section "Stack Machine and Compilation"

theory ASM imports AExp begin

subsection "Stack Machine"

text_raw‹\snip{ASMinstrdef}{0}{1}{%›
datatype instr = LOADI val | LOAD vname | ADD
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
  and folded back again before printing. Internally, they do not
 .›

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›

value "exec [LOADI 5, LOAD ''y'', ADD] <''x'' := 42, ''y'' := 43> [50]"

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 e₁ e₂) = comp e₁ @ comp e₂ @ [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