1 theory BExp imports AExp begin
3 subsection "Boolean Expressions"
5 datatype bexp = B bool | Not bexp | And bexp bexp | Less aexp aexp
7 fun bval :: "bexp \<Rightarrow> state \<Rightarrow> bool" where
9 "bval (Not b) s = (\<not> bval b s)" |
10 "bval (And b1 b2) s = (if bval b1 s then bval b2 s else False)" |
11 "bval (Less a1 a2) s = (aval a1 s < aval a2 s)"
13 value "bval (Less (V ''x'') (Plus (N 3) (V ''y'')))
14 (lookup [(''x'',3),(''y'',1)])"
17 subsection "Optimization"
19 text{* Optimized constructors: *}
21 fun less :: "aexp \<Rightarrow> aexp \<Rightarrow> bexp" where
22 "less (N n1) (N n2) = B(n1 < n2)" |
23 "less a1 a2 = Less a1 a2"
25 lemma [simp]: "bval (less a1 a2) s = (aval a1 s < aval a2 s)"
26 apply(induct a1 a2 rule: less.induct)
30 fun "and" :: "bexp \<Rightarrow> bexp \<Rightarrow> bexp" where
31 "and (B True) b = b" |
32 "and b (B True) = b" |
33 "and (B False) b = B False" |
34 "and b (B False) = B False" |
35 "and b1 b2 = And b1 b2"
37 lemma bval_and[simp]: "bval (and b1 b2) s = (bval b1 s \<and> bval b2 s)"
38 apply(induct b1 b2 rule: and.induct)
42 fun not :: "bexp \<Rightarrow> bexp" where
43 "not (B True) = B False" |
44 "not (B False) = B True" |
47 lemma bval_not[simp]: "bval (not b) s = (~bval b s)"
48 apply(induct b rule: not.induct)
52 text{* Now the overall optimizer: *}
54 fun bsimp :: "bexp \<Rightarrow> bexp" where
55 "bsimp (Less a1 a2) = less (asimp a1) (asimp a2)" |
56 "bsimp (And b1 b2) = and (bsimp b1) (bsimp b2)" |
57 "bsimp (Not b) = not(bsimp b)" |
60 value "bsimp (And (Less (N 0) (N 1)) b)"
62 value "bsimp (And (Less (N 1) (N 0)) (B True))"
64 theorem "bval (bsimp b) s = bval b s"