src/HOLCF/One.thy
 author huffman Mon May 11 08:28:09 2009 -0700 (2009-05-11) changeset 31095 b79d140f6d0b parent 31076 99fe356cbbc2 child 35431 8758fe1fc9f8 permissions -rw-r--r--
simplify fixrec proofs for mutually-recursive definitions; generate better fixpoint induction rules
```     1 (*  Title:      HOLCF/One.thy
```
```     2     Author:     Oscar Slotosch
```
```     3 *)
```
```     4
```
```     5 header {* The unit domain *}
```
```     6
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```     7 theory One
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```     8 imports Lift
```
```     9 begin
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```    10
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```    11 types one = "unit lift"
```
```    12 translations
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```    13   "one" <= (type) "unit lift"
```
```    14
```
```    15 definition
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```    16   ONE :: "one"
```
```    17 where
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```    18   "ONE == Def ()"
```
```    19
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```    20 text {* Exhaustion and Elimination for type @{typ one} *}
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```    21
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```    22 lemma Exh_one: "t = \<bottom> \<or> t = ONE"
```
```    23 unfolding ONE_def by (induct t) simp_all
```
```    24
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```    25 lemma oneE: "\<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: "\<lbrakk>P \<bottom>; P ONE\<rbrakk> \<Longrightarrow> P x"
```
```    29 by (cases x rule: oneE) simp_all
```
```    30
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```    31 lemma dist_below_one [simp]: "\<not> ONE \<sqsubseteq> \<bottom>"
```
```    32 unfolding ONE_def by simp
```
```    33
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```    34 lemma below_ONE [simp]: "x \<sqsubseteq> ONE"
```
```    35 by (induct x rule: one_induct) simp_all
```
```    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
```
```    39
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```    40 lemma ONE_defined [simp]: "ONE \<noteq> \<bottom>"
```
```    41 unfolding ONE_def by simp
```
```    42
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```    43 lemma one_neq_iffs [simp]:
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```    44   "x \<noteq> ONE \<longleftrightarrow> x = \<bottom>"
```
```    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
```
```    49
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```    50 lemma compact_ONE: "compact ONE"
```
```    51 by (rule compact_chfin)
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```    52
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```    53 text {* Case analysis function for type @{typ one} *}
```
```    54
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```    55 definition
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```    56   one_when :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" where
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```    57   "one_when = (\<Lambda> a. strictify\<cdot>(\<Lambda> _. a))"
```
```    58
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```    59 translations
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```    60   "case x of XCONST ONE \<Rightarrow> t" == "CONST one_when\<cdot>t\<cdot>x"
```
```    61   "\<Lambda> (XCONST ONE). t" == "CONST one_when\<cdot>t"
```
```    62
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```    63 lemma one_when1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>"
```
```    64 by (simp add: one_when_def)
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```    65
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```    66 lemma one_when2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t"
```
```    67 by (simp add: one_when_def)
```
```    68
```
```    69 lemma one_when3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x"
```
```    70 by (induct x rule: one_induct) simp_all
```
```    71
```
```    72 end
```