(* 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')←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') ← S. p' = q] ++_f ss ! q) = ss ! q" by (induct S) auto end