datatype aexp = N nat | Deref aexp (\<open>!\<close>) | Plus aexp aexp
fun aval :: "aexp \ state \ nat" where "aval (N n) s = n" | "aval (!a) s = s(aval a s)" | "aval (Plus a\<^sub>1 a\<^sub>2) s = aval a\<^sub>1 s + aval a\<^sub>2 s"
datatype bexp = Bc bool | Not bexp | And bexp bexp | Less aexp aexp
primrec bval :: "bexp \ state \ bool" where "bval (Bc v) _ = v" | "bval (Not b) s = (\ bval b s)" | "bval (And b\<^sub>1 b\<^sub>2) s = (if bval b\<^sub>1 s then bval b\<^sub>2 s else False)" | "bval (Less a\<^sub>1 a\<^sub>2) s = (aval a\<^sub>1 s < aval a\<^sub>2 s)"
datatype
com = SKIP
| Assign aexp aexp (\<open>_ ::= _\<close> [61, 61] 61)
| New aexp aexp
| Seq com com (\<open>_;/ _\<close> [60, 61] 60)
| If bexp com com (\<open>(IF _/ THEN _/ ELSE _)\<close> [0, 0, 61] 61)
| While bexp com (\<open>(WHILE _/ DO _)\<close> [0, 61] 61)
inductive
big_step :: "com \ state \ nat \ state \ nat \ bool" (infix \\\ 55) where
Skip: "(SKIP,sn) \ sn" |
Assign: "(lhs ::= a,s,n) \ (s(aval lhs s := aval a s),n)" |
New: "(New lhs a,s,n) \ (s(aval lhs s := n), n+aval a s)" |
Seq: "\ (c\<^sub>1,sn\<^sub>1) \ sn\<^sub>2; (c\<^sub>2,sn\<^sub>2) \ sn\<^sub>3 \ \
(c\<^sub>1;c\<^sub>2, sn\<^sub>1) \<Rightarrow> sn\<^sub>3" |
IfTrue: "\ bval b s; (c\<^sub>1,s,n) \ tn \ \
(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s,n) \<Rightarrow> tn" |
IfFalse: "\ \bval b s; (c\<^sub>2,s,n) \ tn \ \
(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s,n) \<Rightarrow> tn" |
WhileFalse: "\bval b s \ (WHILE b DO c,s,n) \ (s,n)" |
WhileTrue: "\ bval b s\<^sub>1; (c,s\<^sub>1,n) \ sn\<^sub>2; (WHILE b DO c, sn\<^sub>2) \ sn\<^sub>3 \ \
(WHILE b DO c, s\<^sub>1,n) \<Rightarrow> sn\<^sub>3"
code_pred big_step .
declare [[values_timeout = 3600]]
text\<open>Examples:\<close>
definition "array_sum =
WHILE Less (!(N 0)) (Plus (!(N 1)) (N 1))
DO ( N 2 ::= Plus (!(N 2)) (!(!(N 0)));
N 0 ::= Plus (!(N 0)) (N 1) )"
text\<open>To show the first n variables in a \<^typ>\<open>nat \<Rightarrow> nat\<close> state:\<close> definition "list t n = map t [0 ..< n]"
values "{list t n |t n. (array_sum, nth[3,4,0,3,7],5) \ (t,n)}"
definition "linked_list_sum =
WHILE Less (N 0) (!(N 0))
DO ( N 1 ::= Plus(!(N 1)) (!(!(N 0)));
N 0 ::= !(Plus(!(N 0))(N 1)) )"
values "{list t n |t n. (linked_list_sum, nth[4,0,3,0,7,2],6) \ (t,n)}"
definition "array_init =
New (N 0) (!(N 1)); N 2 ::= !(N 0);
WHILE Less (!(N 2)) (Plus (!(N 0)) (!(N 1)))
DO ( !(N 2) ::= !(N 2);
N 2 ::= Plus (!(N 2)) (N 1) )"
values "{list t n |t n. (array_init, nth[5,2,7],3) \ (t,n)}"
definition "linked_list_init =
WHILE Less (!(N 1)) (!(N 0))
DO ( New (N 3) (N 2);
N 1 ::= Plus (!(N 1)) (N 1);
!(N 3) ::= !(N 1);
Plus (!(N 3)) (N 1) ::= !(N 2);
N 2 ::= !(N 3) )"
values "{list t n |t n. (linked_list_init, nth[2,0,0,0],4) \ (t,n)}"
end
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