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; Boogie program verifier version 2.00, Copyright (c) 2003-2009, Microsoft.
; Command Line Options: -nologo -logPrefix:inline /proverLog:expansion2.sx expansion2.bpl
; Proof obligation: foo
; Z3 command line: c:\Program Files\Microsoft Research\Z3-2.0\bin\z3.exe /si /@ /cex:5 /t:0
; User supplied Z3 options:
; Prover options:
;
(SETPARAMETER MODEL_PARTIAL true)
(SETPARAMETER MODEL_VALUE_COMPLETION false)
(SETPARAMETER MODEL_HIDE_UNUSED_PARTITIONS false)
(SETPARAMETER MODEL_V1 true)
(SETPARAMETER ASYNC_COMMANDS false)
(SETPARAMETER PHASE_SELECTION 0)
(SETPARAMETER RESTART_STRATEGY 0)
(SETPARAMETER RESTART_FACTOR |1.5|)
(SETPARAMETER NNF_SK_HACK true)
(SETPARAMETER QI_EAGER_THRESHOLD 100)
(SETPARAMETER ARITH_RANDOM_INITIAL_VALUE true)
(SETPARAMETER SORT_AND_OR false)
(SETPARAMETER CASE_SPLIT 3)
(SETPARAMETER DELAY_UNITS true)
(SETPARAMETER DELAY_UNITS_THRESHOLD 16)
(SETPARAMETER TYPE_CHECK true)
(SETPARAMETER BV_REFLECT true)
; -------------------------------------------------------------------------
; Boogie 2 universal background predicate for Z3 (Simplify notation with types)
; Copyright (c) 2004-2009, Microsoft Corp.
(DEFTYPE $int :BUILTIN Int)
(DEFTYPE $bool :BUILTIN bool)
(DEFTYPE U)
(DEFTYPE T)
(DEFOP <: U U $bool) ; used for translation with type premisses
(DEFOP <:: T U U $bool) ; used for translation with type arguments
(BG_PUSH (AND
; false is not true
(DISTINCT |@false| |@true|)
; we assume type correctness of the operations here
; a-l>=0 ==> (v ++ w:l)[a:b] = v[a-l:b-l]
(FORALL (v lv w lw lvw a b)
(QID bv:e:c1)
(PATS ($bv_extract ($bv_concat v lv w lw) lvw a b))
(IMPLIES
(>= (- a lw) 0)
(EQ ($bv_extract ($bv_concat v lv w lw) lvw a b) ($bv_extract v lv (- a lw) (- b lw)))))
; b<=l ==> (v ++ w:l)[a:b] = w[a:b]
(FORALL (v lv w lw lvw a b)
(QID bv:e:c2)
(PATS ($bv_extract ($bv_concat v lv w lw) lvw a b))
(IMPLIES
(<= b lw)
(EQ ($bv_extract ($bv_concat v lv w lw) lvw a b) ($bv_extract w lw a b))))
; v:l
; a>=x || b<=y ==> (v[x:l] ++ w ++ v[0:y])[a:b] = v[a:b]
(FORALL (v lv x lxv w lw lwy y a b)
(QID bv:e:c3)
(PATS
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
($bv_concat
w lw
($bv_extract v lv 0 y) y) lwy) lv a b))
(IMPLIES
(AND
(EQ lw (- x y))
(EQ lxv (- lv x))
(EQ lwy (+ w y))
(OR (>= a x) (<= b y)))
(EQ
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
($bv_concat
w lw
($bv_extract v lv 0 y) y) lwy) lv a b)
($bv_extract v lv a b))))
; a>=x ==> (v[x:l] ++ w)[a:b] = v[a:b]
(FORALL (v lv x lxv w a b)
(QID bv:e:c4)
(PATS
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
w x)
lv a b))
(IMPLIES
(AND
(EQ lxv (- lv x))
(>= a x))
(EQ
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
w x)
lv a b)
($bv_extract v lv a b))))
(FORALL (v l)
(QID bv:e:0)
(PATS ($bv_extract v l 0 l))
(EQ ($bv_extract v l 0 l) v))
(FORALL (n)
(QID bv:pow)
(PATS ($pow2 n))
(IMPLIES (> n 0) (EQ ($pow2 n) (* 2 ($pow2 (- n 1))))))
(EQ ($pow2 0) 1)
; 0 <= v < 2^Y ==> 0bvX ++ v[0:Y] == v
(FORALL (v l a b)
(QID bv:e:pow)
(PATS ($bv_concat 0 b ($bv_extract v l 0 a) a))
(IMPLIES
(AND
(<= 0 v)
(< v ($pow2 a))
(EQ l (+ a b)))
(EQ ($bv_concat 0 b ($bv_extract v l 0 a) a) v)))
; X > 0 ==> 0bvX ++ v >= 0
(FORALL (v a b)
(QID bv:e:pos)
(PATS ($bv_concat 0 b v a))
(IMPLIES
(> b 0)
(>= ($bv_concat 0 b v a) 0)))
;; unsound?
; (FORALL (lv w lw)
; (QID bv:c:0)
; (PATS ($bv_concat 0 lv w lw))
; (EQ ($bv_concat 0 lv w lw) w))
;; matching loop
; (FORALL (v l1 a b l2 c d)
; (QID bv:e:e)
; (PATS ($bv_extract ($bv_extract v l1 a b) l2 c d))
; (EQ ($bv_extract ($bv_extract v l1 a b) l2 c d) ($bv_extract v l1 (+ c a) (+ d a))))
; Reflect plus
(FORALL (a b) (PATS (Reflect$Add a b)) (EQ (Reflect$Add a b) (+ a b)))
)) ;; AND, BG_PUSH
; End Boogie universal background predicate
; -------------------------------------------------------------------------
(DEFOP Ctor T $int)
(DEFOP intType T)
(DEFOP boolType T)
(DEFOP int_2_U $int U)
(DEFOP U_2_int U $int)
(DEFOP type U T)
(DEFOP bool_2_U $bool U)
(DEFOP U_2_bool U $bool)
(BG_PUSH
(AND (EQ (Ctor (intType)) 0) (EQ (Ctor (boolType)) 1) (FORALL (arg0 :TYPE $int) (PATS (int_2_U arg0)) (QID typeInv:U_2_int) (EQ (U_2_int (int_2_U arg0)) arg0)) (FORALL (x :TYPE U) (PATS (int_2_U (U_2_int x))) (QID cast:U_2_int) (IMPLIES (EQ (type x) (intType)) (EQ (int_2_U (U_2_int x)) x))) (FORALL (arg0@@0 :TYPE $int) (PATS (int_2_U arg0@@0)) (QID funType:int_2_U) (EQ (type (int_2_U arg0@@0)) (intType))) (FORALL (arg0@@1 :TYPE $bool) (PATS (bool_2_U arg0@@1)) (QID typeInv:U_2_bool) (IFF (U_2_bool (bool_2_U arg0@@1)) arg0@@1)) (FORALL (x@@0 :TYPE U) (PATS (bool_2_U (U_2_bool x@@0))) (QID cast:U_2_bool) (IMPLIES (EQ (type x@@0) (boolType)) (EQ (bool_2_U (U_2_bool x@@0)) x@@0))) (FORALL (arg0@@2 :TYPE $bool) (PATS (bool_2_U arg0@@2)) (QID funType:bool_2_U) (EQ (type (bool_2_U arg0@@2)) (boolType))))
)
(BG_PUSH
(AND (FORALL (x@@1 :TYPE U) (NOPATS (U_2_int x@@1) (U_2_bool x@@1)) (QID bg:subtype-refl) (<: x@@1 x@@1)) (FORALL (x@@2 :TYPE U y :TYPE U z :TYPE U) (PATS (MPAT (<: x@@2 y) (<: y z))) (QID bg:subtype-trans) (LET ((TERM alpha (type x@@2))) (IMPLIES (AND (EQ (type y) alpha) (EQ (type z) alpha)) (IMPLIES (AND (<: x@@2 y) (<: y z)) (<: x@@2 z))))) (FORALL (x@@3 :TYPE U y@@0 :TYPE U) (PATS (MPAT (<: x@@3 y@@0) (<: y@@0 x@@3))) (QID bg:subtype-antisymm) (LET ((TERM alpha@@0 (type x@@3))) (IMPLIES (EQ (type y@@0) alpha@@0) (IMPLIES (AND (<: x@@3 y@@0) (<: y@@0 x@@3)) (EQ x@@3 y@@0))))) (FORALL (z@@0 :TYPE $int) (QID expansio.6:15) (SKOLEMID 2) (IMPLIES (> z@@0 12) (> z@@0 0))) (FORALL (y@@1 :TYPE $int) (QID expansio.7:15) (SKOLEMID 3) (IMPLIES (> (+ y@@1 1) 1) (> y@@1 0))))
)
; Initialized all axioms.
(BG_PUSH
TRUE
)
(LET ((FORMULA anon0_correct (IMPLIES (LBLPOS |+107| TRUE) (IMPLIES TRUE (AND (LBLNEG |@164| (> 12 0)) (IMPLIES (> 12 0) (AND (LBLNEG |@168| (EQ (+ 3 1) 4)) (IMPLIES (EQ (+ 3 1) 4) (IMPLIES TRUE TRUE))))))))) anon0_correct)
; Valid
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