(* *********************************************************************) (* *) (* The Compcert verified compiler *) (* *) (* Xavier Leroy, INRIA Paris-Rocquencourt *) (* *) (* Copyright Institut National de Recherche en Informatique et en *) (* Automatique. All rights reserved. This file is distributed *) (* under the terms of the INRIA Non-Commercial License Agreement. *) (* *) (* *********************************************************************) (** Operators and addressing modes. The abstract syntax and dynamic semantics for the CminorSel, RTL, LTL and Mach languages depend on the following types, defined in this library: - [condition]: boolean conditions for conditional branches; - [operation]: arithmetic and logical operations; - [addressing]: addressing modes for load and store operations. These types are X86-64-specific and correspond roughly to what the processor can compute in one instruction. In other terms, these types reflect the state of the program after instruction selection. For a processor-independent set of operations, see the abstract syntax and dynamic semantics of the Cminor language. *) Require Import BoolEqual. Require Import Coqlib. Require Import AST. Require Import Integers. Require Import Floats. Require Import Values. Require Import Memory. Require Import Globalenvs. Require Import Events. Set Implicit Arguments. (** Conditions (boolean-valued operators). *) Inductive condition : Type := | Ccomp (c: comparison) (**r signed integer comparison *) | Ccompu (c: comparison) (**r unsigned integer comparison *) | Ccompimm (c: comparison) (n: int) (**r signed integer comparison with a constant *) | Ccompuimm (c: comparison) (n: int) (**r unsigned integer comparison with a constant *) | Ccompl (c: comparison) (**r signed 64-bit integer comparison *) | Ccomplu (c: comparison) (**r unsigned 64-bit integer comparison *) | Ccomplimm (c: comparison) (n: int64) (**r signed 64-bit integer comparison with a constant *) | Ccompluimm (c: comparison) (n: int64) (**r unsigned 64-bit integer comparison with a constant *) | Ccompf (c: comparison) (**r 64-bit floating-point comparison *) | Cnotcompf (c: comparison) (**r negation of a floating-point comparison *) | Ccompfs (c: comparison) (**r 32-bit floating-point comparison *) | Cnotcompfs (c: comparison) (**r negation of a floating-point comparison *) | Cmaskzero (n: int) (**r test [(arg & constant) == 0] *) | Cmasknotzero (n: int). (**r test [(arg & constant) != 0] *) (** Addressing modes. [r1], [r2], etc, are the arguments to the addressing. *) Inductive addressing: Type := | Aindexed: Z -> addressing (**r Address is [r1 + offset] *) | Aindexed2: Z -> addressing (**r Address is [r1 + r2 + offset] *) | Ascaled: Z -> Z -> addressing (**r Address is [r1 * scale + offset] *) | Aindexed2scaled: Z -> Z -> addressing (**r Address is [r1 + r2 * scale + offset] *) | Aglobal: ident -> ptrofs -> addressing (**r Address is [symbol + offset] *) | Abased: ident -> ptrofs -> addressing (**r Address is [symbol + offset + r1] *) | Abasedscaled: Z -> ident -> ptrofs -> addressing (**r Address is [symbol + offset + r1 * scale] *) | Ainstack: ptrofs -> addressing. (**r Address is [stack_pointer + offset] *) (** Arithmetic and logical operations. In the descriptions, [rd] is the result of the operation and [r1], [r2], etc, are the arguments. *) Inductive operation : Type := | Omove (**r [rd = r1] *) | Ointconst (n: int) (**r [rd] is set to the given integer constant *) | Olongconst (n: int64) (**r [rd] is set to the given integer constant *) | Ofloatconst (n: float) (**r [rd] is set to the given float constant *) | Osingleconst (n: float32)(**r [rd] is set to the given float constant *) | Oindirectsymbol (id: ident) (**r [rd] is set to the address of the symbol *) (*c 32-bit integer arithmetic: *) | Ocast8signed (**r [rd] is 8-bit sign extension of [r1] *) | Ocast8unsigned (**r [rd] is 8-bit zero extension of [r1] *) | Ocast16signed (**r [rd] is 16-bit sign extension of [r1] *) | Ocast16unsigned (**r [rd] is 16-bit zero extension of [r1] *) | Oneg (**r [rd = - r1] *) | Osub (**r [rd = r1 - r2] *) | Omul (**r [rd = r1 * r2] *) | Omulimm (n: int) (**r [rd = r1 * n] *) | Omulhs (**r [rd = high part of r1 * r2, signed] *) | Omulhu (**r [rd = high part of r1 * r2, unsigned] *) | Odiv (**r [rd = r1 / r2] (signed) *) | Odivu (**r [rd = r1 / r2] (unsigned) *) | Omod (**r [rd = r1 % r2] (signed) *) | Omodu (**r [rd = r1 % r2] (unsigned) *) | Oand (**r [rd = r1 & r2] *) | Oandimm (n: int) (**r [rd = r1 & n] *) | Oor (**r [rd = r1 | r2] *) | Oorimm (n: int) (**r [rd = r1 | n] *) | Oxor (**r [rd = r1 ^ r2] *) | Oxorimm (n: int) (**r [rd = r1 ^ n] *) | Onot (**r [rd = ~r1] *) | Oshl (**r [rd = r1 << r2] *) | Oshlimm (n: int) (**r [rd = r1 << n] *) | Oshr (**r [rd = r1 >> r2] (signed) *) | Oshrimm (n: int) (**r [rd = r1 >> n] (signed) *) | Oshrximm (n: int) (**r [rd = r1 / 2^n] (signed) *) | Oshru (**r [rd = r1 >> r2] (unsigned) *) | Oshruimm (n: int) (**r [rd = r1 >> n] (unsigned) *) | Ororimm (n: int) (**r rotate right immediate *) | Oshldimm (n: int) (**r [rd = r1 << n | r2 >> (32-n)] *) | Olea (a: addressing) (**r effective address *) (*c 64-bit integer arithmetic: *) | Omakelong (**r [rd = r1 << 32 | r2] *) | Olowlong (**r [rd = low-word(r1)] *) | Ohighlong (**r [rd = high-word(r1)] *) | Ocast32signed (**r [rd] is 32-bit sign extension of [r1] *) | Ocast32unsigned (**r [rd] is 32-bit zero extension of [r1] *) | Onegl (**r [rd = - r1] *) | Oaddlimm (n: int64) (**r [rd = r1 + n] *) | Osubl (**r [rd = r1 - r2] *) | Omull (**r [rd = r1 * r2] *) | Omullimm (n: int64) (**r [rd = r1 * n] *) | Omullhs (**r [rd = high part of r1 * r2, signed] *) | Omullhu (**r [rd = high part of r1 * r2, unsigned] *) | Odivl (**r [rd = r1 / r2] (signed) *) | Odivlu (**r [rd = r1 / r2] (unsigned) *) | Omodl (**r [rd = r1 % r2] (signed) *) | Omodlu (**r [rd = r1 % r2] (unsigned) *) | Oandl (**r [rd = r1 & r2] *) | Oandlimm (n: int64) (**r [rd = r1 & n] *) | Oorl (**r [rd = r1 | r2] *) | Oorlimm (n: int64) (**r [rd = r1 | n] *) | Oxorl (**r [rd = r1 ^ r2] *) | Oxorlimm (n: int64) (**r [rd = r1 ^ n] *) | Onotl (**r [rd = ~r1] *) | Oshll (**r [rd = r1 << r2] *) | Oshllimm (n: int) (**r [rd = r1 << n] *) | Oshrl (**r [rd = r1 >> r2] (signed) *) | Oshrlimm (n: int) (**r [rd = r1 >> n] (signed) *) | Oshrxlimm (n: int) (**r [rd = r1 / 2^n] (signed) *) | Oshrlu (**r [rd = r1 >> r2] (unsigned) *) | Oshrluimm (n: int) (**r [rd = r1 >> n] (unsigned) *) | Ororlimm (n: int) (**r rotate right immediate *) | Oleal (a: addressing) (**r effective address *) (*c Floating-point arithmetic: *) | Onegf (**r [rd = - r1] *) | Oabsf (**r [rd = abs(r1)] *) | Oaddf (**r [rd = r1 + r2] *) | Osubf (**r [rd = r1 - r2] *) | Omulf (**r [rd = r1 * r2] *) | Odivf (**r [rd = r1 / r2] *) | Omaxf (**r [rd = max(r1,r2)] *) | Ominf (**r [rd = min(r1,r2)] *) | Onegfs (**r [rd = - r1] *) | Oabsfs (**r [rd = abs(r1)] *) | Oaddfs (**r [rd = r1 + r2] *) | Osubfs (**r [rd = r1 - r2] *) | Omulfs (**r [rd = r1 * r2] *) | Odivfs (**r [rd = r1 / r2] *) | Osingleoffloat (**r [rd] is [r1] truncated to single-precision float *) | Ofloatofsingle (**r [rd] is [r1] extended to double-precision float *) (*c Conversions between int and float: *) | Ointoffloat (**r [rd = signed_int_of_float64(r1)] *) | Ofloatofint (**r [rd = float64_of_signed_int(r1)] *) | Ointofsingle (**r [rd = signed_int_of_float32(r1)] *) | Osingleofint (**r [rd = float32_of_signed_int(r1)] *) | Olongoffloat (**r [rd = signed_long_of_float64(r1)] *) | Ofloatoflong (**r [rd = float64_of_signed_long(r1)] *) | Olongofsingle (**r [rd = signed_long_of_float32(r1)] *) | Osingleoflong (**r [rd = float32_of_signed_long(r1)] *) (*c Boolean tests: *) | Ocmp (cond: condition) (**r [rd = 1] if condition holds, [rd = 0] otherwise. *) | Osel: condition -> typ -> operation. (**r [rd = rs1] if condition holds, [rd = rs2] otherwise. *) (** Comparison functions (used in modules [CSE] and [Allocation]). *) Definition eq_condition (x y: condition) : {x=y} + {x<>y}. Proof. generalize Int.eq_dec Int64.eq_dec; intro. assert (forall (x y: comparison), {x=y}+{x<>y}). decide equality. decide equality. Defined. Definition eq_addressing (x y: addressing) : {x=y} + {x<>y}. Proof. generalize ident_eq Ptrofs.eq_dec zeq; intros. decide equality. Defined. Definition beq_operation: forall (x y: operation), bool. Proof. generalize Int.eq_dec Int64.eq_dec Float.eq_dec Float32.eq_dec ident_eq typ_eq eq_addressing eq_condition; boolean_equality. Defined. Definition eq_operation: forall (x y: operation), {x=y} + {x<>y}. Proof. decidable_equality_from beq_operation. Defined. Global Opaque eq_condition eq_addressing eq_operation. (** Helper function for floating point maximum and minimum operation *) Definition float_max f1 f2 := match Float.compare f1 f2 with | Some Gt => f1 | Some Eq | Some Lt| None => f2 end. Definition maxf (v1 v2: val) : val := match v1, v2 with | Vfloat f1, Vfloat f2 => Vfloat (float_max f1 f2) | _, _ => Vundef end. Definition float_min f1 f2 := match Float.compare f1 f2 with | Some Lt => f1 | Some Eq | Some Gt| None => f2 end. Definition minf (v1 v2: val) : val := match v1, v2 with | Vfloat f1, Vfloat f2 => Vfloat (float_min f1 f2) | _, _ => Vundef end. (** In addressing modes, offsets are 32-bit signed integers, even in 64-bit mode. The following function checks that an addressing mode is valid, i.e. that the offsets are in range. The check always succeeds in 32-bit mode because offsets are always 32-bit integers and are normalized as 32-bit signed integers during code generation (see [Asmgen.normalize_addrmode_32]). Moreover, in 64-bit mode, we use RIP-relative addressing for access to globals. (This is the "small code model" from the x86_64 ELF ABI.) Thus, for addressing global variables, the offset from the variable plus the RIP-relative offset must fit in 32 bits. The "small code model" guarantees that this will fit if the offset is between [-2^24] and [2^24-1], under the assumption that no global variable is bigger than [2^24] bytes. *) Definition offset_in_range (n: Z) : bool := zle Int.min_signed n && zle n Int.max_signed. Definition ptroffset_min := -16777216. (**r [-2^24] *) Definition ptroffset_max := 16777215. (**r [2^24 - 1] *) Definition ptroffset_in_range (n: ptrofs) : bool := let n := Ptrofs.signed n in zle ptroffset_min n && zle n ptroffset_max. Definition addressing_valid (a: addressing) : bool := if Archi.ptr64 then match a with | Aindexed n => offset_in_range n | Aindexed2 n => offset_in_range n | Ascaled sc ofs => offset_in_range ofs | Aindexed2scaled sc ofs => offset_in_range ofs | Aglobal s ofs => ptroffset_in_range ofs | Abased s ofs => ptroffset_in_range ofs | Abasedscaled sc s ofs => ptroffset_in_range ofs | Ainstack ofs => offset_in_range (Ptrofs.signed ofs) end else true. (** * Evaluation functions *) (** Evaluation of conditions, operators and addressing modes applied to lists of values. Return [None] when the computation can trigger an error, e.g. integer division by zero. [eval_condition] returns a boolean, [eval_operation] and [eval_addressing] return a value. *) Definition eval_condition (cond: condition) (vl: list val) (m: mem): option bool := match cond, vl with | Ccomp c, v1 :: v2 :: nil => Val.cmp_bool c v1 v2 | Ccompu c, v1 :: v2 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 v2 | Ccompimm c n, v1 :: nil => Val.cmp_bool c v1 (Vint n) | Ccompuimm c n, v1 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 (Vint n) | Ccompl c, v1 :: v2 :: nil => Val.cmpl_bool c v1 v2 | Ccomplu c, v1 :: v2 :: nil => Val.cmplu_bool (Mem.valid_pointer m) c v1 v2 | Ccomplimm c n, v1 :: nil => Val.cmpl_bool c v1 (Vlong n) | Ccompluimm c n, v1 :: nil => Val.cmplu_bool (Mem.valid_pointer m) c v1 (Vlong n) | Ccompf c, v1 :: v2 :: nil => Val.cmpf_bool c v1 v2 | Cnotcompf c, v1 :: v2 :: nil => option_map negb (Val.cmpf_bool c v1 v2) | Ccompfs c, v1 :: v2 :: nil => Val.cmpfs_bool c v1 v2 | Cnotcompfs c, v1 :: v2 :: nil => option_map negb (Val.cmpfs_bool c v1 v2) | Cmaskzero n, v1 :: nil => Val.maskzero_bool v1 n | Cmasknotzero n, v1 :: nil => option_map negb (Val.maskzero_bool v1 n) | _, _ => None end. Definition eval_addressing32 (F V: Type) (genv: Genv.t F V) (sp: val) (addr: addressing) (vl: list val) : option val := match addr, vl with | Aindexed n, v1::nil => Some (Val.add v1 (Vint (Int.repr n))) | Aindexed2 n, v1::v2::nil => Some (Val.add (Val.add v1 v2) (Vint (Int.repr n))) | Ascaled sc ofs, v1::nil => Some (Val.add (Val.mul v1 (Vint (Int.repr sc))) (Vint (Int.repr ofs))) | Aindexed2scaled sc ofs, v1::v2::nil => Some(Val.add v1 (Val.add (Val.mul v2 (Vint (Int.repr sc))) (Vint (Int.repr ofs)))) | Aglobal s ofs, nil => if Archi.ptr64 then None else Some (Genv.symbol_address genv s ofs) | Abased s ofs, v1::nil => if Archi.ptr64 then None else Some (Val.add (Genv.symbol_address genv s ofs) v1) | Abasedscaled sc s ofs, v1::nil => if Archi.ptr64 then None else Some (Val.add (Genv.symbol_address genv s ofs) (Val.mul v1 (Vint (Int.repr sc)))) | Ainstack ofs, nil => if Archi.ptr64 then None else Some(Val.offset_ptr sp ofs) | _, _ => None end. Definition eval_addressing64 (F V: Type) (genv: Genv.t F V) (sp: val) (addr: addressing) (vl: list val) : option val := match addr, vl with | Aindexed n, v1::nil => Some (Val.addl v1 (Vlong (Int64.repr n))) | Aindexed2 n, v1::v2::nil => Some (Val.addl (Val.addl v1 v2) (Vlong (Int64.repr n))) | Ascaled sc ofs, v1::nil => Some (Val.addl (Val.mull v1 (Vlong (Int64.repr sc))) (Vlong (Int64.repr ofs))) | Aindexed2scaled sc ofs, v1::v2::nil => Some(Val.addl v1 (Val.addl (Val.mull v2 (Vlong (Int64.repr sc))) (Vlong (Int64.repr ofs)))) | Aglobal s ofs, nil => if Archi.ptr64 then Some (Genv.symbol_address genv s ofs) else None | Ainstack ofs, nil => if Archi.ptr64 then Some(Val.offset_ptr sp ofs) else None | _, _ => None end. Definition eval_addressing (F V: Type) (genv: Genv.t F V) (sp: val) (addr: addressing) (vl: list val) : option val := if Archi.ptr64 then eval_addressing64 genv sp addr vl else eval_addressing32 genv sp addr vl. Definition eval_operation (F V: Type) (genv: Genv.t F V) (sp: val) (op: operation) (vl: list val) (m: mem): option val := match op, vl with | Omove, v1::nil => Some v1 | Ointconst n, nil => Some (Vint n) | Olongconst n, nil => Some (Vlong n) | Ofloatconst n, nil => Some (Vfloat n) | Osingleconst n, nil => Some (Vsingle n) | Oindirectsymbol id, nil => Some (Genv.symbol_address genv id Ptrofs.zero) | Ocast8signed, v1 :: nil => Some (Val.sign_ext 8 v1) | Ocast8unsigned, v1 :: nil => Some (Val.zero_ext 8 v1) | Ocast16signed, v1 :: nil => Some (Val.sign_ext 16 v1) | Ocast16unsigned, v1 :: nil => Some (Val.zero_ext 16 v1) | Oneg, v1::nil => Some (Val.neg v1) | Osub, v1::v2::nil => Some (Val.sub v1 v2) | Omul, v1::v2::nil => Some (Val.mul v1 v2) | Omulimm n, v1::nil => Some (Val.mul v1 (Vint n)) | Omulhs, v1::v2::nil => Some (Val.mulhs v1 v2) | Omulhu, v1::v2::nil => Some (Val.mulhu v1 v2) | Odiv, v1::v2::nil => Val.divs v1 v2 | Odivu, v1::v2::nil => Val.divu v1 v2 | Omod, v1::v2::nil => Val.mods v1 v2 | Omodu, v1::v2::nil => Val.modu v1 v2 | Oand, v1::v2::nil => Some(Val.and v1 v2) | Oandimm n, v1::nil => Some (Val.and v1 (Vint n)) | Oor, v1::v2::nil => Some(Val.or v1 v2) | Oorimm n, v1::nil => Some (Val.or v1 (Vint n)) | Oxor, v1::v2::nil => Some(Val.xor v1 v2) | Oxorimm n, v1::nil => Some (Val.xor v1 (Vint n)) | Onot, v1::nil => Some(Val.notint v1) | Oshl, v1::v2::nil => Some (Val.shl v1 v2) | Oshlimm n, v1::nil => Some (Val.shl v1 (Vint n)) | Oshr, v1::v2::nil => Some (Val.shr v1 v2) | Oshrimm n, v1::nil => Some (Val.shr v1 (Vint n)) | Oshrximm n, v1::nil => Val.shrx v1 (Vint n) | Oshru, v1::v2::nil => Some (Val.shru v1 v2) | Oshruimm n, v1::nil => Some (Val.shru v1 (Vint n)) | Ororimm n, v1::nil => Some (Val.ror v1 (Vint n)) | Oshldimm n, v1::v2::nil => Some (Val.or (Val.shl v1 (Vint n)) (Val.shru v2 (Vint (Int.sub Int.iwordsize n)))) | Olea addr, _ => eval_addressing32 genv sp addr vl | Omakelong, v1::v2::nil => Some(Val.longofwords v1 v2) | Olowlong, v1::nil => Some(Val.loword v1) | Ohighlong, v1::nil => Some(Val.hiword v1) | Ocast32signed, v1 :: nil => Some (Val.longofint v1) | Ocast32unsigned, v1 :: nil => Some (Val.longofintu v1) | Onegl, v1::nil => Some (Val.negl v1) | Oaddlimm n, v1::nil => Some (Val.addl v1 (Vlong n)) | Osubl, v1::v2::nil => Some (Val.subl v1 v2) | Omull, v1::v2::nil => Some (Val.mull v1 v2) | Omullimm n, v1::nil => Some (Val.mull v1 (Vlong n)) | Omullhs, v1::v2::nil => Some (Val.mullhs v1 v2) | Omullhu, v1::v2::nil => Some (Val.mullhu v1 v2) | Odivl, v1::v2::nil => Val.divls v1 v2 | Odivlu, v1::v2::nil => Val.divlu v1 v2 | Omodl, v1::v2::nil => Val.modls v1 v2 | Omodlu, v1::v2::nil => Val.modlu v1 v2 | Oandl, v1::v2::nil => Some(Val.andl v1 v2) | Oandlimm n, v1::nil => Some (Val.andl v1 (Vlong n)) | Oorl, v1::v2::nil => Some(Val.orl v1 v2) | Oorlimm n, v1::nil => Some (Val.orl v1 (Vlong n)) | Oxorl, v1::v2::nil => Some(Val.xorl v1 v2) | Oxorlimm n, v1::nil => Some (Val.xorl v1 (Vlong n)) | Onotl, v1::nil => Some(Val.notl v1) | Oshll, v1::v2::nil => Some (Val.shll v1 v2) | Oshllimm n, v1::nil => Some (Val.shll v1 (Vint n)) | Oshrl, v1::v2::nil => Some (Val.shrl v1 v2) | Oshrlimm n, v1::nil => Some (Val.shrl v1 (Vint n)) | Oshrxlimm n, v1::nil => Val.shrxl v1 (Vint n) | Oshrlu, v1::v2::nil => Some (Val.shrlu v1 v2) | Oshrluimm n, v1::nil => Some (Val.shrlu v1 (Vint n)) | Ororlimm n, v1::nil => Some (Val.rorl v1 (Vint n)) | Oleal addr, _ => eval_addressing64 genv sp addr vl | Onegf, v1::nil => Some(Val.negf v1) | Oabsf, v1::nil => Some(Val.absf v1) | Oaddf, v1::v2::nil => Some(Val.addf v1 v2) | Osubf, v1::v2::nil => Some(Val.subf v1 v2) | Omulf, v1::v2::nil => Some(Val.mulf v1 v2) | Odivf, v1::v2::nil => Some(Val.divf v1 v2) | Omaxf, v1::v2::nil => Some (maxf v1 v2) | Ominf, v1::v2::nil => Some (minf v1 v2) | Onegfs, v1::nil => Some(Val.negfs v1) | Oabsfs, v1::nil => Some(Val.absfs v1) | Oaddfs, v1::v2::nil => Some(Val.addfs v1 v2) | Osubfs, v1::v2::nil => Some(Val.subfs v1 v2) | Omulfs, v1::v2::nil => Some(Val.mulfs v1 v2) | Odivfs, v1::v2::nil => Some(Val.divfs v1 v2) | Osingleoffloat, v1::nil => Some(Val.singleoffloat v1) | Ofloatofsingle, v1::nil => Some(Val.floatofsingle v1) | Ointoffloat, v1::nil => Val.intoffloat v1 | Ofloatofint, v1::nil => Val.floatofint v1 | Ointofsingle, v1::nil => Val.intofsingle v1 | Osingleofint, v1::nil => Val.singleofint v1 | Olongoffloat, v1::nil => Val.longoffloat v1 | Ofloatoflong, v1::nil => Val.floatoflong v1 | Olongofsingle, v1::nil => Val.longofsingle v1 | Osingleoflong, v1::nil => Val.singleoflong v1 | Ocmp c, _ => Some(Val.of_optbool (eval_condition c vl m)) | Osel c ty, v1::v2::vl => Some(Val.select (eval_condition c vl m) v1 v2 ty) | _, _ => None end. Remark eval_addressing_Aglobal: forall (F V: Type) (genv: Genv.t F V) sp id ofs, eval_addressing genv sp (Aglobal id ofs) nil = Some (Genv.symbol_address genv id ofs). Proof. intros. unfold eval_addressing, eval_addressing32, eval_addressing64; destruct Archi.ptr64; auto. Qed. Remark eval_addressing_Ainstack: forall (F V: Type) (genv: Genv.t F V) sp ofs, eval_addressing genv sp (Ainstack ofs) nil = Some (Val.offset_ptr sp ofs). Proof. intros. unfold eval_addressing, eval_addressing32, eval_addressing64; destruct Archi.ptr64; auto. Qed. Remark eval_addressing_Ainstack_inv: forall (F V: Type) (genv: Genv.t F V) sp ofs vl v, eval_addressing genv sp (Ainstack ofs) vl = Some v -> vl = nil /\ v = Val.offset_ptr sp ofs. Proof. unfold eval_addressing, eval_addressing32, eval_addressing64; intros; destruct Archi.ptr64; destruct vl; inv H; auto. Qed. Ltac FuncInv := match goal with | H: (match ?x with nil => _ | _ :: _ => _ end = Some _) |- _ => destruct x; simpl in H; FuncInv | H: (match ?v with Vundef => _ | Vint _ => _ | Vfloat _ => _ | Vptr _ _ => _ end = Some _) |- _ => destruct v; simpl in H; FuncInv | H: (if Archi.ptr64 then _ else _) = Some _ |- _ => destruct Archi.ptr64 eqn:?; FuncInv | H: (Some _ = Some _) |- _ => injection H; intros; clear H; FuncInv | H: (None = Some _) |- _ => discriminate H | _ => idtac end. (** * Static typing of conditions, operators and addressing modes. *) Definition type_of_condition (c: condition) : list typ := match c with | Ccomp _ => Tint :: Tint :: nil | Ccompu _ => Tint :: Tint :: nil | Ccompimm _ _ => Tint :: nil | Ccompuimm _ _ => Tint :: nil | Ccompl _ => Tlong :: Tlong :: nil | Ccomplu _ => Tlong :: Tlong :: nil | Ccomplimm _ _ => Tlong :: nil | Ccompluimm _ _ => Tlong :: nil | Ccompf _ => Tfloat :: Tfloat :: nil | Cnotcompf _ => Tfloat :: Tfloat :: nil | Ccompfs _ => Tsingle :: Tsingle :: nil | Cnotcompfs _ => Tsingle :: Tsingle :: nil | Cmaskzero _ => Tint :: nil | Cmasknotzero _ => Tint :: nil end. Definition type_of_addressing_gen (tyA: typ) (addr: addressing): list typ := match addr with | Aindexed _ => tyA :: nil | Aindexed2 _ => tyA :: tyA :: nil | Ascaled _ _ => tyA :: nil | Aindexed2scaled _ _ => tyA :: tyA :: nil | Aglobal _ _ => nil | Abased _ _ => tyA :: nil | Abasedscaled _ _ _ => tyA :: nil | Ainstack _ => nil end. Definition type_of_addressing := type_of_addressing_gen Tptr. Definition type_of_addressing32 := type_of_addressing_gen Tint. Definition type_of_addressing64 := type_of_addressing_gen Tlong. Definition type_of_operation (op: operation) : list typ * typ := match op with | Omove => (nil, Tint) (* treated specially *) | Ointconst _ => (nil, Tint) | Olongconst _ => (nil, Tlong) | Ofloatconst f => (nil, Tfloat) | Osingleconst f => (nil, Tsingle) | Oindirectsymbol _ => (nil, Tptr) | Ocast8signed => (Tint :: nil, Tint) | Ocast8unsigned => (Tint :: nil, Tint) | Ocast16signed => (Tint :: nil, Tint) | Ocast16unsigned => (Tint :: nil, Tint) | Oneg => (Tint :: nil, Tint) | Osub => (Tint :: Tint :: nil, Tint) | Omul => (Tint :: Tint :: nil, Tint) | Omulimm _ => (Tint :: nil, Tint) | Omulhs => (Tint :: Tint :: nil, Tint) | Omulhu => (Tint :: Tint :: nil, Tint) | Odiv => (Tint :: Tint :: nil, Tint) | Odivu => (Tint :: Tint :: nil, Tint) | Omod => (Tint :: Tint :: nil, Tint) | Omodu => (Tint :: Tint :: nil, Tint) | Oand => (Tint :: Tint :: nil, Tint) | Oandimm _ => (Tint :: nil, Tint) | Oor => (Tint :: Tint :: nil, Tint) | Oorimm _ => (Tint :: nil, Tint) | Oxor => (Tint :: Tint :: nil, Tint) | Oxorimm _ => (Tint :: nil, Tint) | Onot => (Tint :: nil, Tint) | Oshl => (Tint :: Tint :: nil, Tint) | Oshlimm _ => (Tint :: nil, Tint) | Oshr => (Tint :: Tint :: nil, Tint) | Oshrimm _ => (Tint :: nil, Tint) | Oshrximm _ => (Tint :: nil, Tint) | Oshru => (Tint :: Tint :: nil, Tint) | Oshruimm _ => (Tint :: nil, Tint) | Ororimm _ => (Tint :: nil, Tint) | Oshldimm _ => (Tint :: Tint :: nil, Tint) | Olea addr => (type_of_addressing32 addr, Tint) | Omakelong => (Tint :: Tint :: nil, Tlong) | Olowlong => (Tlong :: nil, Tint) | Ohighlong => (Tlong :: nil, Tint) | Ocast32signed => (Tint :: nil, Tlong) | Ocast32unsigned => (Tint :: nil, Tlong) | Onegl => (Tlong :: nil, Tlong) | Oaddlimm _ => (Tlong :: nil, Tlong) | Osubl => (Tlong :: Tlong :: nil, Tlong) | Omull => (Tlong :: Tlong :: nil, Tlong) | Omullimm _ => (Tlong :: nil, Tlong) | Omullhs => (Tlong :: Tlong :: nil, Tlong) | Omullhu => (Tlong :: Tlong :: nil, Tlong) | Odivl => (Tlong :: Tlong :: nil, Tlong) | Odivlu => (Tlong :: Tlong :: nil, Tlong) | Omodl => (Tlong :: Tlong :: nil, Tlong) | Omodlu => (Tlong :: Tlong :: nil, Tlong) | Oandl => (Tlong :: Tlong :: nil, Tlong) | Oandlimm _ => (Tlong :: nil, Tlong) | Oorl => (Tlong :: Tlong :: nil, Tlong) | Oorlimm _ => (Tlong :: nil, Tlong) | Oxorl => (Tlong :: Tlong :: nil, Tlong) | Oxorlimm _ => (Tlong :: nil, Tlong) | Onotl => (Tlong :: nil, Tlong) | Oshll => (Tlong :: Tint :: nil, Tlong) | Oshllimm _ => (Tlong :: nil, Tlong) | Oshrl => (Tlong :: Tint :: nil, Tlong) | Oshrlimm _ => (Tlong :: nil, Tlong) | Oshrxlimm _ => (Tlong :: nil, Tlong) | Oshrlu => (Tlong :: Tint :: nil, Tlong) | Oshrluimm _ => (Tlong :: nil, Tlong) | Ororlimm _ => (Tlong :: nil, Tlong) | Oleal addr => (type_of_addressing64 addr, Tlong) | Onegf => (Tfloat :: nil, Tfloat) | Oabsf => (Tfloat :: nil, Tfloat) | Oaddf => (Tfloat :: Tfloat :: nil, Tfloat) | Osubf => (Tfloat :: Tfloat :: nil, Tfloat) | Omulf => (Tfloat :: Tfloat :: nil, Tfloat) | Odivf => (Tfloat :: Tfloat :: nil, Tfloat) | Omaxf => (Tfloat :: Tfloat :: nil, Tfloat) | Ominf => (Tfloat :: Tfloat :: nil, Tfloat) | Onegfs => (Tsingle :: nil, Tsingle) | Oabsfs => (Tsingle :: nil, Tsingle) | Oaddfs => (Tsingle :: Tsingle :: nil, Tsingle) | Osubfs => (Tsingle :: Tsingle :: nil, Tsingle) | Omulfs => (Tsingle :: Tsingle :: nil, Tsingle) | Odivfs => (Tsingle :: Tsingle :: nil, Tsingle) | Osingleoffloat => (Tfloat :: nil, Tsingle) | Ofloatofsingle => (Tsingle :: nil, Tfloat) | Ointoffloat => (Tfloat :: nil, Tint) | Ofloatofint => (Tint :: nil, Tfloat) | Ointofsingle => (Tsingle :: nil, Tint) | Osingleofint => (Tint :: nil, Tsingle) | Olongoffloat => (Tfloat :: nil, Tlong) | Ofloatoflong => (Tlong :: nil, Tfloat) | Olongofsingle => (Tsingle :: nil, Tlong) | Osingleoflong => (Tlong :: nil, Tsingle) | Ocmp c => (type_of_condition c, Tint) | Osel c ty => (ty :: ty :: type_of_condition c, ty) end. (** Weak type soundness results for [eval_operation]: the result values, when defined, are always of the type predicted by [type_of_operation]. *) Section SOUNDNESS. Variable A V: Type. Variable genv: Genv.t A V. Remark type_add: forall v1 v2, Val.has_type (Val.add v1 v2) Tint. Proof. intros. unfold Val.has_type, Val.add. destruct Archi.ptr64, v1, v2; auto. Qed. Remark type_addl: forall v1 v2, Val.has_type (Val.addl v1 v2) Tlong. Proof. intros. unfold Val.has_type, Val.addl. destruct Archi.ptr64, v1, v2; auto. Qed. Lemma type_of_addressing64_sound: forall addr vl sp v, eval_addressing64 genv sp addr vl = Some v -> Val.has_type v Tlong. Proof. intros. destruct addr; simpl in H; FuncInv; subst; simpl; auto using type_addl. - unfold Genv.symbol_address; destruct (Genv.find_symbol genv i); simpl; auto. - destruct sp; simpl; auto. Qed. Lemma type_of_addressing32_sound: forall addr vl sp v, eval_addressing32 genv sp addr vl = Some v -> Val.has_type v Tint. Proof. intros. destruct addr; simpl in H; FuncInv; subst; simpl; auto using type_add. - unfold Genv.symbol_address; destruct (Genv.find_symbol genv i); simpl; auto. - destruct sp; simpl; auto. Qed. Corollary type_of_addressing_sound: forall addr vl sp v, eval_addressing genv sp addr vl = Some v -> Val.has_type v Tptr. Proof. unfold eval_addressing, Tptr; intros. destruct Archi.ptr64; eauto using type_of_addressing64_sound, type_of_addressing32_sound. Qed. Lemma type_of_operation_sound: forall op vl sp v m, op <> Omove -> eval_operation genv sp op vl m = Some v -> Val.has_type v (snd (type_of_operation op)). Proof with (try exact I; try reflexivity). intros. destruct op; simpl in H0; FuncInv; subst; simpl. congruence. exact I. exact I. exact I. exact I. unfold Genv.symbol_address; destruct (Genv.find_symbol genv id)... destruct v0... destruct v0... destruct v0... destruct v0... destruct v0... unfold Val.sub, Val.has_type; destruct Archi.ptr64, v0, v1... destruct (eq_block b b0)... destruct v0; destruct v1... destruct v0... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero); inv H2... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero); inv H2... destruct v0; destruct v1... destruct v0... destruct v0; destruct v1... destruct v0... destruct v0; destruct v1... destruct v0... destruct v0... destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)... destruct v0; simpl... destruct (Int.ltu n Int.iwordsize)... destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)... destruct v0; simpl... destruct (Int.ltu n Int.iwordsize)... destruct v0; simpl in H0; try discriminate. destruct (Int.ltu n (Int.repr 31)); inv H0... destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)... destruct v0; simpl... destruct (Int.ltu n Int.iwordsize)... destruct v0... destruct v0; simpl... destruct (Int.ltu n Int.iwordsize)... destruct v1; simpl... destruct (Int.ltu (Int.sub Int.iwordsize n) Int.iwordsize)... eapply type_of_addressing32_sound; eauto. destruct v0; destruct v1... destruct v0... destruct v0... destruct v0... destruct v0... destruct v0... unfold Val.addl, Val.has_type; destruct Archi.ptr64, v0... unfold Val.subl, Val.has_type; destruct Archi.ptr64, v0, v1... destruct (eq_block b b0)... destruct v0; destruct v1... destruct v0... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int64.eq i0 Int64.zero || Int64.eq i (Int64.repr Int64.min_signed) && Int64.eq i0 Int64.mone); inv H2... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int64.eq i0 Int64.zero); inv H2... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int64.eq i0 Int64.zero || Int64.eq i (Int64.repr Int64.min_signed) && Int64.eq i0 Int64.mone); inv H2... destruct v0; destruct v1; simpl in *; inv H0. destruct (Int64.eq i0 Int64.zero); inv H2... destruct v0; destruct v1... destruct v0... destruct v0; destruct v1... destruct v0... destruct v0; destruct v1... destruct v0... destruct v0... destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int64.iwordsize')... destruct v0; simpl... destruct (Int.ltu n Int64.iwordsize')... destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int64.iwordsize')... destruct v0; simpl... destruct (Int.ltu n Int64.iwordsize')... destruct v0; inv H0. destruct (Int.ltu n (Int.repr 63)); inv H2... destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int64.iwordsize')... destruct v0; simpl... destruct (Int.ltu n Int64.iwordsize')... destruct v0... eapply type_of_addressing64_sound; eauto. destruct v0... destruct v0... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0... destruct v0... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0; destruct v1... destruct v0... destruct v0... destruct v0; simpl in H0; inv H0. destruct (Float.to_int f); inv H2... destruct v0; simpl in H0; inv H0... destruct v0; simpl in H0; inv H0. destruct (Float32.to_int f); inv H2... destruct v0; simpl in H0; inv H0... destruct v0; simpl in H0; inv H0. destruct (Float.to_long f); inv H2... destruct v0; simpl in H0; inv H0... destruct v0; simpl in H0; inv H0. destruct (Float32.to_long f); inv H2... destruct v0; simpl in H0; inv H0... destruct (eval_condition cond vl m); simpl... destruct b... unfold Val.select. destruct (eval_condition c vl m). apply Val.normalize_type. exact I. Qed. Definition is_trapping_op (op : operation) := match op with | Odiv | Odivl | Odivu | Odivlu | Omod | Omodl | Omodu | Omodlu | Oshrximm _ | Oshrxlimm _ | Ointoffloat | Ointofsingle | Olongoffloat | Olongofsingle | Osingleofint | Osingleoflong | Ofloatofint | Ofloatoflong | Olea _ | Oleal _ (* TODO this is suboptimal *) => true | _ => false end. Definition args_of_operation op := if eq_operation op Omove then 1%nat else List.length (fst (type_of_operation op)). Lemma is_trapping_op_sound: forall op vl sp m, is_trapping_op op = false -> (List.length vl) = args_of_operation op -> eval_operation genv sp op vl m <> None. Proof. unfold args_of_operation. destruct op; destruct eq_operation; intros; simpl in *; try congruence. all: try (destruct vl as [ | vh1 vl1]; try discriminate). all: try (destruct vl1 as [ | vh2 vl2]; try discriminate). all: try (destruct vl2 as [ | vh3 vl3]; try discriminate). all: try (destruct vl3 as [ | vh4 vl4]; try discriminate). Qed. End SOUNDNESS. (** * Manipulating and transforming operations *) (** Recognition of move operations. *) Definition is_move_operation (A: Type) (op: operation) (args: list A) : option A := match op, args with | Omove, arg :: nil => Some arg | _, _ => None end. Lemma is_move_operation_correct: forall (A: Type) (op: operation) (args: list A) (a: A), is_move_operation op args = Some a -> op = Omove /\ args = a :: nil. Proof. intros until a. unfold is_move_operation; destruct op; try (intros; discriminate). destruct args. intros; discriminate. destruct args. intros. intuition congruence. intros; discriminate. Qed. (** [negate_condition cond] returns a condition that is logically equivalent to the negation of [cond]. *) Definition negate_condition (cond: condition): condition := match cond with | Ccomp c => Ccomp(negate_comparison c) | Ccompu c => Ccompu(negate_comparison c) | Ccompimm c n => Ccompimm (negate_comparison c) n | Ccompuimm c n => Ccompuimm (negate_comparison c) n | Ccompl c => Ccompl(negate_comparison c) | Ccomplu c => Ccomplu(negate_comparison c) | Ccomplimm c n => Ccomplimm (negate_comparison c) n | Ccompluimm c n => Ccompluimm (negate_comparison c) n | Ccompf c => Cnotcompf c | Cnotcompf c => Ccompf c | Ccompfs c => Cnotcompfs c | Cnotcompfs c => Ccompfs c | Cmaskzero n => Cmasknotzero n | Cmasknotzero n => Cmaskzero n end. Lemma eval_negate_condition: forall cond vl m, eval_condition (negate_condition cond) vl m = option_map negb (eval_condition cond vl m). Proof. intros. destruct cond; simpl. repeat (destruct vl; auto). apply Val.negate_cmp_bool. repeat (destruct vl; auto). apply Val.negate_cmpu_bool. repeat (destruct vl; auto). apply Val.negate_cmp_bool. repeat (destruct vl; auto). apply Val.negate_cmpu_bool. repeat (destruct vl; auto). apply Val.negate_cmpl_bool. repeat (destruct vl; auto). apply Val.negate_cmplu_bool. repeat (destruct vl; auto). apply Val.negate_cmpl_bool. repeat (destruct vl; auto). apply Val.negate_cmplu_bool. repeat (destruct vl; auto). repeat (destruct vl; auto). destruct (Val.cmpf_bool c v v0) as [[]|]; auto. repeat (destruct vl; auto). repeat (destruct vl; auto). destruct (Val.cmpfs_bool c v v0) as [[]|]; auto. destruct vl; auto. destruct vl; auto. destruct vl; auto. destruct vl; auto. destruct (Val.maskzero_bool v n) as [[]|]; auto. Qed. (** Shifting stack-relative references. This is used in [Stacking]. *) Definition shift_stack_addressing (delta: Z) (addr: addressing) := match addr with | Ainstack ofs => Ainstack (Ptrofs.add ofs (Ptrofs.repr delta)) | _ => addr end. Definition shift_stack_operation (delta: Z) (op: operation) := match op with | Olea addr => Olea (shift_stack_addressing delta addr) | Oleal addr => Oleal (shift_stack_addressing delta addr) | _ => op end. Lemma type_shift_stack_addressing: forall delta addr, type_of_addressing (shift_stack_addressing delta addr) = type_of_addressing addr. Proof. intros. destruct addr; auto. Qed. Lemma type_shift_stack_operation: forall delta op, type_of_operation (shift_stack_operation delta op) = type_of_operation op. Proof. intros. destruct op; auto; simpl; decEq; destruct a; auto. Qed. Lemma eval_shift_stack_addressing32: forall F V (ge: Genv.t F V) sp addr vl delta, eval_addressing32 ge (Vptr sp Ptrofs.zero) (shift_stack_addressing delta addr) vl = eval_addressing32 ge (Vptr sp (Ptrofs.repr delta)) addr vl. Proof. intros. assert (A: forall i, Ptrofs.add Ptrofs.zero (Ptrofs.add i (Ptrofs.repr delta)) = Ptrofs.add (Ptrofs.repr delta) i). { intros. rewrite Ptrofs.add_zero_l. apply Ptrofs.add_commut. } destruct addr; simpl; rewrite ?A; reflexivity. Qed. Lemma eval_shift_stack_addressing64: forall F V (ge: Genv.t F V) sp addr vl delta, eval_addressing64 ge (Vptr sp Ptrofs.zero) (shift_stack_addressing delta addr) vl = eval_addressing64 ge (Vptr sp (Ptrofs.repr delta)) addr vl. Proof. intros. assert (A: forall i, Ptrofs.add Ptrofs.zero (Ptrofs.add i (Ptrofs.repr delta)) = Ptrofs.add (Ptrofs.repr delta) i). { intros. rewrite Ptrofs.add_zero_l. apply Ptrofs.add_commut. } destruct addr; simpl; rewrite ?A; reflexivity. Qed. Lemma eval_shift_stack_addressing: forall F V (ge: Genv.t F V) sp addr vl delta, eval_addressing ge (Vptr sp Ptrofs.zero) (shift_stack_addressing delta addr) vl = eval_addressing ge (Vptr sp (Ptrofs.repr delta)) addr vl. Proof. intros. unfold eval_addressing. destruct Archi.ptr64; auto using eval_shift_stack_addressing32, eval_shift_stack_addressing64. Qed. Lemma eval_shift_stack_operation: forall F V (ge: Genv.t F V) sp op vl m delta, eval_operation ge (Vptr sp Ptrofs.zero) (shift_stack_operation delta op) vl m = eval_operation ge (Vptr sp (Ptrofs.repr delta)) op vl m. Proof. intros. destruct op; simpl; auto using eval_shift_stack_addressing32, eval_shift_stack_addressing64. Qed. (** Offset an addressing mode [addr] by a quantity [delta], so that it designates the pointer [delta] bytes past the pointer designated by [addr]. This may be undefined if an offset overflows, in which case [None] is returned. *) Definition offset_addressing_total (addr: addressing) (delta: Z) : addressing := match addr with | Aindexed n => Aindexed (n + delta) | Aindexed2 n => Aindexed2 (n + delta) | Ascaled sc n => Ascaled sc (n + delta) | Aindexed2scaled sc n => Aindexed2scaled sc (n + delta) | Aglobal s n => Aglobal s (Ptrofs.add n (Ptrofs.repr delta)) | Abased s n => Abased s (Ptrofs.add n (Ptrofs.repr delta)) | Abasedscaled sc s n => Abasedscaled sc s (Ptrofs.add n (Ptrofs.repr delta)) | Ainstack n => Ainstack (Ptrofs.add n (Ptrofs.repr delta)) end. Definition offset_addressing (addr: addressing) (delta: Z) : option addressing := let addr' := offset_addressing_total addr delta in if addressing_valid addr' then Some addr' else None. Lemma eval_offset_addressing_total_32: forall (F V: Type) (ge: Genv.t F V) sp addr args delta v, eval_addressing32 ge sp addr args = Some v -> eval_addressing32 ge sp (offset_addressing_total addr delta) args = Some(Val.add v (Vint (Int.repr delta))). Proof. assert (A: forall x y, Int.add (Int.repr x) (Int.repr y) = Int.repr (x + y)). { intros. apply Int.eqm_samerepr; auto with ints. } assert (B: forall delta, Archi.ptr64 = false -> Ptrofs.repr delta = Ptrofs.of_int (Int.repr delta)). { intros; symmetry; auto with ptrofs. } intros. destruct addr; simpl in *; FuncInv; subst; simpl. - rewrite <- A, ! Val.add_assoc; auto. - rewrite <- A, ! Val.add_assoc; auto. - rewrite <- A, ! Val.add_assoc; auto. - rewrite <- A, ! Val.add_assoc; auto. - rewrite B, Genv.shift_symbol_address_32 by auto. auto. - rewrite B, Genv.shift_symbol_address_32 by auto. rewrite ! Val.add_assoc. do 2 f_equal. apply Val.add_commut. - rewrite B, Genv.shift_symbol_address_32 by auto. rewrite ! Val.add_assoc. do 2 f_equal. apply Val.add_commut. - destruct sp; simpl; auto. rewrite Heqb. rewrite Ptrofs.add_assoc. do 4 f_equal. symmetry; auto with ptrofs. Qed. Lemma eval_offset_addressing_total_64: forall (F V: Type) (ge: Genv.t F V) sp addr args delta v, eval_addressing64 ge sp addr args = Some v -> eval_addressing64 ge sp (offset_addressing_total addr delta) args = Some(Val.addl v (Vlong (Int64.repr delta))). Proof. assert (A: forall x y, Int64.add (Int64.repr x) (Int64.repr y) = Int64.repr (x + y)). { intros. apply Int64.eqm_samerepr; auto with ints. } assert (B: forall delta, Archi.ptr64 = true -> Ptrofs.repr delta = Ptrofs.of_int64 (Int64.repr delta)). { intros; symmetry; auto with ptrofs. } intros. destruct addr; simpl in *; FuncInv; subst; simpl. - rewrite <- A, ! Val.addl_assoc; auto. - rewrite <- A, ! Val.addl_assoc; auto. - rewrite <- A, ! Val.addl_assoc; auto. - rewrite <- A, ! Val.addl_assoc; auto. - rewrite B, Genv.shift_symbol_address_64 by auto. auto. - destruct sp; simpl; auto. rewrite Heqb. rewrite Ptrofs.add_assoc. do 4 f_equal. symmetry; auto with ptrofs. Qed. (** The following lemma is used only in [Allocproof] in cases where [Archi.ptr64 = false]. *) Lemma eval_offset_addressing: forall (F V: Type) (ge: Genv.t F V) sp addr args delta addr' v, offset_addressing addr delta = Some addr' -> eval_addressing ge sp addr args = Some v -> Archi.ptr64 = false -> eval_addressing ge sp addr' args = Some(Val.add v (Vint (Int.repr delta))). Proof. intros. unfold offset_addressing in H. destruct (addressing_valid (offset_addressing_total addr delta)); inv H. unfold eval_addressing in *; rewrite H1 in *. apply eval_offset_addressing_total_32; auto. Qed. (** Operations that are so cheap to recompute that CSE should not factor them out. *) Definition is_trivial_op (op: operation) : bool := match op with | Omove => true | Ointconst _ => true | Olongconst _ => true | Olea (Aglobal _ _) => true | Olea (Ainstack _) => true | Oleal (Aglobal _ _) => true | Oleal (Ainstack _) => true | _ => false end. (** Operations that depend on the memory state. *) Definition cond_depends_on_memory (c: condition) : bool := match c with | Ccompu _ => negb Archi.ptr64 | Ccompuimm _ _ => negb Archi.ptr64 | Ccomplu _ => Archi.ptr64 | Ccompluimm _ _ => Archi.ptr64 | _ => false end. Definition op_depends_on_memory (op: operation) : bool := match op with | Ocmp c => cond_depends_on_memory c | Osel c ty => cond_depends_on_memory c | _ => false end. Lemma cond_depends_on_memory_correct: forall c args m1 m2, cond_depends_on_memory c = false -> eval_condition c args m1 = eval_condition c args m2. Proof. intros until m2. destruct c; simpl; intros SF; auto; rewrite ? negb_false_iff in SF; unfold Val.cmpu_bool, Val.cmplu_bool; rewrite SF; reflexivity. Qed. Lemma op_depends_on_memory_correct: forall (F V: Type) (ge: Genv.t F V) sp op args m1 m2, op_depends_on_memory op = false -> eval_operation ge sp op args m1 = eval_operation ge sp op args m2. Proof. intros until m2. destruct op; simpl; try congruence; intros C. - f_equal; f_equal; apply cond_depends_on_memory_correct; auto. - destruct args; auto. destruct args; auto. rewrite (cond_depends_on_memory_correct c args m1 m2 C). auto. Qed. Lemma cond_valid_pointer_eq: forall cond args m1 m2, (forall b z, Mem.valid_pointer m1 b z = Mem.valid_pointer m2 b z) -> eval_condition cond args m1 = eval_condition cond args m2. Proof. intros until m2. intro MEM. destruct cond eqn:COND; simpl; try congruence. all: repeat (destruct args; simpl; try congruence); erewrite cmpu_bool_valid_pointer_eq || erewrite cmplu_bool_valid_pointer_eq; eauto. Qed. Lemma op_valid_pointer_eq: forall (F V: Type) (ge: Genv.t F V) sp op args m1 m2, (forall b z, Mem.valid_pointer m1 b z = Mem.valid_pointer m2 b z) -> eval_operation ge sp op args m1 = eval_operation ge sp op args m2. Proof. intros until m2. destruct op eqn:OP; simpl; try congruence. - intros MEM; destruct cond; simpl; try congruence; repeat (destruct args; simpl; try congruence); erewrite cmpu_bool_valid_pointer_eq || erewrite cmplu_bool_valid_pointer_eq; eauto. - intro MEM; destruct c; simpl; try congruence; repeat (destruct args; simpl; try congruence); erewrite cmpu_bool_valid_pointer_eq || erewrite cmplu_bool_valid_pointer_eq; eauto. Qed. (** Global variables mentioned in an operation or addressing mode *) Definition globals_addressing (addr: addressing) : list ident := match addr with | Aglobal s n => s :: nil | Abased s n => s :: nil | Abasedscaled sc s n => s :: nil | _ => nil end. Definition globals_operation (op: operation) : list ident := match op with | Oindirectsymbol s => s :: nil | Olea addr => globals_addressing addr | Oleal addr => globals_addressing addr | _ => nil end. (** * Invariance and compatibility properties. *) (** [eval_operation] and [eval_addressing] depend on a global environment for resolving references to global symbols. We show that they give the same results if a global environment is replaced by another that assigns the same addresses to the same symbols. *) Section GENV_TRANSF. Variable F1 F2 V1 V2: Type. Variable ge1: Genv.t F1 V1. Variable ge2: Genv.t F2 V2. Hypothesis agree_on_symbols: forall (s: ident), Genv.find_symbol ge2 s = Genv.find_symbol ge1 s. Lemma eval_addressing32_preserved: forall sp addr vl, eval_addressing32 ge2 sp addr vl = eval_addressing32 ge1 sp addr vl. Proof. intros. unfold eval_addressing32, Genv.symbol_address; destruct addr; try rewrite agree_on_symbols; reflexivity. Qed. Lemma eval_addressing64_preserved: forall sp addr vl, eval_addressing64 ge2 sp addr vl = eval_addressing64 ge1 sp addr vl. Proof. intros. unfold eval_addressing64, Genv.symbol_address; destruct addr; try rewrite agree_on_symbols; reflexivity. Qed. Lemma eval_addressing_preserved: forall sp addr vl, eval_addressing ge2 sp addr vl = eval_addressing ge1 sp addr vl. Proof. intros. unfold eval_addressing; destruct Archi.ptr64; auto using eval_addressing32_preserved, eval_addressing64_preserved. Qed. Lemma eval_operation_preserved: forall sp op vl m, eval_operation ge2 sp op vl m = eval_operation ge1 sp op vl m. Proof. intros. unfold eval_operation; destruct op; auto using eval_addressing32_preserved, eval_addressing64_preserved. unfold Genv.symbol_address. rewrite agree_on_symbols. auto. Qed. End GENV_TRANSF. (** Compatibility of the evaluation functions with value injections. *) Section EVAL_COMPAT. Variable F1 F2 V1 V2: Type. Variable ge1: Genv.t F1 V1. Variable ge2: Genv.t F2 V2. Variable f: meminj. Variable m1: mem. Variable m2: mem. Hypothesis valid_pointer_inj: forall b1 ofs b2 delta, f b1 = Some(b2, delta) -> Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true -> Mem.valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true. Hypothesis weak_valid_pointer_inj: forall b1 ofs b2 delta, f b1 = Some(b2, delta) -> Mem.weak_valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true -> Mem.weak_valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true. Hypothesis weak_valid_pointer_no_overflow: forall b1 ofs b2 delta, f b1 = Some(b2, delta) -> Mem.weak_valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true -> 0 <= Ptrofs.unsigned ofs + Ptrofs.unsigned (Ptrofs.repr delta) <= Ptrofs.max_unsigned. Hypothesis valid_different_pointers_inj: forall b1 ofs1 b2 ofs2 b1' delta1 b2' delta2, b1 <> b2 -> Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs1) = true -> Mem.valid_pointer m1 b2 (Ptrofs.unsigned ofs2) = true -> f b1 = Some (b1', delta1) -> f b2 = Some (b2', delta2) -> b1' <> b2' \/ Ptrofs.unsigned (Ptrofs.add ofs1 (Ptrofs.repr delta1)) <> Ptrofs.unsigned (Ptrofs.add ofs2 (Ptrofs.repr delta2)). Ltac InvInject := match goal with | [ H: Val.inject _ (Vint _) _ |- _ ] => inv H; InvInject | [ H: Val.inject _ (Vfloat _) _ |- _ ] => inv H; InvInject | [ H: Val.inject _ (Vptr _ _) _ |- _ ] => inv H; InvInject | [ H: Val.inject_list _ nil _ |- _ ] => inv H; InvInject | [ H: Val.inject_list _ (_ :: _) _ |- _ ] => inv H; InvInject | _ => idtac end. Lemma eval_condition_inj: forall cond vl1 vl2 b, Val.inject_list f vl1 vl2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b. Proof. intros. destruct cond; simpl in H0; FuncInv; InvInject; simpl; auto. - inv H3; inv H2; simpl in H0; inv H0; auto. - eauto 3 using Val.cmpu_bool_inject, Mem.valid_pointer_implies. - inv H3; simpl in H0; inv H0; auto. - eauto 3 using Val.cmpu_bool_inject, Mem.valid_pointer_implies. - inv H3; inv H2; simpl in H0; inv H0; auto. - eauto 3 using Val.cmplu_bool_inject, Mem.valid_pointer_implies. - inv H3; simpl in H0; inv H0; auto. - eauto 3 using Val.cmplu_bool_inject, Mem.valid_pointer_implies. - inv H3; inv H2; simpl in H0; inv H0; auto. - inv H3; inv H2; simpl in H0; inv H0; auto. - inv H3; inv H2; simpl in H0; inv H0; auto. - inv H3; inv H2; simpl in H0; inv H0; auto. - inv H3; try discriminate; auto. - inv H3; try discriminate; auto. Qed. Ltac TrivialExists := match goal with | [ |- exists v2, Some ?v1 = Some v2 /\ Val.inject _ _ v2 ] => exists v1; split; auto | _ => idtac end. Lemma eval_addressing32_inj: forall addr sp1 vl1 sp2 vl2 v1, (forall id ofs, In id (globals_addressing addr) -> Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) -> Val.inject f sp1 sp2 -> Val.inject_list f vl1 vl2 -> eval_addressing32 ge1 sp1 addr vl1 = Some v1 -> exists v2, eval_addressing32 ge2 sp2 addr vl2 = Some v2 /\ Val.inject f v1 v2. Proof. assert (A: forall v1 v2 v1' v2', Val.inject f v1 v1' -> Val.inject f v2 v2' -> Val.inject f (Val.mul v1 v2) (Val.mul v1' v2')). { intros. inv H; simpl; auto. inv H0; auto. } intros. destruct addr; simpl in *; FuncInv; InvInject; TrivialExists; eauto using Val.add_inject, Val.offset_ptr_inject with coqlib. Qed. Lemma eval_addressing64_inj: forall addr sp1 vl1 sp2 vl2 v1, (forall id ofs, In id (globals_addressing addr) -> Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) -> Val.inject f sp1 sp2 -> Val.inject_list f vl1 vl2 -> eval_addressing64 ge1 sp1 addr vl1 = Some v1 -> exists v2, eval_addressing64 ge2 sp2 addr vl2 = Some v2 /\ Val.inject f v1 v2. Proof. assert (A: forall v1 v2 v1' v2', Val.inject f v1 v1' -> Val.inject f v2 v2' -> Val.inject f (Val.mull v1 v2) (Val.mull v1' v2')). { intros. inv H; simpl; auto. inv H0; auto. } intros. destruct addr; simpl in *; FuncInv; InvInject; TrivialExists; eauto using Val.addl_inject, Val.offset_ptr_inject with coqlib. Qed. Lemma eval_addressing_inj: forall addr sp1 vl1 sp2 vl2 v1, (forall id ofs, In id (globals_addressing addr) -> Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) -> Val.inject f sp1 sp2 -> Val.inject_list f vl1 vl2 -> eval_addressing ge1 sp1 addr vl1 = Some v1 -> exists v2, eval_addressing ge2 sp2 addr vl2 = Some v2 /\ Val.inject f v1 v2. Proof. unfold eval_addressing; intros. destruct Archi.ptr64; eauto using eval_addressing32_inj, eval_addressing64_inj. Qed. Lemma eval_addressing_inj_none: forall addr sp1 vl1 sp2 vl2, (forall id ofs, In id (globals_addressing addr) -> Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) -> Val.inject f sp1 sp2 -> Val.inject_list f vl1 vl2 -> eval_addressing ge1 sp1 addr vl1 = None -> eval_addressing ge2 sp2 addr vl2 = None. Proof. intros until vl2. intros Hglobal Hinjsp Hinjvl. destruct addr; simpl in *; inv Hinjvl; trivial; try discriminate; inv H0; trivial; try discriminate; inv H2; trivial; try discriminate. Qed. Lemma eval_operation_inj: forall op sp1 vl1 sp2 vl2 v1, (forall id ofs, In id (globals_operation op) -> Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) -> Val.inject f sp1 sp2 -> Val.inject_list f vl1 vl2 -> eval_operation ge1 sp1 op vl1 m1 = Some v1 -> exists v2, eval_operation ge2 sp2 op vl2 m2 = Some v2 /\ Val.inject f v1 v2. Proof. intros until v1; intros GL; intros. destruct op; simpl in H1; simpl; FuncInv; InvInject; TrivialExists. apply GL; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. apply Val.sub_inject; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2. TrivialExists. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int.eq i0 Int.zero); inv H2. TrivialExists. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int.eq i0 Int.zero || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2. TrivialExists. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int.eq i0 Int.zero); inv H2. TrivialExists. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto. inv H4; simpl; auto. destruct (Int.ltu n Int.iwordsize); auto. inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto. inv H4; simpl; auto. destruct (Int.ltu n Int.iwordsize); auto. inv H4; simpl in H1; try discriminate. simpl. destruct (Int.ltu n (Int.repr 31)); inv H1. TrivialExists. inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto. inv H4; simpl; auto. destruct (Int.ltu n Int.iwordsize); auto. inv H4; simpl; auto. inv H4; simpl; auto. destruct (Int.ltu n Int.iwordsize); auto. inv H2; simpl; auto. destruct (Int.ltu (Int.sub Int.iwordsize n) Int.iwordsize); auto. eapply eval_addressing32_inj; eauto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. apply Val.addl_inject; auto. apply Val.subl_inject; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int64.eq i0 Int64.zero || Int64.eq i (Int64.repr Int64.min_signed) && Int64.eq i0 Int64.mone); inv H2. TrivialExists. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int64.eq i0 Int64.zero); inv H2. TrivialExists. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int64.eq i0 Int64.zero || Int64.eq i (Int64.repr Int64.min_signed) && Int64.eq i0 Int64.mone); inv H2. TrivialExists. inv H4; inv H3; simpl in H1; inv H1. simpl. destruct (Int64.eq i0 Int64.zero); inv H2. TrivialExists. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int64.iwordsize'); auto. inv H4; simpl; auto. destruct (Int.ltu n Int64.iwordsize'); auto. inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int64.iwordsize'); auto. inv H4; simpl; auto. destruct (Int.ltu n Int64.iwordsize'); auto. inv H4; simpl in H1; try discriminate. simpl. destruct (Int.ltu n (Int.repr 63)); inv H1. TrivialExists. inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int64.iwordsize'); auto. inv H4; simpl; auto. destruct (Int.ltu n Int64.iwordsize'); auto. inv H4; simpl; auto. eapply eval_addressing64_inj; eauto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; inv H2; simpl; auto. inv H4; simpl; auto. inv H4; simpl; auto. inv H4; simpl in H1; inv H1. simpl. destruct (Float.to_int f0); simpl in H2; inv H2. exists (Vint i); auto. inv H4; simpl in H1; inv H1. simpl. TrivialExists. inv H4; simpl in H1; inv H1. simpl. destruct (Float32.to_int f0); simpl in H2; inv H2. exists (Vint i); auto. inv H4; simpl in H1; inv H1. simpl. TrivialExists. inv H4; simpl in H1; inv H1. simpl. destruct (Float.to_long f0); simpl in H2; inv H2. exists (Vlong i); auto. inv H4; simpl in H1; inv H1. simpl. TrivialExists. inv H4; simpl in H1; inv H1. simpl. destruct (Float32.to_long f0); simpl in H2; inv H2. exists (Vlong i); auto. inv H4; simpl in H1; inv H1. simpl. TrivialExists. subst v1. destruct (eval_condition cond vl1 m1) eqn:?. exploit eval_condition_inj; eauto. intros EQ; rewrite EQ. destruct b; simpl; constructor. simpl; constructor. apply Val.select_inject; auto. destruct (eval_condition c vl1 m1) eqn:?; auto. right; symmetry; eapply eval_condition_inj; eauto. Qed. End EVAL_COMPAT. (** Compatibility of the evaluation functions with the ``is less defined'' relation over values. *) Section EVAL_LESSDEF. Variable F V: Type. Variable genv: Genv.t F V. Remark valid_pointer_extends: forall m1 m2, Mem.extends m1 m2 -> forall b1 ofs b2 delta, Some(b1, 0) = Some(b2, delta) -> Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true -> Mem.valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true. Proof. intros. inv H0. rewrite Ptrofs.add_zero. eapply Mem.valid_pointer_extends; eauto. Qed. Remark weak_valid_pointer_extends: forall m1 m2, Mem.extends m1 m2 -> forall b1 ofs b2 delta, Some(b1, 0) = Some(b2, delta) -> Mem.weak_valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true -> Mem.weak_valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true. Proof. intros. inv H0. rewrite Ptrofs.add_zero. eapply Mem.weak_valid_pointer_extends; eauto. Qed. Remark weak_valid_pointer_no_overflow_extends: forall m1 b1 ofs b2 delta, Some(b1, 0) = Some(b2, delta) -> Mem.weak_valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true -> 0 <= Ptrofs.unsigned ofs + Ptrofs.unsigned (Ptrofs.repr delta) <= Ptrofs.max_unsigned. Proof. intros. inv H. rewrite Z.add_0_r. apply Ptrofs.unsigned_range_2. Qed. Remark valid_different_pointers_extends: forall m1 b1 ofs1 b2 ofs2 b1' delta1 b2' delta2, b1 <> b2 -> Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs1) = true -> Mem.valid_pointer m1 b2 (Ptrofs.unsigned ofs2) = true -> Some(b1, 0) = Some (b1', delta1) -> Some(b2, 0) = Some (b2', delta2) -> b1' <> b2' \/ Ptrofs.unsigned(Ptrofs.add ofs1 (Ptrofs.repr delta1)) <> Ptrofs.unsigned(Ptrofs.add ofs2 (Ptrofs.repr delta2)). Proof. intros. inv H2; inv H3. auto. Qed. Lemma eval_condition_lessdef: forall cond vl1 vl2 b m1 m2, Val.lessdef_list vl1 vl2 -> Mem.extends m1 m2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b. Proof. intros. eapply eval_condition_inj with (f := fun b => Some(b, 0)) (m1 := m1). apply valid_pointer_extends; auto. apply weak_valid_pointer_extends; auto. apply weak_valid_pointer_no_overflow_extends. apply valid_different_pointers_extends; auto. rewrite <- val_inject_list_lessdef. eauto. auto. Qed. Lemma eval_operation_lessdef: forall sp op vl1 vl2 v1 m1 m2, Val.lessdef_list vl1 vl2 -> Mem.extends m1 m2 -> eval_operation genv sp op vl1 m1 = Some v1 -> exists v2, eval_operation genv sp op vl2 m2 = Some v2 /\ Val.lessdef v1 v2. Proof. intros. rewrite val_inject_list_lessdef in H. assert (exists v2 : val, eval_operation genv sp op vl2 m2 = Some v2 /\ Val.inject (fun b => Some(b, 0)) v1 v2). eapply eval_operation_inj with (m1 := m1) (sp1 := sp). apply valid_pointer_extends; auto. apply weak_valid_pointer_extends; auto. apply weak_valid_pointer_no_overflow_extends. apply valid_different_pointers_extends; auto. intros. apply val_inject_lessdef. auto. apply val_inject_lessdef; auto. eauto. auto. destruct H2 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto. Qed. Lemma eval_addressing_lessdef: forall sp addr vl1 vl2 v1, Val.lessdef_list vl1 vl2 -> eval_addressing genv sp addr vl1 = Some v1 -> exists v2, eval_addressing genv sp addr vl2 = Some v2 /\ Val.lessdef v1 v2. Proof. intros. rewrite val_inject_list_lessdef in H. assert (exists v2 : val, eval_addressing genv sp addr vl2 = Some v2 /\ Val.inject (fun b => Some(b, 0)) v1 v2). eapply eval_addressing_inj with (sp1 := sp). intros. rewrite <- val_inject_lessdef; auto. rewrite <- val_inject_lessdef; auto. eauto. auto. destruct H1 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto. Qed. Lemma eval_addressing_lessdef_none: forall sp addr vl1 vl2, Val.lessdef_list vl1 vl2 -> eval_addressing genv sp addr vl1 = None -> eval_addressing genv sp addr vl2 = None. Proof. intros until vl2. intros Hlessdef Heval1. destruct addr; simpl in *; inv Hlessdef; trivial; try discriminate; inv H0; trivial; try discriminate; inv H2; trivial; try discriminate. Qed. End EVAL_LESSDEF. (** Compatibility of the evaluation functions with memory injections. *) Section EVAL_INJECT. Variable F V: Type. Variable genv: Genv.t F V. Variable f: meminj. Hypothesis globals: meminj_preserves_globals genv f. Variable sp1: block. Variable sp2: block. Variable delta: Z. Hypothesis sp_inj: f sp1 = Some(sp2, delta). Remark symbol_address_inject: forall id ofs, Val.inject f (Genv.symbol_address genv id ofs) (Genv.symbol_address genv id ofs). Proof. intros. unfold Genv.symbol_address. destruct (Genv.find_symbol genv id) eqn:?; auto. exploit (proj1 globals); eauto. intros. econstructor; eauto. rewrite Ptrofs.add_zero; auto. Qed. Lemma eval_condition_inject: forall cond vl1 vl2 b m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_condition cond vl1 m1 = Some b -> eval_condition cond vl2 m2 = Some b. Proof. intros. eapply eval_condition_inj with (f := f) (m1 := m1); eauto. intros; eapply Mem.valid_pointer_inject_val; eauto. intros; eapply Mem.weak_valid_pointer_inject_val; eauto. intros; eapply Mem.weak_valid_pointer_inject_no_overflow; eauto. intros; eapply Mem.different_pointers_inject; eauto. Qed. Lemma eval_addressing_inject: forall addr vl1 vl2 v1, Val.inject_list f vl1 vl2 -> eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = Some v1 -> exists v2, eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = Some v2 /\ Val.inject f v1 v2. Proof. intros. rewrite eval_shift_stack_addressing. eapply eval_addressing_inj with (sp1 := Vptr sp1 Ptrofs.zero); eauto. intros. apply symbol_address_inject. econstructor; eauto. rewrite Ptrofs.add_zero_l; auto. Qed. Lemma eval_addressing_inject_none: forall addr vl1 vl2, Val.inject_list f vl1 vl2 -> eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = None -> eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = None. Proof. intros. rewrite eval_shift_stack_addressing. eapply eval_addressing_inj_none with (sp1 := Vptr sp1 Ptrofs.zero); eauto. intros. apply symbol_address_inject. econstructor; eauto. rewrite Ptrofs.add_zero_l; auto. Qed. Lemma eval_operation_inject: forall op vl1 vl2 v1 m1 m2, Val.inject_list f vl1 vl2 -> Mem.inject f m1 m2 -> eval_operation genv (Vptr sp1 Ptrofs.zero) op vl1 m1 = Some v1 -> exists v2, eval_operation genv (Vptr sp2 Ptrofs.zero) (shift_stack_operation delta op) vl2 m2 = Some v2 /\ Val.inject f v1 v2. Proof. intros. rewrite eval_shift_stack_operation. simpl. eapply eval_operation_inj with (sp1 := Vptr sp1 Ptrofs.zero) (m1 := m1); eauto. intros; eapply Mem.valid_pointer_inject_val; eauto. intros; eapply Mem.weak_valid_pointer_inject_val; eauto. intros; eapply Mem.weak_valid_pointer_inject_no_overflow; eauto. intros; eapply Mem.different_pointers_inject; eauto. intros. apply symbol_address_inject. econstructor; eauto. rewrite Ptrofs.add_zero_l; auto. Qed. End EVAL_INJECT. (** * Handling of builtin arguments *) Definition builtin_arg_ok_1 (A: Type) (ba: builtin_arg A) (c: builtin_arg_constraint) := match c, ba with | OK_all, _ => true | OK_const, (BA_int _ | BA_long _ | BA_float _ | BA_single _) => true | OK_addrstack, BA_addrstack _ => true | OK_addressing, BA_addrstack _ => true | OK_addressing, BA_addrglobal _ _ => true | OK_addressing, BA_addptr (BA _) (BA_int _ | BA_long _) => true | _, _ => false end. Definition builtin_arg_ok (A: Type) (ba: builtin_arg A) (c: builtin_arg_constraint) := match ba with | (BA _ | BA_splitlong (BA _) (BA _)) => true | _ => builtin_arg_ok_1 ba c end.