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theory Pcpodef(* Title: HOLCF/Pcpodef.thy Author: Brian Huffman *) header {* Subtypes of pcpos *} theory Pcpodef imports Adm uses ("Tools/pcpodef_package.ML") begin subsection {* Proving a subtype is a partial order *} text {* A subtype of a partial order is itself a partial order, if the ordering is defined in the standard way. *} setup {* Sign.add_const_constraint (@{const_name Porder.sq_le}, NONE) *} theorem typedef_po: fixes Abs :: "'a::po => 'b::type" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" shows "OFCLASS('b, po_class)" apply (intro_classes, unfold less) apply (rule refl_less) apply (erule (1) trans_less) apply (rule type_definition.Rep_inject [OF type, THEN iffD1]) apply (erule (1) antisym_less) done setup {* Sign.add_const_constraint (@{const_name Porder.sq_le}, SOME @{typ "'a::sq_ord => 'a::sq_ord => bool"}) *} subsection {* Proving a subtype is finite *} lemma typedef_finite_UNIV: fixes Abs :: "'a::type => 'b::type" assumes type: "type_definition Rep Abs A" shows "finite A ==> finite (UNIV :: 'b set)" proof - assume "finite A" hence "finite (Abs ` A)" by (rule finite_imageI) thus "finite (UNIV :: 'b set)" by (simp only: type_definition.Abs_image [OF type]) qed theorem typedef_finite_po: fixes Abs :: "'a::finite_po => 'b::po" assumes type: "type_definition Rep Abs A" shows "OFCLASS('b, finite_po_class)" apply (intro_classes) apply (rule typedef_finite_UNIV [OF type]) apply (rule finite) done subsection {* Proving a subtype is chain-finite *} lemma monofun_Rep: assumes less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" shows "monofun Rep" by (rule monofunI, unfold less) lemmas ch2ch_Rep = ch2ch_monofun [OF monofun_Rep] lemmas ub2ub_Rep = ub2ub_monofun [OF monofun_Rep] theorem typedef_chfin: fixes Abs :: "'a::chfin => 'b::po" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" shows "OFCLASS('b, chfin_class)" apply intro_classes apply (drule ch2ch_Rep [OF less]) apply (drule chfin) apply (unfold max_in_chain_def) apply (simp add: type_definition.Rep_inject [OF type]) done subsection {* Proving a subtype is complete *} text {* A subtype of a cpo is itself a cpo if the ordering is defined in the standard way, and the defining subset is closed with respect to limits of chains. A set is closed if and only if membership in the set is an admissible predicate. *} lemma Abs_inverse_lub_Rep: fixes Abs :: "'a::cpo => 'b::po" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and adm: "adm (λx. x ∈ A)" shows "chain S ==> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))" apply (rule type_definition.Abs_inverse [OF type]) apply (erule admD [OF adm ch2ch_Rep [OF less]]) apply (rule type_definition.Rep [OF type]) done theorem typedef_lub: fixes Abs :: "'a::cpo => 'b::po" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and adm: "adm (λx. x ∈ A)" shows "chain S ==> range S <<| Abs (\<Squnion>i. Rep (S i))" apply (frule ch2ch_Rep [OF less]) apply (rule is_lubI) apply (rule ub_rangeI) apply (simp only: less Abs_inverse_lub_Rep [OF type less adm]) apply (erule is_ub_thelub) apply (simp only: less Abs_inverse_lub_Rep [OF type less adm]) apply (erule is_lub_thelub) apply (erule ub2ub_Rep [OF less]) done lemmas typedef_thelub = typedef_lub [THEN thelubI, standard] theorem typedef_cpo: fixes Abs :: "'a::cpo => 'b::po" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and adm: "adm (λx. x ∈ A)" shows "OFCLASS('b, cpo_class)" proof fix S::"nat => 'b" assume "chain S" hence "range S <<| Abs (\<Squnion>i. Rep (S i))" by (rule typedef_lub [OF type less adm]) thus "∃x. range S <<| x" .. qed subsubsection {* Continuity of @{term Rep} and @{term Abs} *} text {* For any sub-cpo, the @{term Rep} function is continuous. *} theorem typedef_cont_Rep: fixes Abs :: "'a::cpo => 'b::cpo" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and adm: "adm (λx. x ∈ A)" shows "cont Rep" apply (rule contI) apply (simp only: typedef_thelub [OF type less adm]) apply (simp only: Abs_inverse_lub_Rep [OF type less adm]) apply (rule cpo_lubI) apply (erule ch2ch_Rep [OF less]) done text {* For a sub-cpo, we can make the @{term Abs} function continuous only if we restrict its domain to the defining subset by composing it with another continuous function. *} theorem typedef_is_lubI: assumes less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" shows "range (λi. Rep (S i)) <<| Rep x ==> range S <<| x" apply (rule is_lubI) apply (rule ub_rangeI) apply (subst less) apply (erule is_ub_lub) apply (subst less) apply (erule is_lub_lub) apply (erule ub2ub_Rep [OF less]) done theorem typedef_cont_Abs: fixes Abs :: "'a::cpo => 'b::cpo" fixes f :: "'c::cpo => 'a::cpo" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and adm: "adm (λx. x ∈ A)" (* not used *) and f_in_A: "!!x. f x ∈ A" and cont_f: "cont f" shows "cont (λx. Abs (f x))" apply (rule contI) apply (rule typedef_is_lubI [OF less]) apply (simp only: type_definition.Abs_inverse [OF type f_in_A]) apply (erule cont_f [THEN contE]) done subsection {* Proving subtype elements are compact *} theorem typedef_compact: fixes Abs :: "'a::cpo => 'b::cpo" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and adm: "adm (λx. x ∈ A)" shows "compact (Rep k) ==> compact k" proof (unfold compact_def) have cont_Rep: "cont Rep" by (rule typedef_cont_Rep [OF type less adm]) assume "adm (λx. ¬ Rep k \<sqsubseteq> x)" with cont_Rep have "adm (λx. ¬ Rep k \<sqsubseteq> Rep x)" by (rule adm_subst) thus "adm (λx. ¬ k \<sqsubseteq> x)" by (unfold less) qed subsection {* Proving a subtype is pointed *} text {* A subtype of a cpo has a least element if and only if the defining subset has a least element. *} theorem typedef_pcpo_generic: fixes Abs :: "'a::cpo => 'b::cpo" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and z_in_A: "z ∈ A" and z_least: "!!x. x ∈ A ==> z \<sqsubseteq> x" shows "OFCLASS('b, pcpo_class)" apply (intro_classes) apply (rule_tac x="Abs z" in exI, rule allI) apply (unfold less) apply (subst type_definition.Abs_inverse [OF type z_in_A]) apply (rule z_least [OF type_definition.Rep [OF type]]) done text {* As a special case, a subtype of a pcpo has a least element if the defining subset contains @{term ⊥}. *} theorem typedef_pcpo: fixes Abs :: "'a::pcpo => 'b::cpo" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "OFCLASS('b, pcpo_class)" by (rule typedef_pcpo_generic [OF type less UU_in_A], rule minimal) subsubsection {* Strictness of @{term Rep} and @{term Abs} *} text {* For a sub-pcpo where @{term ⊥} is a member of the defining subset, @{term Rep} and @{term Abs} are both strict. *} theorem typedef_Abs_strict: assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "Abs ⊥ = ⊥" apply (rule UU_I, unfold less) apply (simp add: type_definition.Abs_inverse [OF type UU_in_A]) done theorem typedef_Rep_strict: assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "Rep ⊥ = ⊥" apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst]) apply (rule type_definition.Abs_inverse [OF type UU_in_A]) done theorem typedef_Abs_strict_iff: assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "x ∈ A ==> (Abs x = ⊥) = (x = ⊥)" apply (rule typedef_Abs_strict [OF type less UU_in_A, THEN subst]) apply (simp add: type_definition.Abs_inject [OF type] UU_in_A) done theorem typedef_Rep_strict_iff: assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "(Rep x = ⊥) = (x = ⊥)" apply (rule typedef_Rep_strict [OF type less UU_in_A, THEN subst]) apply (simp add: type_definition.Rep_inject [OF type]) done theorem typedef_Abs_defined: assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "[|x ≠ ⊥; x ∈ A|] ==> Abs x ≠ ⊥" by (simp add: typedef_Abs_strict_iff [OF type less UU_in_A]) theorem typedef_Rep_defined: assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "x ≠ ⊥ ==> Rep x ≠ ⊥" by (simp add: typedef_Rep_strict_iff [OF type less UU_in_A]) subsection {* Proving a subtype is flat *} theorem typedef_flat: fixes Abs :: "'a::flat => 'b::pcpo" assumes type: "type_definition Rep Abs A" and less: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y" and UU_in_A: "⊥ ∈ A" shows "OFCLASS('b, flat_class)" apply (intro_classes) apply (unfold less) apply (simp add: type_definition.Rep_inject [OF type, symmetric]) apply (simp add: typedef_Rep_strict [OF type less UU_in_A]) apply (simp add: ax_flat) done subsection {* HOLCF type definition package *} use "Tools/pcpodef_package.ML" end