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 \<Rightarrow> Err
| OK x \<Rightarrow> (case e2 of Err \<Rightarrow> Err | OK y \<Rightarrow> f x y)"
definition le :: "'a ord \ 'a err ord" where "le r e1 e2 == case e2 of Err \<Rightarrow> True |
OK y \<Rightarrow> (case e1 of Err \<Rightarrow> False | OK x \<Rightarrow> 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)"
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 Ok_in_err [iff]: "(OK x \ err A) = (x\A)" by (auto simp add: err_def)
subsection \<open>lift\<close>
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 blast done
lemma Err_lift2 [simp]: "Err +_(lift2 f) x = Err" by (simp add: lift2_def plussub_def)
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 clarify apply simp done
lemma 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
subsection \<open>semilat (err A) (le r) f\<close>
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 \ \<Longrightarrow> x <=_r z" apply (rule OK_le_err_OK [THEN iffD1]) apply (erule subst) apply (simp add: Semilat.ub1 [OF Semilat.intro]) done
lemma semilat_le_err_OK2: "\ x \ A; y \ A; semilat(err A, le r, f); OK x +_f OK y = OK z \ \<Longrightarrow> y <=_r z" apply (rule OK_le_err_OK [THEN iffD1]) apply (erule subst) apply (simp add: Semilat.ub2 [OF Semilat.intro]) done
lemma eq_order_le: "\ x=y; order r \ \ x <=_r y" apply (unfold Semilat.order_def) apply blast done
lemma 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. \<lbrakk> semilat (A,r,f); x +_f y <=_r z; x \<in> A; y \<in> A; z \<in> A \<rbrakk> \<Longrightarrow> x <=_r z \<and> 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 qed
(* FIXME? *) lemma all_bex_swap_lemma [iff]: "(\x. (\y\A. x = f y) \ P x) = (\y\A. P(f y))" by blast
lemma closed_err_Union_lift2I: "\ \A\AS. closed (err A) (lift2 f); AS \ {}; \<forall>A\<in>AS. \<forall>B\<in>AS. A\<noteq>B \<longrightarrow> (\<forall>a\<in>A. \<forall>b\<in>B. a +_f b = Err) \<rbrakk> \<Longrightarrow> closed (err (\<Union>AS)) (lift2 f)" apply (unfold closed_def err_def) apply simp apply clarify apply simp apply fast done
text\<open> If\<^term>\<open>AS = {}\<close> the thm collapses to \<^prop>\<open>order r & closed {Err} f & Err +_f Err = Err\<close>
which may not hold \<close> lemma err_semilat_UnionI: "\ \A\AS. err_semilat(A, r, f); AS \ {}; \<forall>A\<in>AS. \<forall>B\<in>AS. A\<noteq>B \<longrightarrow> (\<forall>a\<in>A. \<forall>b\<in>B. \<not> a <=_r b & a +_f b = Err) \<rbrakk> \<Longrightarrow> err_semilat (\<Union>AS, r, f)" apply (unfold semilat_def sl_def) apply (simp add: closed_err_Union_lift2I) apply (rule conjI) apply blast apply (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 simp apply (rule conjI) apply clarify apply (rename_tac A a u B b) apply (case_tac "A = B") apply simp apply simp apply clarify apply (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 simp apply (rotate_tac -1) apply (case_tac "B = C") apply simp apply (rotate_tac -1) apply simp done
end
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