(* Author: Tobias Nipkow *)

theory Abs_Int1_const

imports Abs_Int1

begin

subsection "Constant Propagation"

datatype const = Const val | Any

fun \<gamma>_const where

"\<gamma>_const (Const n) = {n}" |

"\<gamma>_const (Any) = UNIV"

fun plus_const where

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

"plus_const _ _ = Any"

lemma plus_const_cases: "plus_const a1 a2 =

  (case (a1,a2) of (Const m, Const n) \<Rightarrow> Const(m+n) | _ \<Rightarrow> Any)"

by(auto split: prod.split const.split)

instantiation const :: semilattice

begin

fun le_const where

"_ \<sqsubseteq> Any = True" |

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

"Any \<sqsubseteq> Const _ = False"

fun join_const 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_const_def)

qed

end

interpretation Val_abs

where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const

proof

  case goal1 thus ?case

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

next

  case goal2 show ?case by(simp add: Top_const_def)

next

  case goal3 show ?case by simp

next

  case goal4 thus ?case

    by(auto simp: plus_const_cases split: const.split)

qed

interpretation Abs_Int

where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const

defines AI_const is AI and step_const is step' and aval'_const is aval'

..

subsubsection "Tests"

definition "steps c i = (step_const(top(vars c)) ^^ i) (bot c)"

value "show_acom (steps test1_const 0)"

value "show_acom (steps test1_const 1)"

value "show_acom (steps test1_const 2)"

value "show_acom (steps test1_const 3)"

value "show_acom (the(AI_const test1_const))"

value "show_acom (the(AI_const test2_const))"

value "show_acom (the(AI_const test3_const))"

value "show_acom (steps test4_const 0)"

value "show_acom (steps test4_const 1)"

value "show_acom (steps test4_const 2)"

value "show_acom (steps test4_const 3)"

value "show_acom (steps test4_const 4)"

value "show_acom (the(AI_const test4_const))"

value "show_acom (steps test5_const 0)"

value "show_acom (steps test5_const 1)"

value "show_acom (steps test5_const 2)"

value "show_acom (steps test5_const 3)"

value "show_acom (steps test5_const 4)"

value "show_acom (steps test5_const 5)"

value "show_acom (steps test5_const 6)"

value "show_acom (the(AI_const test5_const))"

value "show_acom (steps test6_const 0)"

value "show_acom (steps test6_const 1)"

value "show_acom (steps test6_const 2)"

value "show_acom (steps test6_const 3)"

value "show_acom (steps test6_const 4)"

value "show_acom (steps test6_const 5)"

value "show_acom (steps test6_const 6)"

value "show_acom (steps test6_const 7)"

value "show_acom (steps test6_const 8)"

value "show_acom (steps test6_const 9)"

value "show_acom (steps test6_const 10)"

value "show_acom (steps test6_const 11)"

value "show_acom (steps test6_const 12)"

value "show_acom (steps test6_const 13)"

value "show_acom (the(AI_const test6_const))"

text{* Monotonicity: *}

interpretation Abs_Int_mono

where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const

proof

  case goal1 thus ?case

    by(auto simp: plus_const_cases split: const.split)

qed

text{* Termination: *}

definition "m_const x = (case x of Const _ \<Rightarrow> 1 | Any \<Rightarrow> 0)"

interpretation Abs_Int_measure

where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const

and m = m_const and h = "1"

proof

  case goal1 thus ?case by(auto simp: m_const_def split: const.splits)

next

  case goal2 thus ?case by(auto simp: m_const_def split: const.splits)

next

  case goal3 thus ?case by(auto simp: m_const_def split: const.splits)

qed

thm AI_Some_measure

end