--- a/src/HOL/HOLCF/Fix.thy Mon Jan 01 21:17:28 2018 +0100
+++ b/src/HOL/HOLCF/Fix.thy Mon Jan 01 23:07:24 2018 +0100
@@ -6,51 +6,50 @@
section \<open>Fixed point operator and admissibility\<close>
theory Fix
-imports Cfun
+ imports Cfun
begin
default_sort pcpo
+
subsection \<open>Iteration\<close>
-primrec iterate :: "nat \<Rightarrow> ('a::cpo \<rightarrow> 'a) \<rightarrow> ('a \<rightarrow> 'a)" where
+primrec iterate :: "nat \<Rightarrow> ('a::cpo \<rightarrow> 'a) \<rightarrow> ('a \<rightarrow> 'a)"
+ where
"iterate 0 = (\<Lambda> F x. x)"
| "iterate (Suc n) = (\<Lambda> F x. F\<cdot>(iterate n\<cdot>F\<cdot>x))"
text \<open>Derive inductive properties of iterate from primitive recursion\<close>
lemma iterate_0 [simp]: "iterate 0\<cdot>F\<cdot>x = x"
-by simp
+ by simp
lemma iterate_Suc [simp]: "iterate (Suc n)\<cdot>F\<cdot>x = F\<cdot>(iterate n\<cdot>F\<cdot>x)"
-by simp
+ by simp
declare iterate.simps [simp del]
lemma iterate_Suc2: "iterate (Suc n)\<cdot>F\<cdot>x = iterate n\<cdot>F\<cdot>(F\<cdot>x)"
-by (induct n) simp_all
+ by (induct n) simp_all
-lemma iterate_iterate:
- "iterate m\<cdot>F\<cdot>(iterate n\<cdot>F\<cdot>x) = iterate (m + n)\<cdot>F\<cdot>x"
-by (induct m) simp_all
+lemma iterate_iterate: "iterate m\<cdot>F\<cdot>(iterate n\<cdot>F\<cdot>x) = iterate (m + n)\<cdot>F\<cdot>x"
+ by (induct m) simp_all
text \<open>The sequence of function iterations is a chain.\<close>
lemma chain_iterate [simp]: "chain (\<lambda>i. iterate i\<cdot>F\<cdot>\<bottom>)"
-by (rule chainI, unfold iterate_Suc2, rule monofun_cfun_arg, rule minimal)
+ by (rule chainI, unfold iterate_Suc2, rule monofun_cfun_arg, rule minimal)
subsection \<open>Least fixed point operator\<close>
-definition
- "fix" :: "('a \<rightarrow> 'a) \<rightarrow> 'a" where
- "fix = (\<Lambda> F. \<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
+definition "fix" :: "('a \<rightarrow> 'a) \<rightarrow> 'a"
+ where "fix = (\<Lambda> F. \<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
text \<open>Binder syntax for @{term fix}\<close>
-abbreviation
- fix_syn :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" (binder "\<mu> " 10) where
- "fix_syn (\<lambda>x. f x) \<equiv> fix\<cdot>(\<Lambda> x. f x)"
+abbreviation fix_syn :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" (binder "\<mu> " 10)
+ where "fix_syn (\<lambda>x. f x) \<equiv> fix\<cdot>(\<Lambda> x. f x)"
notation (ASCII)
fix_syn (binder "FIX " 10)
@@ -60,7 +59,7 @@
text \<open>direct connection between @{term fix} and iteration\<close>
lemma fix_def2: "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
-unfolding fix_def by simp
+ by (simp add: fix_def)
lemma iterate_below_fix: "iterate n\<cdot>f\<cdot>\<bottom> \<sqsubseteq> fix\<cdot>f"
unfolding fix_def2
@@ -72,105 +71,103 @@
\<close>
lemma fix_eq: "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"
-apply (simp add: fix_def2)
-apply (subst lub_range_shift [of _ 1, symmetric])
-apply (rule chain_iterate)
-apply (subst contlub_cfun_arg)
-apply (rule chain_iterate)
-apply simp
-done
+ apply (simp add: fix_def2)
+ apply (subst lub_range_shift [of _ 1, symmetric])
+ apply (rule chain_iterate)
+ apply (subst contlub_cfun_arg)
+ apply (rule chain_iterate)
+ apply simp
+ done
lemma fix_least_below: "F\<cdot>x \<sqsubseteq> x \<Longrightarrow> fix\<cdot>F \<sqsubseteq> x"
-apply (simp add: fix_def2)
-apply (rule lub_below)
-apply (rule chain_iterate)
-apply (induct_tac i)
-apply simp
-apply simp
-apply (erule rev_below_trans)
-apply (erule monofun_cfun_arg)
-done
+ apply (simp add: fix_def2)
+ apply (rule lub_below)
+ apply (rule chain_iterate)
+ apply (induct_tac i)
+ apply simp
+ apply simp
+ apply (erule rev_below_trans)
+ apply (erule monofun_cfun_arg)
+ done
lemma fix_least: "F\<cdot>x = x \<Longrightarrow> fix\<cdot>F \<sqsubseteq> x"
-by (rule fix_least_below, simp)
+ by (rule fix_least_below) simp
lemma fix_eqI:
- assumes fixed: "F\<cdot>x = x" and least: "\<And>z. F\<cdot>z = z \<Longrightarrow> x \<sqsubseteq> z"
+ assumes fixed: "F\<cdot>x = x"
+ and least: "\<And>z. F\<cdot>z = z \<Longrightarrow> x \<sqsubseteq> z"
shows "fix\<cdot>F = x"
-apply (rule below_antisym)
-apply (rule fix_least [OF fixed])
-apply (rule least [OF fix_eq [symmetric]])
-done
+ apply (rule below_antisym)
+ apply (rule fix_least [OF fixed])
+ apply (rule least [OF fix_eq [symmetric]])
+ done
lemma fix_eq2: "f \<equiv> fix\<cdot>F \<Longrightarrow> f = F\<cdot>f"
-by (simp add: fix_eq [symmetric])
+ by (simp add: fix_eq [symmetric])
lemma fix_eq3: "f \<equiv> fix\<cdot>F \<Longrightarrow> f\<cdot>x = F\<cdot>f\<cdot>x"
-by (erule fix_eq2 [THEN cfun_fun_cong])
+ by (erule fix_eq2 [THEN cfun_fun_cong])
lemma fix_eq4: "f = fix\<cdot>F \<Longrightarrow> f = F\<cdot>f"
-apply (erule ssubst)
-apply (rule fix_eq)
-done
+ by (erule ssubst) (rule fix_eq)
lemma fix_eq5: "f = fix\<cdot>F \<Longrightarrow> f\<cdot>x = F\<cdot>f\<cdot>x"
-by (erule fix_eq4 [THEN cfun_fun_cong])
+ by (erule fix_eq4 [THEN cfun_fun_cong])
text \<open>strictness of @{term fix}\<close>
-lemma fix_bottom_iff: "(fix\<cdot>F = \<bottom>) = (F\<cdot>\<bottom> = \<bottom>)"
-apply (rule iffI)
-apply (erule subst)
-apply (rule fix_eq [symmetric])
-apply (erule fix_least [THEN bottomI])
-done
+lemma fix_bottom_iff: "fix\<cdot>F = \<bottom> \<longleftrightarrow> F\<cdot>\<bottom> = \<bottom>"
+ apply (rule iffI)
+ apply (erule subst)
+ apply (rule fix_eq [symmetric])
+ apply (erule fix_least [THEN bottomI])
+ done
lemma fix_strict: "F\<cdot>\<bottom> = \<bottom> \<Longrightarrow> fix\<cdot>F = \<bottom>"
-by (simp add: fix_bottom_iff)
+ by (simp add: fix_bottom_iff)
lemma fix_defined: "F\<cdot>\<bottom> \<noteq> \<bottom> \<Longrightarrow> fix\<cdot>F \<noteq> \<bottom>"
-by (simp add: fix_bottom_iff)
+ by (simp add: fix_bottom_iff)
text \<open>@{term fix} applied to identity and constant functions\<close>
lemma fix_id: "(\<mu> x. x) = \<bottom>"
-by (simp add: fix_strict)
+ by (simp add: fix_strict)
lemma fix_const: "(\<mu> x. c) = c"
-by (subst fix_eq, simp)
+ by (subst fix_eq) simp
+
subsection \<open>Fixed point induction\<close>
-lemma fix_ind: "\<lbrakk>adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (fix\<cdot>F)"
-unfolding fix_def2
-apply (erule admD)
-apply (rule chain_iterate)
-apply (rule nat_induct, simp_all)
-done
+lemma fix_ind: "adm P \<Longrightarrow> P \<bottom> \<Longrightarrow> (\<And>x. P x \<Longrightarrow> P (F\<cdot>x)) \<Longrightarrow> P (fix\<cdot>F)"
+ unfolding fix_def2
+ apply (erule admD)
+ apply (rule chain_iterate)
+ apply (rule nat_induct, simp_all)
+ done
-lemma cont_fix_ind:
- "\<lbrakk>cont F; adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F x)\<rbrakk> \<Longrightarrow> P (fix\<cdot>(Abs_cfun F))"
-by (simp add: fix_ind)
+lemma cont_fix_ind: "cont F \<Longrightarrow> adm P \<Longrightarrow> P \<bottom> \<Longrightarrow> (\<And>x. P x \<Longrightarrow> P (F x)) \<Longrightarrow> P (fix\<cdot>(Abs_cfun F))"
+ by (simp add: fix_ind)
-lemma def_fix_ind:
- "\<lbrakk>f \<equiv> fix\<cdot>F; adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P f"
-by (simp add: fix_ind)
+lemma def_fix_ind: "\<lbrakk>f \<equiv> fix\<cdot>F; adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P f"
+ by (simp add: fix_ind)
lemma fix_ind2:
assumes adm: "adm P"
assumes 0: "P \<bottom>" and 1: "P (F\<cdot>\<bottom>)"
assumes step: "\<And>x. \<lbrakk>P x; P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (F\<cdot>(F\<cdot>x))"
shows "P (fix\<cdot>F)"
-unfolding fix_def2
-apply (rule admD [OF adm chain_iterate])
-apply (rule nat_less_induct)
-apply (case_tac n)
-apply (simp add: 0)
-apply (case_tac nat)
-apply (simp add: 1)
-apply (frule_tac x=nat in spec)
-apply (simp add: step)
-done
+ unfolding fix_def2
+ apply (rule admD [OF adm chain_iterate])
+ apply (rule nat_less_induct)
+ apply (case_tac n)
+ apply (simp add: 0)
+ apply (case_tac nat)
+ apply (simp add: 1)
+ apply (frule_tac x=nat in spec)
+ apply (simp add: step)
+ done
lemma parallel_fix_ind:
assumes adm: "adm (\<lambda>x. P (fst x) (snd x))"
@@ -180,17 +177,17 @@
proof -
from adm have adm': "adm (case_prod P)"
unfolding split_def .
- have "\<And>i. P (iterate i\<cdot>F\<cdot>\<bottom>) (iterate i\<cdot>G\<cdot>\<bottom>)"
- by (induct_tac i, simp add: base, simp add: step)
- hence "\<And>i. case_prod P (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>)"
+ have "P (iterate i\<cdot>F\<cdot>\<bottom>) (iterate i\<cdot>G\<cdot>\<bottom>)" for i
+ by (induct i) (simp add: base, simp add: step)
+ then have "\<And>i. case_prod P (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>)"
by simp
- hence "case_prod P (\<Squnion>i. (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>))"
+ then have "case_prod P (\<Squnion>i. (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>))"
by - (rule admD [OF adm'], simp, assumption)
- hence "case_prod P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>, \<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"
+ then have "case_prod P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>, \<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"
by (simp add: lub_Pair)
- hence "P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>) (\<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"
+ then have "P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>) (\<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"
by simp
- thus "P (fix\<cdot>F) (fix\<cdot>G)"
+ then show "P (fix\<cdot>F) (fix\<cdot>G)"
by (simp add: fix_def2)
qed
@@ -200,7 +197,8 @@
assumes "P \<bottom> \<bottom>"
assumes "\<And>x y. P x y \<Longrightarrow> P (F x) (G y)"
shows "P (fix\<cdot>(Abs_cfun F)) (fix\<cdot>(Abs_cfun G))"
-by (rule parallel_fix_ind, simp_all add: assms)
+ by (rule parallel_fix_ind) (simp_all add: assms)
+
subsection \<open>Fixed-points on product types\<close>
@@ -215,27 +213,35 @@
\<mu> y. snd (F\<cdot>(\<mu> x. fst (F\<cdot>(x, \<mu> y. snd (F\<cdot>(x, y)))), y)))"
(is "fix\<cdot>F = (?x, ?y)")
proof (rule fix_eqI)
- have 1: "fst (F\<cdot>(?x, ?y)) = ?x"
+ have *: "fst (F\<cdot>(?x, ?y)) = ?x"
by (rule trans [symmetric, OF fix_eq], simp)
- have 2: "snd (F\<cdot>(?x, ?y)) = ?y"
+ have "snd (F\<cdot>(?x, ?y)) = ?y"
by (rule trans [symmetric, OF fix_eq], simp)
- from 1 2 show "F\<cdot>(?x, ?y) = (?x, ?y)" by (simp add: prod_eq_iff)
+ with * show "F\<cdot>(?x, ?y) = (?x, ?y)"
+ by (simp add: prod_eq_iff)
next
- fix z assume F_z: "F\<cdot>z = z"
- obtain x y where z: "z = (x,y)" by (rule prod.exhaust)
+ fix z
+ assume F_z: "F\<cdot>z = z"
+ obtain x y where z: "z = (x, y)" by (rule prod.exhaust)
from F_z z have F_x: "fst (F\<cdot>(x, y)) = x" by simp
from F_z z have F_y: "snd (F\<cdot>(x, y)) = y" by simp
let ?y1 = "\<mu> y. snd (F\<cdot>(x, y))"
- have "?y1 \<sqsubseteq> y" by (rule fix_least, simp add: F_y)
- hence "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> fst (F\<cdot>(x, y))"
+ have "?y1 \<sqsubseteq> y"
+ by (rule fix_least) (simp add: F_y)
+ then have "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> fst (F\<cdot>(x, y))"
by (simp add: fst_monofun monofun_cfun)
- hence "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> x" using F_x by simp
- hence 1: "?x \<sqsubseteq> x" by (simp add: fix_least_below)
- hence "snd (F\<cdot>(?x, y)) \<sqsubseteq> snd (F\<cdot>(x, y))"
+ with F_x have "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> x"
+ by simp
+ then have *: "?x \<sqsubseteq> x"
+ by (simp add: fix_least_below)
+ then have "snd (F\<cdot>(?x, y)) \<sqsubseteq> snd (F\<cdot>(x, y))"
by (simp add: snd_monofun monofun_cfun)
- hence "snd (F\<cdot>(?x, y)) \<sqsubseteq> y" using F_y by simp
- hence 2: "?y \<sqsubseteq> y" by (simp add: fix_least_below)
- show "(?x, ?y) \<sqsubseteq> z" using z 1 2 by simp
+ with F_y have "snd (F\<cdot>(?x, y)) \<sqsubseteq> y"
+ by simp
+ then have "?y \<sqsubseteq> y"
+ by (simp add: fix_least_below)
+ with z * show "(?x, ?y) \<sqsubseteq> z"
+ by simp
qed
end