# Theory ASM

section "Stack Machine and Compilation"

theory ASM imports AExp begin

subsection "Stack Machine"

text_raw‹\snip{ASMinstrdef}{0}{1}{%›
text_raw‹}%endsnip›

text_raw‹\snip{ASMstackdef}{1}{2}{%›
type_synonym stack = "val list"
text_raw‹}%endsnip›

text‹\noindent Abbreviations are transparent: they are unfolded after
parsing and folded back again before printing. Internally, they do not
exist.›

text_raw‹\snip{ASMexeconedef}{0}{1}{%›
fun exec1 :: "instr ⇒ state ⇒ stack ⇒ stack" where
"exec1 (LOADI n) _ stk  =  n # stk" |
"exec1 (LOAD x) s stk  =  s(x) # stk" |
"exec1  ADD _ (j # i # stk)  =  (i + j) # stk"
text_raw‹}%endsnip›

text_raw‹\snip{ASMexecdef}{1}{2}{%›
fun exec :: "instr list ⇒ state ⇒ stack ⇒ stack" where
"exec [] _ stk = stk" |
"exec (i#is) s stk = exec is s (exec1 i s stk)"
text_raw‹}%endsnip›

lemma exec_append[simp]:
"exec (is1@is2) s stk = exec is2 s (exec is1 s stk)"
apply(induction is1 arbitrary: stk)
apply (auto)
done

subsection "Compilation"

text_raw‹\snip{ASMcompdef}{0}{2}{%›
fun comp :: "aexp ⇒ instr list" where
"comp (N n) = [LOADI n]" |
"comp (V x) = [LOAD x]" |
"comp (Plus e⇩1 e⇩2) = comp e⇩1 @ comp e⇩2 @ [ADD]"
text_raw‹}%endsnip›

value "comp (Plus (Plus (V ''x'') (N 1)) (V ''z''))"

theorem exec_comp: "exec (comp a) s stk = aval a s # stk"
apply(induction a arbitrary: stk)
apply (auto)
done

end