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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 "base")
(local (include-book "centaur/bitops/ihs-extensions" :dir :system))
(local (include-book "centaur/bitops/signed-byte-p" :dir :system))
(local (include-book "arithmetic/top-with-meta" :dir :system))
;; ======================================================================
(defconst *factorial_recursive*
'(#x85 #xff ;; test %edi,%edi
#xb8 #x01 #x00 #x00 #x00 ;; mov $0x1,%eax
#x74 #x0f ;; je <factorial_recursive+0x18>
#x0f #x1f #x80 #x00 #x00 #x00 #x00 ;; nopl 0x0(%rax)
#x0f #xaf #xc7 ;; imul %edi,%eax
#x83 #xef #x01 ;; sub $0x1,%edi
#x75 #xf8))
(defun-nx factorial-hyps (x86)
(b* ((program-rip 0)
(n (rgfi *rdi* x86)))
(and (x86p x86)
(64-bit-modep x86)
(natp n)
(< n 13)
(equal (ms x86) nil)
(equal (fault x86) nil)
(app-view x86)
(canonical-address-p program-rip)
(canonical-address-p (+ program-rip (len *factorial_recursive*)))
(program-at program-rip *factorial_recursive* x86))))
(acl2::def-model-api
:run x86-run-ALT ;; the run function of the model
:svar x86 ;; name of state variable
:stobjp T ;; and whether it's a stobj
:hyps ((factorial-hyps x86)) ;; invariant to assume about state
:step x86-fetch-decode-execute ;; name of step function
:get-pc (lambda (x86) (xr :rip nil x86)) ;; how to fetch the pc
:put-pc (lambda (v x86) (xw :rip nil v x86)) ;; how to set the pc
;; the ``drivers'' below specify how to dive through updaters (and
;; constructors) and their accessors
:updater-drivers (((XW FLD I :VALUE :BASE)
(XR FLD I :BASE))
((WB N ADDR R-W-X :VALUE :BASE)
(RB N ADDR R-W-X :BASE)))
:constructor-drivers nil
:state-comps-and-types
(((XR :RGF *RDI* X86) (natp (XR :RGF *RDI* X86)))
((XR :RGF *RAX* X86) (natp (XR :RGF *RAX* X86)))
((XR :RIP nil X86) (canonical-address-p (XR :RIP nil X86))))
:callp nil ;; recognizer fn for states with pc on call instruction
:ret-pc nil ;; how to fetch the return pc after a call
:returnp nil ;; recognizer for states with pc on return instruction
:clk+ binary-clk+ ; how to add two clocks
:name-print-base nil ; base to use for pcs appearing in names
; (2, 8, 10, or 16)
; how to generate variable names from state comps
:var-names (((XR :RGF *RDI* X86) "N")
((XR :RGF *RAX* X86) "ACC")
((XR :RIP nil X86) "PC")))
(local (in-theory (e/d* (instruction-decoding-and-spec-rules
rflag-RoWs-enables
x86-operation-mode
shr-spec
shr-spec-32
gpr-and-spec-4
gpr-add-spec-1
gpr-add-spec-4
gpr-add-spec-8
gpr-sub-spec-8
jcc/cmovcc/setcc-spec
imul-spec
imul-spec-32
gpr-sub-spec-4
one-byte-opcode-execute
!rgfi-size
x86-operand-to-reg/mem
wr64
wr32
rr08
rr32
rr64
rml32
rml64
wml32
wml64
rr32
x86-operand-from-modr/m-and-sib-bytes
rime-size
rme-size
wime-size
wme-size
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
subset-p
;; Flags
write-user-rflags
zf-spec)
(unsigned-byte-p
las-to-pas
default-+-2
get-prefixes-opener-lemma-group-1-prefix
get-prefixes-opener-lemma-group-2-prefix
get-prefixes-opener-lemma-group-3-prefix
get-prefixes-opener-lemma-group-4-prefix))))
(encapsulate
()
(local (include-book "centaur/gl/gl" :dir :system))
(defthm-using-gl clk-16-measure-helper
:hyp (and (natp rdi)
(< 0 rdi)
(< rdi 13)
(not (equal (loghead 32 (+ -1 (logext 32 rdi))) 0)))
:concl (o< (loghead 32 (+ -1 (logext 32 rdi)))
(loghead 32 rdi))
:g-bindings (gl::auto-bindings (:int rdi 64)))
(defthm-using-gl low-32-bits-of-rdi
:hyp (and (natp rdi)
(<= 0 rdi)
(< rdi 13))
:concl (equal (equal (loghead 32 rdi) 0)
(equal rdi 0))
:g-bindings (gl::auto-bindings (:int rdi 64)))
(defthm loghead-<=0-of-x
(implies (and (< i 0)
(integerp i))
(equal (loghead i x) 0))
:hints (("Goal" :in-theory (e/d () ())))))
;; The functions clk-16 and sem-16 generated by def-semantics below
;; need to know that the input rdi and rax registers must be greater
;; than 0 in order to terminate. So, I'm grabbing the output from
;; def-semantics and then hand-editing it.
#||
(acl2::def-semantics
:init-pc 0
:focus-regionp (lambda (pc) (and (<= 0 pc) (<= pc 23)))
:root-name nil ; optional - to change the fn names chosen ;
;; :hyps+ ((good-popcount-x86p x86)) ; optional - to strengthen the :hyps of API ;
:annotations nil ; optional - to modify output generated ;
)
||#
(PROGN
(SET-VERIFY-GUARDS-EAGERNESS 0)
(defun-nx
CLK-16 (X86)
(DECLARE (XARGS :STOBJS (X86)
:measure (loghead 32 (logext 32 (rgfi *rdi* x86)))))
(IF
(and (FACTORIAL-HYPS X86)
;; We know the following facts from sem-0.
(< 0 (xr :rgf *rdi* x86))
(< 0 (rgfi *rax* x86)))
(IF
(EQUAL (LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86))))
0)
3
(BINARY-CLK+
3
(CLK-16
(XW
:RGF *RAX*
(LOGHEAD 32
(* (LOGEXT 32 (XR :RGF *RAX* X86))
(LOGEXT 32 (XR :RGF *RDI* X86))))
(XW
:RGF *RDI*
(LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86))))
(XW
:RIP nil 16
(XW
:UNDEF nil (+ 4 (NFIX (XR :UNDEF nil X86)))
(!FLGI
:CF
(LOGHEAD 1
(BOOL->BIT (< (LOGHEAD 32 (XR :RGF *RDI* X86)) 1)))
(!FLGI
:PF
(PF-SPEC32 (LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86)))))
(!FLGI
:AF
(SUB-AF-SPEC32 (LOGHEAD 32 (XR :RGF *RDI* X86))
1)
(!FLGI
:ZF 0
(!FLGI
:SF
(SF-SPEC32 (LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86)))))
(!FLGI :OF
(OF-SPEC32 (+ -1 (LOGEXT 32 (XR :RGF *RDI* X86))))
X86)))))))))))))
0))
(defun-nx
CLK-0 (X86)
(DECLARE (XARGS :STOBJS (X86)))
(IF
(FACTORIAL-HYPS X86)
(IF
(EQUAL (LOGHEAD 32 (XR :RGF *RDI* X86)) 0)
3
(BINARY-CLK+
4
(CLK-16
(XW
:RGF *RAX* 1
(XW
:RIP nil 16
(XW
:UNDEF NIL (+ 1 (NFIX (XR :UNDEF nil X86)))
(!FLGI
:CF 0
(!FLGI
:PF
(PF-SPEC32 (LOGHEAD 32 (XR :RGF *RDI* X86)))
(!FLGI :AF
(LOGHEAD 1
(CREATE-UNDEF (NFIX (XR :UNDEF nil X86))))
(!FLGI :ZF 0
(!FLGI :SF
(SF-SPEC32 (LOGHEAD 32 (XR :RGF *RDI* X86)))
(!FLGI :OF 0 X86))))))))))))
0))
(defun-nx
SEM-16 (X86)
(DECLARE (XARGS :STOBJS (X86)
:measure (loghead 32 (logext 32 (rgfi *rdi* x86)))))
(IF
(and (FACTORIAL-HYPS X86)
;; We know the following facts from sem-0.
(< 0 (xr :rgf *rdi* x86))
(< 0 (rgfi *rax* x86)))
(IF
(EQUAL (LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86))))
0)
(XW
:RGF *RAX*
(LOGHEAD 32
(* (LOGEXT 32 (XR :RGF *RAX* X86))
(LOGEXT 32 (XR :RGF *RDI* X86))))
(XW
:RGF *RDI* 0
(XW
:RIP nil 24
(XW
:UNDEF nil (+ 4 (NFIX (XR :UNDEF nil X86)))
(!FLGI
:CF
(LOGHEAD 1
(BOOL->BIT (< (LOGHEAD 32 (XR :RGF *RDI* X86)) 1)))
(!FLGI
:PF 1
(!FLGI
:AF
(SUB-AF-SPEC32 (LOGHEAD 32 (XR :RGF *RDI* X86))
1)
(!FLGI
:ZF 1
(!FLGI :SF 0
(!FLGI :OF
(OF-SPEC32 (+ -1 (LOGEXT 32 (XR :RGF *RDI* X86))))
X86))))))))))
(SEM-16
(XW
:RGF *RAX*
(LOGHEAD 32
(* (LOGEXT 32 (XR :RGF *RAX* X86))
(LOGEXT 32 (XR :RGF *RDI* X86))))
(XW
:RGF *RDI*
(LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86))))
(XW
:RIP nil 16
(XW
:UNDEF nil (+ 4 (NFIX (XR :UNDEF nil X86)))
(!FLGI
:CF
(LOGHEAD 1
(BOOL->BIT (< (LOGHEAD 32 (XR :RGF *RDI* X86)) 1)))
(!FLGI
:PF
(PF-SPEC32 (LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86)))))
(!FLGI
:AF
(SUB-AF-SPEC32 (LOGHEAD 32 (XR :RGF *RDI* X86))
1)
(!FLGI
:ZF 0
(!FLGI
:SF
(SF-SPEC32 (LOGHEAD 32
(+ -1 (LOGEXT 32 (XR :RGF *RDI* X86)))))
(!FLGI :OF
(OF-SPEC32 (+ -1 (LOGEXT 32 (XR :RGF *RDI* X86))))
X86))))))))))))
X86))
(ACL2::DEFUNM-NX
SEM-0 (X86)
(DECLARE (XARGS :STOBJS (X86)))
(IF
(FACTORIAL-HYPS X86)
(IF
(EQUAL (LOGHEAD 32 (XR :RGF *RDI* X86))
0)
(XW
:RGF *RAX* 1
(XW
:RIP nil 24
(XW
:UNDEF nil (+ 1 (NFIX (XR :UNDEF nil X86)))
(!FLGI :CF 0
(!FLGI :PF 1
(!FLGI :AF
(LOGHEAD 1
(CREATE-UNDEF (NFIX (XR :UNDEF nil X86))))
(!FLGI :ZF 1
(!FLGI :SF 0 (!FLGI :OF 0 X86)))))))))
(SEM-16
(XW
:RGF *RAX* 1
(XW
:RIP nil 16
(XW
:UNDEF nil (+ 1 (NFIX (XR :UNDEF nil X86)))
(!FLGI
:CF 0
(!FLGI
:PF
(PF-SPEC32 (LOGHEAD 32 (XR :RGF *RDI* X86)))
(!FLGI :AF
(LOGHEAD 1
(CREATE-UNDEF (NFIX (XR :UNDEF nil X86))))
(!FLGI :ZF 0
(!FLGI :SF
(SF-SPEC32 (LOGHEAD 32 (XR :RGF *RDI* X86)))
(!FLGI :OF 0 X86)))))))))))
X86))
;; (acl2::why x86-run-opener-not-ms-not-zp-n)
;; (acl2::why x86-fetch-decode-execute-opener)
;; (acl2::why get-prefixes-opener-lemma-no-prefix-byte)
;; (acl2::why one-read-with-rb-from-program-at)
;; (acl2::why program-at-wb-disjoint)
(DEFTHM SEM-16-CORRECT
(IMPLIES (AND (FACTORIAL-HYPS X86)
(EQUAL (XR :RIP nil X86) 16))
(EQUAL (X86-RUN-ALT X86 (CLK-16 X86))
(SEM-16 X86))))
(IN-THEORY (DISABLE CLK-16))
(DEFTHM SEM-0-CORRECT
(IMPLIES (AND (FACTORIAL-HYPS X86)
(EQUAL (XR :RIP nil X86) 0))
(EQUAL (X86-RUN-ALT X86 (CLK-0 X86))
(SEM-0 X86))))
(IN-THEORY (DISABLE CLK-0)))
;; ======================================================================
;; Relating the program and the specification:
(encapsulate
()
(defthm loghead-n-x-<-x
(implies (natp x)
(<= (loghead n x) x))
:hints (("Goal"
:in-theory (e/d* (bitops::ihsext-inductions
bitops::ihsext-recursive-redefs)
())))
:rule-classes :linear)
(defthm loghead-n-1-logext-n-x-<-x
(implies (and (posp x)
(posp n))
(< (loghead n (+ -1 (logext n x))) x))
:hints (("Goal"
:cases ((equal (logcar x) 0))
:in-theory (e/d* (bitops::ihsext-inductions
bitops::ihsext-recursive-redefs)
())))
:rule-classes :linear)
(defun fact-algorithm (n a)
(if (posp n)
(let* ((a (n32 (* (n32-to-i32 a) (n32-to-i32 n))))
(n (n32 (- (n32-to-i32 n) 1))))
(if (not (equal n 0))
(fact-algorithm n a)
a))
1)))
(defthm factorial-loop-result-helper
(implies (and (factorial-hyps x86)
(equal (xr :rip nil x86) 16)
(< 0 (rgfi *rdi* x86))
(< 0 (rgfi *rax* x86)))
(equal (xr :rgf *rax* (sem-16 x86))
(fact-algorithm (rgfi *rdi* x86) (rgfi *rax* x86)))))
(defthm factorial-program-result-helper
(implies (and (factorial-hyps x86)
(equal (xr :rip nil x86) 0))
(equal (xr :rgf *rax* (sem-0 x86))
(fact-algorithm (rgfi *rdi* x86) 1)))
:hints (("Goal" :in-theory (e/d* () ()))))
(defun f (n)
(if (zp n)
1
(* n (f (- n 1)))))
(defun fact-algorithm-simple (n a)
(if (posp n)
(let* ((a (* a n))
(n (- n 1)))
(if (not (equal n 0))
(fact-algorithm-simple n a)
a))
1))
(defthmd fact-algorithm-and-fact-algorithm-simple
(implies (and (natp n)
(< n 13))
(equal (fact-algorithm n 1)
(fact-algorithm-simple n 1)))
:hints (("Goal" :cases ((equal n 0)
(equal n 1)
(equal n 2)
(equal n 3)
(equal n 4)
(equal n 5)
(equal n 6)
(equal n 7)
(equal n 8)
(equal n 9)
(equal n 10)
(equal n 11)
(equal n 12)))))
(defthm fact-algorithm-simple-and-f-1
(implies (and (natp n)
(posp n)
(natp a))
(equal (fact-algorithm-simple n a)
(* a (f n)))))
(defthm fact-algorithm-simple-and-f
(implies (natp n)
(equal (fact-algorithm-simple n 1)
(f n))))
(defthm fact-algorithm-and-f
(implies (and (natp n)
(< n 13))
(equal (fact-algorithm n 1)
(f n)))
:hints (("Goal" :use ((:instance fact-algorithm-and-fact-algorithm-simple)))))
(defthm factorial-program-result
(implies (and (factorial-hyps x86)
(equal (xr :rip nil x86) 0))
(equal (xr :rgf *rax* (x86-run-alt x86 (clk-0 x86)))
(f (rgfi *rdi* x86)))))
;; ----------------------------------------------------------------------
|
