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Theorem uint2bits-add-high
⊦ j < 0 ∧ 0 < i ∧ 2 ^ (i - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i → uint2bits(j + 2 ^ i) = sgn(int2bits(j).rest)
Type: A user defined theorem
Validity: The proof is valid.
Features:
Comment:
Used by:
uint2bits-add-high-01
,
uint2bits-add-high-02
,
Used by units in proved state:
Proof exists?: Yes
View
the proof
Infos about the proof
Proved?: Yes
Interactions: 67
Proof steps: 122
Used lemmas:
int2bits-zero
,
int2bits-falsein-01
,
uint2bits-zero-01
,
int2bits-truein-01
,
int2bits-not-empty
,
uint2bits-2pot-02
,
int2bits-neg-num
,
int2bits-rec-neg
,
int2bits-2pot-01
,
uint2bits-zero
,
int2bits-2pot
,
int2bits-succ
,
int2bits-2pot-02
,
int2bits-neg-03
,
uint2bits-one
,
uint2bits-rec-01
,
Used spec theorems:
bitopsaux
/
succ-zero-03
list-last
/bitlista/
mk-last-01-01
list-last
/bitlista/
mk-last-02
intnat
/
Int2nat-pos
int-abs
/
numadd-01
bitopsaux
/
sgn-mklist-03
int-div
/
div2-add-neg
int-div
/
mod2-add-neg
int-basic2
/
add-c
list
/bitlista/
rest-01
bitopsaux
/
sgn-last-diff
int-div
/
modex
int-abs
/
numlef
int-pot
/
eq-two
int-abs
/
rew-eqzero-01
int-abs
/
rew-eqzero
int-abs
/
Abs-neg
int-pot
/
succ
int-pot
/
pot-dist-pos
Used simplifier rules:
int-div
/
nummult-modls
int-mult
/
add-mult-two
int-mult
/
numls-02
int-mult
/
zero-le
int-mult
/
mods-ls-04
int-abs
/
numlsledel-01
int-mult
/
numls-01
int-basic2
/
add-in
int-abs
/
sub-add-09
int-abs
/
lstwo
int-pot
/
succ
int-mult
/
numle-12
int-mult
/
numls-08
int-mult
/
le-pos
int-mult
/
le-same-01
int-mult
/
numle-09
int-mult
/
dist-rmerge
int-mult
/
ls-pos
int-abs
/
numleledel-01
int-abs
/
elim-sub
int-mult
/
zero-ge-add
int-mult
/
numsucc
int-div
/
nummult-ls-02
int-div
/
mod-two-02
int-div
/
neg-01
int-basic1
/
numlsf
int-mult
/
numls-13
int-mult
/
zero-gr
int-abs
/
numleledel
int-abs
/
lep-02
int-basic1
/
numlslsdel
int-basic1
/
lst
int-div
/
div-elim-pos
int-basic1
/
minus-one
int-abs
/
numls-03
int-abs
/
numle-02
int-pot
/
two-lsz
int-pot
/
eq-one
list
/bitlista/
e-02
int-abs
/
zero-gels-01
int-abs
/
forward-le-04
int-abs
/
forward-le-01
int-abs
/
forward-le-05
int-abs
/
forward-le-02
int-pot
/
pot2-pos
int-pot
/
pot2-pos-01
int-pot
/
pot2-suble
int-pot
/
pot2-subls
int-div
/
nummult-ls
int-mult
/
abs-eqall
int-mult
/
ls-two-02
int-abs
/
zero-gels-04
int-mult
/
ls-two-03
int-pot
/
pot2-pos-02
int-pot
/
pot2-pos-03
int-pot
/
pot2-pos-04
int-mult
/
neq-two
int-div
/
div-mult-03
int-mult
/
pred-rmerge
int-abs
/
minus-add
int-div
/
minus-add
int-abs
/
minus-02
int-abs
/
Abs-pos
int-basic1
/
irref
int-pot
/
pot2-add
int-abs
/
neg-03
int-mult
/
ls-same-01
int-basic2
/
lst-01
int-mult
/
numabs-01
int-abs
/
abs-neg
int-pot
/
two-ls
int-mult
/
numls-03
int-mult
/
modzero-01
int-mult
/
numls-12
int-mult
/
numls-04
int-mult
/
ls-same-02
int-mult
/
succ-rmerge
int-div
/
div-elim2
int-mult
/
zero-ge-add-01
int-div
/
div-two
int-abs
/
neg-02
int-abs
/
subsub
int-mult
/
zero-ge
int-abs
/
numls-02
int-pot
/
zero
int-abs
/
neg-10
int-mult
/
minus-left
int-div
/
div2-neg-odd
int-mult
/
zero-ls
int-mult
/
two-zero
int-mult
/
two-mone
int-basic1
/
forward-pos
int-abs
/
forward-le
int-abs
/
ler
int-abs
/
succ-02
int-abs
/
numsub
int-pot
/
ls-pos-2pot-01
int-pot
/
pot2-ls-pred
int-pot
/
pot2-ls-pred-01
bitopsaux
/
sgn-mklist-05
bitopsaux
/
sgn-empty-01
bitopsaux
/
sgn-mklist-01
list
/bitlista/
app-05
int-div
/
div2-notzero
int-div
/
mod2-neg-odd
int-abs
/
mone-01
bitopsaux
/
invert-zero
bitopsaux
/
inv-one-01
list
/bitlista/
c
bitopsaux
/
succ-zero
intnat
/
abs-succ-02
int-pot
/
eq2
int-abs
/
mone-06
int-pot
/
2pot-leone
int-basic1
/
two-ls
int-abs
/
lels-04
int-div
/
abs
int-abs
/
sub-sub
int-abs
/
bet
int-abs
/
sub-ls-01
int-abs
/
sub-le-01
int-div
/
div-two-zero-01
intnat
/
abs-pred
int-abs
/
lels-03
int-abs
/
numadd-01
int-abs
/
negeq
int-div
/
mod-neg-01
int-abs
/
lep
int-abs
/
forward-lele
int-pot
/
pot2-div2
int-pot
/
pot2-mod2-zero
int-abs
/
neg-05
int-abs
/
neg-13
int-abs
/
neg-15
int-abs
/
neg-06
int-basic2
/
lst-04
int-basic2
/
ls
int-abs
/
forward-lels
nat
/
add-01
nat-basic2
/
z
int-abs
/
sub-eq-01
int-abs
/
numlelsdel-01
list-last
/bitlista/
l
bitopsaux
/
sgn-empty
list
/bitlista/
rest
bitopsaux
/
sgn-two-01
list
/bitlista/
Nil
list-last
/bitlista/
b
list
/bitlista/
app
list-last
/bitlista/
mk-zero
list-last
/bitlista/
mk-butlast
bitopsaux
/
sgn-base-01
bitopsaux
/
sgn-mklist-02
bitopsaux
/
sgn-two-02
list
/bitlista/
app-01
list
/bitlista/
c-07
bitopsaux
/
sgn-base-02
list-last
/bitlista/
mk-last
nat
/
pred-03
list
/bitlista/
e-01
bitopsaux
/
sgn-rec
list
/bitlista/
Last
list-last
/bitlista/
bin
int-abs
/
lels-02
int-basic1
/
lf-03
int-basic1
/
lf-02
list-last
/bitlista/
l-05
intnat
/
abs-zero
list-last
/bitlista/
mk-empty
list
/bitlista/
e
list
/bitlista/
app-02-ant
int-mult
/
commutativity
int-mult
/
associativity
int-mult
/
one
int-mult
/
numsub
int-pot
/
grz
int-basic2
/
zero-02
int-abs
/
numlsf-01
int-basic1
/
numlslsdel-01
int-abs
/
leone
int-abs
/
succ-01
int-abs
/
sub-le
int-basic2
/
lst-03
int-basic2
/
add-zero
int-abs
/
sub-ls
int-abs
/
sub-add-01
nat-basic2
/
c
nat-basic2
/
a
intnat
/
i2i-pos
intnat
/
n2i-add
list
/bitlista/
a
list-last
/bitlista/
mk-len
list
/bitlista/
len-02
list
/bitlista/
len
int-abs
/
lels-01
int-basic1
/
ls-01
int-abs
/
minus-sub
int-abs
/
Sub-def
int-abs
/
sub-eq
int-abs
/
lsone
list
/bitlista/
Two
int-abs
/
zeroone
int-abs
/
anti
int-abs
/
lels
int-basic2
/
ls-02
int-basic2
/
succ-01
int-basic2
/
add-c
int-basic2
/
add-a
int-basic2
/
ls-01
int-basic1
/
trans
int-basic1
/
lf-01
int-abs
/
forward-lsle
int-abs
/
negle-13
int-abs
/
neg-07
int-abs
/
abs-08