(* Title: HOL/MicroJava/DFA/SemilatAlg.thy

Author: Gerwin Klein

Copyright 2002 Technische Universitaet Muenchen

*)

header {* \isaheader{More on Semilattices} *}

theory SemilatAlg

imports Typing_Framework Product

begin

definition lesubstep_type :: "(nat × 's) list => 's ord => (nat × 's) list => bool"

("(_ /<=|_| _)" [50, 0, 51] 50) where

"x <=|r| y ≡ ∀(p,s) ∈ set x. ∃s'. (p,s') ∈ set y ∧ s <=_r s'"

primrec plusplussub :: "'a list => ('a => 'a => 'a) => 'a => 'a" ("(_ /++'__ _)" [65, 1000, 66] 65) where

"[] ++_f y = y"

| "(x#xs) ++_f y = xs ++_f (x +_f y)"

definition bounded :: "'s step_type => nat => bool" where

"bounded step n == !p<n. !s. !(q,t):set(step p s). q<n"

definition pres_type :: "'s step_type => nat => 's set => bool" where

"pres_type step n A == ∀s∈A. ∀p<n. ∀(q,s')∈set (step p s). s' ∈ A"

definition mono :: "'s ord => 's step_type => nat => 's set => bool" where

"mono r step n A ==

∀s p t. s ∈ A ∧ p < n ∧ s <=_r t --> step p s <=|r| step p t"

lemma pres_typeD:

"[| pres_type step n A; s∈A; p<n; (q,s')∈set (step p s) |] ==> s' ∈ A"

by (unfold pres_type_def, blast)

lemma monoD:

"[| mono r step n A; p < n; s∈A; s <=_r t |] ==> step p s <=|r| step p t"

by (unfold mono_def, blast)

lemma boundedD:

"[| bounded step n; p < n; (q,t) : set (step p xs) |] ==> q < n"

by (unfold bounded_def, blast)

lemma lesubstep_type_refl [simp, intro]:

"(!!x. x <=_r x) ==> x <=|r| x"

by (unfold lesubstep_type_def) auto

lemma lesub_step_typeD:

"a <=|r| b ==> (x,y) ∈ set a ==> ∃y'. (x, y') ∈ set b ∧ y <=_r y'"

by (unfold lesubstep_type_def) blast

lemma list_update_le_listI [rule_format]:

"set xs <= A --> set ys <= A --> xs <=[r] ys --> p < size xs -->

x <=_r ys!p --> semilat(A,r,f) --> x∈A -->

xs[p := x +_f xs!p] <=[r] ys"

apply (unfold Listn.le_def lesub_def semilat_def)

apply (simp add: list_all2_conv_all_nth nth_list_update)

done

lemma plusplus_closed: assumes "semilat (A, r, f)" shows

"!!y. [| set x ⊆ A; y ∈ A|] ==> x ++_f y ∈ A" (is "PROP ?P")

proof -

interpret Semilat A r f using assms by (rule Semilat.intro)

show "PROP ?P" proof (induct x)

show "!!y. y ∈ A ==> [] ++_f y ∈ A" by simp

fix y x xs

assume y: "y ∈ A" and xs: "set (x#xs) ⊆ A"

assume IH: "!!y. [| set xs ⊆ A; y ∈ A|] ==> xs ++_f y ∈ A"

from xs obtain x: "x ∈ A" and xs': "set xs ⊆ A" by simp

from x y have "(x +_f y) ∈ A" ..

with xs' have "xs ++_f (x +_f y) ∈ A" by (rule IH)

thus "(x#xs) ++_f y ∈ A" by simp

qed

qed

lemma (in Semilat) pp_ub2:

"!!y. [| set x ⊆ A; y ∈ A|] ==> y <=_r x ++_f y"

proof (induct x)

from semilat show "!!y. y <=_r [] ++_f y" by simp

fix y a l

assume y: "y ∈ A"

assume "set (a#l) ⊆ A"

then obtain a: "a ∈ A" and x: "set l ⊆ A" by simp

assume "!!y. [|set l ⊆ A; y ∈ A|] ==> y <=_r l ++_f y"

hence IH: "!!y. y ∈ A ==> y <=_r l ++_f y" using x .

from a y have "y <=_r a +_f y" ..

also from a y have "a +_f y ∈ A" ..

hence "(a +_f y) <=_r l ++_f (a +_f y)" by (rule IH)

finally have "y <=_r l ++_f (a +_f y)" .

thus "y <=_r (a#l) ++_f y" by simp

qed

lemma (in Semilat) pp_ub1:

shows "!!y. [|set ls ⊆ A; y ∈ A; x ∈ set ls|] ==> x <=_r ls ++_f y"

proof (induct ls)

show "!!y. x ∈ set [] ==> x <=_r [] ++_f y" by simp

fix y s ls

assume "set (s#ls) ⊆ A"

then obtain s: "s ∈ A" and ls: "set ls ⊆ A" by simp

assume y: "y ∈ A"

assume

"!!y. [|set ls ⊆ A; y ∈ A; x ∈ set ls|] ==> x <=_r ls ++_f y"

hence IH: "!!y. x ∈ set ls ==> y ∈ A ==> x <=_r ls ++_f y" using ls .

assume "x ∈ set (s#ls)"

then obtain xls: "x = s ∨ x ∈ set ls" by simp

moreover {

assume xs: "x = s"

from s y have "s <=_r s +_f y" ..

also from s y have "s +_f y ∈ A" ..

with ls have "(s +_f y) <=_r ls ++_f (s +_f y)" by (rule pp_ub2)

finally have "s <=_r ls ++_f (s +_f y)" .

with xs have "x <=_r ls ++_f (s +_f y)" by simp

}

moreover {

assume "x ∈ set ls"

hence "!!y. y ∈ A ==> x <=_r ls ++_f y" by (rule IH)

moreover from s y have "s +_f y ∈ A" ..

ultimately have "x <=_r ls ++_f (s +_f y)" .

}

ultimately

have "x <=_r ls ++_f (s +_f y)" by blast

thus "x <=_r (s#ls) ++_f y" by simp

qed

lemma (in Semilat) pp_lub:

assumes z: "z ∈ A"

shows

"!!y. y ∈ A ==> set xs ⊆ A ==> ∀x ∈ set xs. x <=_r z ==> y <=_r z ==> xs ++_f y <=_r z"

proof (induct xs)

fix y assume "y <=_r z" thus "[] ++_f y <=_r z" by simp

next

fix y l ls assume y: "y ∈ A" and "set (l#ls) ⊆ A"

then obtain l: "l ∈ A" and ls: "set ls ⊆ A" by auto

assume "∀x ∈ set (l#ls). x <=_r z"

then obtain lz: "l <=_r z" and lsz: "∀x ∈ set ls. x <=_r z" by auto

assume "y <=_r z" with lz have "l +_f y <=_r z" using l y z ..

moreover

from l y have "l +_f y ∈ A" ..

moreover

assume "!!y. y ∈ A ==> set ls ⊆ A ==> ∀x ∈ set ls. x <=_r z ==> y <=_r z

==> ls ++_f y <=_r z"

ultimately

have "ls ++_f (l +_f y) <=_r z" using ls lsz by -

thus "(l#ls) ++_f y <=_r z" by simp

qed

lemma ub1':

assumes "semilat (A, r, f)"

shows "[|∀(p,s) ∈ set S. s ∈ A; y ∈ A; (a,b) ∈ set S|]

==> b <=_r map snd [(p', t')\<leftarrow>S. p' = a] ++_f y"

proof -

interpret Semilat A r f using assms by (rule Semilat.intro)

let "b <=_r ?map ++_f y" = ?thesis

assume "y ∈ A"

moreover

assume "∀(p,s) ∈ set S. s ∈ A"

hence "set ?map ⊆ A" by auto

moreover

assume "(a,b) ∈ set S"

hence "b ∈ set ?map" by (induct S, auto)

ultimately

show ?thesis by - (rule pp_ub1)

qed

lemma plusplus_empty:

"∀s'. (q, s') ∈ set S --> s' +_f ss ! q = ss ! q ==>

(map snd [(p', t') \<leftarrow> S. p' = q] ++_f ss ! q) = ss ! q"

by (induct S) auto

end