p1)|] ==> p1=p2" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] IsprodE 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] IsprodE 1), - (hyp_subst_tac 1), - (rtac refl 1), - (hyp_subst_tac 1), - (rtac (strict_Ispair RS sym) 1), - (etac less_sprod2a 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] IsprodE 1), - (hyp_subst_tac 1), - (rtac (strict_Ispair) 1), - (etac less_sprod2a 1), - (hyp_subst_tac 1), - (res_inst_tac [("x1","x"),("y1","xa"),("x","y"),("y","ya")] (arg_cong RS cong) 1), - (rtac antisym_less 1), - (asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct1]) 1), - (asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct1]) 1), - (rtac antisym_less 1), - (asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct2]) 1), - (asm_simp_tac (HOL_ss addsimps [less_sprod2c RS conjunct2]) 1) - ]); - -val trans_less_sprod = prove_goal Sprod1.thy - "[|less_sprod(p1,p2);less_sprod(p2,p3)|] ==> less_sprod(p1,p3)" -(fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] IsprodE 1), - (etac less_sprod1a 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p3")] IsprodE 1), - (hyp_subst_tac 1), - (res_inst_tac [("s","p2"),("t","Ispair(UU,UU)")] subst 1), - (etac less_sprod2b 1), - (atac 1), - (hyp_subst_tac 1), - (res_inst_tac [("Q","p2=Ispair(UU,UU)")] - (excluded_middle RS disjE) 1), - (rtac (defined_Ispair RS less_sprod1b RS ssubst) 1), - (atac 1), - (atac 1), - (rtac conjI 1), - (res_inst_tac [("y","Isfst(p2)")] trans_less 1), - (rtac conjunct1 1), - (rtac (less_sprod1b RS subst) 1), - (rtac defined_Ispair 1), - (atac 1), - (atac 1), - (atac 1), - (rtac conjunct1 1), - (rtac (less_sprod1b RS subst) 1), - (atac 1), - (atac 1), - (res_inst_tac [("y","Issnd(p2)")] trans_less 1), - (rtac conjunct2 1), - (rtac (less_sprod1b RS subst) 1), - (rtac defined_Ispair 1), - (atac 1), - (atac 1), - (atac 1), - (rtac conjunct2 1), - (rtac (less_sprod1b RS subst) 1), - (atac 1), - (atac 1), - (hyp_subst_tac 1), - (res_inst_tac [("s","Ispair(UU,UU)"),("t","Ispair(x,y)")] subst 1), - (etac (less_sprod2b RS sym) 1), - (atac 1) - ]); - - - - - - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod1.thy --- a/src/HOLCF/sprod1.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,22 +0,0 @@ -(* Title: HOLCF/sprod1.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Partial ordering for the strict product -*) - -Sprod1 = Sprod0 + - -consts - less_sprod :: "[('a ** 'b),('a ** 'b)] => bool" - -rules - - less_sprod_def "less_sprod(p1,p2) == @z.\ -\ ( p1=Ispair(UU,UU) --> z = True)\ -\ &(~p1=Ispair(UU,UU) --> z = ( Isfst(p1) << Isfst(p2) &\ -\ Issnd(p1) << Issnd(p2)))" - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod2.ML --- a/src/HOLCF/sprod2.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,274 +0,0 @@ -(* Title: HOLCF/sprod2.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for sprod2.thy -*) - - -open Sprod2; - -(* ------------------------------------------------------------------------ *) -(* access to less_sprod in class po *) -(* ------------------------------------------------------------------------ *) - -val less_sprod3a = prove_goal Sprod2.thy - "p1=Ispair(UU,UU) ==> p1 << p2" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (inst_sprod_po RS ssubst) 1), - (etac less_sprod1a 1) - ]); - - -val less_sprod3b = prove_goal Sprod2.thy - "~p1=Ispair(UU,UU) ==>\ -\ (p1< - [ - (cut_facts_tac prems 1), - (rtac (inst_sprod_po RS ssubst) 1), - (etac less_sprod1b 1) - ]); - -val less_sprod4b = prove_goal Sprod2.thy - "p << Ispair(UU,UU) ==> p = Ispair(UU,UU)" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac less_sprod2b 1), - (etac (inst_sprod_po RS subst) 1) - ]); - -val less_sprod4a = (less_sprod4b RS defined_Ispair_rev); -(* Ispair(?a,?b) << Ispair(UU,UU) ==> ?a = UU | ?b = UU *) - -val less_sprod4c = prove_goal Sprod2.thy - "[|Ispair(xa,ya)<\ -\ xa< - [ - (cut_facts_tac prems 1), - (rtac less_sprod2c 1), - (etac (inst_sprod_po RS subst) 1), - (REPEAT (atac 1)) - ]); - -(* ------------------------------------------------------------------------ *) -(* type sprod is pointed *) -(* ------------------------------------------------------------------------ *) - -val minimal_sprod = prove_goal Sprod2.thy "Ispair(UU,UU)< - [ - (rtac less_sprod3a 1), - (rtac refl 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Ispair is monotone in both arguments *) -(* ------------------------------------------------------------------------ *) - -val monofun_Ispair1 = prove_goalw Sprod2.thy [monofun] "monofun(Ispair)" -(fn prems => - [ - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (res_inst_tac [("Q", - " Ispair(y,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), - (res_inst_tac [("Q", - " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), - (rtac (less_sprod3b RS iffD2) 1), - (atac 1), - (rtac conjI 1), - (rtac (Isfst RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (rtac (Isfst RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (atac 1), - (rtac (Issnd RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (rtac (Issnd RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (rtac refl_less 1), - (etac less_sprod3a 1), - (res_inst_tac [("Q", - " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), - (etac less_sprod3a 2), - (res_inst_tac [("P","Ispair(y,xa) = Ispair(UU,UU)")] notE 1), - (atac 2), - (rtac defined_Ispair 1), - (etac notUU_I 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1) - ]); - - -val monofun_Ispair2 = prove_goalw Sprod2.thy [monofun] "monofun(Ispair(x))" -(fn prems => - [ - (strip_tac 1), - (res_inst_tac [("Q", - " Ispair(x,y) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), - (res_inst_tac [("Q", - " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), - (rtac (less_sprod3b RS iffD2) 1), - (atac 1), - (rtac conjI 1), - (rtac (Isfst RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (rtac (Isfst RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (rtac refl_less 1), - (rtac (Issnd RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (rtac (Issnd RS ssubst) 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac (strict_Ispair_rev RS conjunct2) 1), - (atac 1), - (etac less_sprod3a 1), - (res_inst_tac [("Q", - " Ispair(x,xa) =Ispair(UU,UU)")] (excluded_middle RS disjE) 1), - (etac less_sprod3a 2), - (res_inst_tac [("P","Ispair(x,y) = Ispair(UU,UU)")] notE 1), - (atac 2), - (rtac defined_Ispair 1), - (etac (strict_Ispair_rev RS conjunct1) 1), - (etac notUU_I 1), - (etac (strict_Ispair_rev RS conjunct2) 1) - ]); - -val monofun_Ispair = prove_goal Sprod2.thy - "[|x1< Ispair(x1,y1)< - [ - (cut_facts_tac prems 1), - (rtac trans_less 1), - (rtac (monofun_Ispair1 RS monofunE RS spec RS spec RS mp RS - (less_fun RS iffD1 RS spec)) 1), - (rtac (monofun_Ispair2 RS monofunE RS spec RS spec RS mp) 2), - (atac 1), - (atac 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Isfst and Issnd are monotone *) -(* ------------------------------------------------------------------------ *) - -val monofun_Isfst = prove_goalw Sprod2.thy [monofun] "monofun(Isfst)" -(fn prems => - [ - (strip_tac 1), - (res_inst_tac [("p","x")] IsprodE 1), - (hyp_subst_tac 1), - (rtac trans_less 1), - (rtac minimal 2), - (rtac (strict_Isfst1 RS ssubst) 1), - (rtac refl_less 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","y")] IsprodE 1), - (hyp_subst_tac 1), - (res_inst_tac [("t","Isfst(Ispair(xa,ya))")] subst 1), - (rtac refl_less 2), - (etac (less_sprod4b RS sym RS arg_cong) 1), - (hyp_subst_tac 1), - (rtac (Isfst RS ssubst) 1), - (atac 1), - (atac 1), - (rtac (Isfst RS ssubst) 1), - (atac 1), - (atac 1), - (etac (less_sprod4c RS conjunct1) 1), - (REPEAT (atac 1)) - ]); - -val monofun_Issnd = prove_goalw Sprod2.thy [monofun] "monofun(Issnd)" -(fn prems => - [ - (strip_tac 1), - (res_inst_tac [("p","x")] IsprodE 1), - (hyp_subst_tac 1), - (rtac trans_less 1), - (rtac minimal 2), - (rtac (strict_Issnd1 RS ssubst) 1), - (rtac refl_less 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","y")] IsprodE 1), - (hyp_subst_tac 1), - (res_inst_tac [("t","Issnd(Ispair(xa,ya))")] subst 1), - (rtac refl_less 2), - (etac (less_sprod4b RS sym RS arg_cong) 1), - (hyp_subst_tac 1), - (rtac (Issnd RS ssubst) 1), - (atac 1), - (atac 1), - (rtac (Issnd RS ssubst) 1), - (atac 1), - (atac 1), - (etac (less_sprod4c RS conjunct2) 1), - (REPEAT (atac 1)) - ]); - - -(* ------------------------------------------------------------------------ *) -(* the type 'a ** 'b is a cpo *) -(* ------------------------------------------------------------------------ *) - -val lub_sprod = prove_goal Sprod2.thy -"[|is_chain(S)|] ==> range(S) <<| \ -\ Ispair(lub(range(%i.Isfst(S(i)))),lub(range(%i.Issnd(S(i)))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (res_inst_tac [("t","S(i)")] (surjective_pairing_Sprod RS ssubst) 1), - (rtac monofun_Ispair 1), - (rtac is_ub_thelub 1), - (etac (monofun_Isfst RS ch2ch_monofun) 1), - (rtac is_ub_thelub 1), - (etac (monofun_Issnd RS ch2ch_monofun) 1), - (strip_tac 1), - (res_inst_tac [("t","u")] (surjective_pairing_Sprod RS ssubst) 1), - (rtac monofun_Ispair 1), - (rtac is_lub_thelub 1), - (etac (monofun_Isfst RS ch2ch_monofun) 1), - (etac (monofun_Isfst RS ub2ub_monofun) 1), - (rtac is_lub_thelub 1), - (etac (monofun_Issnd RS ch2ch_monofun) 1), - (etac (monofun_Issnd RS ub2ub_monofun) 1) - ]); - -val thelub_sprod = (lub_sprod RS thelubI); -(* is_chain(?S1) ==> lub(range(?S1)) = *) -(* Ispair(lub(range(%i. Isfst(?S1(i)))),lub(range(%i. Issnd(?S1(i))))) *) - -val cpo_sprod = prove_goal Sprod2.thy - "is_chain(S::nat=>'a**'b)==>? x.range(S)<<| x" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac exI 1), - (etac lub_sprod 1) - ]); - - - - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod2.thy --- a/src/HOLCF/sprod2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,24 +0,0 @@ -(* Title: HOLCF/sprod2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class Instance **::(pcpo,pcpo)po -*) - -Sprod2 = Sprod1 + - -arities "**" :: (pcpo,pcpo)po - -(* Witness for the above arity axiom is sprod1.ML *) - -rules - -(* instance of << for type ['a ** 'b] *) - -inst_sprod_po "(op <<)::['a ** 'b,'a ** 'b]=>bool = less_sprod" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod3.ML --- a/src/HOLCF/sprod3.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,695 +0,0 @@ -(* Title: HOLCF/sprod3.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for Sprod3.thy -*) - -open Sprod3; - -(* ------------------------------------------------------------------------ *) -(* continuity of Ispair, Isfst, Issnd *) -(* ------------------------------------------------------------------------ *) - -val sprod3_lemma1 = prove_goal Sprod3.thy -"[| is_chain(Y); x~= UU; lub(range(Y))~= UU |] ==>\ -\ Ispair(lub(range(Y)),x) =\ -\ Ispair(lub(range(%i. Isfst(Ispair(Y(i),x)))),\ -\ lub(range(%i. Issnd(Ispair(Y(i),x)))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("f1","Ispair")] (arg_cong RS cong) 1), - (rtac lub_equal 1), - (atac 1), - (rtac (monofun_Isfst RS ch2ch_monofun) 1), - (rtac ch2ch_fun 1), - (rtac (monofun_Ispair1 RS ch2ch_monofun) 1), - (atac 1), - (rtac allI 1), - (asm_simp_tac Sprod_ss 1), - (rtac sym 1), - (rtac lub_chain_maxelem 1), - (rtac (monofun_Issnd RS ch2ch_monofun) 1), - (rtac ch2ch_fun 1), - (rtac (monofun_Ispair1 RS ch2ch_monofun) 1), - (atac 1), - (res_inst_tac [("P","%j.~Y(j)=UU")] exE 1), - (rtac (notall2ex RS iffD1) 1), - (res_inst_tac [("Q","lub(range(Y)) = UU")] contrapos 1), - (atac 1), - (rtac chain_UU_I_inverse 1), - (atac 1), - (rtac exI 1), - (etac Issnd2 1), - (rtac allI 1), - (res_inst_tac [("Q","Y(i)=UU")] (excluded_middle RS disjE) 1), - (asm_simp_tac Sprod_ss 1), - (rtac refl_less 1), - (res_inst_tac [("s","UU"),("t","Y(i)")] subst 1), - (etac sym 1), - (asm_simp_tac Sprod_ss 1), - (rtac minimal 1) - ]); - - -val sprod3_lemma2 = prove_goal Sprod3.thy -"[| is_chain(Y); ~ x = UU; lub(range(Y)) = UU |] ==>\ -\ Ispair(lub(range(Y)),x) =\ -\ Ispair(lub(range(%i. Isfst(Ispair(Y(i),x)))),\ -\ lub(range(%i. Issnd(Ispair(Y(i),x)))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1), - (atac 1), - (rtac trans 1), - (rtac strict_Ispair1 1), - (rtac (strict_Ispair RS sym) 1), - (rtac disjI1 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (asm_simp_tac Sprod_ss 1), - (etac (chain_UU_I RS spec) 1), - (atac 1) - ]); - - -val sprod3_lemma3 = prove_goal Sprod3.thy -"[| is_chain(Y); x = UU |] ==>\ -\ Ispair(lub(range(Y)),x) =\ -\ Ispair(lub(range(%i. Isfst(Ispair(Y(i),x)))),\ -\ lub(range(%i. Issnd(Ispair(Y(i),x)))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("s","UU"),("t","x")] ssubst 1), - (atac 1), - (rtac trans 1), - (rtac strict_Ispair2 1), - (rtac (strict_Ispair RS sym) 1), - (rtac disjI1 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (simp_tac Sprod_ss 1) - ]); - - -val contlub_Ispair1 = prove_goal Sprod3.thy "contlub(Ispair)" -(fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (expand_fun_eq RS iffD2) 1), - (strip_tac 1), - (rtac (lub_fun RS thelubI RS ssubst) 1), - (etac (monofun_Ispair1 RS ch2ch_monofun) 1), - (rtac trans 1), - (rtac (thelub_sprod RS sym) 2), - (rtac ch2ch_fun 2), - (etac (monofun_Ispair1 RS ch2ch_monofun) 2), - (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1), - (res_inst_tac - [("Q","lub(range(Y))=UU")] (excluded_middle RS disjE) 1), - (etac sprod3_lemma1 1), - (atac 1), - (atac 1), - (etac sprod3_lemma2 1), - (atac 1), - (atac 1), - (etac sprod3_lemma3 1), - (atac 1) - ]); - -val sprod3_lemma4 = prove_goal Sprod3.thy -"[| is_chain(Y); ~ x = UU; ~ lub(range(Y)) = UU |] ==>\ -\ Ispair(x,lub(range(Y))) =\ -\ Ispair(lub(range(%i. Isfst(Ispair(x,Y(i))))),\ -\ lub(range(%i. Issnd(Ispair(x,Y(i))))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("f1","Ispair")] (arg_cong RS cong) 1), - (rtac sym 1), - (rtac lub_chain_maxelem 1), - (rtac (monofun_Isfst RS ch2ch_monofun) 1), - (rtac (monofun_Ispair2 RS ch2ch_monofun) 1), - (atac 1), - (res_inst_tac [("P","%j.~Y(j)=UU")] exE 1), - (rtac (notall2ex RS iffD1) 1), - (res_inst_tac [("Q","lub(range(Y)) = UU")] contrapos 1), - (atac 1), - (rtac chain_UU_I_inverse 1), - (atac 1), - (rtac exI 1), - (etac Isfst2 1), - (rtac allI 1), - (res_inst_tac [("Q","Y(i)=UU")] (excluded_middle RS disjE) 1), - (asm_simp_tac Sprod_ss 1), - (rtac refl_less 1), - (res_inst_tac [("s","UU"),("t","Y(i)")] subst 1), - (etac sym 1), - (asm_simp_tac Sprod_ss 1), - (rtac minimal 1), - (rtac lub_equal 1), - (atac 1), - (rtac (monofun_Issnd RS ch2ch_monofun) 1), - (rtac (monofun_Ispair2 RS ch2ch_monofun) 1), - (atac 1), - (rtac allI 1), - (asm_simp_tac Sprod_ss 1) - ]); - -val sprod3_lemma5 = prove_goal Sprod3.thy -"[| is_chain(Y); ~ x = UU; lub(range(Y)) = UU |] ==>\ -\ Ispair(x,lub(range(Y))) =\ -\ Ispair(lub(range(%i. Isfst(Ispair(x,Y(i))))),\ -\ lub(range(%i. Issnd(Ispair(x,Y(i))))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1), - (atac 1), - (rtac trans 1), - (rtac strict_Ispair2 1), - (rtac (strict_Ispair RS sym) 1), - (rtac disjI2 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (asm_simp_tac Sprod_ss 1), - (etac (chain_UU_I RS spec) 1), - (atac 1) - ]); - -val sprod3_lemma6 = prove_goal Sprod3.thy -"[| is_chain(Y); x = UU |] ==>\ -\ Ispair(x,lub(range(Y))) =\ -\ Ispair(lub(range(%i. Isfst(Ispair(x,Y(i))))),\ -\ lub(range(%i. Issnd(Ispair(x,Y(i))))))" -(fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("s","UU"),("t","x")] ssubst 1), - (atac 1), - (rtac trans 1), - (rtac strict_Ispair1 1), - (rtac (strict_Ispair RS sym) 1), - (rtac disjI1 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (simp_tac Sprod_ss 1) - ]); - -val contlub_Ispair2 = prove_goal Sprod3.thy "contlub(Ispair(x))" -(fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_sprod RS sym) 2), - (etac (monofun_Ispair2 RS ch2ch_monofun) 2), - (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1), - (res_inst_tac [("Q","lub(range(Y))=UU")] - (excluded_middle RS disjE) 1), - (etac sprod3_lemma4 1), - (atac 1), - (atac 1), - (etac sprod3_lemma5 1), - (atac 1), - (atac 1), - (etac sprod3_lemma6 1), - (atac 1) - ]); - - -val contX_Ispair1 = prove_goal Sprod3.thy "contX(Ispair)" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Ispair1 1), - (rtac contlub_Ispair1 1) - ]); - - -val contX_Ispair2 = prove_goal Sprod3.thy "contX(Ispair(x))" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Ispair2 1), - (rtac contlub_Ispair2 1) - ]); - -val contlub_Isfst = prove_goal Sprod3.thy "contlub(Isfst)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (lub_sprod RS thelubI RS ssubst) 1), - (atac 1), - (res_inst_tac [("Q","lub(range(%i. Issnd(Y(i))))=UU")] - (excluded_middle RS disjE) 1), - (asm_simp_tac Sprod_ss 1), - (res_inst_tac [("s","UU"),("t","lub(range(%i. Issnd(Y(i))))")] - ssubst 1), - (atac 1), - (rtac trans 1), - (asm_simp_tac Sprod_ss 1), - (rtac sym 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (rtac strict_Isfst 1), - (rtac swap 1), - (etac (defined_IsfstIssnd RS conjunct2) 2), - (rtac notnotI 1), - (rtac (chain_UU_I RS spec) 1), - (rtac (monofun_Issnd RS ch2ch_monofun) 1), - (atac 1), - (atac 1) - ]); - - -val contlub_Issnd = prove_goal Sprod3.thy "contlub(Issnd)" -(fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac (lub_sprod RS thelubI RS ssubst) 1), - (atac 1), - (res_inst_tac [("Q","lub(range(%i. Isfst(Y(i))))=UU")] - (excluded_middle RS disjE) 1), - (asm_simp_tac Sprod_ss 1), - (res_inst_tac [("s","UU"),("t","lub(range(%i. Isfst(Y(i))))")] - ssubst 1), - (atac 1), - (asm_simp_tac Sprod_ss 1), - (rtac sym 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (rtac strict_Issnd 1), - (rtac swap 1), - (etac (defined_IsfstIssnd RS conjunct1) 2), - (rtac notnotI 1), - (rtac (chain_UU_I RS spec) 1), - (rtac (monofun_Isfst RS ch2ch_monofun) 1), - (atac 1), - (atac 1) - ]); - - -val contX_Isfst = prove_goal Sprod3.thy "contX(Isfst)" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Isfst 1), - (rtac contlub_Isfst 1) - ]); - -val contX_Issnd = prove_goal Sprod3.thy "contX(Issnd)" -(fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Issnd 1), - (rtac contlub_Issnd 1) - ]); - -(* - -------------------------------------------------------------------------- - more lemmas for Sprod3.thy - - -------------------------------------------------------------------------- -*) - -val spair_eq = prove_goal Sprod3.thy "[|x1=x2;y1=y2|] ==> x1##y1 = x2##y2" - (fn prems => - [ - (cut_facts_tac prems 1), - (fast_tac HOL_cs 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* convert all lemmas to the continuous versions *) -(* ------------------------------------------------------------------------ *) - -val beta_cfun_sprod = prove_goalw Sprod3.thy [spair_def] - "(LAM x y.Ispair(x,y))[a][b] = Ispair(a,b)" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tac 1), - (rtac contX_Ispair2 1), - (rtac contX2contX_CF1L 1), - (rtac contX_Ispair1 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Ispair2 1), - (rtac refl 1) - ]); - -val inject_spair = prove_goalw Sprod3.thy [spair_def] - "[|~aa=UU ; ~ba=UU ; (a##b)=(aa##ba) |] ==> a=aa & b=ba" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac inject_Ispair 1), - (atac 1), - (etac box_equals 1), - (rtac beta_cfun_sprod 1), - (rtac beta_cfun_sprod 1) - ]); - -val inst_sprod_pcpo2 = prove_goalw Sprod3.thy [spair_def] "UU = (UU##UU)" - (fn prems => - [ - (rtac sym 1), - (rtac trans 1), - (rtac beta_cfun_sprod 1), - (rtac sym 1), - (rtac inst_sprod_pcpo 1) - ]); - -val strict_spair = prove_goalw Sprod3.thy [spair_def] - "(a=UU | b=UU) ==> (a##b)=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans 1), - (rtac beta_cfun_sprod 1), - (rtac trans 1), - (rtac (inst_sprod_pcpo RS sym) 2), - (etac strict_Ispair 1) - ]); - -val strict_spair1 = prove_goalw Sprod3.thy [spair_def] "(UU##b) = UU" - (fn prems => - [ - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac trans 1), - (rtac (inst_sprod_pcpo RS sym) 2), - (rtac strict_Ispair1 1) - ]); - -val strict_spair2 = prove_goalw Sprod3.thy [spair_def] "(a##UU) = UU" - (fn prems => - [ - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac trans 1), - (rtac (inst_sprod_pcpo RS sym) 2), - (rtac strict_Ispair2 1) - ]); - -val strict_spair_rev = prove_goalw Sprod3.thy [spair_def] - "~(x##y)=UU ==> ~x=UU & ~y=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac strict_Ispair_rev 1), - (rtac (beta_cfun_sprod RS subst) 1), - (rtac (inst_sprod_pcpo RS subst) 1), - (atac 1) - ]); - -val defined_spair_rev = prove_goalw Sprod3.thy [spair_def] - "(a##b) = UU ==> (a = UU | b = UU)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac defined_Ispair_rev 1), - (rtac (beta_cfun_sprod RS subst) 1), - (rtac (inst_sprod_pcpo RS subst) 1), - (atac 1) - ]); - -val defined_spair = prove_goalw Sprod3.thy [spair_def] - "[|~a=UU; ~b=UU|] ==> ~(a##b) = UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (inst_sprod_pcpo RS ssubst) 1), - (etac defined_Ispair 1), - (atac 1) - ]); - -val Exh_Sprod2 = prove_goalw Sprod3.thy [spair_def] - "z=UU | (? a b. z=(a##b) & ~a=UU & ~b=UU)" - (fn prems => - [ - (rtac (Exh_Sprod RS disjE) 1), - (rtac disjI1 1), - (rtac (inst_sprod_pcpo RS ssubst) 1), - (atac 1), - (rtac disjI2 1), - (etac exE 1), - (etac exE 1), - (rtac exI 1), - (rtac exI 1), - (rtac conjI 1), - (rtac (beta_cfun_sprod RS ssubst) 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - - -val sprodE = prove_goalw Sprod3.thy [spair_def] -"[|p=UU ==> Q;!!x y. [|p=(x##y);~x=UU ; ~y=UU|] ==> Q|] ==> Q" -(fn prems => - [ - (rtac IsprodE 1), - (resolve_tac prems 1), - (rtac (inst_sprod_pcpo RS ssubst) 1), - (atac 1), - (resolve_tac prems 1), - (atac 2), - (atac 2), - (rtac (beta_cfun_sprod RS ssubst) 1), - (atac 1) - ]); - - -val strict_sfst = prove_goalw Sprod3.thy [sfst_def] - "p=UU==>sfst[p]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac strict_Isfst 1), - (rtac (inst_sprod_pcpo RS subst) 1), - (atac 1) - ]); - -val strict_sfst1 = prove_goalw Sprod3.thy [sfst_def,spair_def] - "sfst[UU##y] = UU" - (fn prems => - [ - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac strict_Isfst1 1) - ]); - -val strict_sfst2 = prove_goalw Sprod3.thy [sfst_def,spair_def] - "sfst[x##UU] = UU" - (fn prems => - [ - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac strict_Isfst2 1) - ]); - -val strict_ssnd = prove_goalw Sprod3.thy [ssnd_def] - "p=UU==>ssnd[p]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac strict_Issnd 1), - (rtac (inst_sprod_pcpo RS subst) 1), - (atac 1) - ]); - -val strict_ssnd1 = prove_goalw Sprod3.thy [ssnd_def,spair_def] - "ssnd[UU##y] = UU" - (fn prems => - [ - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac strict_Issnd1 1) - ]); - -val strict_ssnd2 = prove_goalw Sprod3.thy [ssnd_def,spair_def] - "ssnd[x##UU] = UU" - (fn prems => - [ - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac strict_Issnd2 1) - ]); - -val sfst2 = prove_goalw Sprod3.thy [sfst_def,spair_def] - "~y=UU ==>sfst[x##y]=x" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (etac Isfst2 1) - ]); - -val ssnd2 = prove_goalw Sprod3.thy [ssnd_def,spair_def] - "~x=UU ==>ssnd[x##y]=y" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (etac Issnd2 1) - ]); - - -val defined_sfstssnd = prove_goalw Sprod3.thy [sfst_def,ssnd_def,spair_def] - "~p=UU ==> ~sfst[p]=UU & ~ssnd[p]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac defined_IsfstIssnd 1), - (rtac (inst_sprod_pcpo RS subst) 1), - (atac 1) - ]); - - -val surjective_pairing_Sprod2 = prove_goalw Sprod3.thy - [sfst_def,ssnd_def,spair_def] "(sfst[p] ## ssnd[p]) = p" - (fn prems => - [ - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac (surjective_pairing_Sprod RS sym) 1) - ]); - - -val less_sprod5b = prove_goalw Sprod3.thy [sfst_def,ssnd_def,spair_def] - "~p1=UU ==> (p1< - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac less_sprod3b 1), - (rtac (inst_sprod_pcpo RS subst) 1), - (atac 1) - ]); - - -val less_sprod5c = prove_goalw Sprod3.thy [sfst_def,ssnd_def,spair_def] - "[|xa##ya<xa< - [ - (cut_facts_tac prems 1), - (rtac less_sprod4c 1), - (REPEAT (atac 2)), - (rtac (beta_cfun_sprod RS subst) 1), - (rtac (beta_cfun_sprod RS subst) 1), - (atac 1) - ]); - -val lub_sprod2 = prove_goalw Sprod3.thy [sfst_def,ssnd_def,spair_def] -"[|is_chain(S)|] ==> range(S) <<| \ -\ (lub(range(%i.sfst[S(i)])) ## lub(range(%i.ssnd[S(i)])))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun_sprod RS ssubst) 1), - (rtac (beta_cfun RS ext RS ssubst) 1), - (rtac contX_Issnd 1), - (rtac (beta_cfun RS ext RS ssubst) 1), - (rtac contX_Isfst 1), - (rtac lub_sprod 1), - (resolve_tac prems 1) - ]); - - -val thelub_sprod2 = (lub_sprod2 RS thelubI); -(* is_chain(?S1) ==> lub(range(?S1)) = *) -(* (lub(range(%i. sfst[?S1(i)]))## lub(range(%i. ssnd[?S1(i)]))) *) - - - -val ssplit1 = prove_goalw Sprod3.thy [ssplit_def] - "ssplit[f][UU]=UU" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tacR 1), - (rtac (strictify1 RS ssubst) 1), - (rtac refl 1) - ]); - -val ssplit2 = prove_goalw Sprod3.thy [ssplit_def] - "[|~x=UU;~y=UU|] ==> ssplit[f][x##y]=f[x][y]" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tacR 1), - (rtac (strictify2 RS ssubst) 1), - (rtac defined_spair 1), - (resolve_tac prems 1), - (resolve_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (contX_tacR 1), - (rtac (sfst2 RS ssubst) 1), - (resolve_tac prems 1), - (rtac (ssnd2 RS ssubst) 1), - (resolve_tac prems 1), - (rtac refl 1) - ]); - - -val ssplit3 = prove_goalw Sprod3.thy [ssplit_def] - "ssplit[spair][z]=z" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (contX_tacR 1), - (res_inst_tac [("Q","z=UU")] classical2 1), - (hyp_subst_tac 1), - (rtac strictify1 1), - (rtac trans 1), - (rtac strictify2 1), - (atac 1), - (rtac (beta_cfun RS ssubst) 1), - (contX_tacR 1), - (rtac surjective_pairing_Sprod2 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* install simplifier for Sprod *) -(* ------------------------------------------------------------------------ *) - -val Sprod_rews = [strict_spair1,strict_spair2,strict_sfst1,strict_sfst2, - strict_ssnd1,strict_ssnd2,sfst2,ssnd2, - ssplit1,ssplit2]; - -val Sprod_ss = Cfun_ss addsimps Sprod_rews; - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/sprod3.thy --- a/src/HOLCF/sprod3.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,39 +0,0 @@ -(* Title: HOLCF/sprod3.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class instance of ** for class pcpo -*) - -Sprod3 = Sprod2 + - -arities "**" :: (pcpo,pcpo)pcpo (* Witness sprod2.ML *) - -consts - "@spair" :: "'a => 'b => ('a**'b)" ("_##_" [101,100] 100) - "cop @spair" :: "'a -> 'b -> ('a**'b)" ("spair") - (* continuous strict pairing *) - sfst :: "('a**'b)->'a" - ssnd :: "('a**'b)->'b" - ssplit :: "('a->'b->'c)->('a**'b)->'c" - -rules - -inst_sprod_pcpo "UU::'a**'b = Ispair(UU,UU)" -spair_def "spair == (LAM x y.Ispair(x,y))" -sfst_def "sfst == (LAM p.Isfst(p))" -ssnd_def "ssnd == (LAM p.Issnd(p))" -ssplit_def "ssplit == (LAM f. strictify[LAM p.f[sfst[p]][ssnd[p]]])" - -end - -ML - -(* ----------------------------------------------------------------------*) -(* parse translations for the above mixfix *) -(* ----------------------------------------------------------------------*) - -val parse_translation = [("@spair",mk_cinfixtr "@spair")]; - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum0.ML --- a/src/HOLCF/ssum0.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,394 +0,0 @@ -(* Title: HOLCF/ssum0.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory ssum0.thy -*) - -open Ssum0; - -(* ------------------------------------------------------------------------ *) -(* A non-emptyness result for Sssum *) -(* ------------------------------------------------------------------------ *) - -val SsumIl = prove_goalw Ssum0.thy [Ssum_def] "Sinl_Rep(a):Ssum" - (fn prems => - [ - (rtac CollectI 1), - (rtac disjI1 1), - (rtac exI 1), - (rtac refl 1) - ]); - -val SsumIr = prove_goalw Ssum0.thy [Ssum_def] "Sinr_Rep(a):Ssum" - (fn prems => - [ - (rtac CollectI 1), - (rtac disjI2 1), - (rtac exI 1), - (rtac refl 1) - ]); - -val inj_onto_Abs_Ssum = prove_goal Ssum0.thy "inj_onto(Abs_Ssum,Ssum)" -(fn prems => - [ - (rtac inj_onto_inverseI 1), - (etac Abs_Ssum_inverse 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Strictness of Sinr_Rep, Sinl_Rep and Isinl, Isinr *) -(* ------------------------------------------------------------------------ *) - -val strict_SinlSinr_Rep = prove_goalw Ssum0.thy [Sinr_Rep_def,Sinl_Rep_def] - "Sinl_Rep(UU) = Sinr_Rep(UU)" - (fn prems => - [ - (rtac ext 1), - (rtac ext 1), - (rtac ext 1), - (fast_tac HOL_cs 1) - ]); - -val strict_IsinlIsinr = prove_goalw Ssum0.thy [Isinl_def,Isinr_def] - "Isinl(UU) = Isinr(UU)" - (fn prems => - [ - (rtac (strict_SinlSinr_Rep RS arg_cong) 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* distinctness of Sinl_Rep, Sinr_Rep and Isinl, Isinr *) -(* ------------------------------------------------------------------------ *) - -val noteq_SinlSinr_Rep = prove_goalw Ssum0.thy [Sinl_Rep_def,Sinr_Rep_def] - "(Sinl_Rep(a) = Sinr_Rep(b)) ==> a=UU & b=UU" - (fn prems => - [ - (rtac conjI 1), - (res_inst_tac [("Q","a=UU")] classical2 1), - (atac 1), - (rtac ((hd prems) RS fun_cong RS fun_cong RS fun_cong RS iffD2 - RS mp RS conjunct1 RS sym) 1), - (fast_tac HOL_cs 1), - (atac 1), - (res_inst_tac [("Q","b=UU")] classical2 1), - (atac 1), - (rtac ((hd prems) RS fun_cong RS fun_cong RS fun_cong RS iffD1 - RS mp RS conjunct1 RS sym) 1), - (fast_tac HOL_cs 1), - (atac 1) - ]); - - -val noteq_IsinlIsinr = prove_goalw Ssum0.thy [Isinl_def,Isinr_def] - "Isinl(a)=Isinr(b) ==> a=UU & b=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac noteq_SinlSinr_Rep 1), - (etac (inj_onto_Abs_Ssum RS inj_ontoD) 1), - (rtac SsumIl 1), - (rtac SsumIr 1) - ]); - - - -(* ------------------------------------------------------------------------ *) -(* injectivity of Sinl_Rep, Sinr_Rep and Isinl, Isinr *) -(* ------------------------------------------------------------------------ *) - -val inject_Sinl_Rep1 = prove_goalw Ssum0.thy [Sinl_Rep_def] - "(Sinl_Rep(a) = Sinl_Rep(UU)) ==> a=UU" - (fn prems => - [ - (res_inst_tac [("Q","a=UU")] classical2 1), - (atac 1), - (rtac ((hd prems) RS fun_cong RS fun_cong RS fun_cong - RS iffD2 RS mp RS conjunct1 RS sym) 1), - (fast_tac HOL_cs 1), - (atac 1) - ]); - -val inject_Sinr_Rep1 = prove_goalw Ssum0.thy [Sinr_Rep_def] - "(Sinr_Rep(b) = Sinr_Rep(UU)) ==> b=UU" - (fn prems => - [ - (res_inst_tac [("Q","b=UU")] classical2 1), - (atac 1), - (rtac ((hd prems) RS fun_cong RS fun_cong RS fun_cong - RS iffD2 RS mp RS conjunct1 RS sym) 1), - (fast_tac HOL_cs 1), - (atac 1) - ]); - -val inject_Sinl_Rep2 = prove_goalw Ssum0.thy [Sinl_Rep_def] -"[|~a1=UU ; ~a2=UU ; Sinl_Rep(a1)=Sinl_Rep(a2) |] ==> a1=a2" - (fn prems => - [ - (rtac ((nth_elem (2,prems)) RS fun_cong RS fun_cong RS fun_cong - RS iffD1 RS mp RS conjunct1) 1), - (fast_tac HOL_cs 1), - (resolve_tac prems 1) - ]); - -val inject_Sinr_Rep2 = prove_goalw Ssum0.thy [Sinr_Rep_def] -"[|~b1=UU ; ~b2=UU ; Sinr_Rep(b1)=Sinr_Rep(b2) |] ==> b1=b2" - (fn prems => - [ - (rtac ((nth_elem (2,prems)) RS fun_cong RS fun_cong RS fun_cong - RS iffD1 RS mp RS conjunct1) 1), - (fast_tac HOL_cs 1), - (resolve_tac prems 1) - ]); - -val inject_Sinl_Rep = prove_goal Ssum0.thy - "Sinl_Rep(a1)=Sinl_Rep(a2) ==> a1=a2" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("Q","a1=UU")] classical2 1), - (hyp_subst_tac 1), - (rtac (inject_Sinl_Rep1 RS sym) 1), - (etac sym 1), - (res_inst_tac [("Q","a2=UU")] classical2 1), - (hyp_subst_tac 1), - (etac inject_Sinl_Rep1 1), - (etac inject_Sinl_Rep2 1), - (atac 1), - (atac 1) - ]); - -val inject_Sinr_Rep = prove_goal Ssum0.thy - "Sinr_Rep(b1)=Sinr_Rep(b2) ==> b1=b2" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("Q","b1=UU")] classical2 1), - (hyp_subst_tac 1), - (rtac (inject_Sinr_Rep1 RS sym) 1), - (etac sym 1), - (res_inst_tac [("Q","b2=UU")] classical2 1), - (hyp_subst_tac 1), - (etac inject_Sinr_Rep1 1), - (etac inject_Sinr_Rep2 1), - (atac 1), - (atac 1) - ]); - -val inject_Isinl = prove_goalw Ssum0.thy [Isinl_def] -"Isinl(a1)=Isinl(a2)==> a1=a2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac inject_Sinl_Rep 1), - (etac (inj_onto_Abs_Ssum RS inj_ontoD) 1), - (rtac SsumIl 1), - (rtac SsumIl 1) - ]); - -val inject_Isinr = prove_goalw Ssum0.thy [Isinr_def] -"Isinr(b1)=Isinr(b2) ==> b1=b2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac inject_Sinr_Rep 1), - (etac (inj_onto_Abs_Ssum RS inj_ontoD) 1), - (rtac SsumIr 1), - (rtac SsumIr 1) - ]); - -val inject_Isinl_rev = prove_goal Ssum0.thy -"~a1=a2 ==> ~Isinl(a1) = Isinl(a2)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac contrapos 1), - (etac inject_Isinl 2), - (atac 1) - ]); - -val inject_Isinr_rev = prove_goal Ssum0.thy -"~b1=b2 ==> ~Isinr(b1) = Isinr(b2)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac contrapos 1), - (etac inject_Isinr 2), - (atac 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* Exhaustion of the strict sum ++ *) -(* choice of the bottom representation is arbitrary *) -(* ------------------------------------------------------------------------ *) - -val Exh_Ssum = prove_goalw Ssum0.thy [Isinl_def,Isinr_def] - "z=Isinl(UU) | (? a. z=Isinl(a) & ~a=UU) | (? b. z=Isinr(b) & ~b=UU)" - (fn prems => - [ - (rtac (rewrite_rule [Ssum_def] Rep_Ssum RS CollectE) 1), - (etac disjE 1), - (etac exE 1), - (res_inst_tac [("Q","z= Abs_Ssum(Sinl_Rep(UU))")] classical2 1), - (etac disjI1 1), - (rtac disjI2 1), - (rtac disjI1 1), - (rtac exI 1), - (rtac conjI 1), - (rtac (Rep_Ssum_inverse RS sym RS trans) 1), - (etac arg_cong 1), - (res_inst_tac [("Q","Sinl_Rep(a)=Sinl_Rep(UU)")] contrapos 1), - (etac arg_cong 2), - (etac contrapos 1), - (rtac (Rep_Ssum_inverse RS sym RS trans) 1), - (rtac trans 1), - (etac arg_cong 1), - (etac arg_cong 1), - (etac exE 1), - (res_inst_tac [("Q","z= Abs_Ssum(Sinl_Rep(UU))")] classical2 1), - (etac disjI1 1), - (rtac disjI2 1), - (rtac disjI2 1), - (rtac exI 1), - (rtac conjI 1), - (rtac (Rep_Ssum_inverse RS sym RS trans) 1), - (etac arg_cong 1), - (res_inst_tac [("Q","Sinr_Rep(b)=Sinl_Rep(UU)")] contrapos 1), - (hyp_subst_tac 2), - (rtac (strict_SinlSinr_Rep RS sym) 2), - (etac contrapos 1), - (rtac (Rep_Ssum_inverse RS sym RS trans) 1), - (rtac trans 1), - (etac arg_cong 1), - (etac arg_cong 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* elimination rules for the strict sum ++ *) -(* ------------------------------------------------------------------------ *) - -val IssumE = prove_goal Ssum0.thy - "[|p=Isinl(UU) ==> Q ;\ -\ !!x.[|p=Isinl(x); ~x=UU |] ==> Q;\ -\ !!y.[|p=Isinr(y); ~y=UU |] ==> Q|] ==> Q" - (fn prems => - [ - (rtac (Exh_Ssum RS disjE) 1), - (etac disjE 2), - (eresolve_tac prems 1), - (etac exE 1), - (etac conjE 1), - (eresolve_tac prems 1), - (atac 1), - (etac exE 1), - (etac conjE 1), - (eresolve_tac prems 1), - (atac 1) - ]); - -val IssumE2 = prove_goal Ssum0.thy -"[| !!x. [| p = Isinl(x) |] ==> Q; !!y. [| p = Isinr(y) |] ==> Q |] ==>Q" - (fn prems => - [ - (rtac IssumE 1), - (eresolve_tac prems 1), - (eresolve_tac prems 1), - (eresolve_tac prems 1) - ]); - - - - -(* ------------------------------------------------------------------------ *) -(* rewrites for Iwhen *) -(* ------------------------------------------------------------------------ *) - -val Iwhen1 = prove_goalw Ssum0.thy [Iwhen_def] - "Iwhen(f)(g)(Isinl(UU)) = UU" - (fn prems => - [ - (rtac select_equality 1), - (rtac conjI 1), - (fast_tac HOL_cs 1), - (rtac conjI 1), - (strip_tac 1), - (res_inst_tac [("P","a=UU")] notE 1), - (fast_tac HOL_cs 1), - (rtac inject_Isinl 1), - (rtac sym 1), - (fast_tac HOL_cs 1), - (strip_tac 1), - (res_inst_tac [("P","b=UU")] notE 1), - (fast_tac HOL_cs 1), - (rtac inject_Isinr 1), - (rtac sym 1), - (rtac (strict_IsinlIsinr RS subst) 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - - -val Iwhen2 = prove_goalw Ssum0.thy [Iwhen_def] - "~x=UU ==> Iwhen(f)(g)(Isinl(x)) = f[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (fast_tac HOL_cs 2), - (rtac conjI 1), - (strip_tac 1), - (res_inst_tac [("P","x=UU")] notE 1), - (atac 1), - (rtac inject_Isinl 1), - (atac 1), - (rtac conjI 1), - (strip_tac 1), - (rtac cfun_arg_cong 1), - (rtac inject_Isinl 1), - (fast_tac HOL_cs 1), - (strip_tac 1), - (res_inst_tac [("P","Isinl(x) = Isinr(b)")] notE 1), - (fast_tac HOL_cs 2), - (rtac contrapos 1), - (etac noteq_IsinlIsinr 2), - (fast_tac HOL_cs 1) - ]); - -val Iwhen3 = prove_goalw Ssum0.thy [Iwhen_def] - "~y=UU ==> Iwhen(f)(g)(Isinr(y)) = g[y]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (fast_tac HOL_cs 2), - (rtac conjI 1), - (strip_tac 1), - (res_inst_tac [("P","y=UU")] notE 1), - (atac 1), - (rtac inject_Isinr 1), - (rtac (strict_IsinlIsinr RS subst) 1), - (atac 1), - (rtac conjI 1), - (strip_tac 1), - (res_inst_tac [("P","Isinr(y) = Isinl(a)")] notE 1), - (fast_tac HOL_cs 2), - (rtac contrapos 1), - (etac (sym RS noteq_IsinlIsinr) 2), - (fast_tac HOL_cs 1), - (strip_tac 1), - (rtac cfun_arg_cong 1), - (rtac inject_Isinr 1), - (fast_tac HOL_cs 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* instantiate the simplifier *) -(* ------------------------------------------------------------------------ *) - -val Ssum_ss = Cfun_ss addsimps - [(strict_IsinlIsinr RS sym),Iwhen1,Iwhen2,Iwhen3]; - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum0.thy --- a/src/HOLCF/ssum0.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,54 +0,0 @@ -(* Title: HOLCF/ssum0.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Strict sum -*) - -Ssum0 = Cfun3 + - -(* new type for strict sum *) - -types "++" 2 (infixr 10) - -arities "++" :: (pcpo,pcpo)term - -consts - Ssum :: "(['a,'b,bool]=>bool)set" - Sinl_Rep :: "['a,'a,'b,bool]=>bool" - Sinr_Rep :: "['b,'a,'b,bool]=>bool" - Rep_Ssum :: "('a ++ 'b) => (['a,'b,bool]=>bool)" - Abs_Ssum :: "(['a,'b,bool]=>bool) => ('a ++ 'b)" - Isinl :: "'a => ('a ++ 'b)" - Isinr :: "'b => ('a ++ 'b)" - Iwhen :: "('a->'c)=>('b->'c)=>('a ++ 'b)=> 'c" - -rules - - Sinl_Rep_def "Sinl_Rep == (%a.%x y p.\ -\ (~a=UU --> x=a & p))" - - Sinr_Rep_def "Sinr_Rep == (%b.%x y p.\ -\ (~b=UU --> y=b & ~p))" - - Ssum_def "Ssum =={f.(? a.f=Sinl_Rep(a))|(? b.f=Sinr_Rep(b))}" - - (*faking a type definition... *) - (* "++" is isomorphic to Ssum *) - - Rep_Ssum "Rep_Ssum(p):Ssum" - Rep_Ssum_inverse "Abs_Ssum(Rep_Ssum(p)) = p" - Abs_Ssum_inverse "f:Ssum ==> Rep_Ssum(Abs_Ssum(f)) = f" - - (*defining the abstract constants*) - Isinl_def "Isinl(a) == Abs_Ssum(Sinl_Rep(a))" - Isinr_def "Isinr(b) == Abs_Ssum(Sinr_Rep(b))" - - Iwhen_def "Iwhen(f)(g)(s) == @z.\ -\ (s=Isinl(UU) --> z=UU)\ -\ &(!a. ~a=UU & s=Isinl(a) --> z=f[a])\ -\ &(!b. ~b=UU & s=Isinr(b) --> z=g[b])" - -end - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum1.ML --- a/src/HOLCF/ssum1.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,353 +0,0 @@ -(* Title: HOLCF/ssum1.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory ssum1.thy -*) - -open Ssum1; - -local - -fun eq_left s1 s2 = - ( - (res_inst_tac [("s",s1),("t",s2)] (inject_Isinl RS subst) 1) - THEN (rtac trans 1) - THEN (atac 2) - THEN (etac sym 1)); - -fun eq_right s1 s2 = - ( - (res_inst_tac [("s",s1),("t",s2)] (inject_Isinr RS subst) 1) - THEN (rtac trans 1) - THEN (atac 2) - THEN (etac sym 1)); - -fun UU_left s1 = - ( - (res_inst_tac [("t",s1)](noteq_IsinlIsinr RS conjunct1 RS ssubst)1) - THEN (rtac trans 1) - THEN (atac 2) - THEN (etac sym 1)); - -fun UU_right s1 = - ( - (res_inst_tac [("t",s1)](noteq_IsinlIsinr RS conjunct2 RS ssubst)1) - THEN (rtac trans 1) - THEN (atac 2) - THEN (etac sym 1)) - -in - -val less_ssum1a = prove_goalw Ssum1.thy [less_ssum_def] -"[|s1=Isinl(x); s2=Isinl(y)|] ==> less_ssum(s1,s2) = (x << y)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (dtac conjunct1 2), - (dtac spec 2), - (dtac spec 2), - (etac mp 2), - (fast_tac HOL_cs 2), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (eq_left "x" "u"), - (eq_left "y" "xa"), - (rtac refl 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (UU_left "x"), - (UU_right "v"), - (simp_tac Cfun_ss 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (eq_left "x" "u"), - (UU_left "y"), - (rtac iffI 1), - (etac UU_I 1), - (res_inst_tac [("s","x"),("t","UU")] subst 1), - (atac 1), - (rtac refl_less 1), - (strip_tac 1), - (etac conjE 1), - (UU_left "x"), - (UU_right "v"), - (simp_tac Cfun_ss 1) - ]); - - -val less_ssum1b = prove_goalw Ssum1.thy [less_ssum_def] -"[|s1=Isinr(x); s2=Isinr(y)|] ==> less_ssum(s1,s2) = (x << y)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (dtac conjunct2 2), - (dtac conjunct1 2), - (dtac spec 2), - (dtac spec 2), - (etac mp 2), - (fast_tac HOL_cs 2), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (UU_right "x"), - (UU_left "u"), - (simp_tac Cfun_ss 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (eq_right "x" "v"), - (eq_right "y" "ya"), - (rtac refl 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (UU_right "x"), - (UU_left "u"), - (simp_tac Cfun_ss 1), - (strip_tac 1), - (etac conjE 1), - (eq_right "x" "v"), - (UU_right "y"), - (rtac iffI 1), - (etac UU_I 1), - (res_inst_tac [("s","UU"),("t","x")] subst 1), - (etac sym 1), - (rtac refl_less 1) - ]); - - -val less_ssum1c = prove_goalw Ssum1.thy [less_ssum_def] -"[|s1=Isinl(x); s2=Isinr(y)|] ==> less_ssum(s1,s2) = (x = UU)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (eq_left "x" "u"), - (UU_left "xa"), - (rtac iffI 1), - (res_inst_tac [("s","x"),("t","UU")] subst 1), - (atac 1), - (rtac refl_less 1), - (etac UU_I 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (UU_left "x"), - (UU_right "v"), - (simp_tac Cfun_ss 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (eq_left "x" "u"), - (rtac refl 1), - (strip_tac 1), - (etac conjE 1), - (UU_left "x"), - (UU_right "v"), - (simp_tac Cfun_ss 1), - (dtac conjunct2 1), - (dtac conjunct2 1), - (dtac conjunct1 1), - (dtac spec 1), - (dtac spec 1), - (etac mp 1), - (fast_tac HOL_cs 1) - ]); - - -val less_ssum1d = prove_goalw Ssum1.thy [less_ssum_def] -"[|s1=Isinr(x); s2=Isinl(y)|] ==> less_ssum(s1,s2) = (x = UU)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac select_equality 1), - (dtac conjunct2 2), - (dtac conjunct2 2), - (dtac conjunct2 2), - (dtac spec 2), - (dtac spec 2), - (etac mp 2), - (fast_tac HOL_cs 2), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (UU_right "x"), - (UU_left "u"), - (simp_tac Cfun_ss 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (UU_right "ya"), - (eq_right "x" "v"), - (rtac iffI 1), - (etac UU_I 2), - (res_inst_tac [("s","UU"),("t","x")] subst 1), - (etac sym 1), - (rtac refl_less 1), - (rtac conjI 1), - (strip_tac 1), - (etac conjE 1), - (UU_right "x"), - (UU_left "u"), - (simp_tac HOL_ss 1), - (strip_tac 1), - (etac conjE 1), - (eq_right "x" "v"), - (rtac refl 1) - ]) -end; - - -(* ------------------------------------------------------------------------ *) -(* optimize lemmas about less_ssum *) -(* ------------------------------------------------------------------------ *) - -val less_ssum2a = prove_goal Ssum1.thy - "less_ssum(Isinl(x),Isinl(y)) = (x << y)" - (fn prems => - [ - (rtac less_ssum1a 1), - (rtac refl 1), - (rtac refl 1) - ]); - -val less_ssum2b = prove_goal Ssum1.thy - "less_ssum(Isinr(x),Isinr(y)) = (x << y)" - (fn prems => - [ - (rtac less_ssum1b 1), - (rtac refl 1), - (rtac refl 1) - ]); - -val less_ssum2c = prove_goal Ssum1.thy - "less_ssum(Isinl(x),Isinr(y)) = (x = UU)" - (fn prems => - [ - (rtac less_ssum1c 1), - (rtac refl 1), - (rtac refl 1) - ]); - -val less_ssum2d = prove_goal Ssum1.thy - "less_ssum(Isinr(x),Isinl(y)) = (x = UU)" - (fn prems => - [ - (rtac less_ssum1d 1), - (rtac refl 1), - (rtac refl 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* less_ssum is a partial order on ++ *) -(* ------------------------------------------------------------------------ *) - -val refl_less_ssum = prove_goal Ssum1.thy "less_ssum(p,p)" - (fn prems => - [ - (res_inst_tac [("p","p")] IssumE2 1), - (hyp_subst_tac 1), - (rtac (less_ssum2a RS iffD2) 1), - (rtac refl_less 1), - (hyp_subst_tac 1), - (rtac (less_ssum2b RS iffD2) 1), - (rtac refl_less 1) - ]); - -val antisym_less_ssum = prove_goal Ssum1.thy - "[|less_ssum(p1,p2);less_ssum(p2,p1)|] ==> p1=p2" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] IssumE2 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] IssumE2 1), - (hyp_subst_tac 1), - (res_inst_tac [("f","Isinl")] arg_cong 1), - (rtac antisym_less 1), - (etac (less_ssum2a RS iffD1) 1), - (etac (less_ssum2a RS iffD1) 1), - (hyp_subst_tac 1), - (etac (less_ssum2d RS iffD1 RS ssubst) 1), - (etac (less_ssum2c RS iffD1 RS ssubst) 1), - (rtac strict_IsinlIsinr 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p2")] IssumE2 1), - (hyp_subst_tac 1), - (etac (less_ssum2c RS iffD1 RS ssubst) 1), - (etac (less_ssum2d RS iffD1 RS ssubst) 1), - (rtac (strict_IsinlIsinr RS sym) 1), - (hyp_subst_tac 1), - (res_inst_tac [("f","Isinr")] arg_cong 1), - (rtac antisym_less 1), - (etac (less_ssum2b RS iffD1) 1), - (etac (less_ssum2b RS iffD1) 1) - ]); - -val trans_less_ssum = prove_goal Ssum1.thy - "[|less_ssum(p1,p2);less_ssum(p2,p3)|] ==> less_ssum(p1,p3)" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","p1")] IssumE2 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p3")] IssumE2 1), - (hyp_subst_tac 1), - (rtac (less_ssum2a RS iffD2) 1), - (res_inst_tac [("p","p2")] IssumE2 1), - (hyp_subst_tac 1), - (rtac trans_less 1), - (etac (less_ssum2a RS iffD1) 1), - (etac (less_ssum2a RS iffD1) 1), - (hyp_subst_tac 1), - (etac (less_ssum2c RS iffD1 RS ssubst) 1), - (rtac minimal 1), - (hyp_subst_tac 1), - (rtac (less_ssum2c RS iffD2) 1), - (res_inst_tac [("p","p2")] IssumE2 1), - (hyp_subst_tac 1), - (rtac UU_I 1), - (rtac trans_less 1), - (etac (less_ssum2a RS iffD1) 1), - (rtac (antisym_less_inverse RS conjunct1) 1), - (etac (less_ssum2c RS iffD1) 1), - (hyp_subst_tac 1), - (etac (less_ssum2c RS iffD1) 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","p3")] IssumE2 1), - (hyp_subst_tac 1), - (rtac (less_ssum2d RS iffD2) 1), - (res_inst_tac [("p","p2")] IssumE2 1), - (hyp_subst_tac 1), - (etac (less_ssum2d RS iffD1) 1), - (hyp_subst_tac 1), - (rtac UU_I 1), - (rtac trans_less 1), - (etac (less_ssum2b RS iffD1) 1), - (rtac (antisym_less_inverse RS conjunct1) 1), - (etac (less_ssum2d RS iffD1) 1), - (hyp_subst_tac 1), - (rtac (less_ssum2b RS iffD2) 1), - (res_inst_tac [("p","p2")] IssumE2 1), - (hyp_subst_tac 1), - (etac (less_ssum2d RS iffD1 RS ssubst) 1), - (rtac minimal 1), - (hyp_subst_tac 1), - (rtac trans_less 1), - (etac (less_ssum2b RS iffD1) 1), - (etac (less_ssum2b RS iffD1) 1) - ]); - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum1.thy --- a/src/HOLCF/ssum1.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,25 +0,0 @@ -(* Title: HOLCF/ssum1.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Partial ordering for the strict sum ++ -*) - -Ssum1 = Ssum0 + - -consts - - less_ssum :: "[('a ++ 'b),('a ++ 'b)] => bool" - -rules - - less_ssum_def "less_ssum(s1,s2) == (@z.\ -\ (! u x.s1=Isinl(u) & s2=Isinl(x) --> z = (u << x))\ -\ &(! v y.s1=Isinr(v) & s2=Isinr(y) --> z = (v << y))\ -\ &(! u y.s1=Isinl(u) & s2=Isinr(y) --> z = (u = UU))\ -\ &(! v x.s1=Isinr(v) & s2=Isinl(x) --> z = (v = UU)))" - -end - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum2.ML --- a/src/HOLCF/ssum2.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,419 +0,0 @@ -(* Title: HOLCF/ssum2.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for ssum2.thy -*) - -open Ssum2; - -(* ------------------------------------------------------------------------ *) -(* access to less_ssum in class po *) -(* ------------------------------------------------------------------------ *) - -val less_ssum3a = prove_goal Ssum2.thy - "(Isinl(x) << Isinl(y)) = (x << y)" - (fn prems => - [ - (rtac (inst_ssum_po RS ssubst) 1), - (rtac less_ssum2a 1) - ]); - -val less_ssum3b = prove_goal Ssum2.thy - "(Isinr(x) << Isinr(y)) = (x << y)" - (fn prems => - [ - (rtac (inst_ssum_po RS ssubst) 1), - (rtac less_ssum2b 1) - ]); - -val less_ssum3c = prove_goal Ssum2.thy - "(Isinl(x) << Isinr(y)) = (x = UU)" - (fn prems => - [ - (rtac (inst_ssum_po RS ssubst) 1), - (rtac less_ssum2c 1) - ]); - -val less_ssum3d = prove_goal Ssum2.thy - "(Isinr(x) << Isinl(y)) = (x = UU)" - (fn prems => - [ - (rtac (inst_ssum_po RS ssubst) 1), - (rtac less_ssum2d 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* type ssum ++ is pointed *) -(* ------------------------------------------------------------------------ *) - -val minimal_ssum = prove_goal Ssum2.thy "Isinl(UU) << s" - (fn prems => - [ - (res_inst_tac [("p","s")] IssumE2 1), - (hyp_subst_tac 1), - (rtac (less_ssum3a RS iffD2) 1), - (rtac minimal 1), - (hyp_subst_tac 1), - (rtac (strict_IsinlIsinr RS ssubst) 1), - (rtac (less_ssum3b RS iffD2) 1), - (rtac minimal 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Isinl, Isinr are monotone *) -(* ------------------------------------------------------------------------ *) - -val monofun_Isinl = prove_goalw Ssum2.thy [monofun] "monofun(Isinl)" - (fn prems => - [ - (strip_tac 1), - (etac (less_ssum3a RS iffD2) 1) - ]); - -val monofun_Isinr = prove_goalw Ssum2.thy [monofun] "monofun(Isinr)" - (fn prems => - [ - (strip_tac 1), - (etac (less_ssum3b RS iffD2) 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* Iwhen is monotone in all arguments *) -(* ------------------------------------------------------------------------ *) - - -val monofun_Iwhen1 = prove_goalw Ssum2.thy [monofun] "monofun(Iwhen)" - (fn prems => - [ - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (res_inst_tac [("p","xb")] IssumE 1), - (hyp_subst_tac 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1), - (etac monofun_cfun_fun 1), - (asm_simp_tac Ssum_ss 1) - ]); - -val monofun_Iwhen2 = prove_goalw Ssum2.thy [monofun] "monofun(Iwhen(f))" - (fn prems => - [ - (strip_tac 1), - (rtac (less_fun RS iffD2) 1), - (strip_tac 1), - (res_inst_tac [("p","xa")] IssumE 1), - (hyp_subst_tac 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1), - (etac monofun_cfun_fun 1) - ]); - -val monofun_Iwhen3 = prove_goalw Ssum2.thy [monofun] "monofun(Iwhen(f)(g))" - (fn prems => - [ - (strip_tac 1), - (res_inst_tac [("p","x")] IssumE 1), - (hyp_subst_tac 1), - (asm_simp_tac Ssum_ss 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","y")] IssumE 1), - (hyp_subst_tac 1), - (asm_simp_tac Ssum_ss 1), - (res_inst_tac [("P","xa=UU")] notE 1), - (atac 1), - (rtac UU_I 1), - (rtac (less_ssum3a RS iffD1) 1), - (atac 1), - (hyp_subst_tac 1), - (asm_simp_tac Ssum_ss 1), - (rtac monofun_cfun_arg 1), - (etac (less_ssum3a RS iffD1) 1), - (hyp_subst_tac 1), - (res_inst_tac [("s","UU"),("t","xa")] subst 1), - (etac (less_ssum3c RS iffD1 RS sym) 1), - (asm_simp_tac Ssum_ss 1), - (hyp_subst_tac 1), - (res_inst_tac [("p","y")] IssumE 1), - (hyp_subst_tac 1), - (res_inst_tac [("s","UU"),("t","ya")] subst 1), - (etac (less_ssum3d RS iffD1 RS sym) 1), - (asm_simp_tac Ssum_ss 1), - (hyp_subst_tac 1), - (res_inst_tac [("s","UU"),("t","ya")] subst 1), - (etac (less_ssum3d RS iffD1 RS sym) 1), - (asm_simp_tac Ssum_ss 1), - (hyp_subst_tac 1), - (asm_simp_tac Ssum_ss 1), - (rtac monofun_cfun_arg 1), - (etac (less_ssum3b RS iffD1) 1) - ]); - - - - -(* ------------------------------------------------------------------------ *) -(* some kind of exhaustion rules for chains in 'a ++ 'b *) -(* ------------------------------------------------------------------------ *) - - -val ssum_lemma1 = prove_goal Ssum2.thy -"[|~(!i.? x.Y(i::nat)=Isinl(x))|] ==> (? i.! x.~Y(i)=Isinl(x))" - (fn prems => - [ - (cut_facts_tac prems 1), - (fast_tac HOL_cs 1) - ]); - -val ssum_lemma2 = prove_goal Ssum2.thy -"[|(? i.!x.~(Y::nat => 'a++'b)(i::nat)=Isinl(x::'a))|] ==>\ -\ (? i y. (Y::nat => 'a++'b)(i::nat)=Isinr(y::'b) & ~y=UU)" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (res_inst_tac [("p","Y(i)")] IssumE 1), - (dtac spec 1), - (contr_tac 1), - (dtac spec 1), - (contr_tac 1), - (fast_tac HOL_cs 1) - ]); - - -val ssum_lemma3 = prove_goal Ssum2.thy -"[|is_chain(Y);(? i x. Y(i)=Isinr(x) & ~x=UU)|] ==> (!i.? y.Y(i)=Isinr(y))" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (etac exE 1), - (rtac allI 1), - (res_inst_tac [("p","Y(ia)")] IssumE 1), - (rtac exI 1), - (rtac trans 1), - (rtac strict_IsinlIsinr 2), - (atac 1), - (etac exI 2), - (etac conjE 1), - (res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1), - (hyp_subst_tac 2), - (etac exI 2), - (res_inst_tac [("P","x=UU")] notE 1), - (atac 1), - (rtac (less_ssum3d RS iffD1) 1), - (res_inst_tac [("s","Y(i)"),("t","Isinr(x)")] subst 1), - (atac 1), - (res_inst_tac [("s","Y(ia)"),("t","Isinl(xa)")] subst 1), - (atac 1), - (etac (chain_mono RS mp) 1), - (atac 1), - (res_inst_tac [("P","xa=UU")] notE 1), - (atac 1), - (rtac (less_ssum3c RS iffD1) 1), - (res_inst_tac [("s","Y(i)"),("t","Isinr(x)")] subst 1), - (atac 1), - (res_inst_tac [("s","Y(ia)"),("t","Isinl(xa)")] subst 1), - (atac 1), - (etac (chain_mono RS mp) 1), - (atac 1) - ]); - -val ssum_lemma4 = prove_goal Ssum2.thy -"is_chain(Y) ==> (!i.? x.Y(i)=Isinl(x))|(!i.? y.Y(i)=Isinr(y))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac classical2 1), - (etac disjI1 1), - (rtac disjI2 1), - (etac ssum_lemma3 1), - (rtac ssum_lemma2 1), - (etac ssum_lemma1 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* restricted surjectivity of Isinl *) -(* ------------------------------------------------------------------------ *) - -val ssum_lemma5 = prove_goal Ssum2.thy -"z=Isinl(x)==> Isinl((Iwhen (LAM x.x) (LAM y.UU))(z)) = z" - (fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* restricted surjectivity of Isinr *) -(* ------------------------------------------------------------------------ *) - -val ssum_lemma6 = prove_goal Ssum2.thy -"z=Isinr(x)==> Isinr((Iwhen (LAM y.UU) (LAM x.x))(z)) = z" - (fn prems => - [ - (cut_facts_tac prems 1), - (hyp_subst_tac 1), - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* technical lemmas *) -(* ------------------------------------------------------------------------ *) - -val ssum_lemma7 = prove_goal Ssum2.thy -"[|Isinl(x) << z; ~x=UU|] ==> ? y.z=Isinl(y) & ~y=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","z")] IssumE 1), - (hyp_subst_tac 1), - (etac notE 1), - (rtac antisym_less 1), - (etac (less_ssum3a RS iffD1) 1), - (rtac minimal 1), - (fast_tac HOL_cs 1), - (hyp_subst_tac 1), - (rtac notE 1), - (etac (less_ssum3c RS iffD1) 2), - (atac 1) - ]); - -val ssum_lemma8 = prove_goal Ssum2.thy -"[|Isinr(x) << z; ~x=UU|] ==> ? y.z=Isinr(y) & ~y=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (res_inst_tac [("p","z")] IssumE 1), - (hyp_subst_tac 1), - (etac notE 1), - (etac (less_ssum3d RS iffD1) 1), - (hyp_subst_tac 1), - (rtac notE 1), - (etac (less_ssum3d RS iffD1) 2), - (atac 1), - (fast_tac HOL_cs 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* the type 'a ++ 'b is a cpo in three steps *) -(* ------------------------------------------------------------------------ *) - -val lub_ssum1a = prove_goal Ssum2.thy -"[|is_chain(Y);(!i.? x.Y(i)=Isinl(x))|] ==>\ -\ range(Y) <<|\ -\ Isinl(lub(range(%i.(Iwhen (LAM x.x) (LAM y.UU))(Y(i)))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (etac allE 1), - (etac exE 1), - (res_inst_tac [("t","Y(i)")] (ssum_lemma5 RS subst) 1), - (atac 1), - (rtac (monofun_Isinl RS monofunE RS spec RS spec RS mp) 1), - (rtac is_ub_thelub 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (strip_tac 1), - (res_inst_tac [("p","u")] IssumE2 1), - (res_inst_tac [("t","u")] (ssum_lemma5 RS subst) 1), - (atac 1), - (rtac (monofun_Isinl RS monofunE RS spec RS spec RS mp) 1), - (rtac is_lub_thelub 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (etac (monofun_Iwhen3 RS ub2ub_monofun) 1), - (hyp_subst_tac 1), - (rtac (less_ssum3c RS iffD2) 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (res_inst_tac [("p","Y(i)")] IssumE 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 2), - (etac notE 1), - (rtac (less_ssum3c RS iffD1) 1), - (res_inst_tac [("t","Isinl(x)")] subst 1), - (atac 1), - (etac (ub_rangeE RS spec) 1) - ]); - - -val lub_ssum1b = prove_goal Ssum2.thy -"[|is_chain(Y);(!i.? x.Y(i)=Isinr(x))|] ==>\ -\ range(Y) <<|\ -\ Isinr(lub(range(%i.(Iwhen (LAM y.UU) (LAM x.x))(Y(i)))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac is_lubI 1), - (rtac conjI 1), - (rtac ub_rangeI 1), - (rtac allI 1), - (etac allE 1), - (etac exE 1), - (res_inst_tac [("t","Y(i)")] (ssum_lemma6 RS subst) 1), - (atac 1), - (rtac (monofun_Isinr RS monofunE RS spec RS spec RS mp) 1), - (rtac is_ub_thelub 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (strip_tac 1), - (res_inst_tac [("p","u")] IssumE2 1), - (hyp_subst_tac 1), - (rtac (less_ssum3d RS iffD2) 1), - (rtac chain_UU_I_inverse 1), - (rtac allI 1), - (res_inst_tac [("p","Y(i)")] IssumE 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1), - (etac notE 1), - (rtac (less_ssum3d RS iffD1) 1), - (res_inst_tac [("t","Isinr(y)")] subst 1), - (atac 1), - (etac (ub_rangeE RS spec) 1), - (res_inst_tac [("t","u")] (ssum_lemma6 RS subst) 1), - (atac 1), - (rtac (monofun_Isinr RS monofunE RS spec RS spec RS mp) 1), - (rtac is_lub_thelub 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (etac (monofun_Iwhen3 RS ub2ub_monofun) 1) - ]); - - -val thelub_ssum1a = lub_ssum1a RS thelubI; -(* [| is_chain(?Y1); ! i. ? x. ?Y1(i) = Isinl(x) |] ==> *) -(* lub(range(?Y1)) = Isinl(lub(range(%i. Iwhen(LAM x. x,LAM y. UU,?Y1(i)))))*) - -val thelub_ssum1b = lub_ssum1b RS thelubI; -(* [| is_chain(?Y1); ! i. ? x. ?Y1(i) = Isinr(x) |] ==> *) -(* lub(range(?Y1)) = Isinr(lub(range(%i. Iwhen(LAM y. UU,LAM x. x,?Y1(i)))))*) - -val cpo_ssum = prove_goal Ssum2.thy - "is_chain(Y::nat=>'a ++'b) ==> ? x.range(Y) <<|x" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (ssum_lemma4 RS disjE) 1), - (atac 1), - (rtac exI 1), - (etac lub_ssum1a 1), - (atac 1), - (rtac exI 1), - (etac lub_ssum1b 1), - (atac 1) - ]); diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum2.thy --- a/src/HOLCF/ssum2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,23 +0,0 @@ -(* Title: HOLCF/ssum2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class Instance ++::(pcpo,pcpo)po -*) - -Ssum2 = Ssum1 + - -arities "++" :: (pcpo,pcpo)po -(* Witness for the above arity axiom is ssum1.ML *) - -rules - -(* instance of << for type ['a ++ 'b] *) - -inst_ssum_po "(op <<)::['a ++ 'b,'a ++ 'b]=>bool = less_ssum" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum3.ML --- a/src/HOLCF/ssum3.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,728 +0,0 @@ -(* Title: HOLCF/ssum3.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for ssum3.thy -*) - -open Ssum3; - -(* ------------------------------------------------------------------------ *) -(* continuity for Isinl and Isinr *) -(* ------------------------------------------------------------------------ *) - - -val contlub_Isinl = prove_goal Ssum3.thy "contlub(Isinl)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_ssum1a RS sym) 2), - (rtac allI 3), - (rtac exI 3), - (rtac refl 3), - (etac (monofun_Isinl RS ch2ch_monofun) 2), - (res_inst_tac [("Q","lub(range(Y))=UU")] classical2 1), - (res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1), - (atac 1), - (res_inst_tac [("f","Isinl")] arg_cong 1), - (rtac (chain_UU_I_inverse RS sym) 1), - (rtac allI 1), - (res_inst_tac [("s","UU"),("t","Y(i)")] ssubst 1), - (etac (chain_UU_I RS spec ) 1), - (atac 1), - (rtac Iwhen1 1), - (res_inst_tac [("f","Isinl")] arg_cong 1), - (rtac lub_equal 1), - (atac 1), - (rtac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (etac (monofun_Isinl RS ch2ch_monofun) 1), - (rtac allI 1), - (res_inst_tac [("Q","Y(k)=UU")] classical2 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1) - ]); - -val contlub_Isinr = prove_goal Ssum3.thy "contlub(Isinr)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_ssum1b RS sym) 2), - (rtac allI 3), - (rtac exI 3), - (rtac refl 3), - (etac (monofun_Isinr RS ch2ch_monofun) 2), - (res_inst_tac [("Q","lub(range(Y))=UU")] classical2 1), - (res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1), - (atac 1), - ((rtac arg_cong 1) THEN (rtac (chain_UU_I_inverse RS sym) 1)), - (rtac allI 1), - (res_inst_tac [("s","UU"),("t","Y(i)")] ssubst 1), - (etac (chain_UU_I RS spec ) 1), - (atac 1), - (rtac (strict_IsinlIsinr RS subst) 1), - (rtac Iwhen1 1), - ((rtac arg_cong 1) THEN (rtac lub_equal 1)), - (atac 1), - (rtac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (etac (monofun_Isinr RS ch2ch_monofun) 1), - (rtac allI 1), - (res_inst_tac [("Q","Y(k)=UU")] classical2 1), - (asm_simp_tac Ssum_ss 1), - (asm_simp_tac Ssum_ss 1) - ]); - -val contX_Isinl = prove_goal Ssum3.thy "contX(Isinl)" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Isinl 1), - (rtac contlub_Isinl 1) - ]); - -val contX_Isinr = prove_goal Ssum3.thy "contX(Isinr)" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Isinr 1), - (rtac contlub_Isinr 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* continuity for Iwhen in the firts two arguments *) -(* ------------------------------------------------------------------------ *) - -val contlub_Iwhen1 = prove_goal Ssum3.thy "contlub(Iwhen)" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_fun RS sym) 2), - (etac (monofun_Iwhen1 RS ch2ch_monofun) 2), - (rtac (expand_fun_eq RS iffD2) 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_fun RS sym) 2), - (rtac ch2ch_fun 2), - (etac (monofun_Iwhen1 RS ch2ch_monofun) 2), - (rtac (expand_fun_eq RS iffD2) 1), - (strip_tac 1), - (res_inst_tac [("p","xa")] IssumE 1), - (asm_simp_tac Ssum_ss 1), - (rtac (lub_const RS thelubI RS sym) 1), - (asm_simp_tac Ssum_ss 1), - (etac contlub_cfun_fun 1), - (asm_simp_tac Ssum_ss 1), - (rtac (lub_const RS thelubI RS sym) 1) - ]); - -val contlub_Iwhen2 = prove_goal Ssum3.thy "contlub(Iwhen(f))" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (rtac trans 1), - (rtac (thelub_fun RS sym) 2), - (etac (monofun_Iwhen2 RS ch2ch_monofun) 2), - (rtac (expand_fun_eq RS iffD2) 1), - (strip_tac 1), - (res_inst_tac [("p","x")] IssumE 1), - (asm_simp_tac Ssum_ss 1), - (rtac (lub_const RS thelubI RS sym) 1), - (asm_simp_tac Ssum_ss 1), - (rtac (lub_const RS thelubI RS sym) 1), - (asm_simp_tac Ssum_ss 1), - (etac contlub_cfun_fun 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* continuity for Iwhen in its third argument *) -(* ------------------------------------------------------------------------ *) - -(* ------------------------------------------------------------------------ *) -(* first 5 ugly lemmas *) -(* ------------------------------------------------------------------------ *) - -val ssum_lemma9 = prove_goal Ssum3.thy -"[| is_chain(Y); lub(range(Y)) = Isinl(x)|] ==> !i.? x.Y(i)=Isinl(x)" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (res_inst_tac [("p","Y(i)")] IssumE 1), - (etac exI 1), - (etac exI 1), - (res_inst_tac [("P","y=UU")] notE 1), - (atac 1), - (rtac (less_ssum3d RS iffD1) 1), - (etac subst 1), - (etac subst 1), - (etac is_ub_thelub 1) - ]); - - -val ssum_lemma10 = prove_goal Ssum3.thy -"[| is_chain(Y); lub(range(Y)) = Isinr(x)|] ==> !i.? x.Y(i)=Isinr(x)" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (res_inst_tac [("p","Y(i)")] IssumE 1), - (rtac exI 1), - (etac trans 1), - (rtac strict_IsinlIsinr 1), - (etac exI 2), - (res_inst_tac [("P","xa=UU")] notE 1), - (atac 1), - (rtac (less_ssum3c RS iffD1) 1), - (etac subst 1), - (etac subst 1), - (etac is_ub_thelub 1) - ]); - -val ssum_lemma11 = prove_goal Ssum3.thy -"[| is_chain(Y); lub(range(Y)) = Isinl(UU) |] ==>\ -\ Iwhen(f,g,lub(range(Y))) = lub(range(%i. Iwhen(f,g,Y(i))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac Ssum_ss 1), - (rtac (chain_UU_I_inverse RS sym) 1), - (rtac allI 1), - (res_inst_tac [("s","Isinl(UU)"),("t","Y(i)")] subst 1), - (rtac (inst_ssum_pcpo RS subst) 1), - (rtac (chain_UU_I RS spec RS sym) 1), - (atac 1), - (etac (inst_ssum_pcpo RS ssubst) 1), - (asm_simp_tac Ssum_ss 1) - ]); - -val ssum_lemma12 = prove_goal Ssum3.thy -"[| is_chain(Y); lub(range(Y)) = Isinl(x); ~ x = UU |] ==>\ -\ Iwhen(f,g,lub(range(Y))) = lub(range(%i. Iwhen(f,g,Y(i))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac Ssum_ss 1), - (res_inst_tac [("t","x")] subst 1), - (rtac inject_Isinl 1), - (rtac trans 1), - (atac 2), - (rtac (thelub_ssum1a RS sym) 1), - (atac 1), - (etac ssum_lemma9 1), - (atac 1), - (rtac trans 1), - (rtac contlub_cfun_arg 1), - (rtac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (atac 1), - (rtac lub_equal2 1), - (rtac (chain_mono2 RS exE) 1), - (atac 2), - (rtac chain_UU_I_inverse2 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (etac swap 1), - (rtac inject_Isinl 1), - (rtac trans 1), - (etac sym 1), - (etac notnotD 1), - (rtac exI 1), - (strip_tac 1), - (rtac (ssum_lemma9 RS spec RS exE) 1), - (atac 1), - (atac 1), - (res_inst_tac [("t","Y(i)")] ssubst 1), - (atac 1), - (rtac trans 1), - (rtac cfun_arg_cong 1), - (rtac Iwhen2 1), - (res_inst_tac [("P","Y(i)=UU")] swap 1), - (fast_tac HOL_cs 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (res_inst_tac [("t","Y(i)")] ssubst 1), - (atac 1), - (fast_tac HOL_cs 1), - (rtac (Iwhen2 RS ssubst) 1), - (res_inst_tac [("P","Y(i)=UU")] swap 1), - (fast_tac HOL_cs 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (res_inst_tac [("t","Y(i)")] ssubst 1), - (atac 1), - (fast_tac HOL_cs 1), - (simp_tac Cfun_ss 1), - (rtac (monofun_fapp2 RS ch2ch_monofun) 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1) - ]); - - -val ssum_lemma13 = prove_goal Ssum3.thy -"[| is_chain(Y); lub(range(Y)) = Isinr(x); ~ x = UU |] ==>\ -\ Iwhen(f,g,lub(range(Y))) = lub(range(%i. Iwhen(f,g,Y(i))))" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac Ssum_ss 1), - (res_inst_tac [("t","x")] subst 1), - (rtac inject_Isinr 1), - (rtac trans 1), - (atac 2), - (rtac (thelub_ssum1b RS sym) 1), - (atac 1), - (etac ssum_lemma10 1), - (atac 1), - (rtac trans 1), - (rtac contlub_cfun_arg 1), - (rtac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (atac 1), - (rtac lub_equal2 1), - (rtac (chain_mono2 RS exE) 1), - (atac 2), - (rtac chain_UU_I_inverse2 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (etac swap 1), - (rtac inject_Isinr 1), - (rtac trans 1), - (etac sym 1), - (rtac (strict_IsinlIsinr RS subst) 1), - (etac notnotD 1), - (rtac exI 1), - (strip_tac 1), - (rtac (ssum_lemma10 RS spec RS exE) 1), - (atac 1), - (atac 1), - (res_inst_tac [("t","Y(i)")] ssubst 1), - (atac 1), - (rtac trans 1), - (rtac cfun_arg_cong 1), - (rtac Iwhen3 1), - (res_inst_tac [("P","Y(i)=UU")] swap 1), - (fast_tac HOL_cs 1), - (dtac notnotD 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (rtac (strict_IsinlIsinr RS ssubst) 1), - (res_inst_tac [("t","Y(i)")] ssubst 1), - (atac 1), - (fast_tac HOL_cs 1), - (rtac (Iwhen3 RS ssubst) 1), - (res_inst_tac [("P","Y(i)=UU")] swap 1), - (fast_tac HOL_cs 1), - (dtac notnotD 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (rtac (strict_IsinlIsinr RS ssubst) 1), - (res_inst_tac [("t","Y(i)")] ssubst 1), - (atac 1), - (fast_tac HOL_cs 1), - (simp_tac Cfun_ss 1), - (rtac (monofun_fapp2 RS ch2ch_monofun) 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1), - (etac (monofun_Iwhen3 RS ch2ch_monofun) 1) - ]); - - -val contlub_Iwhen3 = prove_goal Ssum3.thy "contlub(Iwhen(f)(g))" - (fn prems => - [ - (rtac contlubI 1), - (strip_tac 1), - (res_inst_tac [("p","lub(range(Y))")] IssumE 1), - (etac ssum_lemma11 1), - (atac 1), - (etac ssum_lemma12 1), - (atac 1), - (atac 1), - (etac ssum_lemma13 1), - (atac 1), - (atac 1) - ]); - -val contX_Iwhen1 = prove_goal Ssum3.thy "contX(Iwhen)" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Iwhen1 1), - (rtac contlub_Iwhen1 1) - ]); - -val contX_Iwhen2 = prove_goal Ssum3.thy "contX(Iwhen(f))" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Iwhen2 1), - (rtac contlub_Iwhen2 1) - ]); - -val contX_Iwhen3 = prove_goal Ssum3.thy "contX(Iwhen(f)(g))" - (fn prems => - [ - (rtac monocontlub2contX 1), - (rtac monofun_Iwhen3 1), - (rtac contlub_Iwhen3 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* continuous versions of lemmas for 'a ++ 'b *) -(* ------------------------------------------------------------------------ *) - -val strict_sinl = prove_goalw Ssum3.thy [sinl_def] "sinl[UU]=UU" - (fn prems => - [ - (simp_tac (Ssum_ss addsimps [contX_Isinl]) 1), - (rtac (inst_ssum_pcpo RS sym) 1) - ]); - -val strict_sinr = prove_goalw Ssum3.thy [sinr_def] "sinr[UU]=UU" - (fn prems => - [ - (simp_tac (Ssum_ss addsimps [contX_Isinr]) 1), - (rtac (inst_ssum_pcpo RS sym) 1) - ]); - -val noteq_sinlsinr = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "sinl[a]=sinr[b] ==> a=UU & b=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac noteq_IsinlIsinr 1), - (etac box_equals 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1) - ]); - -val inject_sinl = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "sinl[a1]=sinl[a2]==> a1=a2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac inject_Isinl 1), - (etac box_equals 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1) - ]); - -val inject_sinr = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "sinr[a1]=sinr[a2]==> a1=a2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac inject_Isinr 1), - (etac box_equals 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1) - ]); - - -val defined_sinl = prove_goal Ssum3.thy - "~x=UU ==> ~sinl[x]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac swap 1), - (rtac inject_sinl 1), - (rtac (strict_sinl RS ssubst) 1), - (etac notnotD 1) - ]); - -val defined_sinr = prove_goal Ssum3.thy - "~x=UU ==> ~sinr[x]=UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac swap 1), - (rtac inject_sinr 1), - (rtac (strict_sinr RS ssubst) 1), - (etac notnotD 1) - ]); - -val Exh_Ssum1 = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "z=UU | (? a. z=sinl[a] & ~a=UU) | (? b. z=sinr[b] & ~b=UU)" - (fn prems => - [ - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (rtac Exh_Ssum 1) - ]); - - -val ssumE = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "[|p=UU ==> Q ;\ -\ !!x.[|p=sinl[x]; ~x=UU |] ==> Q;\ -\ !!y.[|p=sinr[y]; ~y=UU |] ==> Q|] ==> Q" - (fn prems => - [ - (rtac IssumE 1), - (resolve_tac prems 1), - (rtac (inst_ssum_pcpo RS ssubst) 1), - (atac 1), - (resolve_tac prems 1), - (atac 2), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1), - (resolve_tac prems 1), - (atac 2), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1) - ]); - - -val ssumE2 = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "[|!!x.[|p=sinl[x]|] ==> Q;\ -\ !!y.[|p=sinr[y]|] ==> Q|] ==> Q" - (fn prems => - [ - (rtac IssumE2 1), - (resolve_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isinl 1), - (atac 1), - (resolve_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (rtac contX_Isinr 1), - (atac 1) - ]); - -val when1 = prove_goalw Ssum3.thy [when_def,sinl_def,sinr_def] - "when[f][g][UU] = UU" - (fn prems => - [ - (rtac (inst_ssum_pcpo RS ssubst) 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX2contX_CF1L]) 1)), - (simp_tac Ssum_ss 1) - ]); - -val when2 = prove_goalw Ssum3.thy [when_def,sinl_def,sinr_def] - "~x=UU==>when[f][g][sinl[x]] = f[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (asm_simp_tac Ssum_ss 1) - ]); - - - -val when3 = prove_goalw Ssum3.thy [when_def,sinl_def,sinr_def] - "~x=UU==>when[f][g][sinr[x]] = g[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (asm_simp_tac Ssum_ss 1) - ]); - - -val less_ssum4a = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "(sinl[x] << sinl[y]) = (x << y)" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac less_ssum3a 1) - ]); - -val less_ssum4b = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "(sinr[x] << sinr[y]) = (x << y)" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac less_ssum3b 1) - ]); - -val less_ssum4c = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "(sinl[x] << sinr[y]) = (x = UU)" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac less_ssum3c 1) - ]); - -val less_ssum4d = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "(sinr[x] << sinl[y]) = (x = UU)" - (fn prems => - [ - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac less_ssum3d 1) - ]); - -val ssum_chainE = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "is_chain(Y) ==> (!i.? x.Y(i)=sinl[x])|(!i.? y.Y(i)=sinr[y])" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinr,contX_Isinl]) 1), - (etac ssum_lemma4 1) - ]); - - -val thelub_ssum2a = prove_goalw Ssum3.thy [sinl_def,sinr_def,when_def] -"[| is_chain(Y); !i.? x. Y(i) = sinl[x] |] ==>\ -\ lub(range(Y)) = sinl[lub(range(%i. when[LAM x. x][LAM y. UU][Y(i)]))]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ext RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac thelub_ssum1a 1), - (atac 1), - (rtac allI 1), - (etac allE 1), - (etac exE 1), - (rtac exI 1), - (etac box_equals 1), - (rtac refl 1), - (asm_simp_tac (Ssum_ss addsimps [contX_Isinl]) 1) - ]); - -val thelub_ssum2b = prove_goalw Ssum3.thy [sinl_def,sinr_def,when_def] -"[| is_chain(Y); !i.? x. Y(i) = sinr[x] |] ==>\ -\ lub(range(Y)) = sinr[lub(range(%i. when[LAM y. UU][LAM x. x][Y(i)]))]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac (beta_cfun RS ext RS ssubst) 1), - (REPEAT (resolve_tac (contX_lemmas @ [contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3,contX_Isinl,contX_Isinr,contX2contX_CF1L]) 1)), - (rtac thelub_ssum1b 1), - (atac 1), - (rtac allI 1), - (etac allE 1), - (etac exE 1), - (rtac exI 1), - (etac box_equals 1), - (rtac refl 1), - (asm_simp_tac (Ssum_ss addsimps - [contX_Isinr,contX_Isinl,contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3]) 1) - ]); - -val thelub_ssum2a_rev = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "[| is_chain(Y); lub(range(Y)) = sinl[x]|] ==> !i.? x.Y(i)=sinl[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (Ssum_ss addsimps - [contX_Isinr,contX_Isinl,contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3]) 1), - (etac ssum_lemma9 1), - (asm_simp_tac (Ssum_ss addsimps - [contX_Isinr,contX_Isinl,contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3]) 1) - ]); - -val thelub_ssum2b_rev = prove_goalw Ssum3.thy [sinl_def,sinr_def] - "[| is_chain(Y); lub(range(Y)) = sinr[x]|] ==> !i.? x.Y(i)=sinr[x]" - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (Ssum_ss addsimps - [contX_Isinr,contX_Isinl,contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3]) 1), - (etac ssum_lemma10 1), - (asm_simp_tac (Ssum_ss addsimps - [contX_Isinr,contX_Isinl,contX_Iwhen1,contX_Iwhen2, - contX_Iwhen3]) 1) - ]); - -val thelub_ssum3 = prove_goal Ssum3.thy -"is_chain(Y) ==>\ -\ lub(range(Y)) = sinl[lub(range(%i. when[LAM x. x][LAM y. UU][Y(i)]))]\ -\ | lub(range(Y)) = sinr[lub(range(%i. when[LAM y. UU][LAM x. x][Y(i)]))]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac (ssum_chainE RS disjE) 1), - (atac 1), - (rtac disjI1 1), - (etac thelub_ssum2a 1), - (atac 1), - (rtac disjI2 1), - (etac thelub_ssum2b 1), - (atac 1) - ]); - - -val when4 = prove_goal Ssum3.thy - "when[sinl][sinr][z]=z" - (fn prems => - [ - (res_inst_tac [("p","z")] ssumE 1), - (asm_simp_tac (Cfun_ss addsimps [when1,when2,when3]) 1), - (asm_simp_tac (Cfun_ss addsimps [when1,when2,when3]) 1), - (asm_simp_tac (Cfun_ss addsimps [when1,when2,when3]) 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* install simplifier for Ssum *) -(* ------------------------------------------------------------------------ *) - -val Ssum_rews = [strict_sinl,strict_sinr,when1,when2,when3]; -val Ssum_ss = Cfun_ss addsimps Ssum_rews; diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/ssum3.thy --- a/src/HOLCF/ssum3.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,29 +0,0 @@ -(* Title: HOLCF/ssum3.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Class instance of ++ for class pcpo -*) - -Ssum3 = Ssum2 + - -arities "++" :: (pcpo,pcpo)pcpo (* Witness ssum2.ML *) - -consts - sinl :: "'a -> ('a++'b)" - sinr :: "'b -> ('a++'b)" - when :: "('a->'c)->('b->'c)->('a ++ 'b)-> 'c" - -rules - -inst_ssum_pcpo "UU::'a++'b = Isinl(UU)" - -sinl_def "sinl == (LAM x.Isinl(x))" -sinr_def "sinr == (LAM x.Isinr(x))" -when_def "when == (LAM f g s.Iwhen(f)(g)(s))" - -end - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/stream.ML --- a/src/HOLCF/stream.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,791 +0,0 @@ -(* Title: HOLCF/stream.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for stream.thy -*) - -open Stream; - -(* ------------------------------------------------------------------------*) -(* The isomorphisms stream_rep_iso and stream_abs_iso are strict *) -(* ------------------------------------------------------------------------*) - -val stream_iso_strict= stream_rep_iso RS (stream_abs_iso RS - (allI RSN (2,allI RS iso_strict))); - -val stream_rews = [stream_iso_strict RS conjunct1, - stream_iso_strict RS conjunct2]; - -(* ------------------------------------------------------------------------*) -(* Properties of stream_copy *) -(* ------------------------------------------------------------------------*) - -fun prover defs thm = prove_goalw Stream.thy defs thm - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps - (stream_rews @ [stream_abs_iso,stream_rep_iso])) 1) - ]); - -val stream_copy = - [ - prover [stream_copy_def] "stream_copy[f][UU]=UU", - prover [stream_copy_def,scons_def] - "x~=UU ==> stream_copy[f][scons[x][xs]]= scons[x][f[xs]]" - ]; - -val stream_rews = stream_copy @ stream_rews; - -(* ------------------------------------------------------------------------*) -(* Exhaustion and elimination for streams *) -(* ------------------------------------------------------------------------*) - -val Exh_stream = prove_goalw Stream.thy [scons_def] - "s = UU | (? x xs. x~=UU & s = scons[x][xs])" - (fn prems => - [ - (simp_tac HOLCF_ss 1), - (rtac (stream_rep_iso RS subst) 1), - (res_inst_tac [("p","stream_rep[s]")] sprodE 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (asm_simp_tac HOLCF_ss 1), - (res_inst_tac [("p","y")] liftE1 1), - (contr_tac 1), - (rtac disjI2 1), - (rtac exI 1), - (rtac exI 1), - (etac conjI 1), - (asm_simp_tac HOLCF_ss 1) - ]); - -val streamE = prove_goal Stream.thy - "[| s=UU ==> Q; !!x xs.[|s=scons[x][xs];x~=UU|]==>Q|]==>Q" - (fn prems => - [ - (rtac (Exh_stream RS disjE) 1), - (eresolve_tac prems 1), - (etac exE 1), - (etac exE 1), - (resolve_tac prems 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -(* ------------------------------------------------------------------------*) -(* Properties of stream_when *) -(* ------------------------------------------------------------------------*) - -fun prover defs thm = prove_goalw Stream.thy defs thm - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps - (stream_rews @ [stream_abs_iso,stream_rep_iso])) 1) - ]); - - -val stream_when = [ - prover [stream_when_def] "stream_when[f][UU]=UU", - prover [stream_when_def,scons_def] - "x~=UU ==> stream_when[f][scons[x][xs]]= f[x][xs]" - ]; - -val stream_rews = stream_when @ stream_rews; - -(* ------------------------------------------------------------------------*) -(* Rewrites for discriminators and selectors *) -(* ------------------------------------------------------------------------*) - -fun prover defs thm = prove_goalw Stream.thy defs thm - (fn prems => - [ - (simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val stream_discsel = [ - prover [is_scons_def] "is_scons[UU]=UU", - prover [shd_def] "shd[UU]=UU", - prover [stl_def] "stl[UU]=UU" - ]; - -fun prover defs thm = prove_goalw Stream.thy defs thm - (fn prems => - [ - (cut_facts_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val stream_discsel = [ -prover [is_scons_def,shd_def,stl_def] "x~=UU ==> is_scons[scons[x][xs]]=TT", -prover [is_scons_def,shd_def,stl_def] "x~=UU ==> shd[scons[x][xs]]=x", -prover [is_scons_def,shd_def,stl_def] "x~=UU ==> stl[scons[x][xs]]=xs" - ] @ stream_discsel; - -val stream_rews = stream_discsel @ stream_rews; - -(* ------------------------------------------------------------------------*) -(* Definedness and strictness *) -(* ------------------------------------------------------------------------*) - -fun prover contr thm = prove_goal Stream.thy thm - (fn prems => - [ - (res_inst_tac [("P1",contr)] classical3 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (dtac sym 1), - (asm_simp_tac HOLCF_ss 1), - (simp_tac (HOLCF_ss addsimps (prems @ stream_rews)) 1) - ]); - -val stream_constrdef = [ - prover "is_scons[UU] ~= UU" "x~=UU ==> scons[x][xs]~=UU" - ]; - -fun prover defs thm = prove_goalw Stream.thy defs thm - (fn prems => - [ - (simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val stream_constrdef = [ - prover [scons_def] "scons[UU][xs]=UU" - ] @ stream_constrdef; - -val stream_rews = stream_constrdef @ stream_rews; - - -(* ------------------------------------------------------------------------*) -(* Distinctness wrt. << and = *) -(* ------------------------------------------------------------------------*) - - -(* ------------------------------------------------------------------------*) -(* Invertibility *) -(* ------------------------------------------------------------------------*) - -val stream_invert = - [ -prove_goal Stream.thy "[|x1~=UU; y1~=UU;\ -\ scons[x1][x2] << scons[y1][y2]|] ==> x1<< y1 & x2 << y2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac conjI 1), - (dres_inst_tac [("fo5","stream_when[LAM x l.x]")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1), - (dres_inst_tac [("fo5","stream_when[LAM x l.l]")] monofun_cfun_arg 1), - (etac box_less 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1) - ]) - ]; - -(* ------------------------------------------------------------------------*) -(* Injectivity *) -(* ------------------------------------------------------------------------*) - -val stream_inject = - [ -prove_goal Stream.thy "[|x1~=UU; y1~=UU;\ -\ scons[x1][x2] = scons[y1][y2]|] ==> x1= y1 & x2 = y2" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac conjI 1), - (dres_inst_tac [("f","stream_when[LAM x l.x]")] cfun_arg_cong 1), - (etac box_equals 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1), - (dres_inst_tac [("f","stream_when[LAM x l.l]")] cfun_arg_cong 1), - (etac box_equals 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_when) 1) - ]) - ]; - -(* ------------------------------------------------------------------------*) -(* definedness for discriminators and selectors *) -(* ------------------------------------------------------------------------*) - -fun prover thm = prove_goal Stream.thy thm - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac streamE 1), - (contr_tac 1), - (REPEAT (asm_simp_tac (HOLCF_ss addsimps stream_discsel) 1)) - ]); - -val stream_discsel_def = - [ - prover "s~=UU ==> is_scons[s]~=UU", - prover "s~=UU ==> shd[s]~=UU" - ]; - -val stream_rews = stream_discsel_def @ stream_rews; - - -(* ------------------------------------------------------------------------*) -(* Properties stream_take *) -(* ------------------------------------------------------------------------*) - -val stream_take = - [ -prove_goalw Stream.thy [stream_take_def] "stream_take(n)[UU]=UU" - (fn prems => - [ - (res_inst_tac [("n","n")] natE 1), - (asm_simp_tac iterate_ss 1), - (asm_simp_tac iterate_ss 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]), -prove_goalw Stream.thy [stream_take_def] "stream_take(0)[xs]=UU" - (fn prems => - [ - (asm_simp_tac iterate_ss 1) - ])]; - -fun prover thm = prove_goalw Stream.thy [stream_take_def] thm - (fn prems => - [ - (cut_facts_tac prems 1), - (simp_tac iterate_ss 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val stream_take = [ -prover - "x~=UU ==> stream_take(Suc(n))[scons[x][xs]]=scons[x][stream_take(n)[xs]]" - ] @ stream_take; - -val stream_rews = stream_take @ stream_rews; - -(* ------------------------------------------------------------------------*) -(* enhance the simplifier *) -(* ------------------------------------------------------------------------*) - -val stream_copy2 = prove_goal Stream.thy - "stream_copy[f][scons[x][xs]]= scons[x][f[xs]]" - (fn prems => - [ - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val shd2 = prove_goal Stream.thy "shd[scons[x][xs]]=x" - (fn prems => - [ - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val stream_take2 = prove_goal Stream.thy - "stream_take(Suc(n))[scons[x][xs]]=scons[x][stream_take(n)[xs]]" - (fn prems => - [ - (res_inst_tac [("Q","x=UU")] classical2 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val stream_rews = [stream_iso_strict RS conjunct1, - stream_iso_strict RS conjunct2, - hd stream_copy, stream_copy2] - @ stream_when - @ [hd stream_discsel,shd2] @ (tl (tl stream_discsel)) - @ stream_constrdef - @ stream_discsel_def - @ [ stream_take2] @ (tl stream_take); - - -(* ------------------------------------------------------------------------*) -(* take lemma for streams *) -(* ------------------------------------------------------------------------*) - -fun prover reach defs thm = prove_goalw Stream.thy defs thm - (fn prems => - [ - (res_inst_tac [("t","s1")] (reach RS subst) 1), - (res_inst_tac [("t","s2")] (reach RS subst) 1), - (rtac (fix_def2 RS ssubst) 1), - (rtac (contlub_cfun_fun RS ssubst) 1), - (rtac is_chain_iterate 1), - (rtac (contlub_cfun_fun RS ssubst) 1), - (rtac is_chain_iterate 1), - (rtac lub_equal 1), - (rtac (is_chain_iterate RS ch2ch_fappL) 1), - (rtac (is_chain_iterate RS ch2ch_fappL) 1), - (rtac allI 1), - (resolve_tac prems 1) - ]); - -val stream_take_lemma = prover stream_reach [stream_take_def] - "(!!n.stream_take(n)[s1]=stream_take(n)[s2]) ==> s1=s2"; - - -(* ------------------------------------------------------------------------*) -(* Co -induction for streams *) -(* ------------------------------------------------------------------------*) - -val stream_coind_lemma = prove_goalw Stream.thy [stream_bisim_def] -"stream_bisim(R) ==> ! p q.R(p,q) --> stream_take(n)[p]=stream_take(n)[q]" - (fn prems => - [ - (cut_facts_tac prems 1), - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1), - ((etac allE 1) THEN (etac allE 1) THEN (etac (mp RS disjE) 1)), - (atac 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (etac exE 1), - (etac exE 1), - (etac exE 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (REPEAT (etac conjE 1)), - (rtac cfun_arg_cong 1), - (fast_tac HOL_cs 1) - ]); - -val stream_coind = prove_goal Stream.thy "[|stream_bisim(R);R(p,q)|] ==> p = q" - (fn prems => - [ - (rtac stream_take_lemma 1), - (rtac (stream_coind_lemma RS spec RS spec RS mp) 1), - (resolve_tac prems 1), - (resolve_tac prems 1) - ]); - -(* ------------------------------------------------------------------------*) -(* structural induction for admissible predicates *) -(* ------------------------------------------------------------------------*) - -val stream_finite_ind = prove_goal Stream.thy -"[|P(UU);\ -\ !! x s1.[|x~=UU;P(s1)|] ==> P(scons[x][s1])\ -\ |] ==> !s.P(stream_take(n)[s])" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (resolve_tac prems 1), - (rtac allI 1), - (res_inst_tac [("s","s")] streamE 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (resolve_tac prems 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (resolve_tac prems 1), - (atac 1), - (etac spec 1) - ]); - -val stream_finite_ind2 = prove_goalw Stream.thy [stream_finite_def] -"(!!n.P(stream_take(n)[s])) ==> stream_finite(s) -->P(s)" - (fn prems => - [ - (strip_tac 1), - (etac exE 1), - (etac subst 1), - (resolve_tac prems 1) - ]); - -val stream_finite_ind3 = prove_goal Stream.thy -"[|P(UU);\ -\ !! x s1.[|x~=UU;P(s1)|] ==> P(scons[x][s1])\ -\ |] ==> stream_finite(s) --> P(s)" - (fn prems => - [ - (rtac stream_finite_ind2 1), - (rtac (stream_finite_ind RS spec) 1), - (REPEAT (resolve_tac prems 1)), - (REPEAT (atac 1)) - ]); - -val stream_ind = prove_goal Stream.thy -"[|adm(P);\ -\ P(UU);\ -\ !! x s1.[|x~=UU;P(s1)|] ==> P(scons[x][s1])\ -\ |] ==> P(s)" - (fn prems => - [ - (rtac (stream_reach RS subst) 1), - (res_inst_tac [("x","s")] spec 1), - (rtac wfix_ind 1), - (rtac adm_impl_admw 1), - (REPEAT (resolve_tac adm_thms 1)), - (rtac adm_subst 1), - (contX_tacR 1), - (resolve_tac prems 1), - (rtac allI 1), - (rtac (rewrite_rule [stream_take_def] stream_finite_ind) 1), - (REPEAT (resolve_tac prems 1)), - (REPEAT (atac 1)) - ]); - - -(* ------------------------------------------------------------------------*) -(* simplify use of Co-induction *) -(* ------------------------------------------------------------------------*) - -val surjectiv_scons = prove_goal Stream.thy "scons[shd[s]][stl[s]]=s" - (fn prems => - [ - (res_inst_tac [("s","s")] streamE 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - - -val stream_coind_lemma2 = prove_goalw Stream.thy [stream_bisim_def] -"!s1 s2. R(s1, s2)-->shd[s1]=shd[s2] & R(stl[s1],stl[s2]) ==>stream_bisim(R)" - (fn prems => - [ - (cut_facts_tac prems 1), - (strip_tac 1), - (etac allE 1), - (etac allE 1), - (dtac mp 1), - (atac 1), - (etac conjE 1), - (res_inst_tac [("Q","s1 = UU & s2 = UU")] classical2 1), - (rtac disjI1 1), - (fast_tac HOL_cs 1), - (rtac disjI2 1), - (rtac disjE 1), - (etac (de_morgan2 RS ssubst) 1), - (res_inst_tac [("x","shd[s1]")] exI 1), - (res_inst_tac [("x","stl[s1]")] exI 1), - (res_inst_tac [("x","stl[s2]")] exI 1), - (rtac conjI 1), - (eresolve_tac stream_discsel_def 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1), - (eres_inst_tac [("s","shd[s1]"),("t","shd[s2]")] subst 1), - (simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1), - (res_inst_tac [("x","shd[s2]")] exI 1), - (res_inst_tac [("x","stl[s1]")] exI 1), - (res_inst_tac [("x","stl[s2]")] exI 1), - (rtac conjI 1), - (eresolve_tac stream_discsel_def 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1), - (res_inst_tac [("s","shd[s1]"),("t","shd[s2]")] ssubst 1), - (etac sym 1), - (simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1) - ]); - - -(* ------------------------------------------------------------------------*) -(* theorems about finite and infinite streams *) -(* ------------------------------------------------------------------------*) - -(* ----------------------------------------------------------------------- *) -(* 2 lemmas about stream_finite *) -(* ----------------------------------------------------------------------- *) - -val stream_finite_UU = prove_goalw Stream.thy [stream_finite_def] - "stream_finite(UU)" - (fn prems => - [ - (rtac exI 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val inf_stream_not_UU = prove_goal Stream.thy "~stream_finite(s) ==> s ~= UU" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac swap 1), - (dtac notnotD 1), - (hyp_subst_tac 1), - (rtac stream_finite_UU 1) - ]); - -(* ----------------------------------------------------------------------- *) -(* a lemma about shd *) -(* ----------------------------------------------------------------------- *) - -val stream_shd_lemma1 = prove_goal Stream.thy "shd[s]=UU --> s=UU" - (fn prems => - [ - (res_inst_tac [("s","s")] streamE 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (hyp_subst_tac 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - - -(* ----------------------------------------------------------------------- *) -(* lemmas about stream_take *) -(* ----------------------------------------------------------------------- *) - -val stream_take_lemma1 = prove_goal Stream.thy - "!x xs.x~=UU --> \ -\ stream_take(Suc(n))[scons[x][xs]] = scons[x][xs] --> stream_take(n)[xs]=xs" - (fn prems => - [ - (rtac allI 1), - (rtac allI 1), - (rtac impI 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1), - (rtac ((hd stream_inject) RS conjunct2) 1), - (atac 1), - (atac 1), - (atac 1) - ]); - - -val stream_take_lemma2 = prove_goal Stream.thy - "! s2. stream_take(n)[s2] = s2 --> stream_take(Suc(n))[s2]=s2" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1 ), - (hyp_subst_tac 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (rtac allI 1), - (res_inst_tac [("s","s2")] streamE 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1 ), - (subgoal_tac "stream_take(n1)[xs] = xs" 1), - (rtac ((hd stream_inject) RS conjunct2) 2), - (atac 4), - (atac 2), - (atac 2), - (rtac cfun_arg_cong 1), - (fast_tac HOL_cs 1) - ]); - -val stream_take_lemma3 = prove_goal Stream.thy - "!x xs.x~=UU --> \ -\ stream_take(n)[scons[x][xs]] = scons[x][xs] --> stream_take(n)[xs]=xs" - (fn prems => - [ - (nat_ind_tac "n" 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1 ), - (res_inst_tac [("P","scons[x][xs]=UU")] notE 1), - (eresolve_tac stream_constrdef 1), - (etac sym 1), - (strip_tac 1 ), - (rtac (stream_take_lemma2 RS spec RS mp) 1), - (res_inst_tac [("x1.1","x")] ((hd stream_inject) RS conjunct2) 1), - (atac 1), - (atac 1), - (etac (stream_take2 RS subst) 1) - ]); - -val stream_take_lemma4 = prove_goal Stream.thy - "!x xs.\ -\stream_take(n)[xs]=xs --> stream_take(Suc(n))[scons[x][xs]] = scons[x][xs]" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -(* ---- *) - -val stream_take_lemma5 = prove_goal Stream.thy -"!s. stream_take(n)[s]=s --> iterate(n,stl,s)=UU" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac iterate_ss 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1), - (res_inst_tac [("s","s")] streamE 1), - (hyp_subst_tac 1), - (rtac (iterate_Suc2 RS ssubst) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (rtac (iterate_Suc2 RS ssubst) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (etac allE 1), - (etac mp 1), - (hyp_subst_tac 1), - (etac (stream_take_lemma1 RS spec RS spec RS mp RS mp) 1), - (atac 1) - ]); - -val stream_take_lemma6 = prove_goal Stream.thy -"!s.iterate(n,stl,s)=UU --> stream_take(n)[s]=s" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac iterate_ss 1), - (strip_tac 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (rtac allI 1), - (res_inst_tac [("s","s")] streamE 1), - (hyp_subst_tac 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (hyp_subst_tac 1), - (rtac (iterate_Suc2 RS ssubst) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1) - ]); - -val stream_take_lemma7 = prove_goal Stream.thy -"(iterate(n,stl,s)=UU) = (stream_take(n)[s]=s)" - (fn prems => - [ - (rtac iffI 1), - (etac (stream_take_lemma6 RS spec RS mp) 1), - (etac (stream_take_lemma5 RS spec RS mp) 1) - ]); - - -(* ----------------------------------------------------------------------- *) -(* lemmas stream_finite *) -(* ----------------------------------------------------------------------- *) - -val stream_finite_lemma1 = prove_goalw Stream.thy [stream_finite_def] - "stream_finite(xs) ==> stream_finite(scons[x][xs])" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (rtac exI 1), - (etac (stream_take_lemma4 RS spec RS spec RS mp) 1) - ]); - -val stream_finite_lemma2 = prove_goalw Stream.thy [stream_finite_def] - "[|x~=UU; stream_finite(scons[x][xs])|] ==> stream_finite(xs)" - (fn prems => - [ - (cut_facts_tac prems 1), - (etac exE 1), - (rtac exI 1), - (etac (stream_take_lemma3 RS spec RS spec RS mp RS mp) 1), - (atac 1) - ]); - -val stream_finite_lemma3 = prove_goal Stream.thy - "x~=UU ==> stream_finite(scons[x][xs]) = stream_finite(xs)" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac iffI 1), - (etac stream_finite_lemma2 1), - (atac 1), - (etac stream_finite_lemma1 1) - ]); - - -val stream_finite_lemma5 = prove_goalw Stream.thy [stream_finite_def] - "(!n. s1 << s2 --> stream_take(n)[s2] = s2 --> stream_finite(s1))\ -\=(s1 << s2 --> stream_finite(s2) --> stream_finite(s1))" - (fn prems => - [ - (rtac iffI 1), - (fast_tac HOL_cs 1), - (fast_tac HOL_cs 1) - ]); - -val stream_finite_lemma6 = prove_goal Stream.thy - "!s1 s2. s1 << s2 --> stream_take(n)[s2] = s2 --> stream_finite(s1)" - (fn prems => - [ - (nat_ind_tac "n" 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1 ), - (hyp_subst_tac 1), - (dtac UU_I 1), - (hyp_subst_tac 1), - (rtac stream_finite_UU 1), - (rtac allI 1), - (rtac allI 1), - (res_inst_tac [("s","s1")] streamE 1), - (hyp_subst_tac 1), - (strip_tac 1 ), - (rtac stream_finite_UU 1), - (hyp_subst_tac 1), - (res_inst_tac [("s","s2")] streamE 1), - (hyp_subst_tac 1), - (strip_tac 1 ), - (dtac UU_I 1), - (asm_simp_tac(HOLCF_ss addsimps (stream_rews @ [stream_finite_UU])) 1), - (hyp_subst_tac 1), - (simp_tac (HOLCF_ss addsimps stream_rews) 1), - (strip_tac 1 ), - (rtac stream_finite_lemma1 1), - (subgoal_tac "xs << xsa" 1), - (subgoal_tac "stream_take(n1)[xsa] = xsa" 1), - (fast_tac HOL_cs 1), - (res_inst_tac [("x1.1","xa"),("y1.1","xa")] - ((hd stream_inject) RS conjunct2) 1), - (atac 1), - (atac 1), - (atac 1), - (res_inst_tac [("x1.1","x"),("y1.1","xa")] - ((hd stream_invert) RS conjunct2) 1), - (atac 1), - (atac 1), - (atac 1) - ]); - -val stream_finite_lemma7 = prove_goal Stream.thy -"s1 << s2 --> stream_finite(s2) --> stream_finite(s1)" - (fn prems => - [ - (rtac (stream_finite_lemma5 RS iffD1) 1), - (rtac allI 1), - (rtac (stream_finite_lemma6 RS spec RS spec) 1) - ]); - -val stream_finite_lemma8 = prove_goalw Stream.thy [stream_finite_def] -"stream_finite(s) = (? n. iterate(n,stl,s)=UU)" - (fn prems => - [ - (simp_tac (HOL_ss addsimps [stream_take_lemma7]) 1) - ]); - - -(* ----------------------------------------------------------------------- *) -(* admissibility of ~stream_finite *) -(* ----------------------------------------------------------------------- *) - -val adm_not_stream_finite = prove_goalw Stream.thy [adm_def] - "adm(%s. ~ stream_finite(s))" - (fn prems => - [ - (strip_tac 1 ), - (res_inst_tac [("P1","!i. ~ stream_finite(Y(i))")] classical3 1), - (atac 2), - (subgoal_tac "!i.stream_finite(Y(i))" 1), - (fast_tac HOL_cs 1), - (rtac allI 1), - (rtac (stream_finite_lemma7 RS mp RS mp) 1), - (etac is_ub_thelub 1), - (atac 1) - ]); - -(* ----------------------------------------------------------------------- *) -(* alternative prove for admissibility of ~stream_finite *) -(* show that stream_finite(s) = (? n. iterate(n, stl, s) = UU) *) -(* and prove adm. of ~(? n. iterate(n, stl, s) = UU) *) -(* proof uses theorems stream_take_lemma5-7; stream_finite_lemma8 *) -(* ----------------------------------------------------------------------- *) - - -val adm_not_stream_finite2=prove_goal Stream.thy "adm(%s. ~ stream_finite(s))" - (fn prems => - [ - (subgoal_tac "(!s.(~stream_finite(s))=(!n.iterate(n,stl,s)~=UU))" 1), - (etac (adm_cong RS iffD2)1), - (REPEAT(resolve_tac adm_thms 1)), - (rtac contX_iterate2 1), - (rtac allI 1), - (rtac (stream_finite_lemma8 RS ssubst) 1), - (fast_tac HOL_cs 1) - ]); - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/stream.thy --- a/src/HOLCF/stream.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,102 +0,0 @@ -(* Title: HOLCF/stream.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Theory for streams without defined empty stream -*) - -Stream = Dnat2 + - -types stream 1 - -(* ----------------------------------------------------------------------- *) -(* arity axiom is validated by semantic reasoning *) -(* partial ordering is implicit in the isomorphism axioms and their cont. *) - -arities stream::(pcpo)pcpo - -consts - -(* ----------------------------------------------------------------------- *) -(* essential constants *) - -stream_rep :: "('a stream) -> ('a ** ('a stream)u)" -stream_abs :: "('a ** ('a stream)u) -> ('a stream)" - -(* ----------------------------------------------------------------------- *) -(* abstract constants and auxiliary constants *) - -stream_copy :: "('a stream -> 'a stream) ->'a stream -> 'a stream" - -scons :: "'a -> 'a stream -> 'a stream" -stream_when :: "('a -> 'a stream -> 'b) -> 'a stream -> 'b" -is_scons :: "'a stream -> tr" -shd :: "'a stream -> 'a" -stl :: "'a stream -> 'a stream" -stream_take :: "nat => 'a stream -> 'a stream" -stream_finite :: "'a stream => bool" -stream_bisim :: "('a stream => 'a stream => bool) => bool" - -rules - -(* ----------------------------------------------------------------------- *) -(* axiomatization of recursive type 'a stream *) -(* ----------------------------------------------------------------------- *) -(* ('a stream,stream_abs) is the initial F-algebra where *) -(* F is the locally continuous functor determined by domain equation *) -(* X = 'a ** (X)u *) -(* ----------------------------------------------------------------------- *) -(* stream_abs is an isomorphism with inverse stream_rep *) -(* identity is the least endomorphism on 'a stream *) - -stream_abs_iso "stream_rep[stream_abs[x]] = x" -stream_rep_iso "stream_abs[stream_rep[x]] = x" -stream_copy_def "stream_copy == (LAM f. stream_abs oo \ -\ (ssplit[LAM x y. x ## (lift[up oo f])[y]] oo stream_rep))" -stream_reach "(fix[stream_copy])[x]=x" - -(* ----------------------------------------------------------------------- *) -(* properties of additional constants *) -(* ----------------------------------------------------------------------- *) -(* constructors *) - -scons_def "scons == (LAM x l. stream_abs[x##up[l]])" - -(* ----------------------------------------------------------------------- *) -(* discriminator functional *) - -stream_when_def -"stream_when == (LAM f l.ssplit[LAM x l.f[x][lift[ID][l]]][stream_rep[l]])" - -(* ----------------------------------------------------------------------- *) -(* discriminators and selectors *) - -is_scons_def "is_scons == stream_when[LAM x l.TT]" -shd_def "shd == stream_when[LAM x l.x]" -stl_def "stl == stream_when[LAM x l.l]" - -(* ----------------------------------------------------------------------- *) -(* the taker for streams *) - -stream_take_def "stream_take == (%n.iterate(n,stream_copy,UU))" - -(* ----------------------------------------------------------------------- *) - -stream_finite_def "stream_finite == (%s.? n.stream_take(n)[s]=s)" - -(* ----------------------------------------------------------------------- *) -(* definition of bisimulation is determined by domain equation *) -(* simplification and rewriting for abstract constants yields def below *) - -stream_bisim_def "stream_bisim ==\ -\(%R.!s1 s2.\ -\ R(s1,s2) -->\ -\ ((s1=UU & s2=UU) |\ -\ (? x s11 s21. x~=UU & s1=scons[x][s11] & s2 = scons[x][s21] & R(s11,s21))))" - -end - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/stream2.ML --- a/src/HOLCF/stream2.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,43 +0,0 @@ -(* Title: HOLCF/stream2.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for theory Stream2.thy -*) - -open Stream2; - -(* ------------------------------------------------------------------------- *) -(* expand fixed point properties *) -(* ------------------------------------------------------------------------- *) - -val smap_def2 = fix_prover Stream2.thy smap_def - "smap = (LAM f s. stream_when[LAM x l.scons[f[x]][smap[f][l]]][s])"; - - -(* ------------------------------------------------------------------------- *) -(* recursive properties *) -(* ------------------------------------------------------------------------- *) - - -val smap1 = prove_goal Stream2.thy "smap[f][UU] = UU" - (fn prems => - [ - (rtac (smap_def2 RS ssubst) 1), - (simp_tac (HOLCF_ss addsimps stream_when) 1) - ]); - -val smap2 = prove_goal Stream2.thy - "x~=UU ==> smap[f][scons[x][xs]] = scons[f[x]][smap[f][xs]]" - (fn prems => - [ - (cut_facts_tac prems 1), - (rtac trans 1), - (rtac (smap_def2 RS ssubst) 1), - (asm_simp_tac (HOLCF_ss addsimps stream_rews) 1), - (rtac refl 1) - ]); - - -val stream2_rews = [smap1, smap2]; diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/stream2.thy --- a/src/HOLCF/stream2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,29 +0,0 @@ -(* Title: HOLCF/stream2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Additional constants for stream -*) - -Stream2 = Stream + - -consts - -smap :: "('a -> 'b) -> 'a stream -> 'b stream" - -rules - -smap_def - "smap = fix[LAM h f s. stream_when[LAM x l.scons[f[x]][h[f][l]]][s]]" - - -end - - -(* - smap[f][UU] = UU - x~=UU --> smap[f][scons[x][xs]] = scons[f[x]][smap[f][xs]] - -*) - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/test --- a/src/HOLCF/test Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1 +0,0 @@ -Test examples ran successfully diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/tr1.ML --- a/src/HOLCF/tr1.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,164 +0,0 @@ -(* Title: HOLCF/tr1.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for tr1.thy -*) - -open Tr1; - -(* -------------------------------------------------------------------------- *) -(* distinctness for type tr *) -(* -------------------------------------------------------------------------- *) - -val dist_less_tr = [ -prove_goalw Tr1.thy [TT_def] "~TT << UU" - (fn prems => - [ - (rtac classical3 1), - (rtac defined_sinl 1), - (rtac not_less2not_eq 1), - (resolve_tac dist_less_one 1), - (rtac (rep_tr_iso RS subst) 1), - (rtac (rep_tr_iso RS subst) 1), - (rtac cfun_arg_cong 1), - (rtac ((abs_tr_iso RS allI) RS ((rep_tr_iso RS allI) RS iso_strict ) RS conjunct2 RS ssubst) 1), - (etac (eq_UU_iff RS ssubst) 1) - ]), -prove_goalw Tr1.thy [FF_def] "~FF << UU" - (fn prems => - [ - (rtac classical3 1), - (rtac defined_sinr 1), - (rtac not_less2not_eq 1), - (resolve_tac dist_less_one 1), - (rtac (rep_tr_iso RS subst) 1), - (rtac (rep_tr_iso RS subst) 1), - (rtac cfun_arg_cong 1), - (rtac ((abs_tr_iso RS allI) RS ((rep_tr_iso RS allI) RS iso_strict ) RS conjunct2 RS ssubst) 1), - (etac (eq_UU_iff RS ssubst) 1) - ]), -prove_goalw Tr1.thy [FF_def,TT_def] "~TT << FF" - (fn prems => - [ - (rtac classical3 1), - (rtac (less_ssum4c RS iffD1) 2), - (rtac not_less2not_eq 1), - (resolve_tac dist_less_one 1), - (rtac (rep_tr_iso RS subst) 1), - (rtac (rep_tr_iso RS subst) 1), - (etac monofun_cfun_arg 1) - ]), -prove_goalw Tr1.thy [FF_def,TT_def] "~FF << TT" - (fn prems => - [ - (rtac classical3 1), - (rtac (less_ssum4d RS iffD1) 2), - (rtac not_less2not_eq 1), - (resolve_tac dist_less_one 1), - (rtac (rep_tr_iso RS subst) 1), - (rtac (rep_tr_iso RS subst) 1), - (etac monofun_cfun_arg 1) - ]) -]; - -fun prover s = prove_goal Tr1.thy s - (fn prems => - [ - (rtac not_less2not_eq 1), - (resolve_tac dist_less_tr 1) - ]); - -val dist_eq_tr = map prover ["~TT=UU","~FF=UU","~TT=FF"]; -val dist_eq_tr = dist_eq_tr @ (map (fn thm => (thm RS not_sym)) dist_eq_tr); - -(* ------------------------------------------------------------------------ *) -(* Exhaustion and elimination for type tr *) -(* ------------------------------------------------------------------------ *) - -val Exh_tr = prove_goalw Tr1.thy [FF_def,TT_def] "t=UU | t = TT | t = FF" - (fn prems => - [ - (res_inst_tac [("p","rep_tr[t]")] ssumE 1), - (rtac disjI1 1), - (rtac ((abs_tr_iso RS allI) RS ((rep_tr_iso RS allI) RS iso_strict ) - RS conjunct2 RS subst) 1), - (rtac (abs_tr_iso RS subst) 1), - (etac cfun_arg_cong 1), - (rtac disjI2 1), - (rtac disjI1 1), - (rtac (abs_tr_iso RS subst) 1), - (rtac cfun_arg_cong 1), - (etac trans 1), - (rtac cfun_arg_cong 1), - (rtac (Exh_one RS disjE) 1), - (contr_tac 1), - (atac 1), - (rtac disjI2 1), - (rtac disjI2 1), - (rtac (abs_tr_iso RS subst) 1), - (rtac cfun_arg_cong 1), - (etac trans 1), - (rtac cfun_arg_cong 1), - (rtac (Exh_one RS disjE) 1), - (contr_tac 1), - (atac 1) - ]); - - -val trE = prove_goal Tr1.thy - "[| p=UU ==> Q; p = TT ==>Q; p = FF ==>Q|] ==>Q" - (fn prems => - [ - (rtac (Exh_tr RS disjE) 1), - (eresolve_tac prems 1), - (etac disjE 1), - (eresolve_tac prems 1), - (eresolve_tac prems 1) - ]); - - -(* ------------------------------------------------------------------------ *) -(* type tr is flat *) -(* ------------------------------------------------------------------------ *) - -val prems = goalw Tr1.thy [flat_def] "flat(TT)"; -by (rtac allI 1); -by (rtac allI 1); -by (res_inst_tac [("p","x")] trE 1); -by (asm_simp_tac ccc1_ss 1); -by (res_inst_tac [("p","y")] trE 1); -by (asm_simp_tac (ccc1_ss addsimps dist_less_tr) 1); -by (asm_simp_tac (ccc1_ss addsimps dist_less_tr) 1); -by (asm_simp_tac (ccc1_ss addsimps dist_less_tr) 1); -by (res_inst_tac [("p","y")] trE 1); -by (asm_simp_tac (ccc1_ss addsimps dist_less_tr) 1); -by (asm_simp_tac (ccc1_ss addsimps dist_less_tr) 1); -by (asm_simp_tac (ccc1_ss addsimps dist_less_tr) 1); -val flat_tr = result(); - - -(* ------------------------------------------------------------------------ *) -(* properties of tr_when *) -(* ------------------------------------------------------------------------ *) - -fun prover s = prove_goalw Tr1.thy [tr_when_def,TT_def,FF_def] s - (fn prems => - [ - (simp_tac Cfun_ss 1), - (simp_tac (Ssum_ss addsimps [(rep_tr_iso ), - (abs_tr_iso RS allI) RS ((rep_tr_iso RS allI) - RS iso_strict) RS conjunct1]@dist_eq_one)1) - ]); - -val tr_when = map prover [ - "tr_when[x][y][UU] = UU", - "tr_when[x][y][TT] = x", - "tr_when[x][y][FF] = y" - ]; - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/tr1.thy --- a/src/HOLCF/tr1.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,57 +0,0 @@ -(* Title: HOLCF/tr1.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Introduve the domain of truth values tr = {UU,TT,FF} - -This type is introduced using a domain isomorphism. - -The type axiom - - arities tr :: pcpo - -and the continuity of the Isomorphisms are taken for granted. Since the -type is not recursive, it could be easily introduced in a purely conservative -style as it was used for the types sprod, ssum, lift. The definition of the -ordering is canonical using abstraction and representation, but this would take -again a lot of internal constants. It can be easily seen that the axioms below -are consistent. - -Partial Ordering is implicit in isomorphims abs_tr,rep_tr and their continuity - -*) - -Tr1 = One + - -types tr 0 -arities tr :: pcpo - -consts - abs_tr :: "one ++ one -> tr" - rep_tr :: "tr -> one ++ one" - TT :: "tr" - FF :: "tr" - tr_when :: "'c -> 'c -> tr -> 'c" - -rules - - abs_tr_iso "abs_tr[rep_tr[u]] = u" - rep_tr_iso "rep_tr[abs_tr[x]] = x" - - TT_def "TT == abs_tr[sinl[one]]" - FF_def "FF == abs_tr[sinr[one]]" - - tr_when_def "tr_when == (LAM e1 e2 t.when[LAM x.e1][LAM y.e2][rep_tr[t]])" - -end - - - - - - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/tr2.ML --- a/src/HOLCF/tr2.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,82 +0,0 @@ -(* Title: HOLCF/tr2.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for tr2.thy -*) - -open Tr2; - -(* ------------------------------------------------------------------------ *) -(* lemmas about andalso *) -(* ------------------------------------------------------------------------ *) - -fun prover s = prove_goalw Tr2.thy [andalso_def] s - (fn prems => - [ - (simp_tac (ccc1_ss addsimps tr_when) 1) - ]); - -val andalso_thms = map prover [ - "TT andalso y = y", - "FF andalso y = FF", - "UU andalso y = UU" - ]; - -val andalso_thms = andalso_thms @ - [prove_goalw Tr2.thy [andalso_def] "x andalso TT = x" - (fn prems => - [ - (res_inst_tac [("p","x")] trE 1), - (asm_simp_tac (ccc1_ss addsimps tr_when) 1), - (asm_simp_tac (ccc1_ss addsimps tr_when) 1), - (asm_simp_tac (ccc1_ss addsimps tr_when) 1) - ])]; - -(* ------------------------------------------------------------------------ *) -(* lemmas about orelse *) -(* ------------------------------------------------------------------------ *) - -fun prover s = prove_goalw Tr2.thy [orelse_def] s - (fn prems => - [ - (simp_tac (ccc1_ss addsimps tr_when) 1) - ]); - -val orelse_thms = map prover [ - "TT orelse y = TT", - "FF orelse y = y", - "UU orelse y = UU" - ]; - -val orelse_thms = orelse_thms @ - [prove_goalw Tr2.thy [orelse_def] "x orelse FF = x" - (fn prems => - [ - (res_inst_tac [("p","x")] trE 1), - (asm_simp_tac (ccc1_ss addsimps tr_when) 1), - (asm_simp_tac (ccc1_ss addsimps tr_when) 1), - (asm_simp_tac (ccc1_ss addsimps tr_when) 1) - ])]; - - -(* ------------------------------------------------------------------------ *) -(* lemmas about If_then_else_fi *) -(* ------------------------------------------------------------------------ *) - -fun prover s = prove_goalw Tr2.thy [ifte_def] s - (fn prems => - [ - (simp_tac (ccc1_ss addsimps tr_when) 1) - ]); - -val ifte_thms = map prover [ - "If UU then e1 else e2 fi = UU", - "If FF then e1 else e2 fi = e2", - "If TT then e1 else e2 fi = e1"]; - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/tr2.thy --- a/src/HOLCF/tr2.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,56 +0,0 @@ -(* Title: HOLCF/tr2.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Introduce infix if_then_else_fi and boolean connectives andalso, orelse -*) - -Tr2 = Tr1 + - -consts - "@cifte" :: "tr=>'c=>'c=>'c" - ("(3If _/ (then _/ else _) fi)" [60,60,60] 60) - "Icifte" :: "tr->'c->'c->'c" - - "@andalso" :: "tr => tr => tr" ("_ andalso _" [61,60] 60) - "cop @andalso" :: "tr -> tr -> tr" ("trand") - "@orelse" :: "tr => tr => tr" ("_ orelse _" [61,60] 60) - "cop @orelse" :: "tr -> tr -> tr" ("tror") - - -rules - - ifte_def "Icifte == (LAM t e1 e2.tr_when[e1][e2][t])" - andalso_def "trand == (LAM t1 t2.tr_when[t2][FF][t1])" - orelse_def "tror == (LAM t1 t2.tr_when[TT][t2][t1])" - - -end - -ML - -(* ----------------------------------------------------------------------*) -(* translations for the above mixfixes *) -(* ----------------------------------------------------------------------*) - -fun ciftetr ts = - let val Cfapp = Const("fapp",dummyT) in - Cfapp $ - (Cfapp $ - (Cfapp$Const("Icifte",dummyT)$(nth_elem (0,ts)))$ - (nth_elem (1,ts)))$ - (nth_elem (2,ts)) - end; - - -val parse_translation = [("@andalso",mk_cinfixtr "@andalso"), - ("@orelse",mk_cinfixtr "@orelse"), - ("@cifte",ciftetr)]; - -val print_translation = []; - - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/void.ML --- a/src/HOLCF/void.ML Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,69 +0,0 @@ -(* Title: HOLCF/void.ML - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Lemmas for void.thy. - -These lemmas are prototype lemmas for class porder -see class theory porder.thy -*) - -open Void; - -(* ------------------------------------------------------------------------ *) -(* A non-emptyness result for Void *) -(* ------------------------------------------------------------------------ *) - -val VoidI = prove_goalw Void.thy [UU_void_Rep_def,Void_def] - " UU_void_Rep : Void" -(fn prems => - [ - (rtac (mem_Collect_eq RS ssubst) 1), - (rtac refl 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* less_void is a partial ordering on void *) -(* ------------------------------------------------------------------------ *) - -val refl_less_void = prove_goalw Void.thy [ less_void_def ] "less_void(x,x)" -(fn prems => - [ - (fast_tac HOL_cs 1) - ]); - -val antisym_less_void = prove_goalw Void.thy [ less_void_def ] - "[|less_void(x,y); less_void(y,x)|] ==> x = y" -(fn prems => - [ - (cut_facts_tac prems 1), - (rtac (Rep_Void_inverse RS subst) 1), - (etac subst 1), - (rtac (Rep_Void_inverse RS sym) 1) - ]); - -val trans_less_void = prove_goalw Void.thy [ less_void_def ] - "[|less_void(x,y); less_void(y,z)|] ==> less_void(x,z)" -(fn prems => - [ - (cut_facts_tac prems 1), - (fast_tac HOL_cs 1) - ]); - -(* ------------------------------------------------------------------------ *) -(* a technical lemma about void: *) -(* every element in void is represented by UU_void_Rep *) -(* ------------------------------------------------------------------------ *) - -val unique_void = prove_goal Void.thy "Rep_Void(x) = UU_void_Rep" -(fn prems => - [ - (rtac (mem_Collect_eq RS subst) 1), - (fold_goals_tac [Void_def]), - (rtac Rep_Void 1) - ]); - - - - diff -r 717bd79b976f -r f62f9a75f685 src/HOLCF/void.thy --- a/src/HOLCF/void.thy Sat Apr 05 17:03:38 2003 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,48 +0,0 @@ -(* Title: HOLCF/void.thy - ID: $Id$ - Author: Franz Regensburger - Copyright 1993 Technische Universitaet Muenchen - -Definition of type void with partial order. Void is the prototype for -all types in class 'po' - -Type void is defined as a set Void over type bool. -*) - -Void = Holcfb + - -types void 0 - -arities void :: term - -consts - Void :: "bool set" - UU_void_Rep :: "bool" - Rep_Void :: "void => bool" - Abs_Void :: "bool => void" - UU_void :: "void" - less_void :: "[void,void] => bool" - -rules - - (* The unique element in Void is False:bool *) - - UU_void_Rep_def "UU_void_Rep == False" - Void_def "Void == {x. x = UU_void_Rep}" - - (*faking a type definition... *) - (* void is isomorphic to Void *) - - Rep_Void "Rep_Void(x):Void" - Rep_Void_inverse "Abs_Void(Rep_Void(x)) = x" - Abs_Void_inverse "y:Void ==> Rep_Void(Abs_Void(y)) = y" - - (*defining the abstract constants*) - - UU_void_def "UU_void == Abs_Void(UU_void_Rep)" - less_void_def "less_void(x,y) == (Rep_Void(x) = Rep_Void(y))" -end - - - -