# Theory Abs_Int1_parity_ITP

theory Abs_Int1_parity_ITP
imports Abs_Int1_ITP
(* Author: Tobias Nipkow *)

theory Abs_Int1_parity_ITP
imports Abs_Int1_ITP
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"⊑"}. 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 ⊑ y = (y = Either ∨ 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 ⊑ x" by(auto simp: le_parity_def)
next
fix x y z :: parity assume "x ⊑ y" "y ⊑ z" thus "x ⊑ 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 ⊔ y = (if x ⊑ y then y else if y ⊑ x then x else Either)"

definition Top_parity where
"⊤ = 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 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: *}

interpretation Val_abs
where γ = γ_parity and num' = num_parity and plus' = plus_parity
proof txt{* of the locale axioms *}
fix a b :: parity
assume "a ⊑ b" thus "γ_parity a ⊆ γ_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

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: *}

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_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 ⊤ ^^1) (anno None test2_parity))"
value "show_acom ((step_parity ⊤ ^^2) (anno None test2_parity))"
value "show_acom ((step_parity ⊤ ^^3) (anno None test2_parity))"
value "show_acom ((step_parity ⊤ ^^4) (anno None test2_parity))"
value "show_acom ((step_parity ⊤ ^^5) (anno None test2_parity))"
value "show_acom_opt (AI_parity test2_parity)"

subsubsection "Termination"

global_interpretation Abs_Int_mono
where γ = γ_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

definition m_parity :: "parity ⇒ nat" where
"m_parity x = (if x=Either then 0 else 1)"

lemma measure_parity:
"(strict{(x::parity,y). x ⊑ y})^-1 ⊆ measure m_parity"