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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
cwwspthsnon
(
cmpt2
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
crab
(
λ x4 .
wex
(
λ x5 .
wbr
(
cv
x5
)
(
cv
x4
)
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
cspthson
)
)
)
)
(
λ x4 .
co
(
cv
x2
)
(
cv
x3
)
(
co
(
cv
x0
)
(
cv
x1
)
cwwlksnon
)
)
)
)
)
.
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
)
)
)
)
)
)
.
Assume H5:
wceq
cspthson
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
cfv
...
...
)
...
)
)
.
...
■