author | wenzelm |
Tue, 31 May 2005 11:53:12 +0200 | |
changeset 16121 | a80aa66d2271 |
parent 16070 | 4a83dd540b88 |
child 16217 | 96f0c8546265 |
permissions | -rw-r--r-- |
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(* Title: HOLCF/Domain.thy |
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ID: $Id$ |
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Author: Brian Huffman |
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*) |
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|
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header {* Domain package *} |
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|
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theory Domain |
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imports Ssum Sprod One Up |
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files |
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("domain/library.ML") |
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("domain/syntax.ML") |
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("domain/axioms.ML") |
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("domain/theorems.ML") |
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("domain/extender.ML") |
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("domain/interface.ML") |
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begin |
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|
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defaultsort pcpo |
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|
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subsection {* Continuous isomorphisms *} |
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text {* A locale for continuous isomorphisms *} |
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|
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locale iso = |
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fixes abs :: "'a \<rightarrow> 'b" |
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fixes rep :: "'b \<rightarrow> 'a" |
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assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x" |
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assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y" |
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|
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lemma (in iso) swap: "iso rep abs" |
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by (rule iso.intro [OF rep_iso abs_iso]) |
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|
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lemma (in iso) abs_strict: "abs\<cdot>\<bottom> = \<bottom>" |
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proof - |
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have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" .. |
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hence "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg) |
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hence "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp |
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thus ?thesis by (rule UU_I) |
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qed |
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|
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lemma (in iso) rep_strict: "rep\<cdot>\<bottom> = \<bottom>" |
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by (rule iso.abs_strict [OF swap]) |
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|
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lemma (in iso) abs_defin': "abs\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>" |
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proof - |
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assume A: "abs\<cdot>z = \<bottom>" |
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have "z = rep\<cdot>(abs\<cdot>z)" by simp |
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also have "\<dots> = rep\<cdot>\<bottom>" by (simp only: A) |
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also note rep_strict |
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finally show "z = \<bottom>" . |
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qed |
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|
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lemma (in iso) rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>" |
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by (rule iso.abs_defin' [OF swap]) |
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|
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lemma (in iso) abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>" |
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by (erule contrapos_nn, erule abs_defin') |
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|
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lemma (in iso) rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>" |
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by (erule contrapos_nn, erule rep_defin') |
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|
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lemma (in iso) iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)" |
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proof |
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assume "x = abs\<cdot>y" |
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hence "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp |
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thus "rep\<cdot>x = y" by simp |
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next |
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assume "rep\<cdot>x = y" |
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hence "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp |
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thus "x = abs\<cdot>y" by simp |
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qed |
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|
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subsection {* Casedist *} |
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|
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lemma ex_one_defined_iff: |
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"(\<exists>x. P x \<and> x \<noteq> \<bottom>) = P ONE" |
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apply safe |
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apply (rule_tac p=x in oneE) |
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apply simp |
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apply simp |
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apply force |
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done |
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|
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lemma ex_up_defined_iff: |
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"(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))" |
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apply safe |
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apply (rule_tac p=x in upE1) |
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apply simp |
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apply fast |
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apply (force intro!: defined_up) |
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done |
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|
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lemma ex_sprod_defined_iff: |
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"(\<exists>y. P y \<and> y \<noteq> \<bottom>) = |
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(\<exists>x y. (P (:x, y:) \<and> x \<noteq> \<bottom>) \<and> y \<noteq> \<bottom>)" |
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apply safe |
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apply (rule_tac p=y in sprodE) |
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apply simp |
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apply fast |
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apply (force intro!: defined_spair) |
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done |
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|
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lemma ex_sprod_up_defined_iff: |
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"(\<exists>y. P y \<and> y \<noteq> \<bottom>) = |
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(\<exists>x y. P (:up\<cdot>x, y:) \<and> y \<noteq> \<bottom>)" |
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apply safe |
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apply (rule_tac p=y in sprodE) |
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apply simp |
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apply (rule_tac p=x in upE1) |
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apply simp |
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apply fast |
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apply (force intro!: defined_spair) |
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done |
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|
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lemma ex_ssum_defined_iff: |
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"(\<exists>x. P x \<and> x \<noteq> \<bottom>) = |
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((\<exists>x. P (sinl\<cdot>x) \<and> x \<noteq> \<bottom>) \<or> |
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(\<exists>x. P (sinr\<cdot>x) \<and> x \<noteq> \<bottom>))" |
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apply (rule iffI) |
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apply (erule exE) |
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apply (erule conjE) |
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apply (rule_tac p=x in ssumE) |
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apply simp |
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apply (rule disjI1, fast) |
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apply (rule disjI2, fast) |
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apply (erule disjE) |
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apply (force intro: defined_sinl) |
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apply (force intro: defined_sinr) |
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done |
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lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)" |
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by auto |
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lemmas ex_defined_iffs = |
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ex_ssum_defined_iff |
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ex_sprod_up_defined_iff |
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ex_sprod_defined_iff |
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ex_up_defined_iff |
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ex_one_defined_iff |
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text {* Rules for turning exh into casedist *} |
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lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *) |
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by auto |
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|
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lemma exh_casedist1: "((P \<or> Q \<Longrightarrow> R) \<Longrightarrow> S) \<equiv> (\<lbrakk>P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> S)" |
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by rule auto |
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|
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lemma exh_casedist2: "(\<exists>x. P x \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)" |
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by rule auto |
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|
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lemma exh_casedist3: "(P \<and> Q \<Longrightarrow> R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> R)" |
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by rule auto |
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lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3 |
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subsection {* Setting up the package *} |
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ML {* |
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val iso_intro = thm "iso.intro"; |
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val iso_abs_iso = thm "iso.abs_iso"; |
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val iso_rep_iso = thm "iso.rep_iso"; |
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val iso_abs_strict = thm "iso.abs_strict"; |
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val iso_rep_strict = thm "iso.rep_strict"; |
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val iso_abs_defin' = thm "iso.abs_defin'"; |
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val iso_rep_defin' = thm "iso.rep_defin'"; |
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val iso_abs_defined = thm "iso.abs_defined"; |
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val iso_rep_defined = thm "iso.rep_defined"; |
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val iso_iso_swap = thm "iso.iso_swap"; |
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|
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val exh_start = thm "exh_start"; |
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val ex_defined_iffs = thms "ex_defined_iffs"; |
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val exh_casedist0 = thm "exh_casedist0"; |
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val exh_casedists = thms "exh_casedists"; |
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*} |
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|
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end |