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Proofgold Proof
pf
Apply df_trls__df_trlson__df_pths__df_spths__df_pthson__df_spthson__df_clwlks__df_crcts__df_cycls__df_wwlks__df_wwlksn__df_wwlksnon__df_wspthsn__df_wspthsnon__df_clwwlk__df_clwwlkn__df_clwwlknOLD__df_clwwlknon with
wceq
cclwwlk
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
crab
(
λ x1 .
w3a
(
wne
(
cv
x1
)
c0
)
(
wral
(
λ x2 .
wcel
(
cpr
(
cfv
(
cv
x2
)
(
cv
x1
)
)
(
cfv
(
co
(
cv
x2
)
c1
caddc
)
(
cv
x1
)
)
)
(
cfv
(
cv
x0
)
cedg
)
)
(
λ x2 .
co
cc0
(
co
(
cfv
(
cv
x1
)
chash
)
c1
cmin
)
cfzo
)
)
(
wcel
(
cpr
(
cfv
(
cv
x1
)
clsw
)
(
cfv
cc0
(
cv
x1
)
)
)
(
cfv
(
cv
x0
)
cedg
)
)
)
(
λ x1 .
cword
(
cfv
(
cv
x0
)
cvtx
)
)
)
)
.
Assume H0:
wceq
ctrls
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
cwlks
)
)
(
wfun
(
ccnv
(
cv
x1
)
)
)
)
)
)
.
Assume H1:
wceq
ctrlson
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wa
(
wbr
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
cwlkson
)
)
)
(
wbr
(
cv
x3
)
(
cv
x4
)
(
cfv
(
cv
x0
)
ctrls
)
)
)
)
)
)
.
Assume H2:
wceq
cpths
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
w3a
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrls
)
)
(
wfun
(
ccnv
(
cres
(
cv
x2
)
(
co
c1
(
cfv
(
cv
x1
)
chash
)
cfzo
)
)
)
)
(
wceq
(
cin
(
cima
(
cv
x2
)
(
cpr
cc0
(
cfv
(
cv
x1
)
chash
)
)
)
(
cima
(
cv
x2
)
(
co
c1
(
cfv
(
cv
x1
)
chash
)
cfzo
)
)
)
c0
)
)
)
)
.
Assume H3:
wceq
cspths
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrls
)
)
(
wfun
(
ccnv
(
cv
x2
)
)
)
)
)
)
.
Assume H4:
wceq
cpthson
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wa
(
wbr
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrlson
)
)
)
(
wbr
(
cv
x3
)
(
cv
x4
)
(
cfv
(
cv
x0
)
cpths
)
)
)
)
)
)
.
...
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