Abstract syntax and semantics for IA32 assembly language

Require Import Coqlib Maps.
Require Import AST Integers Floats Values Memory Events Globalenvs Smallstep.
Require Import Locations Stacklayout Conventions.

Abstract syntax

Registers.

Integer registers.

Inductive ireg: Type :=
| RAX | RBX | RCX | RDX | RSI | RDI | RBP | RSP
| R8 | R9 | R10 | R11 | R12 | R13 | R14 | R15.

Floating-point registers, i.e. SSE2 registers

Inductive freg: Type :=
| XMM0 | XMM1 | XMM2 | XMM3 | XMM4 | XMM5 | XMM6 | XMM7
| XMM8 | XMM9 | XMM10 | XMM11 | XMM12 | XMM13 | XMM14 | XMM15.

Lemma ireg_eq: forall (x y: ireg), {x=y} + {x<>y}.

Proof.

decide equality. Defined.

Lemma freg_eq: forall (x y: freg), {x=y} + {x<>y}.

Proof.

decide equality. Defined.

Bits of the flags register.

Inductive crbit: Type :=
| ZF | CF | PF | SF | OF.

All registers modeled here.

Inductive preg: Type :=
  | PC: preg (* program counter *)
  | IR: ireg -> preg (* integer register *)
  | FR: freg -> preg (* XMM register *)
  | ST0: preg (* top of FP stack *)
  | CR: crbit -> preg (* bit of the flags register *)
  | RA: preg. (* pseudo-reg representing return address *)

Coercion IR: ireg >-> preg.
Coercion FR: freg >-> preg.
Coercion CR: crbit >-> preg.

Conventional names for stack pointer (SP) and return address (RA)

Notation SP := RSP (only parsing).

Instruction set.

Definition label := positive.

General form of an addressing mode.

Inductive addrmode: Type :=
  | Addrmode (base: option ireg)
             (ofs: option (ireg * Z))
             (const: Z + ident * ptrofs).

Testable conditions (for conditional jumps and more).

Instructions. IA32 instructions accept many combinations of registers, memory references and immediate constants as arguments. Here, we list only the combinations that we actually use. Naming conventions for types:

b: 8 bits
w: 16 bits ("word")
l: 32 bits ("longword")
q: 64 bits ("quadword")
d or sd: FP double precision (64 bits)
s or ss: FP single precision (32 bits)

Naming conventions for operands:

r: integer register operand
f: XMM register operand
m: memory operand
i: immediate integer operand
s: immediate symbol operand
l: immediate label operand
cl: the CL register

For two-operand instructions, the first suffix describes the result (and first argument), the second suffix describes the second argument.

Inductive instruction: Type :=

Moves

  | Pmov_rr (rd: ireg) (r1: ireg) (* mov (integer) *)
  | Pmovl_ri (rd: ireg) (n: int)
  | Pmovq_ri (rd: ireg) (n: int64)
  | Pmov_rs (rd: ireg) (id: ident)
  | Pmovl_rm (rd: ireg) (a: addrmode)
  | Pmovq_rm (rd: ireg) (a: addrmode)
  | Pmovl_mr (a: addrmode) (rs: ireg)
  | Pmovq_mr (a: addrmode) (rs: ireg)
  | Pmovsd_ff (rd: freg) (r1: freg) (* movsd (single 64-bit float) *)
  | Pmovsd_fi (rd: freg) (n: float) (* (pseudo-instruction) *)
  | Pmovsd_fm (rd: freg) (a: addrmode)
  | Pmovsd_mf (a: addrmode) (r1: freg)
  | Pmovss_fi (rd: freg) (n: float32) (* movss (single 32-bit float) *)
  | Pmovss_fm (rd: freg) (a: addrmode)
  | Pmovss_mf (a: addrmode) (r1: freg)
  | Pfldl_m (a: addrmode) (* fld double precision *)
  | Pfstpl_m (a: addrmode) (* fstp double precision *)
  | Pflds_m (a: addrmode) (* fld simple precision *)
  | Pfstps_m (a: addrmode) (* fstp simple precision *)

Moves with conversion

  | Pmovb_mr (a: addrmode) (rs: ireg) (* mov (8-bit int) *)
  | Pmovw_mr (a: addrmode) (rs: ireg) (* mov (16-bit int) *)
  | Pmovzb_rr (rd: ireg) (rs: ireg) (* movzb (8-bit zero-extension) *)
  | Pmovzb_rm (rd: ireg) (a: addrmode)
  | Pmovsb_rr (rd: ireg) (rs: ireg) (* movsb (8-bit sign-extension) *)
  | Pmovsb_rm (rd: ireg) (a: addrmode)
  | Pmovzw_rr (rd: ireg) (rs: ireg) (* movzw (16-bit zero-extension) *)
  | Pmovzw_rm (rd: ireg) (a: addrmode)
  | Pmovsw_rr (rd: ireg) (rs: ireg) (* movsw (16-bit sign-extension) *)
  | Pmovsw_rm (rd: ireg) (a: addrmode)
  | Pmovzl_rr (rd: ireg) (rs: ireg) (* movzl (32-bit zero-extension) *)
  | Pmovsl_rr (rd: ireg) (rs: ireg) (* movsl (32-bit sign-extension) *)
  | Pmovls_rr (rd: ireg)
  | Pcvtsd2ss_ff (rd: freg) (r1: freg) (* conversion to single float *)
  | Pcvtss2sd_ff (rd: freg) (r1: freg) (* conversion to double float *)
  | Pcvttsd2si_rf (rd: ireg) (r1: freg) (* double to signed int *)
  | Pcvtsi2sd_fr (rd: freg) (r1: ireg) (* signed int to double *)
  | Pcvttss2si_rf (rd: ireg) (r1: freg) (* single to signed int *)
  | Pcvtsi2ss_fr (rd: freg) (r1: ireg) (* signed int to single *)
  | Pcvttsd2sl_rf (rd: ireg) (r1: freg) (* double to signed long *)
  | Pcvtsl2sd_fr (rd: freg) (r1: ireg) (* signed long to double *)
  | Pcvttss2sl_rf (rd: ireg) (r1: freg) (* single to signed long *)
  | Pcvtsl2ss_fr (rd: freg) (r1: ireg) (* signed long to single *)

Integer arithmetic

Floating-point arithmetic

Branches and calls

Saving and restoring registers

  | Pmov_rm_a (rd: ireg) (a: addrmode) (* like Pmov_rm, using Many64 chunk *)
  | Pmov_mr_a (a: addrmode) (rs: ireg) (* like Pmov_mr, using Many64 chunk *)
  | Pmovsd_fm_a (rd: freg) (a: addrmode) (* like Pmovsd_fm, using Many64 chunk *)
  | Pmovsd_mf_a (a: addrmode) (r1: freg) (* like Pmovsd_mf, using Many64 chunk *)

Pseudo-instructions

  | Plabel(l: label)
  | Pallocframe(sz: Z)(ofs_ra ofs_link: ptrofs)
  | Pfreeframe(sz: Z)(ofs_ra ofs_link: ptrofs)
  | Pbuiltin(ef: external_function)(args: list (builtin_arg preg))(res: builtin_res preg)

Instructions not generated by Asmgen -- TO CHECK

  | Padcl_ri (rd: ireg) (n: int)
  | Padcl_rr (rd: ireg) (r2: ireg)
  | Paddl_mi (a: addrmode) (n: int)
  | Paddl_rr (rd: ireg) (r2: ireg)
  | Pbsfl (rd: ireg) (r1: ireg)
  | Pbsfq (rd: ireg) (r1: ireg)
  | Pbsrl (rd: ireg) (r1: ireg)
  | Pbsrq (rd: ireg) (r1: ireg)
  | Pbswap64 (rd: ireg)
  | Pbswap32 (rd: ireg)
  | Pbswap16 (rd: ireg)
  | Pcfi_adjust (n: int)
  | Pfmadd132 (rd: freg) (r2: freg) (r3: freg)
  | Pfmadd213 (rd: freg) (r2: freg) (r3: freg)
  | Pfmadd231 (rd: freg) (r2: freg) (r3: freg)
  | Pfmsub132 (rd: freg) (r2: freg) (r3: freg)
  | Pfmsub213 (rd: freg) (r2: freg) (r3: freg)
  | Pfmsub231 (rd: freg) (r2: freg) (r3: freg)
  | Pfnmadd132 (rd: freg) (r2: freg) (r3: freg)
  | Pfnmadd213 (rd: freg) (r2: freg) (r3: freg)
  | Pfnmadd231 (rd: freg) (r2: freg) (r3: freg)
  | Pfnmsub132 (rd: freg) (r2: freg) (r3: freg)
  | Pfnmsub213 (rd: freg) (r2: freg) (r3: freg)
  | Pfnmsub231 (rd: freg) (r2: freg) (r3: freg)
  | Pmaxsd (rd: freg) (r2: freg)
  | Pminsd (rd: freg) (r2: freg)
  | Pmovb_rm (rd: ireg) (a: addrmode)
  | Pmovsq_mr (a: addrmode) (rs: freg)
  | Pmovsq_rm (rd: freg) (a: addrmode)
  | Pmovsb
  | Pmovsw
  | Pmovw_rm (rd: ireg) (ad: addrmode)
  | Prep_movsl
  | Psbbl_rr (rd: ireg) (r2: ireg)
  | Psqrtsd (rd: freg) (r1: freg)
  | Psubl_ri (rd: ireg) (n: int)
  | Psubq_ri (rd: ireg) (n: int64).

Definition code := list instruction.
Record function : Type := mkfunction { fn_sig: signature; fn_code: code }.
Definition fundef := AST.fundef function.
Definition program := AST.program fundef unit.

Operational semantics

Lemma preg_eq: forall (x y: preg), {x=y} + {x<>y}.

Proof.

decide equality. apply ireg_eq. apply freg_eq. decide equality. Defined.

Module PregEq.
Definition t := preg.
Definition eq := preg_eq.
End PregEq.

Module Pregmap := EMap(PregEq).

Definition regset := Pregmap.t val.
Definition genv := Genv.t fundef unit.

Notation "a # b" := (a b) (at level 1, only parsing) : asm.
Notation "a # b <- c" := (Pregmap.set b c a) (at level 1, b at next level) : asm.

Open Scope asm.

Undefining some registers

Fixpoint undef_regs (l: list preg) (rs: regset) : regset :=
  match l with
  | nil => rs
  | r :: l' => undef_regs l' (rs#r <- Vundef)
  end.

Assigning a register pair

Definition set_pair (p: rpair preg) (v: val) (rs: regset) : regset :=
  match p with
  | One r => rs#r <- v
  | Twolong rhi rlo => rs#rhi <- (Val.hiword v) #rlo <- (Val.loword v)
  end.

Assigning the result of a builtin

Fixpoint set_res (res: builtin_res preg) (v: val) (rs: regset) : regset :=
  match res with
  | BR r => rs#r <- v
  | BR_none => rs
  | BR_splitlong hi lo => set_res lo (Val.loword v) (set_res hi (Val.hiword v) rs)
  end.

Section RELSEM.

Looking up instructions in a code sequence by position.

Fixpoint find_instr (pos: Z) (c: code) {struct c} : option instruction :=
  match c with
  | nil => None
  | i :: il => if zeq pos 0 then Some i else find_instr (pos - 1) il
  end.

Position corresponding to a label

Definition is_label (lbl: label) (instr: instruction) : bool :=
  match instr with
  | Plabel lbl' => if peq lbl lbl' then true else false
  | _ => false
  end.

Lemma is_label_correct:
  forall lbl instr,
  if is_label lbl instr then instr = Plabel lbl else instr <> Plabel lbl.

Proof.

intros. destruct instr; simpl; try discriminate.
case (peq lbl l); intro; congruence.
Qed.

Fixpoint label_pos (lbl: label) (pos: Z) (c: code) {struct c} : option Z :=
  match c with
  | nil => None
  | instr :: c' =>
      if is_label lbl instr then Some (pos + 1) else label_pos lbl (pos + 1) c'
  end.

Variable ge: genv.

Evaluating an addressing mode

Definition eval_addrmode32 (a: addrmode) (rs: regset) : val :=
  let '(Addrmode base ofs const) := a in
  Val.add (match base with
             | None => Vint Int.zero
             | Some r => rs r
            end)
  (Val.add (match ofs with
             | None => Vint Int.zero
             | Some(r, sc) =>
                if zeq sc 1
                then rs r
                else Val.mul (rs r) (Vint (Int.repr sc))
             end)
           (match const with
            | inl ofs => Vint (Int.repr ofs)
            | inr(id, ofs) => Genv.symbol_address ge id ofs
            end)).

Definition eval_addrmode64 (a: addrmode) (rs: regset) : val :=
  let '(Addrmode base ofs const) := a in
  Val.addl (match base with
             | None => Vlong Int64.zero
             | Some r => rs r
            end)
  (Val.addl (match ofs with
             | None => Vlong Int64.zero
             | Some(r, sc) =>
                if zeq sc 1
                then rs r
                else Val.mull (rs r) (Vlong (Int64.repr sc))
             end)
           (match const with
            | inl ofs => Vlong (Int64.repr ofs)
            | inr(id, ofs) => Genv.symbol_address ge id ofs
            end)).

Definition eval_addrmode (a: addrmode) (rs: regset) : val :=
  if Archi.ptr64 then eval_addrmode64 a rs else eval_addrmode32 a rs.

Performing a comparison

Integer comparison between x and y:

ZF = 1 if x = y, 0 if x != y
CF = 1 if x <u y, 0 if x >=u y
SF = 1 if x - y is negative, 0 if x - y is positive
OF = 1 if x - y overflows (signed), 0 if not
PF is undefined

Definition compare_ints (x y: val) (rs: regset) (m: mem): regset :=
  rs #ZF <- (Val.cmpu (Mem.valid_pointer m) Ceq x y)
     #CF <- (Val.cmpu (Mem.valid_pointer m) Clt x y)
     #SF <- (Val.negative (Val.sub x y))
     #OF <- (Val.sub_overflow x y)
     #PF <- Vundef.

Definition check_compare_ints (x y: val) (m: mem) : bool :=
  if (Val.cmpu_bool (Mem.valid_pointer m) Ceq x y)
  then true
  else false.

Definition compare_longs (x y: val) (rs: regset) (m: mem): regset :=
  rs #ZF <- (Val.maketotal (Val.cmplu (Mem.valid_pointer m) Ceq x y))
     #CF <- (Val.maketotal (Val.cmplu (Mem.valid_pointer m) Clt x y))
     #SF <- (Val.negativel (Val.subl x y))
     #OF <- (Val.subl_overflow x y)
     #PF <- Vundef.

Floating-point comparison between x and y:

ZF = 1 if x=y or unordered, 0 if x<>y
CF = 1 if x<y or unordered, 0 if x>=y
PF = 1 if unordered, 0 if ordered.
SF and 0F are undefined

Definition compare_floats (vx vy: val) (rs: regset) : regset :=
  match vx, vy with
  | Vfloat x, Vfloat y =>
      rs #ZF <- (Val.of_bool (negb (Float.cmp Cne x y)))
         #CF <- (Val.of_bool (negb (Float.cmp Cge x y)))
         #PF <- (Val.of_bool (negb (Float.cmp Ceq x y || Float.cmp Clt x y || Float.cmp Cgt x y)))
         #SF <- Vundef
         #OF <- Vundef
  | _, _ =>
      undef_regs (CR ZF :: CR CF :: CR PF :: CR SF :: CR OF :: nil) rs
  end.

Definition compare_floats32 (vx vy: val) (rs: regset) : regset :=
  match vx, vy with
  | Vsingle x, Vsingle y =>
      rs #ZF <- (Val.of_bool (negb (Float32.cmp Cne x y)))
         #CF <- (Val.of_bool (negb (Float32.cmp Cge x y)))
         #PF <- (Val.of_bool (negb (Float32.cmp Ceq x y || Float32.cmp Clt x y || Float32.cmp Cgt x y)))
         #SF <- Vundef
         #OF <- Vundef
  | _, _ =>
      undef_regs (CR ZF :: CR CF :: CR PF :: CR SF :: CR OF :: nil) rs
  end.

Testing a condition

Definition eval_testcond (c: testcond) (rs: regset) : option bool :=
  match c with
  | Cond_e =>
      match rs ZF with
      | Vint n => Some (Int.eq n Int.one)
      | _ => None
      end
  | Cond_ne =>
      match rs ZF with
      | Vint n => Some (Int.eq n Int.zero)
      | _ => None
      end
  | Cond_b =>
      match rs CF with
      | Vint n => Some (Int.eq n Int.one)
      | _ => None
      end
  | Cond_be =>
      match rs CF, rs ZF with
      | Vint c, Vint z => Some (Int.eq c Int.one || Int.eq z Int.one)
      | _, _ => None
      end
  | Cond_ae =>
      match rs CF with
      | Vint n => Some (Int.eq n Int.zero)
      | _ => None
      end
  | Cond_a =>
      match rs CF, rs ZF with
      | Vint c, Vint z => Some (Int.eq c Int.zero && Int.eq z Int.zero)
      | _, _ => None
      end
  | Cond_l =>
      match rs OF, rs SF with
      | Vint o, Vint s => Some (Int.eq (Int.xor o s) Int.one)
      | _, _ => None
      end
  | Cond_le =>
      match rs OF, rs SF, rs ZF with
      | Vint o, Vint s, Vint z => Some (Int.eq (Int.xor o s) Int.one || Int.eq z Int.one)
      | _, _, _ => None
      end
  | Cond_ge =>
      match rs OF, rs SF with
      | Vint o, Vint s => Some (Int.eq (Int.xor o s) Int.zero)
      | _, _ => None
      end
  | Cond_g =>
      match rs OF, rs SF, rs ZF with
      | Vint o, Vint s, Vint z => Some (Int.eq (Int.xor o s) Int.zero && Int.eq z Int.zero)
      | _, _, _ => None
      end
  | Cond_p =>
      match rs PF with
      | Vint n => Some (Int.eq n Int.one)
      | _ => None
      end
  | Cond_np =>
      match rs PF with
      | Vint n => Some (Int.eq n Int.zero)
      | _ => None
      end
  end.

The semantics is purely small-step and defined as a function from the current state (a register set + a memory state) to either Next rs' m' where rs' and m' are the updated register set and memory state after execution of the instruction at rs#PC, or Stuck if the processor is stuck.

Inductive outcome: Type :=
| Next: regset -> mem -> outcome
| Stuck: outcome.

Manipulations over the PC register: continuing with the next instruction (nextinstr) or branching to a label (goto_label). nextinstr_nf is a variant of nextinstr that sets condition flags to Vundef in addition to incrementing the PC.

Definition nextinstr (rs: regset) :=
  rs#PC <- (Val.offset_ptr rs#PC Ptrofs.one).

Definition nextinstr_nf (rs: regset) : regset :=
  nextinstr (undef_regs (CR ZF :: CR CF :: CR PF :: CR SF :: CR OF :: nil) rs).

Definition goto_label (f: function) (lbl: label) (rs: regset) (m: mem) :=
  match label_pos lbl 0 (fn_code f) with
  | None => Stuck
  | Some pos =>
      match rs#PC with
      | Vptr b ofs => Next (rs#PC <- (Vptr b (Ptrofs.repr pos))) m
      | _ => Stuck
    end
  end.

Auxiliaries for memory accesses.

Definition exec_load (chunk: memory_chunk) (m: mem)
                     (a: addrmode) (rs: regset) (rd: preg) :=
  match Mem.loadv chunk m (eval_addrmode a rs) with
  | Some v => Next (nextinstr_nf (rs#rd <- v)) m
  | None => Stuck
  end.

Definition exec_store (chunk: memory_chunk) (m: mem)
                      (a: addrmode) (rs: regset) (r1: preg)
                      (destroyed: list preg) :=
  match Mem.storev chunk m (eval_addrmode a rs) (rs r1) with
  | Some m' => Next (nextinstr_nf (undef_regs destroyed rs)) m'
  | None => Stuck
  end.

Execution of a single instruction i in initial state rs and m. Return updated state. For instructions that correspond to actual IA32 instructions, the cases are straightforward transliterations of the informal descriptions given in the IA32 reference manuals. For pseudo-instructions, refer to the informal descriptions given above. Note that we set to Vundef the registers used as temporaries by the expansions of the pseudo-instructions, so that the IA32 code we generate cannot use those registers to hold values that must survive the execution of the pseudo-instruction. Concerning condition flags, the comparison instructions set them accurately; other instructions (moves, lea) preserve them; and all other instruction set those flags to Vundef, to reflect the fact that the processor updates some or all of those flags, but we do not need to model this precisely.

Definition exec_instr (f: function) (i: instruction) (rs: regset) (m: mem) : outcome :=
match i with

Moves

  | Pmov_rr rd r1 =>
      Next (nextinstr (rs#rd <- (rs r1))) m
  | Pmovl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Vint n))) m
  | Pmovq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Vlong n))) m
  | Pmov_rs rd id =>
      Next (nextinstr_nf (rs#rd <- (Genv.symbol_address ge id Ptrofs.zero))) m
  | Pmovl_rm rd a =>
      exec_load Mint32 m a rs rd
  | Pmovq_rm rd a =>
      exec_load Mint64 m a rs rd
  | Pmovl_mr a r1 =>
      exec_store Mint32 m a rs r1 nil
  | Pmovq_mr a r1 =>
      exec_store Mint64 m a rs r1 nil
  | Pmovsd_ff rd r1 =>
      Next (nextinstr (rs#rd <- (rs r1))) m
  | Pmovsd_fi rd n =>
      Next (nextinstr (rs#rd <- (Vfloat n))) m
  | Pmovsd_fm rd a =>
      exec_load Mfloat64 m a rs rd
  | Pmovsd_mf a r1 =>
      exec_store Mfloat64 m a rs r1 nil
  | Pmovss_fi rd n =>
      Next (nextinstr (rs#rd <- (Vsingle n))) m
  | Pmovss_fm rd a =>
      exec_load Mfloat32 m a rs rd
  | Pmovss_mf a r1 =>
      exec_store Mfloat32 m a rs r1 nil
  | Pfldl_m a =>
      exec_load Mfloat64 m a rs ST0
  | Pfstpl_m a =>
      exec_store Mfloat64 m a rs ST0 (ST0 :: nil)
  | Pflds_m a =>
      exec_load Mfloat32 m a rs ST0
  | Pfstps_m a =>
      exec_store Mfloat32 m a rs ST0 (ST0 :: nil)

Moves with conversion

  | Pmovb_mr a r1 =>
      exec_store Mint8unsigned m a rs r1 nil
  | Pmovw_mr a r1 =>
      exec_store Mint16unsigned m a rs r1 nil
  | Pmovzb_rr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.zero_ext 8 rs#r1))) m
  | Pmovzb_rm rd a =>
      exec_load Mint8unsigned m a rs rd
  | Pmovsb_rr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.sign_ext 8 rs#r1))) m
  | Pmovsb_rm rd a =>
      exec_load Mint8signed m a rs rd
  | Pmovzw_rr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.zero_ext 16 rs#r1))) m
  | Pmovzw_rm rd a =>
      exec_load Mint16unsigned m a rs rd
  | Pmovsw_rr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.sign_ext 16 rs#r1))) m
  | Pmovsw_rm rd a =>
      exec_load Mint16signed m a rs rd
  | Pmovzl_rr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.longofintu rs#r1))) m
  | Pmovsl_rr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.longofint rs#r1))) m
  | Pmovls_rr rd =>
      Next (nextinstr (rs#rd <- (Val.loword rs#rd))) m
  | Pcvtsd2ss_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.singleoffloat rs#r1))) m
  | Pcvtss2sd_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.floatofsingle rs#r1))) m
  | Pcvttsd2si_rf rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.intoffloat rs#r1)))) m
  | Pcvtsi2sd_fr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.floatofint rs#r1)))) m
  | Pcvttss2si_rf rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.intofsingle rs#r1)))) m
  | Pcvtsi2ss_fr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.singleofint rs#r1)))) m
  | Pcvttsd2sl_rf rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.longoffloat rs#r1)))) m
  | Pcvtsl2sd_fr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.floatoflong rs#r1)))) m
  | Pcvttss2sl_rf rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.longofsingle rs#r1)))) m
  | Pcvtsl2ss_fr rd r1 =>
      Next (nextinstr (rs#rd <- (Val.maketotal (Val.singleoflong rs#r1)))) m

Integer arithmetic

  | Pleal rd a =>
      Next (nextinstr (rs#rd <- (eval_addrmode32 a rs))) m
  | Pleaq rd a =>
      Next (nextinstr (rs#rd <- (eval_addrmode64 a rs))) m
  | Pnegl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.neg rs#rd))) m
  | Pnegq rd =>
      Next (nextinstr_nf (rs#rd <- (Val.negl rs#rd))) m
  | Paddl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.add rs#rd (Vint n)))) m
  | Paddq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.addl rs#rd (Vlong n)))) m
  | Psubl_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.sub rs#rd rs#r1))) m
  | Psubq_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.subl rs#rd rs#r1))) m
  | Pimull_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.mul rs#rd rs#r1))) m
  | Pimulq_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.mull rs#rd rs#r1))) m
  | Pimull_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.mul rs#rd (Vint n)))) m
  | Pimulq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.mull rs#rd (Vlong n)))) m
  | Pimull_r r1 =>
      Next (nextinstr_nf (rs#RAX <- (Val.mul rs#RAX rs#r1)
                            #RDX <- (Val.mulhs rs#RAX rs#r1))) m
  | Pimulq_r r1 =>
      Next (nextinstr_nf (rs#RAX <- (Val.mull rs#RAX rs#r1)
                            #RDX <- (Val.mullhs rs#RAX rs#r1))) m
  | Pmull_r r1 =>
      Next (nextinstr_nf (rs#RAX <- (Val.mul rs#RAX rs#r1)
                            #RDX <- (Val.mulhu rs#RAX rs#r1))) m
  | Pmulq_r r1 =>
      Next (nextinstr_nf (rs#RAX <- (Val.mull rs#RAX rs#r1)
                            #RDX <- (Val.mullhu rs#RAX rs#r1))) m
  | Pcltd =>
      Next (nextinstr_nf (rs#RDX <- (Val.shr rs#RAX (Vint (Int.repr 31))))) m
  | Pcqto =>
      Next (nextinstr_nf (rs#RDX <- (Val.shrl rs#RAX (Vint (Int.repr 63))))) m
  | Pdivl r1 =>
      match rs#RDX, rs#RAX, rs#r1 with
      | Vint nh, Vint nl, Vint d =>
          match Int.divmodu2 nh nl d with
          | Some(q, r) => Next (nextinstr_nf (rs#RAX <- (Vint q) #RDX <- (Vint r))) m
          | None => Stuck
          end
      | _, _, _ => Stuck
      end
  | Pdivq r1 =>
      match rs#RDX, rs#RAX, rs#r1 with
      | Vlong nh, Vlong nl, Vlong d =>
          match Int64.divmodu2 nh nl d with
          | Some(q, r) => Next (nextinstr_nf (rs#RAX <- (Vlong q) #RDX <- (Vlong r))) m
          | None => Stuck
          end
      | _, _, _ => Stuck
      end
  | Pidivl r1 =>
      match rs#RDX, rs#RAX, rs#r1 with
      | Vint nh, Vint nl, Vint d =>
          match Int.divmods2 nh nl d with
          | Some(q, r) => Next (nextinstr_nf (rs#RAX <- (Vint q) #RDX <- (Vint r))) m
          | None => Stuck
          end
      | _, _, _ => Stuck
      end
  | Pidivq r1 =>
      match rs#RDX, rs#RAX, rs#r1 with
      | Vlong nh, Vlong nl, Vlong d =>
          match Int64.divmods2 nh nl d with
          | Some(q, r) => Next (nextinstr_nf (rs#RAX <- (Vlong q) #RDX <- (Vlong r))) m
          | None => Stuck
          end
      | _, _, _ => Stuck
      end
  | Pandl_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.and rs#rd rs#r1))) m
  | Pandq_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.andl rs#rd rs#r1))) m
  | Pandl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.and rs#rd (Vint n)))) m
  | Pandq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.andl rs#rd (Vlong n)))) m
  | Porl_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.or rs#rd rs#r1))) m
  | Porq_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.orl rs#rd rs#r1))) m
  | Porl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.or rs#rd (Vint n)))) m
  | Porq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.orl rs#rd (Vlong n)))) m
  | Pxorl_r rd =>
      Next (nextinstr_nf (rs#rd <- Vzero)) m
  | Pxorq_r rd =>
      Next (nextinstr_nf (rs#rd <- (Vlong Int64.zero))) m
  | Pxorl_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.xor rs#rd rs#r1))) m
  | Pxorq_rr rd r1 =>
      Next (nextinstr_nf (rs#rd <- (Val.xorl rs#rd rs#r1))) m
  | Pxorl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.xor rs#rd (Vint n)))) m
  | Pxorq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.xorl rs#rd (Vlong n)))) m
  | Pnotl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.notint rs#rd))) m
  | Pnotq rd =>
      Next (nextinstr_nf (rs#rd <- (Val.notl rs#rd))) m
  | Psall_rcl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.shl rs#rd rs#RCX))) m
  | Psalq_rcl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.shll rs#rd rs#RCX))) m
  | Psall_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.shl rs#rd (Vint n)))) m
  | Psalq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.shll rs#rd (Vint n)))) m
  | Pshrl_rcl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.shru rs#rd rs#RCX))) m
  | Pshrq_rcl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.shrlu rs#rd rs#RCX))) m
  | Pshrl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.shru rs#rd (Vint n)))) m
  | Pshrq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.shrlu rs#rd (Vint n)))) m
  | Psarl_rcl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.shr rs#rd rs#RCX))) m
  | Psarq_rcl rd =>
      Next (nextinstr_nf (rs#rd <- (Val.shrl rs#rd rs#RCX))) m
  | Psarl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.shr rs#rd (Vint n)))) m
  | Psarq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.shrl rs#rd (Vint n)))) m
  | Pshld_ri rd r1 n =>
      Next (nextinstr_nf
              (rs#rd <- (Val.or (Val.shl rs#rd (Vint n))
                                (Val.shru rs#r1 (Vint (Int.sub Int.iwordsize n)))))) m
  | Prorl_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.ror rs#rd (Vint n)))) m
  | Prorq_ri rd n =>
      Next (nextinstr_nf (rs#rd <- (Val.rorl rs#rd (Vint n)))) m
  | Pcmpl_rr r1 r2 =>
    if check_compare_ints (rs r1) (rs r2) m
    then Next (nextinstr (compare_ints (rs r1) (rs r2) rs m)) m
    else Stuck
  | Pcmpq_rr r1 r2 =>
      Next (nextinstr (compare_longs (rs r1) (rs r2) rs m)) m
  | Pcmpl_ri r1 n =>
    if check_compare_ints (rs r1) (Vint n) m
    then Next (nextinstr (compare_ints (rs r1) (Vint n) rs m)) m
    else Stuck
  | Pcmpq_ri r1 n =>
      Next (nextinstr (compare_longs (rs r1) (Vlong n) rs m)) m
  | Ptestl_rr r1 r2 =>
      Next (nextinstr (compare_ints (Val.and (rs r1) (rs r2)) Vzero rs m)) m
  | Ptestq_rr r1 r2 =>
      Next (nextinstr (compare_longs (Val.andl (rs r1) (rs r2)) (Vlong Int64.zero) rs m)) m
  | Ptestl_ri r1 n =>
      Next (nextinstr (compare_ints (Val.and (rs r1) (Vint n)) Vzero rs m)) m
  | Ptestq_ri r1 n =>
      Next (nextinstr (compare_longs (Val.andl (rs r1) (Vlong n)) (Vlong Int64.zero) rs m)) m
  | Pcmov c rd r1 =>
      match eval_testcond c rs with
      | Some true => Next (nextinstr (rs#rd <- (rs#r1))) m
      | Some false => Next (nextinstr rs) m
      | None => Next (nextinstr (rs#rd <- Vundef)) m
      end
  | Psetcc c rd =>
      Next (nextinstr (rs#rd <- (Val.of_optbool (eval_testcond c rs)))) m

Arithmetic operations over double-precision floats

  | Paddd_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.addf rs#rd rs#r1))) m
  | Psubd_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.subf rs#rd rs#r1))) m
  | Pmuld_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.mulf rs#rd rs#r1))) m
  | Pdivd_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.divf rs#rd rs#r1))) m
  | Pnegd rd =>
      Next (nextinstr (rs#rd <- (Val.negf rs#rd))) m
  | Pabsd rd =>
      Next (nextinstr (rs#rd <- (Val.absf rs#rd))) m
  | Pcomisd_ff r1 r2 =>
      Next (nextinstr (compare_floats (rs r1) (rs r2) rs)) m
  | Pxorpd_f rd =>
      Next (nextinstr_nf (rs#rd <- (Vfloat Float.zero))) m

Arithmetic operations over single-precision floats

  | Padds_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.addfs rs#rd rs#r1))) m
  | Psubs_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.subfs rs#rd rs#r1))) m
  | Pmuls_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.mulfs rs#rd rs#r1))) m
  | Pdivs_ff rd r1 =>
      Next (nextinstr (rs#rd <- (Val.divfs rs#rd rs#r1))) m
  | Pnegs rd =>
      Next (nextinstr (rs#rd <- (Val.negfs rs#rd))) m
  | Pabss rd =>
      Next (nextinstr (rs#rd <- (Val.absfs rs#rd))) m
  | Pcomiss_ff r1 r2 =>
      Next (nextinstr (compare_floats32 (rs r1) (rs r2) rs)) m
  | Pxorps_f rd =>
      Next (nextinstr_nf (rs#rd <- (Vsingle Float32.zero))) m

Branches and calls

  | Pjmp_l lbl =>
      goto_label f lbl rs m
  | Pjmp_s id sg =>
      Next (rs#PC <- (Genv.symbol_address ge id Ptrofs.zero)) m
  | Pjmp_r r sg =>
      Next (rs#PC <- (rs r)) m
  | Pjcc cond lbl =>
      match eval_testcond cond rs with
      | Some true => goto_label f lbl rs m
      | Some false => Next (nextinstr rs) m
      | None => Stuck
      end
  | Pjcc2 cond1 cond2 lbl =>
      match eval_testcond cond1 rs, eval_testcond cond2 rs with
      | Some true, Some true => goto_label f lbl rs m
      | Some _, Some _ => Next (nextinstr rs) m
      | _, _ => Stuck
      end
  | Pjmptbl r tbl =>
      match rs#r with
      | Vint n =>
          match list_nth_z tbl (Int.unsigned n) with
          | None => Stuck
          | Some lbl => goto_label f lbl (rs #RAX <- Vundef #RDX <- Vundef) m
          end
      | _ => Stuck
      end
  | Pcall_s id sg =>
      Next (rs#RA <- (Val.offset_ptr rs#PC Ptrofs.one) #PC <- (Genv.symbol_address ge id Ptrofs.zero)) m
  | Pcall_r r sg =>
      Next (rs#RA <- (Val.offset_ptr rs#PC Ptrofs.one) #PC <- (rs r)) m
  | Pret =>
      Next (rs#PC <- (rs#RA)) m

Saving and restoring registers

  | Pmov_rm_a rd a =>
      exec_load (if Archi.ptr64 then Many64 else Many32) m a rs rd
  | Pmov_mr_a a r1 =>
      exec_store (if Archi.ptr64 then Many64 else Many32) m a rs r1 nil
  | Pmovsd_fm_a rd a =>
      exec_load Many64 m a rs rd
  | Pmovsd_mf_a a r1 =>
      exec_store Many64 m a rs r1 nil

Pseudo-instructions

  | Plabel lbl =>
      Next (nextinstr rs) m
  | Pallocframe sz ofs_ra ofs_link =>
      let (m1, stk) := Mem.alloc m 0 sz in
      let sp := Vptr stk Ptrofs.zero in
      match Mem.storev Mptr m1 (Val.offset_ptr sp ofs_link) rs#RSP with
      | None => Stuck
      | Some m2 =>
          match Mem.storev Mptr m2 (Val.offset_ptr sp ofs_ra) rs#RA with
          | None => Stuck
          | Some m3 => Next (nextinstr (rs #RAX <- (rs#RSP) #RSP <- sp)) m3
          end
      end
  | Pfreeframe sz ofs_ra ofs_link =>
      match Mem.loadv Mptr m (Val.offset_ptr rs#RSP ofs_ra) with
      | None => Stuck
      | Some ra =>
          match Mem.loadv Mptr m (Val.offset_ptr rs#RSP ofs_link) with
          | None => Stuck
          | Some sp =>
              match rs#RSP with
              | Vptr stk ofs =>
                  match Mem.free m stk 0 sz with
                  | None => Stuck
                  | Some m' => Next (nextinstr (rs#RSP <- sp #RA <- ra)) m'
                  end
              | _ => Stuck
              end
          end
      end
  | Pbuiltin ef args res =>
      Stuck (* treated specially below *)

The following instructions and directives are not generated directly by Asmgen, so we do not model them.

Translation of the LTL/Linear/Mach view of machine registers to the Asm view.

Extract the values of the arguments of an external call. We exploit the calling conventions from module Conventions, except that we use machine registers instead of locations.

Inductive extcall_arg (rs: regset) (m: mem): loc -> val -> Prop :=
  | extcall_arg_reg: forall r,
      extcall_arg rs m (R r) (rs (preg_of r))
  | extcall_arg_stack: forall ofs ty bofs v,
      bofs = Stacklayout.fe_ofs_arg + 4 * ofs ->
      Mem.loadv (chunk_of_type ty) m
                (Val.offset_ptr (rs (IR RSP)) (Ptrofs.repr bofs)) = Some v ->
      extcall_arg rs m (S Outgoing ofs ty) v.

Inductive extcall_arg_pair (rs: regset) (m: mem): rpair loc -> val -> Prop :=
  | extcall_arg_one: forall l v,
      extcall_arg rs m l v ->
      extcall_arg_pair rs m (One l) v
  | extcall_arg_twolong: forall hi lo vhi vlo,
      extcall_arg rs m hi vhi ->
      extcall_arg rs m lo vlo ->
      extcall_arg_pair rs m (Twolong hi lo) (Val.longofwords vhi vlo).

Definition extcall_arguments
    (rs: regset) (m: mem) (sg: signature) (args: list val) : Prop :=
  list_forall2 (extcall_arg_pair rs m) (loc_arguments sg) args.

Definition loc_external_result (sg: signature) : rpair preg :=
  map_rpair preg_of (loc_result sg).

Execution of the instruction at rs#PC.

Inductive state: Type :=
  | State: regset -> mem -> state.

Inductive step: state -> trace -> state -> Prop :=
  | exec_step_internal:
      forall b ofs f i rs m rs' m',
      rs PC = Vptr b ofs ->
      Genv.find_funct_ptr ge b = Some (Internal f) ->
      find_instr (Ptrofs.unsigned ofs) f.(fn_code) = Some i ->
      exec_instr f i rs m = Next rs' m' ->
      step (State rs m) E0 (State rs' m')
  | exec_step_builtin:
      forall b ofs f ef args res rs m vargs t vres rs' m',
      rs PC = Vptr b ofs ->
      Genv.find_funct_ptr ge b = Some (Internal f) ->
      find_instr (Ptrofs.unsigned ofs) f.(fn_code) = Some (Pbuiltin ef args res) ->
      eval_builtin_args ge rs (rs RSP) m args vargs ->
      external_call ef ge vargs m t vres m' ->
      rs' = nextinstr_nf
             (set_res res vres
               (undef_regs (map preg_of (destroyed_by_builtin ef)) rs)) ->
      step (State rs m) t (State rs' m')
  | exec_step_external:
      forall b ef args res rs m t rs' m',
      rs PC = Vptr b Ptrofs.zero ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      extcall_arguments rs m (ef_sig ef) args ->
      external_call ef ge args m t res m' ->
      rs' = (set_pair (loc_external_result (ef_sig ef)) res rs) #PC <- (rs RA) ->
      step (State rs m) t (State rs' m').

End RELSEM.

Execution of whole programs.

Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall m0,
      Genv.init_mem p = Some m0 ->
      let ge := Genv.globalenv p in
      let rs0 :=
        (Pregmap.init Vundef)
        # PC <- (Genv.symbol_address ge p.(prog_main) Ptrofs.zero)
        # RA <- Vnullptr
        # RSP <- Vnullptr in
      initial_state p (State rs0 m0).

Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall rs m r,
      rs#PC = Vnullptr ->
      rs#RAX = Vint r ->
      final_state (State rs m) r.

Definition semantics (p: program) :=
  Semantics step (initial_state p) final_state (Genv.globalenv p).

Determinacy of the Asm semantics.

Remark extcall_arguments_determ:
forall rs m sg args1 args2,
extcall_arguments rs m sg args1 -> extcall_arguments rs m sg args2 -> args1 = args2.

Proof.

  intros until m.
  assert (A: forall l v1 v2,
             extcall_arg rs m l v1 -> extcall_arg rs m l v2 -> v1 = v2).
  { intros. inv H; inv H0; congruence. }
  assert (B: forall p v1 v2,
             extcall_arg_pair rs m p v1 -> extcall_arg_pair rs m p v2 -> v1 = v2).
  { intros. inv H; inv H0.
    eapply A; eauto.
    f_equal; eapply A; eauto. }
  assert (C: forall ll vl1, list_forall2 (extcall_arg_pair rs m) ll vl1 ->
             forall vl2, list_forall2 (extcall_arg_pair rs m) ll vl2 -> vl1 = vl2).
  {
    induction 1; intros vl2 EA; inv EA.
    auto.
    f_equal; eauto. }
  intros. eapply C; eauto.
Qed.

Lemma semantics_determinate: forall p, determinate (semantics p).

Proof.

Ltac Equalities :=
  match goal with
  | [ H1: ?a = ?b, H2: ?a = ?c |- _ ] =>
      rewrite H1 in H2; inv H2; Equalities
  | _ => idtac
  end.
  intros; constructor; simpl; intros.
- (* determ *)
  inv H; inv H0; Equalities.
+ split. constructor. auto.
+ discriminate.
+ discriminate.
+ assert (vargs0 = vargs) by (eapply eval_builtin_args_determ; eauto). subst vargs0.
  exploit external_call_determ. eexact H5. eexact H11. intros [A B].
  split. auto. intros. destruct B; auto. subst. auto.
+ assert (args0 = args) by (eapply extcall_arguments_determ; eauto). subst args0.
  exploit external_call_determ. eexact H4. eexact H9. intros [A B].
  split. auto. intros. destruct B; auto. subst. auto.
- (* trace length *)
  red; intros; inv H; simpl.
  omega.
  eapply external_call_trace_length; eauto.
  eapply external_call_trace_length; eauto.
- (* initial states *)
  inv H; inv H0. f_equal. congruence.
- (* final no step *)
  assert (NOTNULL: forall b ofs, Vnullptr <> Vptr b ofs).
  { intros; unfold Vnullptr; destruct Archi.ptr64; congruence. }
  inv H. red; intros; red; intros. inv H; rewrite H0 in *; eelim NOTNULL; eauto.
- (* final states *)
  inv H; inv H0. congruence.
Qed.

Classification functions for processor registers (used in Asmgenproof).

Module Asm

Abstract syntax

Registers.

Instruction set.

Operational semantics