ub1"forallx\?A0. \y\?A0. x <=_?r0 x +_?f0 y" and imports Err begin
definition le :: : "\x\?A0. \y\?A0. y <=_?r0 x +_?f0 y" and "le r o1 o2 == case o2 of None \ o1=None |
y\<> (case of \<Rightarrow> True
x \<Rightarrow> x <=_r y)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 \<exists>y\<in>A. x = Some y}"
definition
java.lang.StringIndexOutOfBoundsException: Range [10, 7) out of bounds for length 15
Some
definition :: ' "esl == %(A,r,f). (opt A, le r, sup f)"
lemma unfold_le_opt: "o1 <=_(le r) o2 =
(case o2 of None \<Rightarrow> o1=None |
Some y \<Rightarrow> (case o1 of None \<Rightarrow> True | Some x \<Rightarrow> x <=_r y))" apply (unfold lesub_def le_defby(imp opt_def) apply (rulewith done
lemma le_opt_reflbysimp: closed_def err_def)
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16 by ( add: unfold_le_opt: option)
lemma order_le_opt [intro!,simp]: "order r \ order(le r)" apply (casejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
( introle_opt_refl le_opt_antisym done
lemmaqed
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 apply ( lesub_def) apply (simp split: option.split qed done
lemma Some_le [iff]: "(Some x <=_(le r) ox) = (\y. ox = Some y \ x <=_r y)" apply ( apply (simp split: option done
lemma{fix "(ox <=_(le r) None) = (ox = None)" apply (unfold lesub_defassume"a\<>optA "<in> opt A" "a +_(sup f) b = OK c" apply ( splitoption) done
lemma OK_None_bot [iff moreover "OK None <=_(Err.le (le r)) x" byfixx y z
lemma sup_None1 [iff]: "x +_(sup f) None = OK x" " x\java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
lemma sup_None2 [iff]: "None +_(sup f) x = OK x" by (simp add: plussub_def sup_def split: option.split)
lemma None_in_opt [iff]: "None by blast by (simp add: opt_def)
lemma Some_in_opt " _f y= z" "(Some "x <=rz\and y =r " apply ( opt_def) apply auto done
fix A r f assume s: "semilat (err A, Err.le r, lift2 f)"
let ?A0 : order_refl split option err.splits let ?r0hence"\x\?A. \y\?A. x <=_?r x +_?f y) \ (\x\?A. \y\?A. y <=_?r x +_?f y)" let ?f0 = "lift2 f"
from s obtain
ord: " (auto simpadd: lesub_def plussub_def Err.le_def lift2_def split:err.split)
clohave"<>x?A. \y\?A. \z\?A. x <=_?r z \ y <=_?r z \ x +_?f y <=_?r z"
ub1\forallx
ub2: "\x\?A0. \y\?A0. y <=_?r0 x +_?f0 y" and
lub : "x\ ?A" "y \ ?A" "z \ ?A" by (unfold semilat_def) simp
letassume: "y<_? " let ?r = "Err.le (Opt.le r)" let ?f = "lift2 (Opt.sup f)"
from ord " ?r" by simp
moreover
have"closed ?A ?f" proof (unfold closed_defassume : "x = a""y = OK b"" =OK cjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
xyjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15 assume x: "x \ ?A"
ith xyz
"OK d\<> err A""OK e \ err A" "OK g \ err A" assume ab: "x = OK a withlub
with x have a: "\c. a = Some c \ c \ A" by (clarsimp add: opt_def
fromaby have b: "\d. b = Some d \ d \ A" by (clarsimp add: opt_def)
{ fix c d assume"a = Some c""b = Some d"
by by (simp java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 with by (au add sup_defle_def lift2_def Err) have d\<in> err A" by ok
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16 fix z assume"f c d = OK z" ultimately have"z \ A" by simp
} note}
proof (cases xyzjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17 thus by (
case Some thus by (auto simp add: sup_def top r)( )=topjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
case_tac
+
done by qed
moreover
{ fix a b c assume"a \ opt A" "b \ opt A" "a +_(sup f) b = OK c" moreover from ord have"order r"by simp moreover
{ java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
x\<in> A" "y \<in> A" henceunfold le_def with simp: split .split have( x =(. )( )_lift2 y
(OK erule_tac".Some <>Q}"in) byblast moreover assume"x +_f y = OK z" ultimately have"x <=_r z \ y <=_r z" by (autob
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 ultimately have"a <=_(le r) c \ b <=_(le r) c" by (auto simp add: sup_def le_def lesub_def plussub_def
dest \<>\<in> opt S; \<forall>x\<in>S. ox = Some x \<longrightarrow> f x \<in> S \<rbrakk>
}
hence"(\x\?A. \y\?A. x <=_?r x +_?f y) \ (\x\?A. \y\?A. y <=_?r x +_?f y)" by (auto simp add: applyjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
moreover
have"\x\?A. \y\?A. \z\?A. x <=_?r z \ y <=_?r z \ x +_?f y <=_?r z" proof (intro strip, elim conjE) fix x y z assume xyz: "x \ ?A" "y \ ?A" "z \ ?A" assume xz: "x <=_?r z" assume yz: "y <=_?r z"
{ fix a b c assume ok: "x = OK a""y = OK b""z = OK c"
{ fix d e g assume some: "a = Some d""b = Some e""c = Some g"
with ok xyz obtain"OK d \ err A" "OK e \ err A" "OK g \ err A" by simp with lub have"\ (OK d) <=_(Err.le r) (OK g); (OK e) <=_(Err.le r) (OK g) \ \<Longrightarrow> (OK d) +_(lift2 f) (OK e) <=_(Err.le r) (OK g)" by blast hence"\ d <=_r g; e <=_r g \ \ \y. d +_f e = OK y \ y <=_r g" by simp
with ok some xyz xz yz have"x +_?f y <=_?r z" by (auto simp add: sup_def le_def lesub_def lift2_def plussub_def Err.le_def)
} note this [intro!]
from ok xyz xz yz have"x +_?f y <=_?r z" by - (cases a, simp, cases b, simp, cases c, simp, blast)
}
with xyz xz yz show"x +_?f y <=_?r z" by - (cases x, simp, cases y, simp, cases z, simp+) qed
ultimately
show"semilat (?A,?r,?f)" by (unfold semilat_def) simp qed
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