# Theory Err

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theory Err
imports Semilat
`(*  Title:      HOL/MicroJava/DFA/Err.thy    Author:     Tobias Nipkow    Copyright   2000 TUM*)header {* \isaheader{The Error Type} *}theory Errimports Semilatbegindatatype 'a err = Err | OK 'atype_synonym 'a ebinop = "'a => 'a => 'a err"type_synonym 'a esl = "'a set * 'a ord * 'a ebinop"primrec ok_val :: "'a err => 'a" where  "ok_val (OK x) = x"definition lift :: "('a => 'b err) => ('a err => 'b err)" where"lift f e == case e of Err => Err | OK x => f x"definition lift2 :: "('a => 'b => 'c err) => 'a err => 'b err => 'c err" where"lift2 f e1 e2 == case e1 of Err  => Err          | OK x => (case e2 of Err => Err | OK y => f x y)"definition le :: "'a ord => 'a err ord" where"le r e1 e2 ==        case e2 of Err => True |                   OK y => (case e1 of Err => False | OK x => x <=_r y)"definition sup :: "('a => 'b => 'c) => ('a err => 'b err => 'c err)" where"sup f == lift2(%x y. OK(x +_f y))"definition err :: "'a set => 'a err set" where"err A == insert Err {x . ? y:A. x = OK y}"definition esl :: "'a sl => 'a esl" where"esl == %(A,r,f). (A,r, %x y. OK(f x y))"definition sl :: "'a esl => 'a err sl" where"sl == %(A,r,f). (err A, le r, lift2 f)"abbreviation  err_semilat :: "'a esl => bool"  where "err_semilat L == semilat(Err.sl L)"primrec strict :: "('a => 'b err) => ('a err => 'b err)" where  "strict f Err    = Err"| "strict f (OK x) = f x"lemma strict_Some [simp]:   "(strict f x = OK y) = (∃ z. x = OK z ∧ f z = OK y)"  by (cases x, auto)lemma not_Err_eq:  "(x ≠ Err) = (∃a. x = OK a)"   by (cases x) autolemma not_OK_eq:  "(∀y. x ≠ OK y) = (x = Err)"  by (cases x) auto  lemma unfold_lesub_err:  "e1 <=_(le r) e2 == le r e1 e2"  by (simp add: lesub_def)lemma le_err_refl:  "!x. x <=_r x ==> e <=_(Err.le r) e"apply (unfold lesub_def Err.le_def)apply (simp split: err.split)done lemma le_err_trans [rule_format]:  "order r ==> e1 <=_(le r) e2 --> e2 <=_(le r) e3 --> e1 <=_(le r) e3"apply (unfold unfold_lesub_err le_def)apply (simp split: err.split)apply (blast intro: order_trans)donelemma le_err_antisym [rule_format]:  "order r ==> e1 <=_(le r) e2 --> e2 <=_(le r) e1 --> e1=e2"apply (unfold unfold_lesub_err le_def)apply (simp split: err.split)apply (blast intro: order_antisym)done lemma OK_le_err_OK:  "(OK x <=_(le r) OK y) = (x <=_r y)"  by (simp add: unfold_lesub_err le_def)lemma order_le_err [iff]:  "order(le r) = order r"apply (rule iffI) apply (subst Semilat.order_def) apply (blast dest: order_antisym OK_le_err_OK [THEN iffD2]              intro: order_trans OK_le_err_OK [THEN iffD1])apply (subst Semilat.order_def)apply (blast intro: le_err_refl le_err_trans le_err_antisym             dest: order_refl)done lemma le_Err [iff]:  "e <=_(le r) Err"  by (simp add: unfold_lesub_err le_def)lemma Err_le_conv [iff]: "Err <=_(le r) e  = (e = Err)"  by (simp add: unfold_lesub_err le_def  split: err.split)lemma le_OK_conv [iff]:  "e <=_(le r) OK x  =  (? y. e = OK y & y <=_r x)"  by (simp add: unfold_lesub_err le_def split: err.split)lemma OK_le_conv: "OK x <=_(le r) e  =  (e = Err | (? y. e = OK y & x <=_r y))"  by (simp add: unfold_lesub_err le_def split: err.split)lemma top_Err [iff]: "top (le r) Err";  by (simp add: top_def)lemma OK_less_conv [rule_format, iff]:  "OK x <_(le r) e = (e=Err | (? y. e = OK y & x <_r y))"  by (simp add: lesssub_def lesub_def le_def split: err.split)lemma not_Err_less [rule_format, iff]:  "~(Err <_(le r) x)"  by (simp add: lesssub_def lesub_def le_def split: err.split)lemma semilat_errI [intro]:  assumes semilat: "semilat (A, r, f)"  shows "semilat(err A, Err.le r, lift2(%x y. OK(f x y)))"  apply(insert semilat)  apply (unfold semilat_Def closed_def plussub_def lesub_def     lift2_def Err.le_def err_def)  apply (simp split: err.split)  donelemma err_semilat_eslI_aux:  assumes semilat: "semilat (A, r, f)"  shows "err_semilat(esl(A,r,f))"  apply (unfold sl_def esl_def)  apply (simp add: semilat_errI[OF semilat])  donelemma err_semilat_eslI [intro, simp]: "!!L. semilat L ==> err_semilat(esl L)"by(simp add: err_semilat_eslI_aux split_tupled_all)lemma acc_err [simp, intro!]:  "acc r ==> acc(le r)"apply (unfold acc_def lesub_def le_def lesssub_def)apply (simp add: wf_eq_minimal split: err.split)apply clarifyapply (case_tac "Err : Q") apply blastapply (erule_tac x = "{a . OK a : Q}" in allE)apply (case_tac "x") apply fastapply blastdone lemma Err_in_err [iff]: "Err : err A"  by (simp add: err_def)lemma Ok_in_err [iff]: "(OK x : err A) = (x:A)"  by (auto simp add: err_def)section {* lift *}lemma lift_in_errI:  "[| e : err S; !x:S. e = OK x --> f x : err S |] ==> lift f e : err S"apply (unfold lift_def)apply (simp split: err.split)apply blastdone lemma Err_lift2 [simp]:   "Err +_(lift2 f) x = Err"  by (simp add: lift2_def plussub_def)lemma lift2_Err [simp]:   "x +_(lift2 f) Err = Err"  by (simp add: lift2_def plussub_def split: err.split)lemma OK_lift2_OK [simp]:  "OK x +_(lift2 f) OK y = x +_f y"  by (simp add: lift2_def plussub_def split: err.split)section {* sup *}lemma Err_sup_Err [simp]:  "Err +_(Err.sup f) x = Err"  by (simp add: plussub_def Err.sup_def Err.lift2_def)lemma Err_sup_Err2 [simp]:  "x +_(Err.sup f) Err = Err"  by (simp add: plussub_def Err.sup_def Err.lift2_def split: err.split)lemma Err_sup_OK [simp]:  "OK x +_(Err.sup f) OK y = OK(x +_f y)"  by (simp add: plussub_def Err.sup_def Err.lift2_def)lemma Err_sup_eq_OK_conv [iff]:  "(Err.sup f ex ey = OK z) = (? x y. ex = OK x & ey = OK y & f x y = z)"apply (unfold Err.sup_def lift2_def plussub_def)apply (rule iffI) apply (simp split: err.split_asm)apply clarifyapply simpdonelemma Err_sup_eq_Err [iff]:  "(Err.sup f ex ey = Err) = (ex=Err | ey=Err)"apply (unfold Err.sup_def lift2_def plussub_def)apply (simp split: err.split)done section {* semilat (err A) (le r) f *}lemma semilat_le_err_Err_plus [simp]:  "[| x: err A; semilat(err A, le r, f) |] ==> Err +_f x = Err"  by (blast intro: Semilat.le_iff_plus_unchanged [OF Semilat.intro, THEN iffD1]                   Semilat.le_iff_plus_unchanged2 [OF Semilat.intro, THEN iffD1])lemma semilat_le_err_plus_Err [simp]:  "[| x: err A; semilat(err A, le r, f) |] ==> x +_f Err = Err"  by (blast intro: Semilat.le_iff_plus_unchanged [OF Semilat.intro, THEN iffD1]                   Semilat.le_iff_plus_unchanged2 [OF Semilat.intro, THEN iffD1])lemma semilat_le_err_OK1:  "[| x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z |]   ==> x <=_r z";apply (rule OK_le_err_OK [THEN iffD1])apply (erule subst)apply (simp add: Semilat.ub1 [OF Semilat.intro])donelemma semilat_le_err_OK2:  "[| x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z |]   ==> y <=_r z"apply (rule OK_le_err_OK [THEN iffD1])apply (erule subst)apply (simp add: Semilat.ub2 [OF Semilat.intro])donelemma eq_order_le:  "[| x=y; order r |] ==> x <=_r y"apply (unfold Semilat.order_def)apply blastdonelemma OK_plus_OK_eq_Err_conv [simp]:  assumes "x:A" and "y:A" and "semilat(err A, le r, fe)"  shows "((OK x) +_fe (OK y) = Err) = (~(? z:A. x <=_r z & y <=_r z))"proof -  have plus_le_conv3: "!!A x y z f r.     [| semilat (A,r,f); x +_f y <=_r z; x:A; y:A; z:A |]     ==> x <=_r z ∧ y <=_r z"    by (rule Semilat.plus_le_conv [OF Semilat.intro, THEN iffD1])  from assms show ?thesis  apply (rule_tac iffI)   apply clarify   apply (drule OK_le_err_OK [THEN iffD2])   apply (drule OK_le_err_OK [THEN iffD2])   apply (drule Semilat.lub [OF Semilat.intro, of _ _ _ "OK x" _ "OK y"])        apply assumption       apply assumption      apply simp     apply simp    apply simp   apply simp  apply (case_tac "(OK x) +_fe (OK y)")   apply assumption  apply (rename_tac z)  apply (subgoal_tac "OK z: err A")  apply (drule eq_order_le)    apply (erule Semilat.orderI [OF Semilat.intro])   apply (blast dest: plus_le_conv3)   apply (erule subst)  apply (blast intro: Semilat.closedI [OF Semilat.intro] closedD)  done qedsection {* semilat (err(Union AS)) *}(* FIXME? *)lemma all_bex_swap_lemma [iff]:  "(!x. (? y:A. x = f y) --> P x) = (!y:A. P(f y))"  by blastlemma closed_err_Union_lift2I:   "[| !A:AS. closed (err A) (lift2 f); AS ~= {};       !A:AS.!B:AS. A~=B --> (!a:A.!b:B. a +_f b = Err) |]   ==> closed (err(Union AS)) (lift2 f)"apply (unfold closed_def err_def)apply simpapply clarifyapply simpapply fastdone text {*   If @{term "AS = {}"} the thm collapses to  @{prop "order r & closed {Err} f & Err +_f Err = Err"}  which may not hold *}lemma err_semilat_UnionI:  "[| !A:AS. err_semilat(A, r, f); AS ~= {};       !A:AS.!B:AS. A~=B --> (!a:A.!b:B. ~ a <=_r b & a +_f b = Err) |]   ==> err_semilat(Union AS, r, f)"apply (unfold semilat_def sl_def)apply (simp add: closed_err_Union_lift2I)apply (rule conjI) apply blastapply (simp add: err_def)apply (rule conjI) apply clarify apply (rename_tac A a u B b) apply (case_tac "A = B")  apply simp apply simpapply (rule conjI) apply clarify apply (rename_tac A a u B b) apply (case_tac "A = B")  apply simp apply simpapply clarifyapply (rename_tac A ya yb B yd z C c a b)apply (case_tac "A = B") apply (case_tac "A = C")  apply simp apply (rotate_tac -1) apply simpapply (rotate_tac -1)apply (case_tac "B = C") apply simpapply (rotate_tac -1)apply simpdone end`