1(*  Title:      HOL/HOLCF/Sprod.thy
2    Author:     Franz Regensburger
3    Author:     Brian Huffman
4*)
5
6section \<open>The type of strict products\<close>
7
8theory Sprod
9  imports Cfun
10begin
11
12default_sort pcpo
13
14
15subsection \<open>Definition of strict product type\<close>
16
17definition "sprod = {p::'a \<times> 'b. p = \<bottom> \<or> (fst p \<noteq> \<bottom> \<and> snd p \<noteq> \<bottom>)}"
18
19pcpodef ('a, 'b) sprod  ("(_ \<otimes>/ _)" [21,20] 20) = "sprod :: ('a \<times> 'b) set"
20  by (simp_all add: sprod_def)
21
22instance sprod :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
23  by (rule typedef_chfin [OF type_definition_sprod below_sprod_def])
24
25type_notation (ASCII)
26  sprod  (infixr "**" 20)
27
28
29subsection \<open>Definitions of constants\<close>
30
31definition sfst :: "('a ** 'b) \<rightarrow> 'a"
32  where "sfst = (\<Lambda> p. fst (Rep_sprod p))"
33
34definition ssnd :: "('a ** 'b) \<rightarrow> 'b"
35  where "ssnd = (\<Lambda> p. snd (Rep_sprod p))"
36
37definition spair :: "'a \<rightarrow> 'b \<rightarrow> ('a ** 'b)"
38  where "spair = (\<Lambda> a b. Abs_sprod (seq\<cdot>b\<cdot>a, seq\<cdot>a\<cdot>b))"
39
40definition ssplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a ** 'b) \<rightarrow> 'c"
41  where "ssplit = (\<Lambda> f p. seq\<cdot>p\<cdot>(f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p)))"
42
43syntax "_stuple" :: "[logic, args] \<Rightarrow> logic"  ("(1'(:_,/ _:'))")
44translations
45  "(:x, y, z:)" \<rightleftharpoons> "(:x, (:y, z:):)"
46  "(:x, y:)" \<rightleftharpoons> "CONST spair\<cdot>x\<cdot>y"
47
48translations
49  "\<Lambda>(CONST spair\<cdot>x\<cdot>y). t" \<rightleftharpoons> "CONST ssplit\<cdot>(\<Lambda> x y. t)"
50
51
52subsection \<open>Case analysis\<close>
53
54lemma spair_sprod: "(seq\<cdot>b\<cdot>a, seq\<cdot>a\<cdot>b) \<in> sprod"
55  by (simp add: sprod_def seq_conv_if)
56
57lemma Rep_sprod_spair: "Rep_sprod (:a, b:) = (seq\<cdot>b\<cdot>a, seq\<cdot>a\<cdot>b)"
58  by (simp add: spair_def cont_Abs_sprod Abs_sprod_inverse spair_sprod)
59
60lemmas Rep_sprod_simps =
61  Rep_sprod_inject [symmetric] below_sprod_def
62  prod_eq_iff below_prod_def
63  Rep_sprod_strict Rep_sprod_spair
64
65lemma sprodE [case_names bottom spair, cases type: sprod]:
66  obtains "p = \<bottom>" | x y where "p = (:x, y:)" and "x \<noteq> \<bottom>" and "y \<noteq> \<bottom>"
67  using Rep_sprod [of p] by (auto simp add: sprod_def Rep_sprod_simps)
68
69lemma sprod_induct [case_names bottom spair, induct type: sprod]:
70  "\<lbrakk>P \<bottom>; \<And>x y. \<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> P (:x, y:)\<rbrakk> \<Longrightarrow> P x"
71  by (cases x) simp_all
72
73
74subsection \<open>Properties of \emph{spair}\<close>
75
76lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>"
77  by (simp add: Rep_sprod_simps)
78
79lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>"
80  by (simp add: Rep_sprod_simps)
81
82lemma spair_bottom_iff [simp]: "(:x, y:) = \<bottom> \<longleftrightarrow> x = \<bottom> \<or> y = \<bottom>"
83  by (simp add: Rep_sprod_simps seq_conv_if)
84
85lemma spair_below_iff: "(:a, b:) \<sqsubseteq> (:c, d:) \<longleftrightarrow> a = \<bottom> \<or> b = \<bottom> \<or> (a \<sqsubseteq> c \<and> b \<sqsubseteq> d)"
86  by (simp add: Rep_sprod_simps seq_conv_if)
87
88lemma spair_eq_iff: "(:a, b:) = (:c, d:) \<longleftrightarrow> a = c \<and> b = d \<or> (a = \<bottom> \<or> b = \<bottom>) \<and> (c = \<bottom> \<or> d = \<bottom>)"
89  by (simp add: Rep_sprod_simps seq_conv_if)
90
91lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>"
92  by simp
93
94lemma spair_strict_rev: "(:x, y:) \<noteq> \<bottom> \<Longrightarrow> x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>"
95  by simp
96
97lemma spair_defined: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<noteq> \<bottom>"
98  by simp
99
100lemma spair_defined_rev: "(:x, y:) = \<bottom> \<Longrightarrow> x = \<bottom> \<or> y = \<bottom>"
101  by simp
102
103lemma spair_below: "x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> (:x, y:) \<sqsubseteq> (:a, b:) \<longleftrightarrow> x \<sqsubseteq> a \<and> y \<sqsubseteq> b"
104  by (simp add: spair_below_iff)
105
106lemma spair_eq: "x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> (:x, y:) = (:a, b:) \<longleftrightarrow> x = a \<and> y = b"
107  by (simp add: spair_eq_iff)
108
109lemma spair_inject: "x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> (:x, y:) = (:a, b:) \<Longrightarrow> x = a \<and> y = b"
110  by (rule spair_eq [THEN iffD1])
111
112lemma inst_sprod_pcpo2: "\<bottom> = (:\<bottom>, \<bottom>:)"
113  by simp
114
115lemma sprodE2: "(\<And>x y. p = (:x, y:) \<Longrightarrow> Q) \<Longrightarrow> Q"
116  by (cases p) (simp only: inst_sprod_pcpo2, simp)
117
118
119subsection \<open>Properties of \emph{sfst} and \emph{ssnd}\<close>
120
121lemma sfst_strict [simp]: "sfst\<cdot>\<bottom> = \<bottom>"
122  by (simp add: sfst_def cont_Rep_sprod Rep_sprod_strict)
123
124lemma ssnd_strict [simp]: "ssnd\<cdot>\<bottom> = \<bottom>"
125  by (simp add: ssnd_def cont_Rep_sprod Rep_sprod_strict)
126
127lemma sfst_spair [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>(:x, y:) = x"
128  by (simp add: sfst_def cont_Rep_sprod Rep_sprod_spair)
129
130lemma ssnd_spair [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>(:x, y:) = y"
131  by (simp add: ssnd_def cont_Rep_sprod Rep_sprod_spair)
132
133lemma sfst_bottom_iff [simp]: "sfst\<cdot>p = \<bottom> \<longleftrightarrow> p = \<bottom>"
134  by (cases p) simp_all
135
136lemma ssnd_bottom_iff [simp]: "ssnd\<cdot>p = \<bottom> \<longleftrightarrow> p = \<bottom>"
137  by (cases p) simp_all
138
139lemma sfst_defined: "p \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>p \<noteq> \<bottom>"
140  by simp
141
142lemma ssnd_defined: "p \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>p \<noteq> \<bottom>"
143  by simp
144
145lemma spair_sfst_ssnd: "(:sfst\<cdot>p, ssnd\<cdot>p:) = p"
146  by (cases p) simp_all
147
148lemma below_sprod: "x \<sqsubseteq> y \<longleftrightarrow> sfst\<cdot>x \<sqsubseteq> sfst\<cdot>y \<and> ssnd\<cdot>x \<sqsubseteq> ssnd\<cdot>y"
149  by (simp add: Rep_sprod_simps sfst_def ssnd_def cont_Rep_sprod)
150
151lemma eq_sprod: "x = y \<longleftrightarrow> sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y"
152  by (auto simp add: po_eq_conv below_sprod)
153
154lemma sfst_below_iff: "sfst\<cdot>x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> (:y, ssnd\<cdot>x:)"
155  by (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp, simp add: below_sprod)
156
157lemma ssnd_below_iff: "ssnd\<cdot>x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> (:sfst\<cdot>x, y:)"
158  by (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp, simp add: below_sprod)
159
160
161subsection \<open>Compactness\<close>
162
163lemma compact_sfst: "compact x \<Longrightarrow> compact (sfst\<cdot>x)"
164  by (rule compactI) (simp add: sfst_below_iff)
165
166lemma compact_ssnd: "compact x \<Longrightarrow> compact (ssnd\<cdot>x)"
167  by (rule compactI) (simp add: ssnd_below_iff)
168
169lemma compact_spair: "compact x \<Longrightarrow> compact y \<Longrightarrow> compact (:x, y:)"
170  by (rule compact_sprod) (simp add: Rep_sprod_spair seq_conv_if)
171
172lemma compact_spair_iff: "compact (:x, y:) \<longleftrightarrow> x = \<bottom> \<or> y = \<bottom> \<or> (compact x \<and> compact y)"
173  apply (safe elim!: compact_spair)
174     apply (drule compact_sfst, simp)
175    apply (drule compact_ssnd, simp)
176   apply simp
177  apply simp
178  done
179
180
181subsection \<open>Properties of \emph{ssplit}\<close>
182
183lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>"
184  by (simp add: ssplit_def)
185
186lemma ssplit2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> ssplit\<cdot>f\<cdot>(:x, y:) = f\<cdot>x\<cdot>y"
187  by (simp add: ssplit_def)
188
189lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z"
190  by (cases z) simp_all
191
192
193subsection \<open>Strict product preserves flatness\<close>
194
195instance sprod :: (flat, flat) flat
196proof
197  fix x y :: "'a \<otimes> 'b"
198  assume "x \<sqsubseteq> y"
199  then show "x = \<bottom> \<or> x = y"
200    apply (induct x, simp)
201    apply (induct y, simp)
202    apply (simp add: spair_below_iff flat_below_iff)
203    done
204qed
205
206end
207