Theory C_like

theory C_like
imports Main
theory C_like imports Main begin

subsection "A C-like Language"

type_synonym state = "nat => nat"

datatype aexp = N nat | Deref aexp ("!") | Plus aexp aexp

fun aval :: "aexp => state => nat" where
"aval (N n) s = n" |
"aval (!a) s = s(aval a s)" |
"aval (Plus a1 a2) s = aval a1 s + aval a2 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 b1 b2) s = (if bval b1 s then bval b2 s else False)" |
"bval (Less a1 a2) s = (aval a1 s < aval a2 s)"


datatype
com = SKIP
| Assign aexp aexp ("_ ::= _" [61, 61] 61)
| New aexp aexp
| Seq com com ("_;/ _" [60, 61] 60)
| If bexp com com ("(IF _/ THEN _/ ELSE _)" [0, 0, 61] 61)
| While bexp com ("(WHILE _/ DO _)" [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: "[| (c1,sn1) => sn2; (c2,sn2) => sn3 |] ==>
(c1;c2, sn1) => sn3"
|

IfTrue: "[| bval b s; (c1,s,n) => tn |] ==>
(IF b THEN c1 ELSE c2, s,n) => tn"
|
IfFalse: "[| ¬bval b s; (c2,s,n) => tn |] ==>
(IF b THEN c1 ELSE c2, s,n) => tn"
|

WhileFalse: "¬bval b s ==> (WHILE b DO c,s,n) => (s,n)" |
WhileTrue:
"[| bval b s1; (c,s1,n) => sn2; (WHILE b DO c, sn2) => sn3 |] ==>
(WHILE b DO c, s1,n) => sn3"


code_pred big_step .

declare [[values_timeout = 3600]]

text{* Examples: *}

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 {* To show the first n variables in a @{typ "nat => nat"} state: *}
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