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|
; X86ISA Library
; Note: The license below is based on the template at:
; http://opensource.org/licenses/BSD-3-Clause
; Copyright (C) 2015, Regents of the University of Texas
; All rights reserved.
; Redistribution and use in source and binary forms, with or without
; modification, are permitted provided that the following conditions are
; met:
; o Redistributions of source code must retain the above copyright
; notice, this list of conditions and the following disclaimer.
; o Redistributions in binary form must reproduce the above copyright
; notice, this list of conditions and the following disclaimer in the
; documentation and/or other materials provided with the distribution.
; o Neither the name of the copyright holders nor the names of its
; contributors may be used to endorse or promote products derived
; from this software without specific prior written permission.
; THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
; "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
; LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
; A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
; HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
; SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
; LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
; DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
; THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
; (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
; OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
; Original Author(s):
; Shilpi Goel <shigoel@cs.utexas.edu>
(in-package "X86ISA")
(include-book "app-view/user-level-memory-utils" :dir :proof-utils :ttags :all)
(local (include-book "centaur/gl/gl" :dir :system))
(local (include-book "centaur/bitops/ihs-extensions" :dir :system))
(local (include-book "centaur/bitops/signed-byte-p" :dir :system))
;; ======================================================================
;; bool isPowerOfTwo? (long int v) {
;; bool f;
;; f = v && !(v & (v - 1));
;; return f;
;; }
;; GCC O2 Optimization
;; 0000000100000ec0 <_powerOfTwo>:
;; 100000ec0: 55 push %rbp
;; 100000ec1: 48 89 e5 mov %rsp,%rbp
;; 100000ec4: 48 85 ff test %rdi,%rdi
;; 100000ec7: 74 0c je 100000ed5 <_powerOfTwo+0x15>
;; 100000ec9: 48 8d 47 ff lea -0x1(%rdi),%rax
;; 100000ecd: 48 85 c7 test %rax,%rdi
;; 100000ed0: 0f 94 c0 sete %al
;; 100000ed3: eb 02 jmp 100000ed7 <_powerOfTwo+0x17>
;; 100000ed5: 31 c0 xor %eax,%eax
;; 100000ed7: 5d pop %rbp
;; 100000ed8: c3 retq
;; 100000ed9: 0f 1f 80 00 00 00 00 nopl 0x0(%rax)
(defconst *program*
'(
#x55 ;; push %rbp
#x48 #x89 #xe5 ;; mov %rsp,%rbp
#x48 #x85 #xff ;; test %rdi,%rdi
#x74 #x0c ;; je 100000ed5 <_powerOfTwo+0x15>
#x48 #x8d #x47 #xff ;; lea -0x1(%rdi),%rax
#x48 #x85 #xc7 ;; test %rax,%rdi
#x0f #x94 #xc0 ;; sete %al
#xeb #x02 ;; jmp 100000ed7 <_powerOfTwo+0x17>
#x31 #xc0 ;; xor %eax,%eax
#x5d ;; pop %rbp
#xc3 ;; retq
))
(defconst *l* (len *program*))
(encapsulate
()
(local (include-book "arithmetic-5/top" :dir :system))
(defun power-of-2-p (x)
(if (zp x)
nil
(if (equal x 1)
t
(if (zp (mod x 2))
(power-of-2-p (floor x 2))
nil))))
(defun compute-log-2 (x count)
(declare (xargs :measure (log-2-measure x)
:guard (natp count)))
;; Find the number n such that x approx.== 2^n.
(if (natp x)
(if (<= x 1)
count
(compute-log-2 (* 1/2 x) (1+ count)))
count))
(defthm integerp-compute-log-2
(implies (integerp count)
(integerp (compute-log-2 x count)))
:rule-classes :type-prescription)
(defthm natp-compute-log-2
(implies (natp count)
(natp (compute-log-2 x count)))
:rule-classes :type-prescription)
(local
(defthm compute-log-2-and-expt
(implies
;; natp would lead to a stronger rule, but I want a weak one here.
(posp count)
(equal (expt 2 (compute-log-2 x count))
(expt 2 (+ 1 (compute-log-2 x (1- count))))))))
(defthm if-power-of-2-p-returns-t-then-x-is-indeed-a-power-of-2
;; If (power-of-2-p x) is t, then there exists a natural number n
;; such that x is equal to (expt 2 n).
(implies (and (power-of-2-p x)
(natp x))
(equal (expt 2 (compute-log-2 x 0)) x)))
(defthm power-of-2-p-returns-t-for-x=2^n
;; Every number x == 2^n, where n is a natural number,
;; satisfies power-of-2-p.
(implies (and (natp n)
(equal x (expt 2 n)))
(power-of-2-p x))
:hints (("Goal" :in-theory (e/d* (expt) ())))))
(local
(def-gl-thm program-effects-helper-1
:hyp (and (unsigned-byte-p 64 x)
(power-of-2-p x))
:concl (equal (logand x (+ -1 x)) 0)
:g-bindings
`((x (:g-number ,(increasing-list 0 1 65))))))
(local
(def-gl-thm program-effects-helper-2
:hyp (and (unsigned-byte-p 64 x)
(not (equal x 0))
(not (power-of-2-p x)))
:concl (equal (equal (logand x (+ -1 x)) 0) nil)
:g-bindings
`((x (:g-number ,(increasing-list 0 1 65))))))
(local
(def-gl-thm program-effects-helper-3
:hyp (and (signed-byte-p 64 x)
(power-of-2-p (loghead 64 x)))
:concl (equal (logand (loghead 64 x)
(loghead 64 (+ -1 x)))
0)
:g-bindings
`((x (:g-number ,(increasing-list 0 1 65))))))
(local
(def-gl-thm program-effects-helper-4
:hyp (and (signed-byte-p 64 x)
(< 0 (loghead 64 x))
(not (power-of-2-p (loghead 64 x))))
:concl (not (equal (logand (loghead 64 x)
(loghead 64 (+ -1 x)))
0))
:g-bindings
`((x (:g-number ,(increasing-list 0 1 65))))))
(local
(def-gl-thm power-of-2-p-result-helper-1
:hyp (unsigned-byte-p 64 x)
:concl (equal (loghead 8 (logior 1 (logext 64 (bitops::logsquash 8 (+ -1 x)))))
1)
:g-bindings
`((x (:g-number ,(increasing-list 0 1 65))))))
(local
(def-gl-thm power-of-2-p-result-helper-2
:hyp (signed-byte-p 64 x)
:concl (signed-byte-p 64 (logior 1 (bitops::logsquash 8 (logext 64 (+ -1 x)))))
:g-bindings
`((x (:g-number ,(increasing-list 0 1 65))))))
(local
(def-gl-thm power-of-2-p-result-helper-3
:hyp (signed-byte-p 64 x)
:concl (signed-byte-p 64 (bitops::logsquash 8 (logext 64 (+ -1 x))))
:g-bindings
`((x (:g-number ,(increasing-list 0 1 65))))))
;; ======================================================================
(defun-nx preconditions (x86)
(and
;; The x86 state is well-formed.
(x86p x86)
;; Alignment checking is turned off.
(not (alignment-checking-enabled-p x86))
;; The model is operating in 64-bit mode.
(64-bit-modep x86)
;; The model is operating in the application-level view.
(app-view x86)
;; The program is located at linear addresses ranging from (rip
;; x86) to (+ -1 (len *program*) (rip x86)).
(program-at (rip x86) *program* x86)
;; The addresses where the program is located are canonical.
(canonical-address-p (rip x86))
(canonical-address-p (+ (len *program*) (rip x86)))
;; The initial state is error-free.
(equal (ms x86) nil)
(equal (fault x86) nil)
;; Stack addresses are canonical.
(canonical-address-p (+ -8 (xr :rgf *rsp* x86)))
(canonical-address-p (xr :rgf *rsp* x86))
(canonical-address-p (+ 8 (xr :rgf *rsp* x86)))
;; Return address is canonical.
(canonical-address-p
(logext 64 (mv-nth 1 (rb 8 (xr :rgf *rsp* x86) :r x86))))
;; Program and stack are disjoint.
(separate
:x *l* (xr :rip nil x86)
:w 8 (+ -8 (xr :rgf *rsp* x86)))
(unsigned-byte-p 64 (rr64 *rdi* x86))))
(defun is-power-of-2-clock () 10)
(defun is-not-power-of-2-clock (x)
(if (zp x) 7 10))
(defthm power-of-2-p-result
(implies (and (preconditions x86)
(power-of-2-p (rr64 *rdi* x86)))
(equal (loghead 8 (xr :rgf *rax* (x86-run (is-power-of-2-clock) x86)))
1))
:hints (("Goal"
:do-not-induct t
:in-theory (e/d* (instruction-decoding-and-spec-rules
rflag-RoWs-enables
x86-operation-mode
jcc/cmovcc/setcc-spec
gpr-and-spec-8
gpr-xor-spec-4
one-byte-opcode-execute
!rgfi-size
x86-operand-to-reg/mem
wr08
wr32
wr64
rr32
rr64
rime-size
rme-size
wime-size
wme-size
rml32
rml64
wml32
wml64
rr32
x86-operand-from-modr/m-and-sib-bytes
check-instruction-length
riml-size
riml32
n32-to-i32
n64-to-i64
riml08
two-byte-opcode-decode-and-execute
x86-effective-addr-when-64-bit-modep
x86-effective-addr-32/64
;; Flags
write-user-rflags
zf-spec)
(canonical-address-p
signed-byte-p)))))
(defthm not-power-of-2-p-result
(implies (and (preconditions x86)
(not (power-of-2-p (rr64 *rdi* x86))))
(equal (loghead
8
(xr :rgf *rax*
(x86-run (is-not-power-of-2-clock (rr64 *rdi* x86)) x86)))
0))
:hints (("Goal"
:do-not-induct t
:cases (equal (rgfi *rdi* x86) 0)
:in-theory (e/d* (instruction-decoding-and-spec-rules
rflag-RoWs-enables
x86-operation-mode
jcc/cmovcc/setcc-spec
gpr-and-spec-8
gpr-xor-spec-4
one-byte-opcode-execute
!rgfi-size
x86-operand-to-reg/mem
wr08
wr32
wr64
rr32
rr64
rime-size
rme-size
wime-size
wme-size
rml32
rml64
wml32
wml64
rr32
x86-operand-from-modr/m-and-sib-bytes
check-instruction-length
riml-size
riml32
n32-to-i32
n64-to-i64
riml08
two-byte-opcode-decode-and-execute
x86-effective-addr-when-64-bit-modep
x86-effective-addr-32/64
;; Flags
write-user-rflags
zf-spec)
(canonical-address-p
not
signed-byte-p)))))
;; ======================================================================
|
