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Proofgold Proof
pf
Apply df_conngr__df_eupth__df_frgr__df_plig__df_grpo__df_gid__df_ginv__df_gdiv__df_ablo__df_vc__df_nv__df_va__df_ba__df_sm__df_0v__df_vs__df_nmcv__df_ims with
wceq
cnmcv
c2nd
.
Assume H0:
wceq
cconngr
(
cab
(
λ x0 .
wsbc
(
λ x1 .
wral
(
λ x2 .
wral
(
λ x3 .
wex
(
λ x4 .
wex
(
λ x5 .
wbr
(
cv
x4
)
(
cv
x5
)
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x0
)
cpthson
)
)
)
)
)
(
λ x3 .
cv
x1
)
)
(
λ x2 .
cv
x1
)
)
(
cfv
(
cv
x0
)
cvtx
)
)
)
.
Assume H1:
wceq
ceupth
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrls
)
)
(
wfo
(
co
cc0
(
cfv
(
cv
x1
)
chash
)
cfzo
)
(
cdm
(
cfv
(
cv
x0
)
ciedg
)
)
(
cv
x1
)
)
)
)
)
.
Assume H2:
wceq
cfrgr
(
cab
(
λ x0 .
wa
(
wcel
(
cv
x0
)
cusgr
)
(
wsbc
(
λ x1 .
wsbc
(
λ x2 .
wral
(
λ x3 .
wral
(
λ x4 .
wreu
(
λ x5 .
wss
(
cpr
(
cpr
(
cv
x5
)
(
cv
x3
)
)
(
cpr
(
cv
x5
)
(
cv
x4
)
)
)
(
cv
x2
)
)
(
λ x5 .
cv
x1
)
)
(
λ x4 .
cdif
(
cv
x1
)
(
csn
(
cv
x3
)
)
)
)
(
λ x3 .
cv
x1
)
)
(
cfv
(
cv
x0
)
cedg
)
)
(
cfv
(
cv
x0
)
cvtx
)
)
)
)
.
Assume H3:
wceq
cplig
(
cab
(
λ x0 .
w3a
(
wral
(
λ x1 .
wral
(
λ x2 .
wne
(
cv
x1
)
(
cv
x2
)
⟶
wreu
(
λ x3 .
wa
(
wcel
(
cv
x1
)
(
cv
x3
)
)
(
wcel
(
cv
x2
)
(
cv
x3
)
)
)
(
λ x3 .
cv
x0
)
)
(
λ x2 .
cuni
(
cv
x0
)
)
)
(
λ x1 .
cuni
(
cv
x0
)
)
)
(
wral
(
λ x1 .
wrex
(
λ x2 .
wrex
(
λ x3 .
w3a
(
wne
(
cv
x2
)
(
cv
x3
)
)
(
wcel
(
cv
x2
)
(
cv
x1
)
)
(
wcel
(
cv
x3
)
(
cv
x1
)
)
)
(
λ x3 .
cuni
(
cv
x0
)
)
)
(
λ x2 .
cuni
(
cv
x0
)
)
)
(
λ x1 .
cv
x0
)
)
(
wrex
(
λ x1 .
wrex
(
λ x2 .
wrex
(
λ x3 .
wral
(
λ x4 .
wn
(
w3a
(
wcel
(
cv
x1
)
(
cv
x4
)
)
(
wcel
(
cv
x2
)
(
cv
x4
)
)
(
wcel
(
cv
x3
)
(
cv
x4
)
)
)
)
(
λ x4 .
cv
x0
)
)
(
λ x3 .
cuni
(
cv
x0
)
)
)
(
λ x2 .
cuni
(
cv
x0
)
)
)
(
λ x1 .
cuni
(
cv
x0
)
)
)
)
)
.
Assume H4:
wceq
cgr
(
cab
(
λ x0 .
wex
(
λ x1 .
w3a
(
wf
(
cxp
(
cv
...
)
...
)
...
...
)
...
...
)
)
)
.
...
■