# HG changeset patch
# User wenzelm
# Date 1181752217 -7200
# Node ID 53317a1ec8b2d456049f3d27c2afdf5c1c8b2972
# Parent 45cd7db985b38ca2d698436b901f0b0172a0f214
tuned proofs: avoid implicit prems;
tuned;
diff -r 45cd7db985b3 -r 53317a1ec8b2 src/HOLCF/Domain.thy
--- a/src/HOLCF/Domain.thy Wed Jun 13 18:30:16 2007 +0200
+++ b/src/HOLCF/Domain.thy Wed Jun 13 18:30:17 2007 +0200
@@ -11,6 +11,7 @@
defaultsort pcpo
+
subsection {* Continuous isomorphisms *}
text {* A locale for continuous isomorphisms *}
@@ -20,41 +21,42 @@
fixes rep :: "'b \ 'a"
assumes abs_iso [simp]: "rep\(abs\x) = x"
assumes rep_iso [simp]: "abs\(rep\y) = y"
+begin
-lemma (in iso) swap: "iso rep abs"
-by (rule iso.intro [OF rep_iso abs_iso])
+lemma swap: "iso rep abs"
+ by (rule iso.intro [OF rep_iso abs_iso])
-lemma (in iso) abs_less: "(abs\x \ abs\y) = (x \ y)"
+lemma abs_less: "(abs\x \ abs\y) = (x \ y)"
proof
assume "abs\x \ abs\y"
- hence "rep\(abs\x) \ rep\(abs\y)" by (rule monofun_cfun_arg)
- thus "x \ y" by simp
+ then have "rep\(abs\x) \ rep\(abs\y)" by (rule monofun_cfun_arg)
+ then show "x \ y" by simp
next
assume "x \ y"
- thus "abs\x \ abs\y" by (rule monofun_cfun_arg)
+ then show "abs\x \ abs\y" by (rule monofun_cfun_arg)
qed
-lemma (in iso) rep_less: "(rep\x \ rep\y) = (x \ y)"
-by (rule iso.abs_less [OF swap])
+lemma rep_less: "(rep\x \ rep\y) = (x \ y)"
+ by (rule iso.abs_less [OF swap])
-lemma (in iso) abs_eq: "(abs\x = abs\y) = (x = y)"
-by (simp add: po_eq_conv abs_less)
+lemma abs_eq: "(abs\x = abs\y) = (x = y)"
+ by (simp add: po_eq_conv abs_less)
-lemma (in iso) rep_eq: "(rep\x = rep\y) = (x = y)"
-by (rule iso.abs_eq [OF swap])
+lemma rep_eq: "(rep\x = rep\y) = (x = y)"
+ by (rule iso.abs_eq [OF swap])
-lemma (in iso) abs_strict: "abs\\ = \"
+lemma abs_strict: "abs\\ = \"
proof -
have "\ \ rep\\" ..
- hence "abs\\ \ abs\(rep\\)" by (rule monofun_cfun_arg)
- hence "abs\\ \ \" by simp
- thus ?thesis by (rule UU_I)
+ then have "abs\\ \ abs\(rep\\)" by (rule monofun_cfun_arg)
+ then have "abs\\ \ \" by simp
+ then show ?thesis by (rule UU_I)
qed
-lemma (in iso) rep_strict: "rep\\ = \"
-by (rule iso.abs_strict [OF swap])
+lemma rep_strict: "rep\\ = \"
+ by (rule iso.abs_strict [OF swap])
-lemma (in iso) abs_defin': "abs\x = \ \ x = \"
+lemma abs_defin': "abs\x = \ \ x = \"
proof -
have "x = rep\(abs\x)" by simp
also assume "abs\x = \"
@@ -62,49 +64,52 @@
finally show "x = \" .
qed
-lemma (in iso) rep_defin': "rep\z = \ \ z = \"
-by (rule iso.abs_defin' [OF swap])
+lemma rep_defin': "rep\z = \ \ z = \"
+ by (rule iso.abs_defin' [OF swap])
-lemma (in iso) abs_defined: "z \ \ \ abs\z \ \"
-by (erule contrapos_nn, erule abs_defin')
+lemma abs_defined: "z \ \ \ abs\z \ \"
+ by (erule contrapos_nn, erule abs_defin')
-lemma (in iso) rep_defined: "z \ \ \ rep\z \ \"
-by (rule iso.abs_defined [OF iso.swap])
+lemma rep_defined: "z \ \ \ rep\z \ \"
+ by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
-lemma (in iso) abs_defined_iff: "(abs\x = \) = (x = \)"
-by (auto elim: abs_defin' intro: abs_strict)
+lemma abs_defined_iff: "(abs\x = \) = (x = \)"
+ by (auto elim: abs_defin' intro: abs_strict)
-lemma (in iso) rep_defined_iff: "(rep\x = \) = (x = \)"
-by (rule iso.abs_defined_iff [OF iso.swap])
+lemma rep_defined_iff: "(rep\x = \) = (x = \)"
+ by (rule iso.abs_defined_iff [OF iso.swap]) (rule iso_axioms)
lemma (in iso) compact_abs_rev: "compact (abs\x) \ compact x"
proof (unfold compact_def)
assume "adm (\y. \ abs\x \ y)"
with cont_Rep_CFun2
have "adm (\y. \ abs\x \ abs\y)" by (rule adm_subst)
- thus "adm (\y. \ x \ y)" using abs_less by simp
+ then show "adm (\y. \ x \ y)" using abs_less by simp
qed
-lemma (in iso) compact_rep_rev: "compact (rep\x) \ compact x"
-by (rule iso.compact_abs_rev [OF iso.swap])
+lemma compact_rep_rev: "compact (rep\x) \ compact x"
+ by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
-lemma (in iso) compact_abs: "compact x \ compact (abs\x)"
-by (rule compact_rep_rev, simp)
+lemma compact_abs: "compact x \ compact (abs\x)"
+ by (rule compact_rep_rev) simp
-lemma (in iso) compact_rep: "compact x \ compact (rep\x)"
-by (rule iso.compact_abs [OF iso.swap])
+lemma compact_rep: "compact x \ compact (rep\x)"
+ by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
-lemma (in iso) iso_swap: "(x = abs\y) = (rep\x = y)"
+lemma iso_swap: "(x = abs\y) = (rep\x = y)"
proof
assume "x = abs\y"
- hence "rep\x = rep\(abs\y)" by simp
- thus "rep\x = y" by simp
+ then have "rep\x = rep\(abs\y)" by simp
+ then show "rep\x = y" by simp
next
assume "rep\x = y"
- hence "abs\(rep\x) = abs\y" by simp
- thus "x = abs\y" by simp
+ then have "abs\(rep\x) = abs\y" by simp
+ then show "x = abs\y" by simp
qed
+end
+
+
subsection {* Casedist *}
lemma ex_one_defined_iff:
@@ -114,7 +119,7 @@
apply simp
apply simp
apply force
-done
+ done
lemma ex_up_defined_iff:
"(\x. P x \ x \ \) = (\x. P (up\x))"
@@ -123,7 +128,7 @@
apply simp
apply fast
apply (force intro!: up_defined)
-done
+ done
lemma ex_sprod_defined_iff:
"(\y. P y \ y \ \) =
@@ -133,7 +138,7 @@
apply simp
apply fast
apply (force intro!: spair_defined)
-done
+ done
lemma ex_sprod_up_defined_iff:
"(\y. P y \ y \ \) =
@@ -145,7 +150,7 @@
apply simp
apply fast
apply (force intro!: spair_defined)
-done
+ done
lemma ex_ssum_defined_iff:
"(\x. P x \ x \ \) =
@@ -161,10 +166,10 @@
apply (erule disjE)
apply force
apply force
-done
+ done
lemma exh_start: "p = \ \ (\x. p = x \ x \ \)"
-by auto
+ by auto
lemmas ex_defined_iffs =
ex_ssum_defined_iff
@@ -176,16 +181,16 @@
text {* Rules for turning exh into casedist *}
lemma exh_casedist0: "\R; R \ P\ \ P" (* like make_elim *)
-by auto
+ by auto
lemma exh_casedist1: "((P \ Q \ R) \ S) \ (\P \ R; Q \ R\ \ S)"
-by rule auto
+ by rule auto
lemma exh_casedist2: "(\x. P x \ Q) \ (\x. P x \ Q)"
-by rule auto
+ by rule auto
lemma exh_casedist3: "(P \ Q \ R) \ (P \ Q \ R)"
-by rule auto
+ by rule auto
lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3