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diff --git a/ia32/Op.v b/ia32/Op.v deleted file mode 100644 index eb2fd110..00000000 --- a/ia32/Op.v +++ /dev/null @@ -1,1457 +0,0 @@ -(* *********************************************************************) -(* *) -(* 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 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] *) - | 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. *) - -(** 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 eq_operation (x y: operation): {x=y} + {x<>y}. -Proof. - generalize Int.eq_dec Int64.eq_dec Float.eq_dec Float32.eq_dec; intros. - decide equality. - apply ident_eq. - apply eq_addressing. - apply eq_addressing. - apply eq_condition. -Defined. - -Global Opaque eq_condition eq_addressing eq_operation. - -(** 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. *) - -Definition offset_in_range (n: Z) : bool := zle Int.min_signed n && zle n Int.max_signed. - -Definition addressing_valid (a: addressing) : bool := - 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 => true - | Abased s ofs => true - | Abasedscaled sc s ofs => true - | Ainstack ofs => offset_in_range (Ptrofs.signed ofs) - end. - -(** * 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) - | 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)) - | _, _ => 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; try discriminate H; FuncInv - | H: (match ?v with Vundef => _ | Vint _ => _ | Vfloat _ => _ | Vptr _ _ => _ end = Some _) |- _ => - destruct v; simpl in H; try discriminate H; FuncInv - | H: (if Archi.ptr64 then _ else _) = Some _ |- _ => - destruct Archi.ptr64 eqn:?; try discriminate H; 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) - | 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) - 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). - 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)... - unfold Val.has_type, Tptr; destruct Archi.ptr64; auto. - destruct v0... - destruct v0... - destruct v0... - destruct v0... - destruct v0... - destruct v0; destruct v1; simpl... - unfold Val.has_type; destruct Archi.ptr64; auto. - unfold Val.has_type; destruct Archi.ptr64; auto. destruct (eq_block b b0); auto. - 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... - destruct v0; simpl... unfold Val.has_type; destruct Archi.ptr64; auto. - destruct v0; destruct v1; simpl... - unfold Val.has_type; destruct Archi.ptr64; auto. - unfold Val.has_type; destruct Archi.ptr64; simpl; auto. destruct (eq_block b b0); auto. - 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 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... -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. destruct addr; simpl; auto. - destruct vl; auto. destruct Archi.ptr64 eqn:SF; auto. - do 2 f_equal. rewrite Ptrofs.add_zero_l. apply Ptrofs.add_commut. -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. destruct addr; simpl; auto. - destruct vl; auto. destruct Archi.ptr64 eqn:SF; auto. - do 2 f_equal. rewrite Ptrofs.add_zero_l. apply Ptrofs.add_commut. -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 op_depends_on_memory (op: operation) : bool := - match op with - | Ocmp (Ccompu _) => negb Archi.ptr64 - | Ocmp (Ccompuimm _ _) => negb Archi.ptr64 - | Ocmp (Ccomplu _) => Archi.ptr64 - | Ocmp (Ccompluimm _ _) => Archi.ptr64 - | _ => false - end. - -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. - destruct cond; simpl; intros SF; auto; rewrite ? negb_false_iff in SF; - unfold Val.cmpu_bool, Val.cmplu_bool; rewrite SF; reflexivity. -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. - unfold Genv.symbol_address. rewrite agree_on_symbols. auto. - apply eval_addressing32_preserved. - apply eval_addressing64_preserved. -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_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; 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. -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 Zplus_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. - -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_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. |