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Proofgold Proposition
∀ x0 : ο .
(
wceq
cmgm2
(
cab
(
λ x1 .
wbr
(
cfv
(
cv
x1
)
cplusg
)
(
cfv
(
cv
x1
)
cbs
)
ccllaw
)
)
⟶
wceq
ccmgm2
(
crab
(
λ x1 .
wbr
(
cfv
(
cv
x1
)
cplusg
)
(
cfv
(
cv
x1
)
cbs
)
ccomlaw
)
(
λ x1 .
cmgm2
)
)
⟶
wceq
csgrp2
(
crab
(
λ x1 .
wbr
(
cfv
(
cv
x1
)
cplusg
)
(
cfv
(
cv
x1
)
cbs
)
casslaw
)
(
λ x1 .
cmgm2
)
)
⟶
wceq
ccsgrp2
(
crab
(
λ x1 .
wbr
(
cfv
(
cv
x1
)
cplusg
)
(
cfv
(
cv
x1
)
cbs
)
ccomlaw
)
(
λ x1 .
csgrp2
)
)
⟶
wceq
crng
(
crab
(
λ x1 .
wa
(
wcel
(
cfv
(
cv
x1
)
cmgp
)
csgrp
)
(
wsbc
(
λ x2 .
wsbc
(
λ x3 .
wsbc
(
λ x4 .
wral
(
λ x5 .
wral
(
λ x6 .
wral
(
λ x7 .
wa
(
wceq
(
co
(
cv
x5
)
(
co
(
cv
x6
)
(
cv
x7
)
(
cv
x3
)
)
(
cv
x4
)
)
(
co
(
co
(
cv
x5
)
(
cv
x6
)
(
cv
x4
)
)
(
co
(
cv
x5
)
(
cv
x7
)
(
cv
x4
)
)
(
cv
x3
)
)
)
(
wceq
(
co
(
co
(
cv
x5
)
(
cv
x6
)
(
cv
x3
)
)
(
cv
x7
)
(
cv
x4
)
)
(
co
(
co
(
cv
x5
)
(
cv
x7
)
(
cv
x4
)
)
(
co
(
cv
x6
)
(
cv
x7
)
(
cv
x4
)
)
(
cv
x3
)
)
)
)
(
λ x7 .
cv
x2
)
)
(
λ x6 .
cv
x2
)
)
(
λ x5 .
cv
x2
)
)
(
cfv
(
cv
x1
)
cmulr
)
)
(
cfv
(
cv
x1
)
cplusg
)
)
(
cfv
(
cv
x1
)
cbs
)
)
)
(
λ x1 .
cabl
)
)
⟶
wceq
crngh
(
cmpt2
(
λ x1 x2 .
crng
)
(
λ x1 x2 .
crng
)
(
λ x1 x2 .
csb
(
cfv
(
cv
x1
)
cbs
)
(
λ x3 .
csb
(
cfv
(
cv
x2
)
cbs
)
(
λ x4 .
crab
(
λ x5 .
wral
(
λ x6 .
wral
(
λ x7 .
wa
(
wceq
(
cfv
(
co
(
cv
x6
)
(
cv
x7
)
(
cfv
(
cv
x1
)
cplusg
)
)
(
cv
x5
)
)
(
co
(
cfv
(
cv
x6
)
(
cv
x5
)
)
(
cfv
(
cv
x7
)
(
cv
x5
)
)
(
cfv
(
cv
x2
)
cplusg
)
)
)
(
wceq
(
cfv
(
co
(
cv
x6
)
(
cv
x7
)
(
cfv
(
cv
x1
)
cmulr
)
)
(
cv
x5
)
)
(
co
(
cfv
(
cv
x6
)
(
cv
x5
)
)
(
cfv
(
cv
x7
)
(
cv
x5
)
)
(
cfv
(
cv
x2
)
cmulr
)
)
)
)
(
λ x7 .
cv
x3
)
)
(
λ x6 .
cv
x3
)
)
(
λ x5 .
co
(
cv
x4
)
(
cv
x3
)
cmap
)
)
)
)
)
⟶
wceq
crngs
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
crab
(
λ x3 .
wcel
(
ccnv
(
cv
x3
)
)
(
co
(
cv
x2
)
(
cv
x1
)
crngh
)
)
(
λ x3 .
co
(
cv
x1
)
(
cv
x2
)
crngh
)
)
)
⟶
wceq
crngc
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
co
(
cfv
(
cv
x1
)
cestrc
)
(
cres
crngh
(
cxp
(
cin
(
cv
x1
)
crng
)
(
cin
(
cv
x1
)
crng
)
)
)
cresc
)
)
⟶
wceq
crngcALTV
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
csb
(
cin
(
cv
x1
)
crng
)
(
λ x2 .
ctp
(
cop
(
cfv
cnx
cbs
)
(
cv
x2
)
)
(
cop
(
cfv
cnx
chom
)
(
cmpt2
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
co
(
cv
x3
)
(
cv
x4
)
crngh
)
)
)
(
cop
(
cfv
cnx
cco
)
(
cmpt2
(
λ x3 x4 .
cxp
(
cv
x2
)
(
cv
x2
)
)
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
cmpt2
(
λ x5 x6 .
co
(
cfv
(
cv
x3
)
c2nd
)
(
cv
x4
)
crngh
)
(
λ x5 x6 .
co
(
cfv
(
cv
x3
)
c1st
)
(
cfv
(
cv
x3
)
c2nd
)
crngh
)
(
λ x5 x6 .
ccom
(
cv
x5
)
(
cv
x6
)
)
)
)
)
)
)
)
⟶
wceq
cringc
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
co
(
cfv
(
cv
x1
)
cestrc
)
(
cres
crh
(
cxp
(
cin
(
cv
x1
)
crg
)
(
cin
(
cv
x1
)
crg
)
)
)
cresc
)
)
⟶
wceq
cringcALTV
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
csb
(
cin
(
cv
x1
)
crg
)
(
λ x2 .
ctp
(
cop
(
cfv
cnx
cbs
)
(
cv
x2
)
)
(
cop
(
cfv
cnx
chom
)
(
cmpt2
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
co
(
cv
x3
)
(
cv
x4
)
crh
)
)
)
(
cop
(
cfv
cnx
cco
)
(
cmpt2
(
λ x3 x4 .
cxp
(
cv
x2
)
(
cv
x2
)
)
(
λ x3 x4 .
cv
x2
)
(
λ x3 x4 .
cmpt2
(
λ x5 x6 .
co
(
cfv
(
cv
x3
)
c2nd
)
(
cv
x4
)
crh
)
(
λ x5 x6 .
co
(
cfv
(
cv
x3
)
c1st
)
(
cfv
(
cv
x3
)
c2nd
)
crh
)
(
λ x5 x6 .
ccom
(
cv
x5
)
(
cv
x6
)
)
)
)
)
)
)
)
⟶
wceq
cdmatalt
(
cmpt2
(
λ x1 x2 .
cfn
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
csb
(
co
(
cv
x1
)
(
cv
x2
)
cmat
)
(
λ x3 .
co
(
cv
x3
)
(
crab
(
λ x4 .
wral
(
λ x5 .
wral
(
λ x6 .
wne
(
cv
x5
)
(
cv
x6
)
⟶
wceq
(
co
(
cv
x5
)
(
cv
x6
)
(
cv
x4
)
)
(
cfv
(
cv
x2
)
c0g
)
)
(
λ x6 .
cv
x1
)
)
(
λ x5 .
cv
x1
)
)
(
λ x4 .
cfv
(
cv
x3
)
cbs
)
)
cress
)
)
)
⟶
wceq
cscmatalt
(
cmpt2
(
λ x1 x2 .
cfn
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
csb
(
co
(
cv
x1
)
(
cv
x2
)
cmat
)
(
λ x3 .
co
(
cv
x3
)
(
crab
(
λ x4 .
wrex
(
λ x5 .
wral
(
λ x6 .
wral
(
λ x7 .
wceq
(
co
(
cv
x6
)
(
cv
x7
)
(
cv
x4
)
)
(
cif
(
wceq
(
cv
x6
)
(
cv
x7
)
)
(
cv
x5
)
(
cfv
(
cv
x2
)
c0g
)
)
)
(
λ x7 .
cv
x1
)
)
(
λ x6 .
cv
x1
)
)
(
λ x5 .
cfv
(
cv
x2
)
cbs
)
)
(
λ x4 .
cfv
(
cv
x3
)
cbs
)
)
cress
)
)
)
⟶
wceq
clinc
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cmpt2
(
λ x2 x3 .
co
(
cfv
(
cfv
(
cv
x1
)
csca
)
cbs
)
(
cv
x3
)
cmap
)
(
λ x2 x3 .
cpw
(
cfv
(
cv
x1
)
cbs
)
)
(
λ x2 x3 .
co
(
cv
x1
)
(
cmpt
(
λ x4 .
cv
x3
)
(
λ x4 .
co
(
cfv
(
cv
x4
)
(
cv
x2
)
)
(
cv
x4
)
(
cfv
(
cv
x1
)
cvsca
)
)
)
cgsu
)
)
)
⟶
wceq
clinco
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cpw
(
cfv
(
cv
x1
)
cbs
)
)
(
λ x1 x2 .
crab
(
λ x3 .
wrex
(
λ x4 .
wa
(
wbr
(
cv
x4
)
(
cfv
(
cfv
(
cv
x1
)
csca
)
c0g
)
cfsupp
)
(
wceq
(
cv
x3
)
(
co
(
cv
x4
)
(
cv
x2
)
(
cfv
(
cv
x1
)
clinc
)
)
)
)
(
λ x4 .
co
(
cfv
(
cfv
(
cv
x1
)
csca
)
cbs
)
(
cv
x2
)
cmap
)
)
(
λ x3 .
cfv
(
cv
x1
)
cbs
)
)
)
⟶
wceq
clininds
(
copab
(
λ x1 x2 .
wa
(
wcel
(
cv
x1
)
(
cpw
(
cfv
(
cv
x2
)
cbs
)
)
)
(
wral
(
λ x3 .
wa
(
wbr
(
cv
x3
)
(
cfv
(
cfv
(
cv
x2
)
csca
)
c0g
)
cfsupp
)
(
wceq
(
co
(
cv
x3
)
(
cv
x1
)
(
cfv
(
cv
x2
)
clinc
)
)
(
cfv
(
cv
x2
)
c0g
)
)
⟶
wral
(
λ x4 .
wceq
(
cfv
(
cv
x4
)
(
cv
x3
)
)
(
cfv
(
cfv
(
cv
x2
)
csca
)
c0g
)
)
(
λ x4 .
cv
x1
)
)
(
λ x3 .
co
(
cfv
(
cfv
(
cv
x2
)
csca
)
cbs
)
(
cv
x1
)
cmap
)
)
)
)
⟶
wceq
clindeps
(
copab
(
λ x1 x2 .
wn
(
wbr
(
cv
x1
)
(
cv
x2
)
clininds
)
)
)
⟶
wceq
cfdiv
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cres
(
co
(
cv
x1
)
(
cv
x2
)
(
cof
cdiv
)
)
(
co
(
cv
x2
)
cc0
csupp
)
)
)
⟶
x0
)
⟶
x0
type
prop
theory
SetMM
name
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
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