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Theorem bxor-flip-one-bit
2 ^ (j + 1) < i ∧ 0 ≤ i ∧ 0 ≤ j, boolvar ↔ int2bits(i)[# int2bits(i) - i→n(j) +1] ⊦ (int2bits(bxor(i, 2 ^ j))[# int2bits(i) - i→n(j) +1] ↔ ¬ boolvar) ∧ int2bits(bxor(i, 2 ^ j))[# int2bits(i) - i→n(j) +1, boolvar] = int2bits(i)
Type: A user defined theorem
Validity: The proof is valid.
Features:
Comment:
Used by:
Used by units in proved state:
Proof exists?: Yes
View
the proof
Infos about the proof
Proved?: Yes
Interactions: 64
Proof steps: 123
Used lemmas:
bitsxor-zeros-01
,
bitsxor-first
,
bitsxor-rec-02
,
bitsxor-rec-01
,
bitsxor-append-same-size
,
bitsxor-zerol
,
bitsxor-empty
,
bxor-def
,
Used spec theorems:
list
/bitlista/
split-rest
int2bits
/
int2bits-lslen
list-del
/bitlista/
get-zero
int2bits
/
bits2int-ge-zero
int2bits
/
int2bits-leneq-pos
int-pot
/
ls-pos-2pot
list-del
/bitlista/
get-append-less
list-del
/bitlista/
at-last
list
/bitlista/
len
int2bits
/
int2bits-rec-pos-01
int2bits
/
int2bits-one-bit
list
/bitlista/
firsteq-01
int2bits
/
bits2bits-def
list-data
/bitlista/
lf-01
bitopsaux
/
sgn-base-05
int-pot
/
succ
int2bits
/
int2bits-lslen-12
list
/bitlista/
first-01
int2bits
/
int2bits-leading-sgn
list-del
/bitlista/
put-app-left
list-del
/bitlista/
put-last
Used simplifier rules:
nat-basic1
/
lf-02
nat-basic1
/
lf-01
intnat
/
labs
int-abs
/
absle
int-basic1
/
irref
list-del
/bitlista/
get-app
nat-basic2
/
eq-01
list
/bitlista/
in-13
list
/bitlista/
first
list
/bitlista/
eq-04
nat-basic2
/
one-03
list
/bitlista/
first-02
list
/bitlista/
e-03
list
/bitlista/
eq-05
list
/bitlista/
app-01
nat
/
add-num-01
nat
/
le-num-02
nat-basic2
/
eq
nat
/
le-add
intnat
/
rax-04
list-del
/bitlista/
get-zero-01
bitopsaux
/
sgn-base-02
int2bits
/
int2bits-first-05
list-last
/bitlista/
l-06
list-del
/bitlista/
get-last-03
list
/bitlista/
c-08
list
/bitlista/
c-07
nat-basic2
/
z-01
nat
/
add-num-03
bitopsaux
/
sgn-rec
list
/bitlista/
in-12
list
/bitlista/
in-11
list
/bitlista/
inapp-08
list
/bitlista/
ni
nat-basic2
/
one
int2bits
/
int2bits-first-03
nat-basic2
/
one-02
nat-basic2
/
oz
list
/bitlista/
rest-01
list
/bitlista/
first-01
nat-basic2
/
z
nat
/
add-01
int-abs
/
minus-add
int-abs
/
sub-eq
int-div
/
div-elim-pospos
bitopsaux
/
sgn-two-01
list-del
/bitlista/
one-03
list-del
/bitlista/
get-numrec
bitopsaux
/
sgn-two-03
int-div
/
div-two-zero
int-div
/
mod-two-06
list
/bitlista/
app-06
list
/bitlista/
in-append
list
/bitlista/
in-14
list
/bitlista/
c-03
list-data
/bitlista/
elim-list
list
/bitlista/
c
list
/bitlista/
in-17
list
/bitlista/
in-15
list-last
/bitlista/
last-elim
list
/bitlista/
e-02
list-last
/bitlista/
last-def
list-last
/bitlista/
butlast-def
list-last
/bitlista/
mk-len
nat
/
ler
nat
/
lels
nat-basic1
/
one-01
nat
/
ls-01
nat-basic2
/
s
nat-basic1
/
ref-1
int-abs
/
ler
int-abs
/
succ-02
int-abs
/
numsub
int-abs
/
sub-add-01
int-abs
/
sub-ls
int-basic1
/
two-ls
int-abs
/
leone
int-abs
/
forward-lels
intnat
/
i2i-pos-01
int-basic1
/
lst-01
int2bits
/
int2bits-len-one
bitlista
/
mklist-num
list-last
/bitlista/
mk-zero
list
/bitlista/
app
list
/bitlista/
Last
list
/bitlista/
e-01
list
/bitlista/
app-02-ant
list
/bitlista/
e
int2bits
/
bits2int-base
int2bits
/
bits2bits-def
int2bits
/
int2bits-first-02
int2bits
/
int2bits-not-empty
list-data
/bitlista/
size-01
list
/bitlista/
Nil
list-data
/bitlista/
l
nat-basic2
/
a
nat
/
cancel-01
list
/bitlista/
len
int-basic1
/
lst
int-pot
/
ls-pos-2pot-01
int-basic2
/
ls
int-abs
/
numleledel-01
int-abs
/
numle-02
int-abs
/
numls-02
int-abs
/
numadd-01
int-abs
/
elim-sub
int-basic1
/
numlslsdel-01
int-abs
/
sub-ls-01
int-pot
/
pot2-pos-01
int-abs
/
sub-le-01
int-pot
/
pot2-pos
int-basic1
/
forward-pos
intnat
/
i2i-pos
int-abs
/
lels
int-abs
/
anti
int2bits
/
int2bits-zero
list
/bitlista/
len-02
nat-basic1
/
one
int-basic1
/
ls-01
intnat
/
num-i2n-eq
int-mult
/
zero-ls
int-pot
/
two-lsz
int-mult
/
associativity
int-mult
/
commutativity
int-abs
/
forward-le-05
int-abs
/
lels-02
intnat
/
abs-zero
nat-basic1
/
f
nat
/
lels-01
list
/bitlista/
a
int-abs
/
lels-01
int-abs
/
lels-04
int2bits
/
int2bits-2pot
int-abs
/
zero-gels-04
int-abs
/
forward-le-01
nat-basic2
/
c
nat-basic2
/
succ
int-basic2
/
add-c
int-basic2
/
add-a
int-pot
/
succ