(** * Abstract syntax *)

(** Like Linear, the Mach language is organized as lists of instructions
  operating over machine registers, with default fall-through behaviour
  and explicit labels and branch instructions.

  The main difference with Linear lies in the instructions used to
  access the activation record.  Mach has three such instructions:
  [Mgetstack] and [Msetstack] to read and write within the activation
  record for the current function, at a given word offset and with a
  given type; and [Mgetparam], to read within the activation record of
  the caller.

  These instructions implement a more concrete view of the activation
  record than the the [Lgetstack] and [Lsetstack] instructions of
  Linear: actual offsets are used instead of abstract stack slots, and the
  distinction between the caller's frame and the callee's frame is
  made explicit. *)

Definition label := positive.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Inductive instruction: Type :=
  | Mgetstack: ptrofs -> typ -> mreg -> instruction
  | Msetstack: mreg -> ptrofs -> typ -> instruction
  | Mgetparam: ptrofs -> typ -> mreg -> instruction
  | Mop: operation -> list mreg -> mreg -> instruction
  | Mload: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mstore: memory_chunk -> addressing -> list mreg -> mreg -> instruction
  | Mcall: signature -> mreg + ident -> instruction
  | Mtailcall: signature -> mreg + ident -> instruction
  | Mbuiltin: external_function -> list (builtin_arg mreg) -> builtin_res mreg -> instruction
  | Mlabel: label -> instruction
  | Mgoto: label -> instruction
  | Mcond: condition -> list mreg -> label -> instruction
  | Mjumptable: mreg -> list label -> instruction
  | Mreturn: instruction.Definition code := list instruction.Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.Record function: Type := mkfunction
  { fn_sig: signature;
    fn_code: code;
    fn_stacksize: Z;
    fn_link_ofs: ptrofs;
    fn_retaddr_ofs: ptrofs }.Definition fundef := AST.fundef function.Definition program := AST.program fundef unit.Definition funsig (fd: fundef) :=
  match fd with
  | Internal f => fn_sig f
  | External ef => ef_sig ef
  end.Definition genv := Genv.t fundef unit.(** * Operational semantics *)

(** The semantics for Mach is close to that of [Linear]: they differ only
  on the interpretation of stack slot accesses.  In Mach, these
  accesses are interpreted as memory accesses relative to the
  stack pointer.  More precisely:
- [Mgetstack ofs ty r] is a memory load at offset [ofs * 4] relative
  to the stack pointer.
- [Msetstack r ofs ty] is a memory store at offset [ofs * 4] relative
  to the stack pointer.
- [Mgetparam ofs ty r] is a memory load at offset [ofs * 4]
  relative to the pointer found at offset 0 from the stack pointer.
  The semantics maintain a linked structure of activation records,
  with the current record containing a pointer to the record of the
  caller function at offset 0.

In addition to this linking of activation records, the
semantics also make provisions for storing a back link at offset
[f.(fn_link_ofs)] from the stack pointer, and a return address at
offset [f.(fn_retaddr_ofs)].  The latter stack location will be used
by the Asm code generated by [Asmgen] to save the return address into
the caller at the beginning of a function, then restore it and jump to
it at the end of a function.  The Mach concrete semantics does not
attach any particular meaning to the pointer stored in this reserved
location, but makes sure that it is preserved during execution of a
function.  The [return_address_offset] parameter is used to guess the
value of the return address that the Asm code generated later will
store in the reserved location.
*)

Definition load_stack (m: mem) (sp: val) (ty: typ) (ofs: ptrofs) :=
  Mem.loadv (chunk_of_type ty) m (Val.offset_ptr sp ofs).Definition store_stack (m: mem) (sp: val) (ty: typ) (ofs: ptrofs) (v: val) :=
  Mem.storev (chunk_of_type ty) m (Val.offset_ptr sp ofs) v.Definition t := mreg.Definition eq := mreg_eq.Module Regmap := EMap(RegEq).Definition regset := Regmap.t val.Notation "a ## b" := (List.map a b) (at level 1).Notation "a # b <- c" := (Regmap.set b c a) (at level 1, b at next level).Fixpoint undef_regs (rl: list mreg) (rs: regset) {struct rl} : regset :=
  match rl with
  | nil => rs
  | r1 :: rl' => Regmap.set r1 Vundef (undef_regs rl' rs)
  end.

Lemma undef_regs_same: forall r rl rs, In r rl -> undef_regs rl rs r = Vundef.