(* Title: HOLCF/pcpo.ML
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
Lemmas for pcpo.thy
*)
open Pcpo;
(* ------------------------------------------------------------------------ *)
(* derive the old rule minimal *)
(* ------------------------------------------------------------------------ *)
qed_goalw "UU_least" thy [ UU_def ] "!z. UU << z"
(fn prems => [
(rtac (select_eq_Ex RS iffD2) 1),
(rtac least 1)]);
bind_thm("minimal",UU_least RS spec);
(* ------------------------------------------------------------------------ *)
(* in cpo's everthing equal to THE lub has lub properties for every chain *)
(* ------------------------------------------------------------------------ *)
qed_goal "thelubE" thy
"[| is_chain(S);lub(range(S)) = (l::'a::cpo)|] ==> range(S) <<| l "
(fn prems =>
[
(cut_facts_tac prems 1),
(hyp_subst_tac 1),
(rtac lubI 1),
(etac cpo 1)
]);
(* ------------------------------------------------------------------------ *)
(* Properties of the lub *)
(* ------------------------------------------------------------------------ *)
bind_thm ("is_ub_thelub", cpo RS lubI RS is_ub_lub);
(* is_chain(?S1) ==> ?S1(?x) << lub(range(?S1)) *)
bind_thm ("is_lub_thelub", cpo RS lubI RS is_lub_lub);
(* [| is_chain(?S5); range(?S5) <| ?x1 |] ==> lub(range(?S5)) << ?x1 *)
qed_goal "maxinch_is_thelub" thy "is_chain Y ==> \
\ max_in_chain i Y = (lub(range(Y)) = ((Y i)::'a::cpo))"
(fn prems =>
[
cut_facts_tac prems 1,
rtac iffI 1,
fast_tac (HOL_cs addSIs [thelubI,lub_finch1]) 1,
rewtac max_in_chain_def,
safe_tac (HOL_cs addSIs [antisym_less]),
fast_tac (HOL_cs addSEs [chain_mono3]) 1,
dtac sym 1,
fast_tac ((HOL_cs addSEs [is_ub_thelub]) addss !simpset) 1
]);
(* ------------------------------------------------------------------------ *)
(* the << relation between two chains is preserved by their lubs *)
(* ------------------------------------------------------------------------ *)
qed_goal "lub_mono" thy
"[|is_chain(C1::(nat=>'a::cpo));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 *)
(* ------------------------------------------------------------------------ *)
qed_goal "lub_equal" thy
"[| is_chain(C1::(nat=>'a::cpo));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 *)
(* ------------------------------------------------------------------------ *)
qed_goal "lub_mono2" thy
"[|? j.!i. j<i --> X(i::nat)=Y(i);is_chain(X::nat=>'a::cpo);is_chain(Y)|]\
\ ==> lub(range(X))<<lub(range(Y))"
(fn prems =>
[
(rtac exE 1),
(resolve_tac prems 1),
(rtac is_lub_thelub 1),
(resolve_tac prems 1),
(rtac ub_rangeI 1),
(strip_tac 1),
(case_tac "x<i" 1),
(res_inst_tac [("s","Y(i)"),("t","X(i)")] subst 1),
(rtac sym 1),
(Fast_tac 1),
(rtac is_ub_thelub 1),
(resolve_tac prems 1),
(res_inst_tac [("y","X(Suc(x))")] trans_less 1),
(rtac (chain_mono RS mp) 1),
(resolve_tac prems 1),
(rtac (not_less_eq RS subst) 1),
(atac 1),
(res_inst_tac [("s","Y(Suc(x))"),("t","X(Suc(x))")] subst 1),
(rtac sym 1),
(Asm_simp_tac 1),
(rtac is_ub_thelub 1),
(resolve_tac prems 1)
]);
qed_goal "lub_equal2" thy
"[|? j.!i. j<i --> X(i)=Y(i);is_chain(X::nat=>'a::cpo);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,
(Step_tac 1),
Safe_tac,
(rtac sym 1),
(Fast_tac 1)
]);
qed_goal "lub_mono3" thy "[|is_chain(Y::nat=>'a::cpo);is_chain(X);\
\! i. ? j. Y(i)<< X(j)|]==> lub(range(Y))<<lub(range(X))"
(fn prems =>
[
(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_sym = prove_goal thy "(UU = x) = (x = UU)" (fn _ => [
Fast_tac 1]);
qed_goal "eq_UU_iff" thy "(x=UU)=(x<<UU)"
(fn prems =>
[
(rtac iffI 1),
(hyp_subst_tac 1),
(rtac refl_less 1),
(rtac antisym_less 1),
(atac 1),
(rtac minimal 1)
]);
qed_goal "UU_I" thy "x << UU ==> x = UU"
(fn prems =>
[
(stac eq_UU_iff 1),
(resolve_tac prems 1)
]);
qed_goal "not_less2not_eq" thy "~(x::'a::po)<<y ==> ~x=y"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac classical2 1),
(atac 1),
(hyp_subst_tac 1),
(rtac refl_less 1)
]);
qed_goal "chain_UU_I" 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),
(etac subst 1),
(etac is_ub_thelub 1)
]);
qed_goal "chain_UU_I_inverse" 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 exI 1),
(etac spec 1),
(rtac allI 1),
(rtac (antisym_less_inverse RS conjunct1) 1),
(etac spec 1)
]);
qed_goal "chain_UU_I_inverse2" thy
"~lub(range(Y::(nat=>'a::pcpo)))=UU ==> ? i.~ Y(i)=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac (not_all RS iffD1) 1),
(rtac swap 1),
(rtac chain_UU_I_inverse 2),
(etac notnotD 2),
(atac 1)
]);
qed_goal "notUU_I" thy "[| x<<y; ~x=UU |] ==> ~y=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac contrapos 1),
(rtac UU_I 1),
(hyp_subst_tac 1),
(atac 1)
]);
qed_goal "chain_mono2" thy
"[|? j.~Y(j)=UU;is_chain(Y::nat=>'a::pcpo)|]\
\ ==> ? j.!i. j<i-->~Y(i)=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
Safe_tac,
(Step_tac 1),
(strip_tac 1),
(rtac notUU_I 1),
(atac 2),
(etac (chain_mono RS mp) 1),
(atac 1)
]);
(**************************************)
(* some properties for chfin and flat *)
(**************************************)
(* ------------------------------------------------------------------------ *)
(* flat types are chain_finite *)
(* ------------------------------------------------------------------------ *)
qed_goalw "flat_imp_chain_finite" thy [max_in_chain_def]
"!Y::nat=>'a::flat. is_chain Y-->(? n. max_in_chain n Y)"
(fn _ =>
[
(strip_tac 1),
(case_tac "!i. Y(i)=UU" 1),
(res_inst_tac [("x","0")] exI 1),
(Asm_simp_tac 1),
(Asm_full_simp_tac 1),
(etac exE 1),
(res_inst_tac [("x","i")] exI 1),
(strip_tac 1),
(dres_inst_tac [("x","i"),("y","j")] chain_mono 1),
(etac (le_imp_less_or_eq RS disjE) 1),
Safe_tac,
(dtac (ax_flat RS spec RS spec RS mp) 1),
(Fast_tac 1)
]);
(* flat subclass of chfin --> adm_flat not needed *)
qed_goal "flat_eq" thy "(a::'a::flat) ~= UU ==> a << b = (a = b)"
(fn prems=>
[
cut_facts_tac prems 1,
safe_tac (HOL_cs addSIs [refl_less]),
dtac (ax_flat RS spec RS spec RS mp) 1,
fast_tac (HOL_cs addSIs [refl_less,ax_flat RS spec RS spec RS mp]) 1
]);
qed_goal "chfin2finch" thy
"is_chain (Y::nat=>'a::chfin) ==> finite_chain Y"
(fn prems =>
[
cut_facts_tac prems 1,
fast_tac (HOL_cs addss
(!simpset addsimps [chfin,finite_chain_def])) 1
]);
(* ------------------------------------------------------------------------ *)
(* lemmata for improved admissibility introdution rule *)
(* ------------------------------------------------------------------------ *)
qed_goal "infinite_chain_adm_lemma" Porder.thy
"[|is_chain Y; !i. P (Y i); \
\ (!!Y. [| is_chain Y; !i. P (Y i); ~ finite_chain Y |] ==> P (lub (range Y)))\
\ |] ==> P (lub (range Y))"
(fn prems => [
cut_facts_tac prems 1,
case_tac "finite_chain Y" 1,
eresolve_tac prems 2, atac 2, atac 2,
rewtac finite_chain_def,
safe_tac HOL_cs,
etac (lub_finch1 RS thelubI RS ssubst) 1, atac 1, etac spec 1]);
qed_goal "increasing_chain_adm_lemma" Porder.thy
"[|is_chain Y; !i. P (Y i); \
\ (!!Y. [| is_chain Y; !i. P (Y i); !i. ? j. i < j & Y i ~= Y j & Y i << Y j|]\
\ ==> P (lub (range Y))) |] ==> P (lub (range Y))"
(fn prems => [
cut_facts_tac prems 1,
etac infinite_chain_adm_lemma 1, atac 1, etac thin_rl 1,
rewtac finite_chain_def,
safe_tac HOL_cs,
etac swap 1,
rewtac max_in_chain_def,
resolve_tac prems 1, atac 1, atac 1,
fast_tac (HOL_cs addDs [le_imp_less_or_eq]
addEs [chain_mono RS mp]) 1]);