src/HOL/IMP/Abs_Int0_parity.thy

+theory Abs_Int0_parity
+imports Abs_Int0
+begin
+
+subsection "Parity Analysis"
+
+datatype parity = Even | Odd | Either
+
+text{* Instantiation of class @{class preord} with type @{typ parity}: *}
+
+instantiation parity :: preord
+begin
+
+text{* First the definition of the interface function @{text"\<sqsubseteq>"}. Note that
+the header of the definition must refer to the ascii name @{const le} of the
+constants as @{text le_parity} and the definition is named @{text
+le_parity_def}.  Inside the definition the symbolic names can be used. *}
+
+definition le_parity where
+"x \<sqsubseteq> y = (y = Either \<or> x=y)"
+
+text{* 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 \<sqsubseteq> x" by(auto simp: le_parity_def)
+next
+  fix x y z :: parity assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
+    by(auto simp: le_parity_def)
+qed
+
+end
+
+text{* Instantiation of class @{class SL_top} with type @{typ parity}: *}
+
+instantiation parity :: SL_top
+begin
+
+
+definition join_parity where
+"x \<squnion> y = (if x \<sqsubseteq> y then y else if y \<sqsubseteq> x then x else Either)"
+
+definition Top_parity where
+"\<top> = Either"
+
+text{* Now the instance proof. This time we take a lazy shortcut: we do not
+write out the proof obligations but use the @{text goali} primitive to refer
+to the assumptions of subgoal i and @{text "case?"} to refer to the
+conclusion of subgoal i. The class axioms are presented in the same order as
+in the class definition. *}
+
+instance
+proof
+  case goal1 (*join1*) show ?case by(auto simp: le_parity_def join_parity_def)
+next
+  case goal2 (*join2*) show ?case by(auto simp: le_parity_def join_parity_def)
+next
+  case goal3 (*join least*) thus ?case by(auto simp: le_parity_def join_parity_def)
+next
+  case goal4 (*Top*) show ?case by(auto simp: le_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 \<gamma>_parity :: "parity \<Rightarrow> val set" where
+"\<gamma>_parity Even = {i. i mod 2 = 0}" |
+"\<gamma>_parity Odd  = {i. i mod 2 = 1}" |
+"\<gamma>_parity Either = UNIV"
+
+fun num_parity :: "val \<Rightarrow> parity" where
+"num_parity i = (if i mod 2 = 0 then Even else Odd)"
+
+fun plus_parity :: "parity \<Rightarrow> parity \<Rightarrow> 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: *}
+
+interpretation Val_abs
+where \<gamma> = \<gamma>_parity and num' = num_parity and plus' = plus_parity
+defines aval_parity is aval'
+proof txt{* of the locale axioms *}
+  fix a b :: parity
+  assume "a \<sqsubseteq> b" thus "\<gamma>_parity a \<subseteq> \<gamma>_parity b"
+    by(auto simp: le_parity_def)
+next txt{* The rest in the lazy, implicit way *}
+  case goal2 show ?case by(auto simp: Top_parity_def)
+next
+  case goal3 show ?case by auto
+next
+  txt{* Warning: this subproof refers to the names @{text a1} and @{text a2}
+  from the statement of the axiom. *}
+  case goal4 thus ?case
+  proof(cases a1 a2 rule: parity.exhaust[case_product parity.exhaust])
+  qed (auto simp add:mod_add_eq)
+qed
+
+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 AI: *}
+
+interpretation Abs_Int
+where \<gamma> = \<gamma>_parity and num' = num_parity and plus' = plus_parity
+defines step_parity is step' and AI_parity is AI
+proof qed
+
+
+subsubsection "Tests"
+
+definition "test1_parity =
+  ''x'' ::= N 1;
+  WHILE Less (V ''x'') (N 100) DO ''x'' ::= Plus (V ''x'') (N 2)"
+
+value "show_acom_opt (AI_parity test1_parity)"
+
+definition "test2_parity =
+  ''x'' ::= N 1;
+  WHILE Less (V ''x'') (N 100) DO ''x'' ::= Plus (V ''x'') (N 3)"
+
+value "show_acom ((step_parity \<top> ^^1) (anno None test2_parity))"
+value "show_acom ((step_parity \<top> ^^2) (anno None test2_parity))"
+value "show_acom ((step_parity \<top> ^^3) (anno None test2_parity))"
+value "show_acom ((step_parity \<top> ^^4) (anno None test2_parity))"
+value "show_acom ((step_parity \<top> ^^5) (anno None test2_parity))"
+value "show_acom_opt (AI_parity test2_parity)"
+
+
+subsubsection "Termination"
+
+interpretation Abs_Int_mono
+where \<gamma> = \<gamma>_parity and num' = num_parity and plus' = plus_parity
+proof
+  case goal1 thus ?case
+  proof(cases a1 a2 b1 b2
+   rule: parity.exhaust[case_product parity.exhaust[case_product parity.exhaust[case_product parity.exhaust]]]) (* FIXME - UGLY! *)
+  qed (auto simp add:le_parity_def)
+qed
+
+
+definition m_parity :: "parity \<Rightarrow> nat" where
+"m_parity x = (if x=Either then 0 else 1)"
+
+lemma measure_parity:
+  "(strict{(x::parity,y). x \<sqsubseteq> y})^-1 \<subseteq> measure m_parity"
+by(auto simp add: m_parity_def le_parity_def)
+
+lemma measure_parity_eq:
+  "\<forall>x y::parity. x \<sqsubseteq> y \<and> y \<sqsubseteq> x \<longrightarrow> m_parity x = m_parity y"
+by(auto simp add: m_parity_def le_parity_def)
+
+lemma AI_parity_Some: "\<exists>c'. AI_parity c = Some c'"
+by(rule AI_Some_measure[OF measure_parity measure_parity_eq])
+
+end