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(* Author: Tobias Nipkow *)
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theory AbsInt0_const
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imports AbsInt0
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begin
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subsection "Constant Propagation"
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datatype cval = Const val | Any
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fun rep_cval where
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"rep_cval (Const n) = {n}" |
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"rep_cval (Any) = UNIV"
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fun plus_cval where
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"plus_cval (Const m) (Const n) = Const(m+n)" |
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"plus_cval _ _ = Any"
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instantiation cval :: SL_top
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begin
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fun le_cval where
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"_ \<sqsubseteq> Any = True" |
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"Const n \<sqsubseteq> Const m = (n=m)" |
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"Any \<sqsubseteq> Const _ = False"
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fun join_cval where
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"Const m \<squnion> Const n = (if n=m then Const m else Any)" |
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"_ \<squnion> _ = Any"
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definition "Top = Any"
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instance
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proof
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case goal1 thus ?case by (cases x) simp_all
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next
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case goal2 thus ?case by(cases z, cases y, cases x, simp_all)
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next
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case goal3 thus ?case by(cases x, cases y, simp_all)
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next
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case goal4 thus ?case by(cases y, cases x, simp_all)
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next
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case goal5 thus ?case by(cases z, cases y, cases x, simp_all)
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next
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case goal6 thus ?case by(simp add: Top_cval_def)
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qed
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end
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interpretation Rep rep_cval
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proof
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case goal1 thus ?case
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by(cases a, cases b, simp, simp, cases b, simp, simp)
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qed
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interpretation Val_abs rep_cval Const plus_cval
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proof
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case goal1 show ?case by simp
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next
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case goal2 thus ?case
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by(cases a1, cases a2, simp, simp, cases a2, simp, simp)
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qed
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interpretation Abs_Int rep_cval Const plus_cval
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defines AI_const is AI
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and aval'_const is aval'
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..
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text{* Straight line code: *}
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definition "test1_const =
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''y'' ::= N 7;
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''z'' ::= Plus (V ''y'') (N 2);
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''y'' ::= Plus (V ''x'') (N 0)"
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value [code] "list (AI_const test1_const Top)"
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text{* Conditional: *}
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definition "test2_const =
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IF Less (N 41) (V ''x'') THEN ''x'' ::= N 5 ELSE ''x'' ::= N 5"
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value "list (AI_const test2_const Top)"
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text{* Conditional, test is ignored: *}
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definition "test3_const =
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''x'' ::= N 42;
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IF Less (N 41) (V ''x'') THEN ''x'' ::= N 5 ELSE ''x'' ::= N 6"
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value "list (AI_const test3_const Top)"
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text{* While: *}
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definition "test4_const =
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''x'' ::= N 0; WHILE B True DO ''x'' ::= N 0"
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value [code] "list (AI_const test4_const Top)"
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text{* While, test is ignored: *}
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definition "test5_const =
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''x'' ::= N 0; WHILE Less (V ''x'') (N 1) DO ''x'' ::= N 1"
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value [code] "list (AI_const test5_const Top)"
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text{* Iteration is needed: *}
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definition "test6_const =
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''x'' ::= N 0; ''y'' ::= N 0; ''z'' ::= N 2;
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WHILE Less (V ''x'') (N 1) DO (''x'' ::= V ''y''; ''y'' ::= V ''z'')"
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value [code] "list (AI_const test6_const Top)"
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text{* More iteration would be needed: *}
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definition "test7_const =
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''x'' ::= N 0; ''y'' ::= N 0; ''z'' ::= N 0; ''u'' ::= N 3;
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WHILE B True DO (''x'' ::= V ''y''; ''y'' ::= V ''z''; ''z'' ::= V ''u'')"
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value [code] "list (AI_const test7_const Top)"
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end
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