author huffman Fri Jun 20 23:01:09 2008 +0200 (2008-06-20) changeset 27310 d0229bc6c461 parent 27309 c74270fd72a8 child 27311 aa28b1d33866
simplify profinite class axioms
 src/HOLCF/Bifinite.thy file | annotate | diff | revisions src/HOLCF/ConvexPD.thy file | annotate | diff | revisions src/HOLCF/Cprod.thy file | annotate | diff | revisions src/HOLCF/Lift.thy file | annotate | diff | revisions src/HOLCF/LowerPD.thy file | annotate | diff | revisions src/HOLCF/Sprod.thy file | annotate | diff | revisions src/HOLCF/Ssum.thy file | annotate | diff | revisions src/HOLCF/Up.thy file | annotate | diff | revisions src/HOLCF/UpperPD.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOLCF/Bifinite.thy	Fri Jun 20 22:51:50 2008 +0200
1.2 +++ b/src/HOLCF/Bifinite.thy	Fri Jun 20 23:01:09 2008 +0200
1.3 @@ -13,7 +13,7 @@
1.4
1.5  class profinite = cpo +
1.6    fixes approx :: "nat \<Rightarrow> 'a \<rightarrow> 'a"
1.7 -  assumes chain_approx_app: "chain (\<lambda>i. approx i\<cdot>x)"
1.8 +  assumes chain_approx [simp]: "chain approx"
1.9    assumes lub_approx_app [simp]: "(\<Squnion>i. approx i\<cdot>x) = x"
1.10    assumes approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
1.11    assumes finite_fixes_approx: "finite {x. approx i\<cdot>x = x}"
1.12 @@ -27,13 +27,6 @@
1.13  apply (clarify, erule subst, rule exI, rule refl)
1.14  done
1.15
1.16 -lemma chain_approx [simp]: "chain approx"
1.17 -apply (rule chainI)
1.18 -apply (rule less_cfun_ext)
1.19 -apply (rule chainE)
1.20 -apply (rule chain_approx_app)
1.21 -done
1.22 -
1.23  lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda> x. x)"
1.24  by (rule ext_cfun, simp add: contlub_cfun_fun)
1.25
```
```     2.1 --- a/src/HOLCF/ConvexPD.thy	Fri Jun 20 22:51:50 2008 +0200
2.2 +++ b/src/HOLCF/ConvexPD.thy	Fri Jun 20 23:01:09 2008 +0200
2.3 @@ -194,7 +194,7 @@
2.4
2.5  instance
2.6  apply (intro_classes, unfold approx_convex_pd_def)
2.8 +apply (rule convex_pd.chain_completion_approx)
2.9  apply (rule convex_pd.lub_completion_approx)
2.10  apply (rule convex_pd.completion_approx_idem)
2.11  apply (rule convex_pd.finite_fixes_completion_approx)
```
```     3.1 --- a/src/HOLCF/Cprod.thy	Fri Jun 20 22:51:50 2008 +0200
3.2 +++ b/src/HOLCF/Cprod.thy	Fri Jun 20 23:01:09 2008 +0200
3.3 @@ -351,7 +351,7 @@
3.4
3.5  instance proof
3.6    fix i :: nat and x :: "'a \<times> 'b"
3.7 -  show "chain (\<lambda>i. approx i\<cdot>x)"
3.8 +  show "chain (approx :: nat \<Rightarrow> 'a \<times> 'b \<rightarrow> 'a \<times> 'b)"
3.9      unfolding approx_cprod_def by simp
3.10    show "(\<Squnion>i. approx i\<cdot>x) = x"
3.11      unfolding approx_cprod_def
```
```     4.1 --- a/src/HOLCF/Lift.thy	Fri Jun 20 22:51:50 2008 +0200
4.2 +++ b/src/HOLCF/Lift.thy	Fri Jun 20 23:01:09 2008 +0200
4.3 @@ -137,6 +137,13 @@
4.4  apply (rule cont_lift_case1)
4.5  done
4.6
4.7 +lemma FLIFT_mono:
4.8 +  "(\<And>x. f x \<sqsubseteq> g x) \<Longrightarrow> (FLIFT x. f x) \<sqsubseteq> (FLIFT x. g x)"
4.9 +apply (rule monofunE [where f=flift1])
4.10 +apply (rule cont2mono [OF cont_flift1])
4.12 +done
4.13 +
4.14  lemma cont2cont_flift1 [simp]:
4.15    "\<lbrakk>\<And>y. cont (\<lambda>x. f x y)\<rbrakk> \<Longrightarrow> cont (\<lambda>x. FLIFT y. f x y)"
4.16  apply (rule cont_flift1 [THEN cont2cont_app3])
4.17 @@ -204,9 +211,9 @@
4.18
4.19  instance proof
4.20    fix x :: "'a lift"
4.21 -  show "chain (\<lambda>i. approx i\<cdot>x)"
4.22 +  show "chain (approx :: nat \<Rightarrow> 'a lift \<rightarrow> 'a lift)"
4.23      unfolding approx_lift_def
4.24 -    by (rule chainI, cases x, simp_all)
4.25 +    by (rule chainI, simp add: FLIFT_mono)
4.26  next
4.27    fix x :: "'a lift"
4.28    show "(\<Squnion>i. approx i\<cdot>x) = x"
```
```     5.1 --- a/src/HOLCF/LowerPD.thy	Fri Jun 20 22:51:50 2008 +0200
5.2 +++ b/src/HOLCF/LowerPD.thy	Fri Jun 20 23:01:09 2008 +0200
5.3 @@ -149,7 +149,7 @@
5.4
5.5  instance
5.6  apply (intro_classes, unfold approx_lower_pd_def)
5.8 +apply (rule lower_pd.chain_completion_approx)
5.9  apply (rule lower_pd.lub_completion_approx)
5.10  apply (rule lower_pd.completion_approx_idem)
5.11  apply (rule lower_pd.finite_fixes_completion_approx)
```
```     6.1 --- a/src/HOLCF/Sprod.thy	Fri Jun 20 22:51:50 2008 +0200
6.2 +++ b/src/HOLCF/Sprod.thy	Fri Jun 20 23:01:09 2008 +0200
6.3 @@ -73,7 +73,7 @@
6.4    Rep_Sprod_inject [symmetric] less_Sprod_def
6.5    Rep_Sprod_strict Rep_Sprod_spair
6.6
6.7 -lemma Exh_Sprod2:
6.8 +lemma Exh_Sprod:
6.9    "z = \<bottom> \<or> (\<exists>a b. z = (:a, b:) \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
6.10  apply (insert Rep_Sprod [of z])
6.11  apply (simp add: Rep_Sprod_simps eq_cprod)
6.12 @@ -85,7 +85,7 @@
6.13
6.14  lemma sprodE [cases type: **]:
6.15    "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x y. \<lbrakk>p = (:x, y:); x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
6.16 -by (cut_tac z=p in Exh_Sprod2, auto)
6.17 +by (cut_tac z=p in Exh_Sprod, auto)
6.18
6.19  lemma sprod_induct [induct type: **]:
6.20    "\<lbrakk>P \<bottom>; \<And>x y. \<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> P (:x, y:)\<rbrakk> \<Longrightarrow> P x"
6.21 @@ -222,11 +222,14 @@
6.22  subsection {* Strict product preserves flatness *}
6.23
6.24  instance "**" :: (flat, flat) flat
6.25 -apply (intro_classes, clarify)
6.26 -apply (rule_tac p=x in sprodE, simp)
6.27 -apply (rule_tac p=y in sprodE, simp)
6.28 -apply (simp add: flat_less_iff spair_less)
6.29 -done
6.30 +proof
6.31 +  fix x y :: "'a \<otimes> 'b"
6.32 +  assume "x \<sqsubseteq> y" thus "x = \<bottom> \<or> x = y"
6.33 +    apply (induct x, simp)
6.34 +    apply (induct y, simp)
6.35 +    apply (simp add: spair_less_iff flat_less_iff)
6.36 +    done
6.37 +qed
6.38
6.39  subsection {* Strict product is a bifinite domain *}
6.40
6.41 @@ -239,7 +242,7 @@
6.42
6.43  instance proof
6.44    fix i :: nat and x :: "'a \<otimes> 'b"
6.45 -  show "chain (\<lambda>i. approx i\<cdot>x)"
6.46 +  show "chain (approx :: nat \<Rightarrow> 'a \<otimes> 'b \<rightarrow> 'a \<otimes> 'b)"
6.47      unfolding approx_sprod_def by simp
6.48    show "(\<Squnion>i. approx i\<cdot>x) = x"
6.49      unfolding approx_sprod_def
6.50 @@ -249,7 +252,7 @@
6.51      by (simp add: ssplit_def strictify_conv_if)
6.52    have "Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x} \<subseteq> {x. approx i\<cdot>x = x}"
6.53      unfolding approx_sprod_def
6.54 -    apply (clarify, rule_tac p=x in sprodE)
6.55 +    apply (clarify, case_tac x)
6.57      apply (simp add: Rep_Sprod_spair spair_eq_iff)
6.58      done
```
```     7.1 --- a/src/HOLCF/Ssum.thy	Fri Jun 20 22:51:50 2008 +0200
7.2 +++ b/src/HOLCF/Ssum.thy	Fri Jun 20 23:01:09 2008 +0200
7.3 @@ -231,7 +231,7 @@
7.4
7.5  instance proof
7.6    fix i :: nat and x :: "'a \<oplus> 'b"
7.7 -  show "chain (\<lambda>i. approx i\<cdot>x)"
7.8 +  show "chain (approx :: nat \<Rightarrow> 'a \<oplus> 'b \<rightarrow> 'a \<oplus> 'b)"
7.9      unfolding approx_ssum_def by simp
7.10    show "(\<Squnion>i. approx i\<cdot>x) = x"
7.11      unfolding approx_ssum_def
7.12 @@ -241,7 +241,7 @@
7.13    have "{x::'a \<oplus> 'b. approx i\<cdot>x = x} \<subseteq>
7.14          (\<lambda>x. sinl\<cdot>x) ` {x. approx i\<cdot>x = x} \<union>
7.15          (\<lambda>x. sinr\<cdot>x) ` {x. approx i\<cdot>x = x}"
7.16 -    by (rule subsetI, rule_tac p=x in ssumE2, simp, simp)
7.17 +    by (rule subsetI, case_tac x rule: ssumE2, simp, simp)
7.18    thus "finite {x::'a \<oplus> 'b. approx i\<cdot>x = x}"
7.19      by (rule finite_subset,
7.20          intro finite_UnI finite_imageI finite_fixes_approx)
```
```     8.1 --- a/src/HOLCF/Up.thy	Fri Jun 20 22:51:50 2008 +0200
8.2 +++ b/src/HOLCF/Up.thy	Fri Jun 20 23:01:09 2008 +0200
8.3 @@ -320,7 +320,7 @@
8.4
8.5  instance proof
8.6    fix i :: nat and x :: "'a u"
8.7 -  show "chain (\<lambda>i. approx i\<cdot>x)"
8.8 +  show "chain (approx :: nat \<Rightarrow> 'a u \<rightarrow> 'a u)"
8.9      unfolding approx_up_def by simp
8.10    show "(\<Squnion>i. approx i\<cdot>x) = x"
8.11      unfolding approx_up_def
8.12 @@ -331,7 +331,7 @@
8.13    have "{x::'a u. approx i\<cdot>x = x} \<subseteq>
8.14          insert \<bottom> ((\<lambda>x. up\<cdot>x) ` {x::'a. approx i\<cdot>x = x})"
8.15      unfolding approx_up_def
8.16 -    by (rule subsetI, rule_tac p=x in upE, simp_all)
8.17 +    by (rule subsetI, case_tac x, simp_all)
8.18    thus "finite {x::'a u. approx i\<cdot>x = x}"
8.19      by (rule finite_subset, simp add: finite_fixes_approx)
8.20  qed
```
```     9.1 --- a/src/HOLCF/UpperPD.thy	Fri Jun 20 22:51:50 2008 +0200
9.2 +++ b/src/HOLCF/UpperPD.thy	Fri Jun 20 23:01:09 2008 +0200
9.3 @@ -147,7 +147,7 @@
9.4
9.5  instance
9.6  apply (intro_classes, unfold approx_upper_pd_def)
9.8 +apply (rule upper_pd.chain_completion_approx)
9.9  apply (rule upper_pd.lub_completion_approx)
9.10  apply (rule upper_pd.completion_approx_idem)
9.11  apply (rule upper_pd.finite_fixes_completion_approx)
```