src/HOLCF/ex/Domain_Proofs.thy
author wenzelm
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(*  Title:      HOLCF/ex/Domain_Proofs.thy
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    Author:     Brian Huffman
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*)
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header {* Internal domain package proofs done manually *}
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theory Domain_Proofs
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imports HOLCF
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begin
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default_sort rep
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(*
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The definitions and proofs below are for the following recursive
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datatypes:
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domain 'a foo = Foo1 | Foo2 (lazy 'a) (lazy "'a bar")
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   and 'a bar = Bar (lazy "'a baz \<rightarrow> tr")
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   and 'a baz = Baz (lazy "'a foo convex_pd \<rightarrow> tr")
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*)
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(********************************************************************)
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subsection {* Step 1: Define the new type combinators *}
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text {* Start with the one-step non-recursive version *}
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definition
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  foo_bar_baz_deflF ::
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    "TypeRep \<rightarrow> TypeRep \<times> TypeRep \<times> TypeRep \<rightarrow> TypeRep \<times> TypeRep \<times> TypeRep"
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where
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  "foo_bar_baz_deflF = (\<Lambda> a. Abs_CFun (\<lambda>(t1, t2, t3). 
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    ( ssum_defl\<cdot>REP(one)\<cdot>(sprod_defl\<cdot>(u_defl\<cdot>a)\<cdot>(u_defl\<cdot>t2))
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    , u_defl\<cdot>(cfun_defl\<cdot>t3\<cdot>REP(tr))
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    , u_defl\<cdot>(cfun_defl\<cdot>(convex_defl\<cdot>t1)\<cdot>REP(tr)))))"
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lemma foo_bar_baz_deflF_beta:
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  "foo_bar_baz_deflF\<cdot>a\<cdot>t =
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    ( ssum_defl\<cdot>REP(one)\<cdot>(sprod_defl\<cdot>(u_defl\<cdot>a)\<cdot>(u_defl\<cdot>(fst (snd t))))
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    , u_defl\<cdot>(cfun_defl\<cdot>(snd (snd t))\<cdot>REP(tr))
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    , u_defl\<cdot>(cfun_defl\<cdot>(convex_defl\<cdot>(fst t))\<cdot>REP(tr)))"
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unfolding foo_bar_baz_deflF_def
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by (simp add: split_def)
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text {* Individual type combinators are projected from the fixed point. *}
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definition foo_defl :: "TypeRep \<rightarrow> TypeRep"
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where "foo_defl = (\<Lambda> a. fst (fix\<cdot>(foo_bar_baz_deflF\<cdot>a)))"
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definition bar_defl :: "TypeRep \<rightarrow> TypeRep"
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where "bar_defl = (\<Lambda> a. fst (snd (fix\<cdot>(foo_bar_baz_deflF\<cdot>a))))"
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definition baz_defl :: "TypeRep \<rightarrow> TypeRep"
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where "baz_defl = (\<Lambda> a. snd (snd (fix\<cdot>(foo_bar_baz_deflF\<cdot>a))))"
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lemma defl_apply_thms:
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  "foo_defl\<cdot>a = fst (fix\<cdot>(foo_bar_baz_deflF\<cdot>a))"
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  "bar_defl\<cdot>a = fst (snd (fix\<cdot>(foo_bar_baz_deflF\<cdot>a)))"
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  "baz_defl\<cdot>a = snd (snd (fix\<cdot>(foo_bar_baz_deflF\<cdot>a)))"
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unfolding foo_defl_def bar_defl_def baz_defl_def by simp_all
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text {* Unfold rules for each combinator. *}
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lemma foo_defl_unfold:
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  "foo_defl\<cdot>a = ssum_defl\<cdot>REP(one)\<cdot>(sprod_defl\<cdot>(u_defl\<cdot>a)\<cdot>(u_defl\<cdot>(bar_defl\<cdot>a)))"
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unfolding defl_apply_thms by (subst fix_eq, simp add: foo_bar_baz_deflF_beta)
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lemma bar_defl_unfold: "bar_defl\<cdot>a = u_defl\<cdot>(cfun_defl\<cdot>(baz_defl\<cdot>a)\<cdot>REP(tr))"
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unfolding defl_apply_thms by (subst fix_eq, simp add: foo_bar_baz_deflF_beta)
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lemma baz_defl_unfold: "baz_defl\<cdot>a = u_defl\<cdot>(cfun_defl\<cdot>(convex_defl\<cdot>(foo_defl\<cdot>a))\<cdot>REP(tr))"
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unfolding defl_apply_thms by (subst fix_eq, simp add: foo_bar_baz_deflF_beta)
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text "The automation for the previous steps will be quite similar to
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how the fixrec package works."
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(********************************************************************)
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subsection {* Step 2: Define types, prove class instances *}
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text {* Use @{text pcpodef} with the appropriate type combinator. *}
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pcpodef (open) 'a foo = "{x. x ::: foo_defl\<cdot>REP('a)}"
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by (simp_all add: adm_in_deflation)
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pcpodef (open) 'a bar = "{x. x ::: bar_defl\<cdot>REP('a)}"
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by (simp_all add: adm_in_deflation)
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pcpodef (open) 'a baz = "{x. x ::: baz_defl\<cdot>REP('a)}"
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by (simp_all add: adm_in_deflation)
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text {* Prove rep instance using lemma @{text typedef_rep_class}. *}
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instantiation foo :: (rep) rep
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begin
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definition emb_foo :: "'a foo \<rightarrow> udom"
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where "emb_foo \<equiv> (\<Lambda> x. Rep_foo x)"
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definition prj_foo :: "udom \<rightarrow> 'a foo"
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where "prj_foo \<equiv> (\<Lambda> y. Abs_foo (cast\<cdot>(foo_defl\<cdot>REP('a))\<cdot>y))"
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definition approx_foo :: "nat \<Rightarrow> 'a foo \<rightarrow> 'a foo"
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where "approx_foo \<equiv> repdef_approx Rep_foo Abs_foo (foo_defl\<cdot>REP('a))"
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instance
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apply (rule typedef_rep_class)
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apply (rule type_definition_foo)
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apply (rule below_foo_def)
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apply (rule emb_foo_def)
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apply (rule prj_foo_def)
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apply (rule approx_foo_def)
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done
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end
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instantiation bar :: (rep) rep
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begin
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definition emb_bar :: "'a bar \<rightarrow> udom"
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where "emb_bar \<equiv> (\<Lambda> x. Rep_bar x)"
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definition prj_bar :: "udom \<rightarrow> 'a bar"
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where "prj_bar \<equiv> (\<Lambda> y. Abs_bar (cast\<cdot>(bar_defl\<cdot>REP('a))\<cdot>y))"
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definition approx_bar :: "nat \<Rightarrow> 'a bar \<rightarrow> 'a bar"
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where "approx_bar \<equiv> repdef_approx Rep_bar Abs_bar (bar_defl\<cdot>REP('a))"
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instance
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apply (rule typedef_rep_class)
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apply (rule type_definition_bar)
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apply (rule below_bar_def)
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apply (rule emb_bar_def)
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apply (rule prj_bar_def)
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apply (rule approx_bar_def)
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done
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end
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instantiation baz :: (rep) rep
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begin
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definition emb_baz :: "'a baz \<rightarrow> udom"
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where "emb_baz \<equiv> (\<Lambda> x. Rep_baz x)"
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definition prj_baz :: "udom \<rightarrow> 'a baz"
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where "prj_baz \<equiv> (\<Lambda> y. Abs_baz (cast\<cdot>(baz_defl\<cdot>REP('a))\<cdot>y))"
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definition approx_baz :: "nat \<Rightarrow> 'a baz \<rightarrow> 'a baz"
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where "approx_baz \<equiv> repdef_approx Rep_baz Abs_baz (baz_defl\<cdot>REP('a))"
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instance
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apply (rule typedef_rep_class)
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apply (rule type_definition_baz)
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apply (rule below_baz_def)
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apply (rule emb_baz_def)
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apply (rule prj_baz_def)
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apply (rule approx_baz_def)
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done
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end
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text {* Prove REP rules using lemma @{text typedef_REP}. *}
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lemma REP_foo: "REP('a foo) = foo_defl\<cdot>REP('a)"
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apply (rule typedef_REP)
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apply (rule type_definition_foo)
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apply (rule below_foo_def)
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apply (rule emb_foo_def)
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apply (rule prj_foo_def)
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done
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lemma REP_bar: "REP('a bar) = bar_defl\<cdot>REP('a)"
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apply (rule typedef_REP)
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apply (rule type_definition_bar)
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apply (rule below_bar_def)
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apply (rule emb_bar_def)
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apply (rule prj_bar_def)
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done
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lemma REP_baz: "REP('a baz) = baz_defl\<cdot>REP('a)"
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apply (rule typedef_REP)
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apply (rule type_definition_baz)
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apply (rule below_baz_def)
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apply (rule emb_baz_def)
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apply (rule prj_baz_def)
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done
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text {* Prove REP equations using type combinator unfold lemmas. *}
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lemma REP_foo': "REP('a foo) = REP(one \<oplus> 'a\<^sub>\<bottom> \<otimes> ('a bar)\<^sub>\<bottom>)"
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unfolding REP_foo REP_bar REP_baz REP_simps
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by (rule foo_defl_unfold)
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lemma REP_bar': "REP('a bar) = REP(('a baz \<rightarrow> tr)\<^sub>\<bottom>)"
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unfolding REP_foo REP_bar REP_baz REP_simps
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by (rule bar_defl_unfold)
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lemma REP_baz': "REP('a baz) = REP(('a foo convex_pd \<rightarrow> tr)\<^sub>\<bottom>)"
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unfolding REP_foo REP_bar REP_baz REP_simps REP_convex
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by (rule baz_defl_unfold)
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(********************************************************************)
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subsection {* Step 3: Define rep and abs functions *}
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text {* Define them all using @{text coerce}! *}
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definition foo_rep :: "'a foo \<rightarrow> one \<oplus> ('a\<^sub>\<bottom> \<otimes> ('a bar)\<^sub>\<bottom>)"
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where "foo_rep \<equiv> coerce"
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definition foo_abs :: "one \<oplus> ('a\<^sub>\<bottom> \<otimes> ('a bar)\<^sub>\<bottom>) \<rightarrow> 'a foo"
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where "foo_abs \<equiv> coerce"
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definition bar_rep :: "'a bar \<rightarrow> ('a baz \<rightarrow> tr)\<^sub>\<bottom>"
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where "bar_rep \<equiv> coerce"
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definition bar_abs :: "('a baz \<rightarrow> tr)\<^sub>\<bottom> \<rightarrow> 'a bar"
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where "bar_abs \<equiv> coerce"
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definition baz_rep :: "'a baz \<rightarrow> ('a foo convex_pd \<rightarrow> tr)\<^sub>\<bottom>"
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where "baz_rep \<equiv> coerce"
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definition baz_abs :: "('a foo convex_pd \<rightarrow> tr)\<^sub>\<bottom> \<rightarrow> 'a baz"
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where "baz_abs \<equiv> coerce"
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text {* Prove isomorphism rules. *}
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lemma foo_abs_iso: "foo_rep\<cdot>(foo_abs\<cdot>x) = x"
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by (rule domain_abs_iso [OF REP_foo' foo_abs_def foo_rep_def])
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lemma foo_rep_iso: "foo_abs\<cdot>(foo_rep\<cdot>x) = x"
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by (rule domain_rep_iso [OF REP_foo' foo_abs_def foo_rep_def])
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lemma bar_abs_iso: "bar_rep\<cdot>(bar_abs\<cdot>x) = x"
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by (rule domain_abs_iso [OF REP_bar' bar_abs_def bar_rep_def])
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lemma bar_rep_iso: "bar_abs\<cdot>(bar_rep\<cdot>x) = x"
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by (rule domain_rep_iso [OF REP_bar' bar_abs_def bar_rep_def])
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lemma baz_abs_iso: "baz_rep\<cdot>(baz_abs\<cdot>x) = x"
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by (rule domain_abs_iso [OF REP_baz' baz_abs_def baz_rep_def])
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lemma baz_rep_iso: "baz_abs\<cdot>(baz_rep\<cdot>x) = x"
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by (rule domain_rep_iso [OF REP_baz' baz_abs_def baz_rep_def])
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text {* Prove isodefl rules using @{text isodefl_coerce}. *}
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lemma isodefl_foo_abs:
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  "isodefl d t \<Longrightarrow> isodefl (foo_abs oo d oo foo_rep) t"
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by (rule isodefl_abs_rep [OF REP_foo' foo_abs_def foo_rep_def])
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lemma isodefl_bar_abs:
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  "isodefl d t \<Longrightarrow> isodefl (bar_abs oo d oo bar_rep) t"
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by (rule isodefl_abs_rep [OF REP_bar' bar_abs_def bar_rep_def])
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lemma isodefl_baz_abs:
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  "isodefl d t \<Longrightarrow> isodefl (baz_abs oo d oo baz_rep) t"
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by (rule isodefl_abs_rep [OF REP_baz' baz_abs_def baz_rep_def])
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(********************************************************************)
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subsection {* Step 4: Define map functions, prove isodefl property *}
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text {* Start with the one-step non-recursive version. *}
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text {* Note that the type of the map function depends on which
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variables are used in positive and negative positions. *}
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definition
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  foo_bar_baz_mapF ::
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    "('a \<rightarrow> 'b) \<rightarrow>
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     ('a foo \<rightarrow> 'b foo) \<times> ('a bar \<rightarrow> 'b bar) \<times> ('b baz \<rightarrow> 'a baz) \<rightarrow>
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     ('a foo \<rightarrow> 'b foo) \<times> ('a bar \<rightarrow> 'b bar) \<times> ('b baz \<rightarrow> 'a baz)"
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where
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  "foo_bar_baz_mapF = (\<Lambda> f. Abs_CFun (\<lambda>(d1, d2, d3).
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    (
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      foo_abs oo
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        ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>f)\<cdot>(u_map\<cdot>d2))
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          oo foo_rep
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    ,
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      bar_abs oo u_map\<cdot>(cfun_map\<cdot>d3\<cdot>ID) oo bar_rep
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    ,
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      baz_abs oo u_map\<cdot>(cfun_map\<cdot>(convex_map\<cdot>d1)\<cdot>ID) oo baz_rep
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    )))"
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lemma foo_bar_baz_mapF_beta:
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  "foo_bar_baz_mapF\<cdot>f\<cdot>d =
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    (
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      foo_abs oo
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        ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>f)\<cdot>(u_map\<cdot>(fst (snd d))))
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          oo foo_rep
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    ,
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      bar_abs oo u_map\<cdot>(cfun_map\<cdot>(snd (snd d))\<cdot>ID) oo bar_rep
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    ,
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      baz_abs oo u_map\<cdot>(cfun_map\<cdot>(convex_map\<cdot>(fst d))\<cdot>ID) oo baz_rep
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    )"
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unfolding foo_bar_baz_mapF_def
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by (simp add: split_def)
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text {* Individual map functions are projected from the fixed point. *}
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definition foo_map :: "('a \<rightarrow> 'b) \<rightarrow> ('a foo \<rightarrow> 'b foo)"
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where "foo_map = (\<Lambda> f. fst (fix\<cdot>(foo_bar_baz_mapF\<cdot>f)))"
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definition bar_map :: "('a \<rightarrow> 'b) \<rightarrow> ('a bar \<rightarrow> 'b bar)"
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where "bar_map = (\<Lambda> f. fst (snd (fix\<cdot>(foo_bar_baz_mapF\<cdot>f))))"
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definition baz_map :: "('a \<rightarrow> 'b) \<rightarrow> ('b baz \<rightarrow> 'a baz)"
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where "baz_map = (\<Lambda> f. snd (snd (fix\<cdot>(foo_bar_baz_mapF\<cdot>f))))"
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lemma map_apply_thms:
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  "foo_map\<cdot>f = fst (fix\<cdot>(foo_bar_baz_mapF\<cdot>f))"
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  "bar_map\<cdot>f = fst (snd (fix\<cdot>(foo_bar_baz_mapF\<cdot>f)))"
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  "baz_map\<cdot>f = snd (snd (fix\<cdot>(foo_bar_baz_mapF\<cdot>f)))"
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unfolding foo_map_def bar_map_def baz_map_def by simp_all
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text {* Prove isodefl rules for all map functions simultaneously. *}
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lemma isodefl_foo_bar_baz:
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  assumes isodefl_d: "isodefl d t"
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  shows
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  "isodefl (foo_map\<cdot>d) (foo_defl\<cdot>t) \<and>
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  isodefl (bar_map\<cdot>d) (bar_defl\<cdot>t) \<and>
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  isodefl (baz_map\<cdot>d) (baz_defl\<cdot>t)"
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unfolding map_apply_thms defl_apply_thms
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 apply (rule parallel_fix_ind)
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   apply (intro adm_conj adm_isodefl cont2cont_fst cont2cont_snd cont_id)
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  apply (simp only: fst_strict snd_strict isodefl_bottom simp_thms)
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 apply (simp only: foo_bar_baz_mapF_beta
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                   foo_bar_baz_deflF_beta
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                   fst_conv snd_conv)
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 apply (elim conjE)
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 apply (intro
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  conjI
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  isodefl_foo_abs
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  isodefl_bar_abs
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  isodefl_baz_abs
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  isodefl_ssum isodefl_sprod isodefl_ID_REP
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  isodefl_u isodefl_convex isodefl_cfun
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  isodefl_d
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 )
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 apply assumption+
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done
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lemmas isodefl_foo = isodefl_foo_bar_baz [THEN conjunct1]
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lemmas isodefl_bar = isodefl_foo_bar_baz [THEN conjunct2, THEN conjunct1]
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lemmas isodefl_baz = isodefl_foo_bar_baz [THEN conjunct2, THEN conjunct2]
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text {* Prove map ID lemmas, using isodefl_REP_imp_ID *}
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   353
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lemma foo_map_ID: "foo_map\<cdot>ID = ID"
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apply (rule isodefl_REP_imp_ID)
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apply (subst REP_foo)
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apply (rule isodefl_foo)
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apply (rule isodefl_ID_REP)
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done
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   360
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lemma bar_map_ID: "bar_map\<cdot>ID = ID"
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apply (rule isodefl_REP_imp_ID)
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apply (subst REP_bar)
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apply (rule isodefl_bar)
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   365
apply (rule isodefl_ID_REP)
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done
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lemma baz_map_ID: "baz_map\<cdot>ID = ID"
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apply (rule isodefl_REP_imp_ID)
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apply (subst REP_baz)
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apply (rule isodefl_baz)
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apply (rule isodefl_ID_REP)
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done
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(********************************************************************)
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subsection {* Step 5: Define take functions, prove lub-take lemmas *}
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definition
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  foo_bar_baz_takeF ::
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    "('a foo \<rightarrow> 'a foo) \<times> ('a bar \<rightarrow> 'a bar) \<times> ('a baz \<rightarrow> 'a baz) \<rightarrow>
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     ('a foo \<rightarrow> 'a foo) \<times> ('a bar \<rightarrow> 'a bar) \<times> ('a baz \<rightarrow> 'a baz)"
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where
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  "foo_bar_baz_takeF = (\<Lambda> p.
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    ( foo_abs oo
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        ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>ID)\<cdot>(u_map\<cdot>(fst (snd p))))
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          oo foo_rep
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    , bar_abs oo
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        u_map\<cdot>(cfun_map\<cdot>(snd (snd p))\<cdot>ID) oo bar_rep
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    , baz_abs oo
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        u_map\<cdot>(cfun_map\<cdot>(convex_map\<cdot>(fst p))\<cdot>ID) oo baz_rep
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   392
    ))"
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diff changeset
   393
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   394
lemma foo_bar_baz_takeF_beta:
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   395
  "foo_bar_baz_takeF\<cdot>p =
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   396
    ( foo_abs oo
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   397
        ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>ID)\<cdot>(u_map\<cdot>(fst (snd p))))
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   398
          oo foo_rep
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   399
    , bar_abs oo
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   400
        u_map\<cdot>(cfun_map\<cdot>(snd (snd p))\<cdot>ID) oo bar_rep
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diff changeset
   401
    , baz_abs oo
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huffman
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diff changeset
   402
        u_map\<cdot>(cfun_map\<cdot>(convex_map\<cdot>(fst p))\<cdot>ID) oo baz_rep
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huffman
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diff changeset
   403
    )"
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huffman
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diff changeset
   404
unfolding foo_bar_baz_takeF_def by (rule beta_cfun, simp)
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huffman
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diff changeset
   405
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diff changeset
   406
definition
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   407
  foo_take :: "nat \<Rightarrow> 'a foo \<rightarrow> 'a foo"
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diff changeset
   408
where
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   409
  "foo_take = (\<lambda>n. fst (iterate n\<cdot>foo_bar_baz_takeF\<cdot>\<bottom>))"
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huffman
parents: 35493
diff changeset
   410
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diff changeset
   411
definition
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diff changeset
   412
  bar_take :: "nat \<Rightarrow> 'a bar \<rightarrow> 'a bar"
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diff changeset
   413
where
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
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diff changeset
   414
  "bar_take = (\<lambda>n. fst (snd (iterate n\<cdot>foo_bar_baz_takeF\<cdot>\<bottom>)))"
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diff changeset
   415
36132
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diff changeset
   416
definition
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diff changeset
   417
  baz_take :: "nat \<Rightarrow> 'a baz \<rightarrow> 'a baz"
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diff changeset
   418
where
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   419
  "baz_take = (\<lambda>n. snd (snd (iterate n\<cdot>foo_bar_baz_takeF\<cdot>\<bottom>)))"
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diff changeset
   420
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diff changeset
   421
lemma chain_take_thms: "chain foo_take" "chain bar_take" "chain baz_take"
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diff changeset
   422
unfolding foo_take_def bar_take_def baz_take_def
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diff changeset
   423
by (intro ch2ch_fst ch2ch_snd chain_iterate)+
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diff changeset
   424
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diff changeset
   425
lemma take_0_thms: "foo_take 0 = \<bottom>" "bar_take 0 = \<bottom>" "baz_take 0 = \<bottom>"
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diff changeset
   426
unfolding foo_take_def bar_take_def baz_take_def
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huffman
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diff changeset
   427
by (simp only: iterate_0 fst_strict snd_strict)+
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diff changeset
   428
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diff changeset
   429
lemma take_Suc_thms:
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diff changeset
   430
  "foo_take (Suc n) =
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diff changeset
   431
    foo_abs oo ssum_map\<cdot>ID\<cdot>(sprod_map\<cdot>(u_map\<cdot>ID)\<cdot>(u_map\<cdot>(bar_take n))) oo foo_rep"
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huffman
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diff changeset
   432
  "bar_take (Suc n) =
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diff changeset
   433
    bar_abs oo u_map\<cdot>(cfun_map\<cdot>(baz_take n)\<cdot>ID) oo bar_rep"
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huffman
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diff changeset
   434
  "baz_take (Suc n) =
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huffman
parents: 35493
diff changeset
   435
    baz_abs oo u_map\<cdot>(cfun_map\<cdot>(convex_map\<cdot>(foo_take n))\<cdot>ID) oo baz_rep"
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   436
unfolding foo_take_def bar_take_def baz_take_def
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
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diff changeset
   437
by (simp only: iterate_Suc foo_bar_baz_takeF_beta fst_conv snd_conv)+
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
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diff changeset
   438
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diff changeset
   439
lemma lub_take_lemma:
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diff changeset
   440
  "(\<Squnion>n. foo_take n, \<Squnion>n. bar_take n, \<Squnion>n. baz_take n)
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huffman
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diff changeset
   441
    = (foo_map\<cdot>(ID::'a \<rightarrow> 'a), bar_map\<cdot>(ID::'a \<rightarrow> 'a), baz_map\<cdot>(ID::'a \<rightarrow> 'a))"
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   442
apply (simp only: thelub_Pair [symmetric] ch2ch_Pair chain_take_thms)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   443
apply (simp only: map_apply_thms pair_collapse)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   444
apply (simp only: fix_def2)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   445
apply (rule lub_eq)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   446
apply (rule nat.induct)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   447
apply (simp only: iterate_0 Pair_strict take_0_thms)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   448
apply (simp only: iterate_Suc Pair_fst_snd_eq fst_conv snd_conv
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   449
                  foo_bar_baz_mapF_beta take_Suc_thms simp_thms)
33781
c7d32e726bb9 avoid using csplit; define copy functions exactly like the current domain package
huffman
parents: 33779
diff changeset
   450
done
33591
51091e1041a7 HOLCF example: domain package proofs done manually
huffman
parents:
diff changeset
   451
36132
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   452
lemma lub_foo_take: "(\<Squnion>n. foo_take n) = ID"
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   453
apply (rule trans [OF _ foo_map_ID])
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   454
using lub_take_lemma
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   455
apply (elim Pair_inject)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   456
apply assumption
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   457
done
33591
51091e1041a7 HOLCF example: domain package proofs done manually
huffman
parents:
diff changeset
   458
36132
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   459
lemma lub_bar_take: "(\<Squnion>n. bar_take n) = ID"
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   460
apply (rule trans [OF _ bar_map_ID])
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   461
using lub_take_lemma
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   462
apply (elim Pair_inject)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   463
apply assumption
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   464
done
33591
51091e1041a7 HOLCF example: domain package proofs done manually
huffman
parents:
diff changeset
   465
36132
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   466
lemma lub_baz_take: "(\<Squnion>n. baz_take n) = ID"
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   467
apply (rule trans [OF _ baz_map_ID])
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   468
using lub_take_lemma
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   469
apply (elim Pair_inject)
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   470
apply assumption
6afa012a8f5c bring HOLCF/ex/Domain_Proofs.thy up to date
huffman
parents: 35493
diff changeset
   471
done
33591
51091e1041a7 HOLCF example: domain package proofs done manually
huffman
parents:
diff changeset
   472
51091e1041a7 HOLCF example: domain package proofs done manually
huffman
parents:
diff changeset
   473
end