3 *) |
3 *) |
4 |
4 |
5 section \<open>The unit domain\<close> |
5 section \<open>The unit domain\<close> |
6 |
6 |
7 theory One |
7 theory One |
8 imports Lift |
8 imports Lift |
9 begin |
9 begin |
10 |
10 |
11 type_synonym |
11 type_synonym one = "unit lift" |
12 one = "unit lift" |
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13 |
12 |
14 translations |
13 translations |
15 (type) "one" <= (type) "unit lift" |
14 (type) "one" \<leftharpoondown> (type) "unit lift" |
16 |
15 |
17 definition ONE :: "one" |
16 definition ONE :: "one" |
18 where "ONE == Def ()" |
17 where "ONE \<equiv> Def ()" |
19 |
18 |
20 text \<open>Exhaustion and Elimination for type @{typ one}\<close> |
19 text \<open>Exhaustion and Elimination for type @{typ one}\<close> |
21 |
20 |
22 lemma Exh_one: "t = \<bottom> \<or> t = ONE" |
21 lemma Exh_one: "t = \<bottom> \<or> t = ONE" |
23 unfolding ONE_def by (induct t) simp_all |
22 by (induct t) (simp_all add: ONE_def) |
24 |
23 |
25 lemma oneE [case_names bottom ONE]: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = ONE \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" |
24 lemma oneE [case_names bottom ONE]: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = ONE \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" |
26 unfolding ONE_def by (induct p) simp_all |
25 by (induct p) (simp_all add: ONE_def) |
27 |
26 |
28 lemma one_induct [case_names bottom ONE]: "\<lbrakk>P \<bottom>; P ONE\<rbrakk> \<Longrightarrow> P x" |
27 lemma one_induct [case_names bottom ONE]: "P \<bottom> \<Longrightarrow> P ONE \<Longrightarrow> P x" |
29 by (cases x rule: oneE) simp_all |
28 by (cases x rule: oneE) simp_all |
30 |
29 |
31 lemma dist_below_one [simp]: "ONE \<notsqsubseteq> \<bottom>" |
30 lemma dist_below_one [simp]: "ONE \<notsqsubseteq> \<bottom>" |
32 unfolding ONE_def by simp |
31 by (simp add: ONE_def) |
33 |
32 |
34 lemma below_ONE [simp]: "x \<sqsubseteq> ONE" |
33 lemma below_ONE [simp]: "x \<sqsubseteq> ONE" |
35 by (induct x rule: one_induct) simp_all |
34 by (induct x rule: one_induct) simp_all |
36 |
35 |
37 lemma ONE_below_iff [simp]: "ONE \<sqsubseteq> x \<longleftrightarrow> x = ONE" |
36 lemma ONE_below_iff [simp]: "ONE \<sqsubseteq> x \<longleftrightarrow> x = ONE" |
38 by (induct x rule: one_induct) simp_all |
37 by (induct x rule: one_induct) simp_all |
39 |
38 |
40 lemma ONE_defined [simp]: "ONE \<noteq> \<bottom>" |
39 lemma ONE_defined [simp]: "ONE \<noteq> \<bottom>" |
41 unfolding ONE_def by simp |
40 by (simp add: ONE_def) |
42 |
41 |
43 lemma one_neq_iffs [simp]: |
42 lemma one_neq_iffs [simp]: |
44 "x \<noteq> ONE \<longleftrightarrow> x = \<bottom>" |
43 "x \<noteq> ONE \<longleftrightarrow> x = \<bottom>" |
45 "ONE \<noteq> x \<longleftrightarrow> x = \<bottom>" |
44 "ONE \<noteq> x \<longleftrightarrow> x = \<bottom>" |
46 "x \<noteq> \<bottom> \<longleftrightarrow> x = ONE" |
45 "x \<noteq> \<bottom> \<longleftrightarrow> x = ONE" |
47 "\<bottom> \<noteq> x \<longleftrightarrow> x = ONE" |
46 "\<bottom> \<noteq> x \<longleftrightarrow> x = ONE" |
48 by (induct x rule: one_induct) simp_all |
47 by (induct x rule: one_induct) simp_all |
49 |
48 |
50 lemma compact_ONE: "compact ONE" |
49 lemma compact_ONE: "compact ONE" |
51 by (rule compact_chfin) |
50 by (rule compact_chfin) |
52 |
51 |
53 text \<open>Case analysis function for type @{typ one}\<close> |
52 text \<open>Case analysis function for type @{typ one}\<close> |
54 |
53 |
55 definition |
54 definition one_case :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" |
56 one_case :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" where |
55 where "one_case = (\<Lambda> a x. seq\<cdot>x\<cdot>a)" |
57 "one_case = (\<Lambda> a x. seq\<cdot>x\<cdot>a)" |
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58 |
56 |
59 translations |
57 translations |
60 "case x of XCONST ONE \<Rightarrow> t" == "CONST one_case\<cdot>t\<cdot>x" |
58 "case x of XCONST ONE \<Rightarrow> t" \<rightleftharpoons> "CONST one_case\<cdot>t\<cdot>x" |
61 "case x of XCONST ONE :: 'a \<Rightarrow> t" => "CONST one_case\<cdot>t\<cdot>x" |
59 "case x of XCONST ONE :: 'a \<Rightarrow> t" \<rightharpoonup> "CONST one_case\<cdot>t\<cdot>x" |
62 "\<Lambda> (XCONST ONE). t" == "CONST one_case\<cdot>t" |
60 "\<Lambda> (XCONST ONE). t" \<rightleftharpoons> "CONST one_case\<cdot>t" |
63 |
61 |
64 lemma one_case1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>" |
62 lemma one_case1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>" |
65 by (simp add: one_case_def) |
63 by (simp add: one_case_def) |
66 |
64 |
67 lemma one_case2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t" |
65 lemma one_case2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t" |
68 by (simp add: one_case_def) |
66 by (simp add: one_case_def) |
69 |
67 |
70 lemma one_case3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x" |
68 lemma one_case3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x" |
71 by (induct x rule: one_induct) simp_all |
69 by (induct x rule: one_induct) simp_all |
72 |
70 |
73 end |
71 end |