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Proofgold Proof
pf
Apply df_mgm2__df_cmgm2__df_sgrp2__df_csgrp2__df_rng0__df_rnghomo__df_rngisom__df_rngc__df_rngcALTV__df_ringc__df_ringcALTV__df_dmatalt__df_scmatalt__df_linc__df_lco__df_lininds__df_lindeps__df_fdiv with
wceq
clininds
(
copab
(
λ x0 x1 .
wa
(
wcel
(
cv
x0
)
(
cpw
(
cfv
(
cv
x1
)
cbs
)
)
)
(
wral
(
λ x2 .
wa
(
wbr
(
cv
x2
)
(
cfv
(
cfv
(
cv
x1
)
csca
)
c0g
)
cfsupp
)
(
wceq
(
co
(
cv
x2
)
(
cv
x0
)
(
cfv
(
cv
x1
)
clinc
)
)
(
cfv
(
cv
x1
)
c0g
)
)
⟶
wral
(
λ x3 .
wceq
(
cfv
(
cv
x3
)
(
cv
x2
)
)
(
cfv
(
cfv
(
cv
x1
)
csca
)
c0g
)
)
(
λ x3 .
cv
x0
)
)
(
λ x2 .
co
(
cfv
(
cfv
(
cv
x1
)
csca
)
cbs
)
(
cv
x0
)
cmap
)
)
)
)
.
Assume H0:
wceq
cmgm2
(
cab
(
λ x0 .
wbr
(
cfv
(
cv
x0
)
cplusg
)
(
cfv
(
cv
x0
)
cbs
)
ccllaw
)
)
.
Assume H1:
wceq
ccmgm2
(
crab
(
λ x0 .
wbr
(
cfv
(
cv
x0
)
cplusg
)
(
cfv
(
cv
x0
)
cbs
)
ccomlaw
)
(
λ x0 .
cmgm2
)
)
.
Assume H2:
wceq
csgrp2
(
crab
(
λ x0 .
wbr
(
cfv
(
cv
x0
)
cplusg
)
(
cfv
(
cv
x0
)
cbs
)
casslaw
)
(
λ x0 .
cmgm2
)
)
.
Assume H3:
wceq
ccsgrp2
(
crab
(
λ x0 .
wbr
(
cfv
(
cv
x0
)
cplusg
)
(
cfv
(
cv
x0
)
cbs
)
ccomlaw
)
(
λ x0 .
csgrp2
)
)
.
Assume H4:
wceq
crng
(
crab
(
λ x0 .
wa
(
wcel
(
cfv
(
cv
x0
)
cmgp
)
csgrp
)
(
wsbc
(
λ x1 .
wsbc
(
λ x2 .
wsbc
(
λ x3 .
wral
(
λ x4 .
wral
(
λ x5 .
wral
(
λ x6 .
wa
(
wceq
(
co
(
cv
x4
)
(
co
(
cv
x5
)
(
cv
x6
)
(
cv
x2
)
)
(
cv
x3
)
)
(
co
(
co
(
cv
x4
)
(
cv
x5
)
(
cv
x3
)
)
(
co
(
cv
x4
)
(
cv
x6
)
(
cv
x3
)
)
(
cv
x2
)
)
)
(
wceq
(
co
(
co
(
cv
x4
)
(
cv
x5
)
(
cv
x2
)
)
(
cv
x6
)
(
cv
x3
)
)
(
co
(
co
(
cv
x4
)
(
cv
x6
)
(
cv
x3
)
)
(
co
(
cv
x5
)
(
cv
x6
)
(
cv
x3
)
)
(
cv
x2
)
)
)
)
(
λ x6 .
cv
x1
)
)
(
λ x5 .
cv
x1
)
)
(
λ x4 .
cv
x1
)
)
(
cfv
(
cv
x0
)
cmulr
)
)
(
cfv
(
cv
x0
)
cplusg
)
)
(
cfv
(
cv
x0
)
cbs
)
)
)
(
λ x0 .
cabl
)
)
.
Assume H5:
wceq
crngh
(
cmpt2
(
λ x0 x1 .
crng
)
(
λ x0 x1 .
crng
)
(
λ x0 x1 .
csb
(
cfv
(
cv
...
)
...
)
...
)
)
.
...
■