src/HOL/HOLCF/One.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 62175 8ffc4d0e652d child 67312 0d25e02759b7 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/HOLCF/One.thy
```
```     2     Author:     Oscar Slotosch
```
```     3 *)
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```     4
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```     5 section \<open>The unit domain\<close>
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```     6
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```     7 theory One
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```     8 imports Lift
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```     9 begin
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```    10
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```    11 type_synonym
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```    12   one = "unit lift"
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```    13
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```    14 translations
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```    15   (type) "one" <= (type) "unit lift"
```
```    16
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```    17 definition ONE :: "one"
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```    18   where "ONE == Def ()"
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```    19
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```    20 text \<open>Exhaustion and Elimination for type @{typ one}\<close>
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```    21
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```    22 lemma Exh_one: "t = \<bottom> \<or> t = ONE"
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```    23 unfolding ONE_def by (induct t) simp_all
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```    24
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```    25 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
```
```    27
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```    28 lemma one_induct [case_names bottom ONE]: "\<lbrakk>P \<bottom>; P ONE\<rbrakk> \<Longrightarrow> P x"
```
```    29 by (cases x rule: oneE) simp_all
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```    30
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```    31 lemma dist_below_one [simp]: "ONE \<notsqsubseteq> \<bottom>"
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```    32 unfolding ONE_def by simp
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```    33
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```    34 lemma below_ONE [simp]: "x \<sqsubseteq> ONE"
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```    35 by (induct x rule: one_induct) simp_all
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```    36
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```    37 lemma ONE_below_iff [simp]: "ONE \<sqsubseteq> x \<longleftrightarrow> x = ONE"
```
```    38 by (induct x rule: one_induct) simp_all
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```    39
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```    40 lemma ONE_defined [simp]: "ONE \<noteq> \<bottom>"
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```    41 unfolding ONE_def by simp
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```    42
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```    43 lemma one_neq_iffs [simp]:
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```    44   "x \<noteq> ONE \<longleftrightarrow> x = \<bottom>"
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```    45   "ONE \<noteq> x \<longleftrightarrow> x = \<bottom>"
```
```    46   "x \<noteq> \<bottom> \<longleftrightarrow> x = ONE"
```
```    47   "\<bottom> \<noteq> x \<longleftrightarrow> x = ONE"
```
```    48 by (induct x rule: one_induct) simp_all
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```    49
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```    50 lemma compact_ONE: "compact ONE"
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```    51 by (rule compact_chfin)
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```    52
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```    53 text \<open>Case analysis function for type @{typ one}\<close>
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```    54
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```    55 definition
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```    56   one_case :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" where
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```    57   "one_case = (\<Lambda> a x. seq\<cdot>x\<cdot>a)"
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```    58
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```    59 translations
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```    60   "case x of XCONST ONE \<Rightarrow> t" == "CONST one_case\<cdot>t\<cdot>x"
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```    61   "case x of XCONST ONE :: 'a \<Rightarrow> t" => "CONST one_case\<cdot>t\<cdot>x"
```
```    62   "\<Lambda> (XCONST ONE). t" == "CONST one_case\<cdot>t"
```
```    63
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```    64 lemma one_case1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>"
```
```    65 by (simp add: one_case_def)
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```    66
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```    67 lemma one_case2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t"
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```    68 by (simp add: one_case_def)
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```    69
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```    70 lemma one_case3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x"
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```    71 by (induct x rule: one_induct) simp_all
```
```    72
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```    73 end
```