(* Title: HOL/MicroJava/DFA/SemilatAlg.thy Author: Gerwin Klein Copyright 2002 Technische Universitaet Muenchen *) section {* 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