(* Author: Tobias Nipkow *) theory Abs_Int1_parity imports Abs_Int1 begin subsection "Parity Analysis" datatype parity = Even | Odd | Either text‹Instantiation of class @{class order} with type @{typ parity}:› instantiation parity :: order begin text‹First the definition of the interface function @{text"≤"}. Note that the header of the definition must refer to the ascii name @{const less_eq} of the constants as @{text less_eq_parity} and the definition is named @{text less_eq_parity_def}. Inside the definition the symbolic names can be used.› definition less_eq_parity where "x ≤ y = (y = Either ∨ x=y)" text‹We also need @{text"<"}, which is defined canonically:› definition less_parity where "x < y = (x ≤ y ∧ ¬ y ≤ (x::parity))" text‹\noindent(The type annotation is necessary to fix the type of the polymorphic predicates.) Now the instance proof, i.e.\ the proof that the definition fulfills the axioms (assumptions) of the class. The initial proof-step generates the necessary proof obligations.› instance proof fix x::parity show "x ≤ x" by(auto simp: less_eq_parity_def) next fix x y z :: parity assume "x ≤ y" "y ≤ z" thus "x ≤ z" by(auto simp: less_eq_parity_def) next fix x y :: parity assume "x ≤ y" "y ≤ x" thus "x = y" by(auto simp: less_eq_parity_def) next fix x y :: parity show "(x < y) = (x ≤ y ∧ ¬ y ≤ x)" by(rule less_parity_def) qed end text‹Instantiation of class @{class semilattice_sup_top} with type @{typ parity}:› instantiation parity :: semilattice_sup_top begin definition sup_parity where "x ⊔ y = (if x = y then x else Either)" definition top_parity where "⊤ = Either" text‹Now the instance proof. This time we take a shortcut with the help of proof method @{text goal_cases}: it creates cases 1 ... n for the subgoals 1 ... n; in case i, i is also the name of the assumptions of subgoal i and @{text "case?"} refers to the conclusion of subgoal i. The class axioms are presented in the same order as in the class definition.› instance proof (standard, goal_cases) case 1 (*sup1*) show ?case by(auto simp: less_eq_parity_def sup_parity_def) next case 2 (*sup2*) show ?case by(auto simp: less_eq_parity_def sup_parity_def) next case 3 (*sup least*) thus ?case by(auto simp: less_eq_parity_def sup_parity_def) next case 4 (*top*) show ?case by(auto simp: less_eq_parity_def top_parity_def) qed end text‹Now we define the functions used for instantiating the abstract interpretation locales. Note that the Isabelle terminology is \emph{interpretation}, not \emph{instantiation} of locales, but we use instantiation to avoid confusion with abstract interpretation.› fun γ_parity :: "parity ⇒ val set" where "γ_parity Even = {i. i mod 2 = 0}" | "γ_parity Odd = {i. i mod 2 = 1}" | "γ_parity Either = UNIV" fun num_parity :: "val ⇒ parity" where "num_parity i = (if i mod 2 = 0 then Even else Odd)" fun plus_parity :: "parity ⇒ parity ⇒ parity" where "plus_parity Even Even = Even" | "plus_parity Odd Odd = Even" | "plus_parity Even Odd = Odd" | "plus_parity Odd Even = Odd" | "plus_parity Either y = Either" | "plus_parity x Either = Either" text‹First we instantiate the abstract value interface and prove that the functions on type @{typ parity} have all the necessary properties:› global_interpretation Val_semilattice where γ = γ_parity and num' = num_parity and plus' = plus_parity proof (standard, goal_cases) txt‹subgoals are the locale axioms› case 1 thus ?case by(auto simp: less_eq_parity_def) next case 2 show ?case by(auto simp: top_parity_def) next case 3 show ?case by auto next case (4 _ a1 _ a2) thus ?case by (induction a1 a2 rule: plus_parity.induct) (auto simp add: mod_add_eq [symmetric]) qed text‹In case 4 we needed to refer to particular variables. Writing (i x y z) fixes the names of the variables in case i to be x, y and z in the left-to-right order in which the variables occur in the subgoal. Underscores are anonymous placeholders for variable names we don't care to fix.› text‹Instantiating the abstract interpretation locale requires no more proofs (they happened in the instatiation above) but delivers the instantiated abstract interpreter which we call @{text AI_parity}:› global_interpretation Abs_Int where γ = γ_parity and num' = num_parity and plus' = plus_parity defines aval_parity = aval' and step_parity = step' and AI_parity = AI .. subsubsection "Tests" definition "test1_parity = ''x'' ::= N 1;; WHILE Less (V ''x'') (N 100) DO ''x'' ::= Plus (V ''x'') (N 2)" value "show_acom (the(AI_parity test1_parity))" definition "test2_parity = ''x'' ::= N 1;; WHILE Less (V ''x'') (N 100) DO ''x'' ::= Plus (V ''x'') (N 3)" definition "steps c i = ((step_parity ⊤) ^^ i) (bot c)" value "show_acom (steps test2_parity 0)" value "show_acom (steps test2_parity 1)" value "show_acom (steps test2_parity 2)" value "show_acom (steps test2_parity 3)" value "show_acom (steps test2_parity 4)" value "show_acom (steps test2_parity 5)" value "show_acom (steps test2_parity 6)" value "show_acom (the(AI_parity test2_parity))" subsubsection "Termination" global_interpretation Abs_Int_mono where γ = γ_parity and num' = num_parity and plus' = plus_parity proof (standard, goal_cases) case (1 _ a1 _ a2) thus ?case by(induction a1 a2 rule: plus_parity.induct) (auto simp add:less_eq_parity_def) qed definition m_parity :: "parity ⇒ nat" where "m_parity x = (if x = Either then 0 else 1)" global_interpretation Abs_Int_measure where γ = γ_parity and num' = num_parity and plus' = plus_parity and m = m_parity and h = "1" proof (standard, goal_cases) case 1 thus ?case by(auto simp add: m_parity_def less_eq_parity_def) next case 2 thus ?case by(auto simp add: m_parity_def less_eq_parity_def less_parity_def) qed thm AI_Some_measure end