theory C_like imports Main begin
subsection "A C-like Language"
type_synonym state = "nat \<Rightarrow> nat"
datatype aexp = N nat | Deref aexp ("!") | Plus aexp aexp
fun aval :: "aexp \<Rightarrow> state \<Rightarrow> nat" where
"aval (N n) s = n" |
"aval (!a) s = s(aval a s)" |
"aval (Plus a\<^isub>1 a\<^isub>2) s = aval a\<^isub>1 s + aval a\<^isub>2 s"
datatype bexp = B bool | Not bexp | And bexp bexp | Less aexp aexp
primrec bval :: "bexp \<Rightarrow> state \<Rightarrow> bool" where
"bval (B bv) _ = bv" |
"bval (Not b) s = (\<not> bval b s)" |
"bval (And b\<^isub>1 b\<^isub>2) s = (if bval b\<^isub>1 s then bval b\<^isub>2 s else False)" |
"bval (Less a\<^isub>1 a\<^isub>2) s = (aval a\<^isub>1 s < aval a\<^isub>2 s)"
datatype
com = SKIP
| Assign aexp aexp ("_ ::= _" [61, 61] 61)
| New aexp aexp
| Semi 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 \<times> state \<times> nat \<Rightarrow> state \<times> nat \<Rightarrow> bool" (infix "\<Rightarrow>" 55)
where
Skip: "(SKIP,sn) \<Rightarrow> sn" |
Assign: "(lhs ::= a,s,n) \<Rightarrow> (s(aval lhs s := aval a s),n)" |
New: "(New lhs a,s,n) \<Rightarrow> (s(aval lhs s := n), n+aval a s)" |
Semi: "\<lbrakk> (c\<^isub>1,sn\<^isub>1) \<Rightarrow> sn\<^isub>2; (c\<^isub>2,sn\<^isub>2) \<Rightarrow> sn\<^isub>3 \<rbrakk> \<Longrightarrow>
(c\<^isub>1;c\<^isub>2, sn\<^isub>1) \<Rightarrow> sn\<^isub>3" |
IfTrue: "\<lbrakk> bval b s; (c\<^isub>1,s,n) \<Rightarrow> tn \<rbrakk> \<Longrightarrow>
(IF b THEN c\<^isub>1 ELSE c\<^isub>2, s,n) \<Rightarrow> tn" |
IfFalse: "\<lbrakk> \<not>bval b s; (c\<^isub>2,s,n) \<Rightarrow> tn \<rbrakk> \<Longrightarrow>
(IF b THEN c\<^isub>1 ELSE c\<^isub>2, s,n) \<Rightarrow> tn" |
WhileFalse: "\<not>bval b s \<Longrightarrow> (WHILE b DO c,s,n) \<Rightarrow> (s,n)" |
WhileTrue:
"\<lbrakk> bval b s\<^isub>1; (c,s\<^isub>1,n) \<Rightarrow> sn\<^isub>2; (WHILE b DO c, sn\<^isub>2) \<Rightarrow> sn\<^isub>3 \<rbrakk> \<Longrightarrow>
(WHILE b DO c, s\<^isub>1,n) \<Rightarrow> sn\<^isub>3"
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 \<Rightarrow> 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) \<Rightarrow> (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) \<Rightarrow> (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) \<Rightarrow> (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) \<Rightarrow> (t,n)}"
end