(* Author: Tobias Nipkow *)

theory Abs_Int_den0_const

imports Abs_Int_den0

begin

subsection "Constant Propagation"

datatype cval = Const val | Any

fun rep_cval where

"rep_cval (Const n) = {n}" |

"rep_cval (Any) = UNIV"

fun plus_cval where

"plus_cval (Const m) (Const n) = Const(m+n)" |

"plus_cval _ _ = Any"

instantiation cval :: SL_top

begin

fun le_cval where

"_ \<sqsubseteq> Any = True" |

"Const n \<sqsubseteq> Const m = (n=m)" |

"Any \<sqsubseteq> Const _ = False"

fun join_cval where

"Const m \<squnion> Const n = (if n=m then Const m else Any)" |

"_ \<squnion> _ = Any"

definition "Top = Any"

instance

proof

  case goal1 thus ?case by (cases x) simp_all

next

  case goal2 thus ?case by(cases z, cases y, cases x, simp_all)

next

  case goal3 thus ?case by(cases x, cases y, simp_all)

next

  case goal4 thus ?case by(cases y, cases x, simp_all)

next

  case goal5 thus ?case by(cases z, cases y, cases x, simp_all)

next

  case goal6 thus ?case by(simp add: Top_cval_def)

qed

end

interpretation Rep rep_cval

proof

  case goal1 thus ?case

    by(cases a, cases b, simp, simp, cases b, simp, simp)

qed

interpretation Val_abs rep_cval Const plus_cval

proof

  case goal1 show ?case by simp

next

  case goal2 thus ?case

    by(cases a1, cases a2, simp, simp, cases a2, simp, simp)

qed

interpretation Abs_Int rep_cval Const plus_cval "(iter' 3)"

defines AI_const is AI

and aval'_const is aval'

proof qed (auto simp: iter'_pfp_above)

text{* Straight line code: *}

definition "test1_const =

 ''y'' ::= N 7;

 ''z'' ::= Plus (V ''y'') (N 2);

 ''y'' ::= Plus (V ''x'') (N 0)"

text{* Conditional: *}

definition "test2_const =

 IF Less (N 41) (V ''x'') THEN ''x'' ::= N 5 ELSE ''x'' ::= N 5"

text{* Conditional, test is ignored: *}

definition "test3_const =

 ''x'' ::= N 42;

 IF Less (N 41) (V ''x'') THEN ''x'' ::= N 5 ELSE ''x'' ::= N 6"

text{* While: *}

definition "test4_const =

 ''x'' ::= N 0; WHILE B True DO ''x'' ::= N 0"

text{* While, test is ignored: *}

definition "test5_const =

 ''x'' ::= N 0; WHILE Less (V ''x'') (N 1) DO ''x'' ::= N 1"

text{* Iteration is needed: *}

definition "test6_const =

  ''x'' ::= N 0; ''y'' ::= N 0; ''z'' ::= N 2;

  WHILE Less (V ''x'') (N 1) DO (''x'' ::= V ''y''; ''y'' ::= V ''z'')"

text{* More iteration would be needed: *}

definition "test7_const =

  ''x'' ::= N 0; ''y'' ::= N 0; ''z'' ::= N 0; ''u'' ::= N 3;

  WHILE Less (V ''x'') (N 1)

  DO (''x'' ::= V ''y''; ''y'' ::= V ''z''; ''z'' ::= V ''u'')"

value [code] "list (AI_const test1_const Top)"

value [code] "list (AI_const test2_const Top)"

value [code] "list (AI_const test3_const Top)"

value [code] "list (AI_const test4_const Top)"

value [code] "list (AI_const test5_const Top)"

value [code] "list (AI_const test6_const Top)"

value [code] "list (AI_const test7_const Top)"

end