author  huffman 
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permissions  rwrr 
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(* Title: HOL/HOLCF/Cprod.thy 
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Author: Franz Regensburger 
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*) 
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header {* The cpo of cartesian products *} 
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theory Cprod 
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imports Cfun 
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begin 
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default_sort cpo 
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subsection {* Continuous case function for unit type *} 
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definition 
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unit_when :: "'a \<rightarrow> unit \<rightarrow> 'a" where 
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"unit_when = (\<Lambda> a _. a)" 
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translations 
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"\<Lambda>(). t" == "CONST unit_when\<cdot>t" 
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lemma unit_when [simp]: "unit_when\<cdot>a\<cdot>u = a" 

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by (simp add: unit_when_def) 

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subsection {* Continuous version of split function *} 
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definition 
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csplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a * 'b) \<rightarrow> 'c" where 
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"csplit = (\<Lambda> f p. f\<cdot>(fst p)\<cdot>(snd p))" 
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translations 
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"\<Lambda>(CONST Pair x y). t" == "CONST csplit\<cdot>(\<Lambda> x y. t)" 
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abbreviation 
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cfst :: "'a \<times> 'b \<rightarrow> 'a" where 
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"cfst \<equiv> Abs_cfun fst" 
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abbreviation 
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csnd :: "'a \<times> 'b \<rightarrow> 'b" where 
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"csnd \<equiv> Abs_cfun snd" 
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subsection {* Convert all lemmas to the continuous versions *} 
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lemma csplit1 [simp]: "csplit\<cdot>f\<cdot>\<bottom> = f\<cdot>\<bottom>\<cdot>\<bottom>" 
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by (simp add: csplit_def) 
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lemma csplit_Pair [simp]: "csplit\<cdot>f\<cdot>(x, y) = f\<cdot>x\<cdot>y" 
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by (simp add: csplit_def) 
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end 