src/HOLCF/Pcpo.ML
author nipkow
Wed, 19 Jan 1994 17:35:01 +0100
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permissions -rw-r--r--
Franz Regensburger's Higher-Order Logic of Computable Functions embedding LCF in HOL.
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(*  Title: 	HOLCF/pcpo.ML
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    ID:         $Id$
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    Author: 	Franz Regensburger
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    Copyright   1993 Technische Universitaet Muenchen
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Lemmas for pcpo.thy
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*)
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open Pcpo;
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(* ------------------------------------------------------------------------ *)
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(* in pcpo's everthing equal to THE lub has lub properties for every chain  *)
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(* ------------------------------------------------------------------------ *)
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val thelubE = prove_goal  Pcpo.thy 
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	"[| is_chain(S);lub(range(S)) = l::'a::pcpo|] ==> range(S) <<| l "
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(fn prems =>
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	[
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	(cut_facts_tac prems 1), 
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	(hyp_subst_tac 1),
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	(rtac lubI 1),
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	(etac cpo 1)
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	]);
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(* ------------------------------------------------------------------------ *)
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(* Properties of the lub                                                    *)
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(* ------------------------------------------------------------------------ *)
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val is_ub_thelub = (cpo RS lubI RS is_ub_lub);
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(* is_chain(?S1) ==> ?S1(?x) << lub(range(?S1))                             *)
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val is_lub_thelub = (cpo RS lubI RS is_lub_lub);
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(* [| is_chain(?S5); range(?S5) <| ?x1 |] ==> lub(range(?S5)) << ?x1        *)
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(* ------------------------------------------------------------------------ *)
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(* the << relation between two chains is preserved by their lubs            *)
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(* ------------------------------------------------------------------------ *)
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val lub_mono = prove_goal Pcpo.thy 
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	"[|is_chain(C1::(nat=>'a::pcpo));is_chain(C2); ! k. C1(k) << C2(k)|]\
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\           ==> lub(range(C1)) << lub(range(C2))"
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(fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(etac is_lub_thelub 1),
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	(rtac ub_rangeI 1),
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	(rtac allI 1),
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	(rtac trans_less 1),
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	(etac spec 1),
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	(etac is_ub_thelub 1)
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	]);
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(* ------------------------------------------------------------------------ *)
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(* the = relation between two chains is preserved by their lubs            *)
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(* ------------------------------------------------------------------------ *)
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val lub_equal = prove_goal Pcpo.thy
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"[| is_chain(C1::(nat=>'a::pcpo));is_chain(C2);!k.C1(k)=C2(k)|]\
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\	==> lub(range(C1))=lub(range(C2))"
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(fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(rtac antisym_less 1),
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	(rtac lub_mono 1),
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	(atac 1),
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	(atac 1),
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	(strip_tac 1),
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	(rtac (antisym_less_inverse RS conjunct1) 1),
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	(etac spec 1),
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	(rtac lub_mono 1),
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	(atac 1),
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	(atac 1),
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	(strip_tac 1),
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	(rtac (antisym_less_inverse RS conjunct2) 1),
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	(etac spec 1)
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	]);
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(* ------------------------------------------------------------------------ *)
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(* more results about mono and = of lubs of chains                          *)
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(* ------------------------------------------------------------------------ *)
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val lub_mono2 = prove_goal Pcpo.thy 
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"[|? j.!i. j<i --> X(i::nat)=Y(i);is_chain(X::nat=>'a::pcpo);is_chain(Y)|]\
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\ ==> lub(range(X))<<lub(range(Y))"
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 (fn prems =>
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	[
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	(rtac  exE 1),
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	(resolve_tac prems 1),
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	(rtac is_lub_thelub 1),
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	(resolve_tac prems 1),
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	(rtac ub_rangeI 1),
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	(strip_tac 1),
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	(res_inst_tac [("Q","x<i")] classical2 1),
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	(res_inst_tac [("s","Y(i)"),("t","X(i)")] subst 1),
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	(rtac sym 1),
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	(fast_tac HOL_cs 1),
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	(rtac is_ub_thelub 1),
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	(resolve_tac prems 1),
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	(res_inst_tac [("y","X(Suc(x))")] trans_less 1),
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	(rtac (chain_mono RS mp) 1),
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	(resolve_tac prems 1),
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	(rtac (not_less_eq RS subst) 1),
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	(atac 1),
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	(res_inst_tac [("s","Y(Suc(x))"),("t","X(Suc(x))")] subst 1),
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	(rtac sym 1),
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	(asm_simp_tac nat_ss 1),
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	(rtac is_ub_thelub 1),
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	(resolve_tac prems 1)
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	]);
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val lub_equal2 = prove_goal Pcpo.thy 
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"[|? j.!i. j<i --> X(i)=Y(i);is_chain(X::nat=>'a::pcpo);is_chain(Y)|]\
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\ ==> lub(range(X))=lub(range(Y))"
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 (fn prems =>
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	[
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	(rtac antisym_less 1),
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	(rtac lub_mono2 1),
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	(REPEAT (resolve_tac prems 1)),
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	(cut_facts_tac prems 1),
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	(rtac lub_mono2 1),
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	(safe_tac HOL_cs),
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	(step_tac HOL_cs 1),
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	(safe_tac HOL_cs),
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	(rtac sym 1),
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	(fast_tac HOL_cs 1)
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	]);
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val lub_mono3 = prove_goal Pcpo.thy "[|is_chain(Y::nat=>'a::pcpo);is_chain(X);\
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\! i. ? j. Y(i)<< X(j)|]==> lub(range(Y))<<lub(range(X))"
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 (fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(rtac is_lub_thelub 1),
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	(atac 1),
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	(rtac ub_rangeI 1),
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	(strip_tac 1),
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	(etac allE 1),
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	(etac exE 1),
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	(rtac trans_less 1),
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	(rtac is_ub_thelub 2),
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	(atac 2),
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	(atac 1)
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	]);
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(* ------------------------------------------------------------------------ *)
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(* usefull lemmas about UU                                                  *)
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(* ------------------------------------------------------------------------ *)
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val eq_UU_iff = prove_goal Pcpo.thy "(x=UU)=(x<<UU)"
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 (fn prems =>
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	[
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	(rtac iffI 1),
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	(hyp_subst_tac 1),
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	(rtac refl_less 1),
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	(rtac antisym_less 1),
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	(atac 1),
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	(rtac minimal 1)
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	]);
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val UU_I = prove_goal Pcpo.thy "x << UU ==> x = UU"
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 (fn prems =>
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	[
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	(rtac (eq_UU_iff RS ssubst) 1),
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	(resolve_tac prems 1)
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	]);
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val not_less2not_eq = prove_goal Pcpo.thy "~x<<y ==> ~x=y"
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 (fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(rtac classical3 1),
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	(atac 1),
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	(hyp_subst_tac 1),
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	(rtac refl_less 1)
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	]);
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val chain_UU_I = prove_goal Pcpo.thy
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	"[|is_chain(Y);lub(range(Y))=UU|] ==> ! i.Y(i)=UU"
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(fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(rtac allI 1),
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	(rtac antisym_less 1),
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	(rtac minimal 2),
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	(res_inst_tac [("t","UU")] subst 1),
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	(atac 1),
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	(etac is_ub_thelub 1)
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	]);
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val chain_UU_I_inverse = prove_goal Pcpo.thy 
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	"!i.Y(i::nat)=UU ==> lub(range(Y::(nat=>'a::pcpo)))=UU"
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(fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(rtac lub_chain_maxelem 1),
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	(rtac is_chainI 1),
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	(rtac allI 1),
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	(res_inst_tac [("s","UU"),("t","Y(i)")] subst 1),
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	(rtac sym 1),
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	(etac spec 1),
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	(rtac minimal 1),
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	(rtac exI 1),
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	(etac spec 1),
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	(rtac allI 1),
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	(rtac (antisym_less_inverse RS conjunct1) 1),
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	(etac spec 1)
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	]);
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val chain_UU_I_inverse2 = prove_goal Pcpo.thy 
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	"~lub(range(Y::(nat=>'a::pcpo)))=UU ==> ? i.~ Y(i)=UU"
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 (fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(rtac (notall2ex RS iffD1) 1),
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	(rtac swap 1),
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	(rtac chain_UU_I_inverse 2),
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	(etac notnotD 2),
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	(atac 1)
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	]);
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val notUU_I = prove_goal Pcpo.thy "[| x<<y; ~x=UU |] ==> ~y=UU"
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(fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(etac contrapos 1),
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	(rtac UU_I 1),
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	(hyp_subst_tac 1),
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	(atac 1)
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	]);
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val chain_mono2 = prove_goal Pcpo.thy 
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"[|? j.~Y(j)=UU;is_chain(Y::nat=>'a::pcpo)|]\
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\ ==> ? j.!i.j<i-->~Y(i)=UU"
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 (fn prems =>
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	[
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	(cut_facts_tac prems 1),
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	(safe_tac HOL_cs),
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	(step_tac HOL_cs 1),
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	(strip_tac 1),
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	(rtac notUU_I 1),
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	(atac 2),
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	(etac (chain_mono RS mp) 1),
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	(atac 1)
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	]);
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(* ------------------------------------------------------------------------ *)
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(* uniqueness in void                                                       *)
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(* ------------------------------------------------------------------------ *)
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val unique_void2 = prove_goal Pcpo.thy "x::void=UU"
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 (fn prems =>
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	[
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	(rtac (inst_void_pcpo RS ssubst) 1),
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	(rtac (Rep_Void_inverse RS subst) 1),
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	(rtac (Rep_Void_inverse RS subst) 1),
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	(rtac arg_cong 1),
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	(rtac box_equals 1),
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	(rtac refl 1),
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	(rtac (unique_void RS sym) 1),
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	(rtac (unique_void RS sym) 1)
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	]);
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