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Proofgold Proof
pf
Apply df_pj__df_hil__df_obs__df_dsmm__df_frlm__df_uvc__df_lindf__df_linds__df_mamu__df_mat__df_dmat__df_scmat__df_mvmul__df_marrep__df_marepv__df_subma__df_mdet__df_madu with
wceq
cmmul
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
csb
(
cfv
(
cfv
(
cv
x1
)
c1st
)
c1st
)
(
λ x2 .
csb
(
cfv
(
cfv
(
cv
x1
)
c1st
)
c2nd
)
(
λ x3 .
csb
(
cfv
(
cv
x1
)
c2nd
)
(
λ x4 .
cmpt2
(
λ x5 x6 .
co
(
cfv
(
cv
x0
)
cbs
)
(
cxp
(
cv
x2
)
(
cv
x3
)
)
cmap
)
(
λ x5 x6 .
co
(
cfv
(
cv
x0
)
cbs
)
(
cxp
(
cv
x3
)
(
cv
x4
)
)
cmap
)
(
λ x5 x6 .
cmpt2
(
λ x7 x8 .
cv
x2
)
(
λ x7 x8 .
cv
x4
)
(
λ x7 x8 .
co
(
cv
x0
)
(
cmpt
(
λ x9 .
cv
x3
)
(
λ x9 .
co
(
co
(
cv
x7
)
(
cv
x9
)
(
cv
x5
)
)
(
co
(
cv
x9
)
(
cv
x8
)
(
cv
x6
)
)
(
cfv
(
cv
x0
)
cmulr
)
)
)
cgsu
)
)
)
)
)
)
)
.
Assume H0:
wceq
cpj
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cin
(
cmpt
(
λ x1 .
cfv
(
cv
x0
)
clss
)
(
λ x1 .
co
(
cv
x1
)
(
cfv
(
cv
x1
)
(
cfv
(
cv
x0
)
cocv
)
)
(
cfv
(
cv
x0
)
cpj1
)
)
)
(
cxp
cvv
(
co
(
cfv
(
cv
x0
)
cbs
)
(
cfv
(
cv
x0
)
cbs
)
cmap
)
)
)
)
.
Assume H1:
wceq
chs
(
crab
(
λ x0 .
wceq
(
cdm
(
cfv
(
cv
x0
)
cpj
)
)
(
cfv
(
cv
x0
)
ccss
)
)
(
λ x0 .
cphl
)
)
.
Assume H2:
wceq
cobs
(
cmpt
(
λ x0 .
cphl
)
(
λ x0 .
crab
(
λ x1 .
wa
(
wral
(
λ x2 .
wral
(
λ x3 .
wceq
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x0
)
cip
)
)
(
cif
(
wceq
(
cv
x2
)
(
cv
x3
)
)
(
cfv
(
cfv
(
cv
x0
)
csca
)
cur
)
(
cfv
(
cfv
(
cv
x0
)
csca
)
c0g
)
)
)
(
λ x3 .
cv
x1
)
)
(
λ x2 .
cv
x1
)
)
(
wceq
(
cfv
(
cv
x1
)
(
cfv
(
cv
x0
)
cocv
)
)
(
csn
(
cfv
(
cv
x0
)
c0g
)
)
)
)
(
λ x1 .
cpw
(
cfv
(
cv
x0
)
cbs
)
)
)
)
.
Assume H3:
wceq
cdsmm
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
co
(
co
(
cv
x0
)
(
cv
x1
)
cprds
)
(
crab
(
λ x2 .
wcel
(
crab
(
λ x3 .
wne
(
cfv
(
cv
x3
)
(
cv
x2
)
)
(
cfv
(
cfv
(
cv
x3
)
(
cv
x1
)
)
c0g
)
)
(
λ x3 .
cdm
(
cv
...
)
)
)
...
)
...
)
...
)
)
.
...
■