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Proofgold Proposition
∀ x0 : ο .
(
wceq
ccoda
(
ccom
c2nd
c1st
)
⟶
wceq
choma
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cmpt
(
λ x2 .
cxp
(
cfv
(
cv
x1
)
cbs
)
(
cfv
(
cv
x1
)
cbs
)
)
(
λ x2 .
cxp
(
csn
(
cv
x2
)
)
(
cfv
(
cv
x2
)
(
cfv
(
cv
x1
)
chom
)
)
)
)
)
⟶
wceq
carw
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cuni
(
crn
(
cfv
(
cv
x1
)
choma
)
)
)
)
⟶
wceq
cida
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cmpt
(
λ x2 .
cfv
(
cv
x1
)
cbs
)
(
λ x2 .
cotp
(
cv
x2
)
(
cv
x2
)
(
cfv
(
cv
x2
)
(
cfv
(
cv
x1
)
ccid
)
)
)
)
)
⟶
wceq
ccoa
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
carw
)
(
λ x2 x3 .
crab
(
λ x4 .
wceq
(
cfv
(
cv
x4
)
ccoda
)
(
cfv
(
cv
x2
)
cdoma
)
)
(
λ x4 .
cfv
(
cv
x1
)
carw
)
)
(
λ x2 x3 .
cotp
(
cfv
(
cv
x3
)
cdoma
)
(
cfv
(
cv
x2
)
ccoda
)
(
co
(
cfv
(
cv
x2
)
c2nd
)
(
cfv
(
cv
x3
)
c2nd
)
(
co
(
cop
(
cfv
(
cv
x3
)
cdoma
)
(
cfv
(
cv
x2
)
cdoma
)
)
(
cfv
(
cv
x2
)
ccoda
)
(
cfv
(
cv
x1
)
cco
)
)
)
)
)
)
⟶
wceq
csetc
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
ctp
(
cop
(
cfv
cnx
cbs
)
(
cv
x1
)
)
(
cop
(
cfv
cnx
chom
)
(
cmpt2
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
co
(
cv
x3
)
(
cv
x2
)
cmap
)
)
)
(
cop
(
cfv
cnx
cco
)
(
cmpt2
(
λ x2 x3 .
cxp
(
cv
x1
)
(
cv
x1
)
)
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
cmpt2
(
λ x4 x5 .
co
(
cv
x3
)
(
cfv
(
cv
x2
)
c2nd
)
cmap
)
(
λ x4 x5 .
co
(
cfv
(
cv
x2
)
c2nd
)
(
cfv
(
cv
x2
)
c1st
)
cmap
)
(
λ x4 x5 .
ccom
(
cv
x4
)
(
cv
x5
)
)
)
)
)
)
)
⟶
wceq
ccatc
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
csb
(
cin
(
cv
x1
)
ccat
)
(
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
)
cfunc
)
)
)
(
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
)
cfunc
)
(
λ x5 x6 .
cfv
(
cv
x3
)
cfunc
)
(
λ x5 x6 .
co
(
cv
x5
)
(
cv
x6
)
ccofu
)
)
)
)
)
)
)
⟶
wceq
cestrc
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
ctp
(
cop
(
cfv
cnx
cbs
)
(
cv
x1
)
)
(
cop
(
cfv
cnx
chom
)
(
cmpt2
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
co
(
cfv
(
cv
x3
)
cbs
)
(
cfv
(
cv
x2
)
cbs
)
cmap
)
)
)
(
cop
(
cfv
cnx
cco
)
(
cmpt2
(
λ x2 x3 .
cxp
(
cv
x1
)
(
cv
x1
)
)
(
λ x2 x3 .
cv
x1
)
(
λ x2 x3 .
cmpt2
(
λ x4 x5 .
co
(
cfv
(
cv
x3
)
cbs
)
(
cfv
(
cfv
(
cv
x2
)
c2nd
)
cbs
)
cmap
)
(
λ x4 x5 .
co
(
cfv
(
cfv
(
cv
x2
)
c2nd
)
cbs
)
(
cfv
(
cfv
(
cv
x2
)
c1st
)
cbs
)
cmap
)
(
λ x4 x5 .
ccom
(
cv
x4
)
(
cv
x5
)
)
)
)
)
)
)
⟶
wceq
cxpc
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
csb
(
cxp
(
cfv
(
cv
x1
)
cbs
)
(
cfv
(
cv
x2
)
cbs
)
)
(
λ x3 .
csb
(
cmpt2
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cxp
(
co
(
cfv
(
cv
x4
)
c1st
)
(
cfv
(
cv
x5
)
c1st
)
(
cfv
(
cv
x1
)
chom
)
)
(
co
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x5
)
c2nd
)
(
cfv
(
cv
x2
)
chom
)
)
)
)
(
λ x4 .
ctp
(
cop
(
cfv
cnx
cbs
)
(
cv
x3
)
)
(
cop
(
cfv
cnx
chom
)
(
cv
x4
)
)
(
cop
(
cfv
cnx
cco
)
(
cmpt2
(
λ x5 x6 .
cxp
(
cv
x3
)
(
cv
x3
)
)
(
λ x5 x6 .
cv
x3
)
(
λ x5 x6 .
cmpt2
(
λ x7 x8 .
co
(
cfv
(
cv
x5
)
c2nd
)
(
cv
x6
)
(
cv
x4
)
)
(
λ x7 x8 .
cfv
(
cv
x5
)
(
cv
x4
)
)
(
λ x7 x8 .
cop
(
co
(
cfv
(
cv
x7
)
c1st
)
(
cfv
(
cv
x8
)
c1st
)
(
co
(
cop
(
cfv
(
cfv
(
cv
x5
)
c1st
)
c1st
)
(
cfv
(
cfv
(
cv
x5
)
c2nd
)
c1st
)
)
(
cfv
(
cv
x6
)
c1st
)
(
cfv
(
cv
x1
)
cco
)
)
)
(
co
(
cfv
(
cv
x7
)
c2nd
)
(
cfv
(
cv
x8
)
c2nd
)
(
co
(
cop
(
cfv
(
cfv
(
cv
x5
)
c1st
)
c2nd
)
(
cfv
(
cfv
(
cv
x5
)
c2nd
)
c2nd
)
)
(
cfv
(
cv
x6
)
c2nd
)
(
cfv
(
cv
x2
)
cco
)
)
)
)
)
)
)
)
)
)
)
⟶
wceq
c1stf
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
csb
(
cxp
(
cfv
(
cv
x1
)
cbs
)
(
cfv
(
cv
x2
)
cbs
)
)
(
λ x3 .
cop
(
cres
c1st
(
cv
x3
)
)
(
cmpt2
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cres
c1st
(
co
(
cv
x4
)
(
cv
x5
)
(
cfv
(
co
(
cv
x1
)
(
cv
x2
)
cxpc
)
chom
)
)
)
)
)
)
)
⟶
wceq
c2ndf
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
csb
(
cxp
(
cfv
(
cv
x1
)
cbs
)
(
cfv
(
cv
x2
)
cbs
)
)
(
λ x3 .
cop
(
cres
c2nd
(
cv
x3
)
)
(
cmpt2
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cres
c2nd
(
co
(
cv
x4
)
(
cv
x5
)
(
cfv
(
co
(
cv
x1
)
(
cv
x2
)
cxpc
)
chom
)
)
)
)
)
)
)
⟶
wceq
cprf
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
csb
(
cdm
(
cfv
(
cv
x1
)
c1st
)
)
(
λ x3 .
cop
(
cmpt
(
λ x4 .
cv
x3
)
(
λ x4 .
cop
(
cfv
(
cv
x4
)
(
cfv
(
cv
x1
)
c1st
)
)
(
cfv
(
cv
x4
)
(
cfv
(
cv
x2
)
c1st
)
)
)
)
(
cmpt2
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cv
x3
)
(
λ x4 x5 .
cmpt
(
λ x6 .
cdm
(
co
(
cv
x4
)
(
cv
x5
)
(
cfv
(
cv
x1
)
c2nd
)
)
)
(
λ x6 .
cop
(
cfv
(
cv
x6
)
(
co
(
cv
x4
)
(
cv
x5
)
(
cfv
(
cv
x1
)
c2nd
)
)
)
(
cfv
(
cv
x6
)
(
co
(
cv
x4
)
(
cv
x5
)
(
cfv
(
cv
x2
)
c2nd
)
)
)
)
)
)
)
)
)
⟶
wceq
cevlf
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
cop
(
cmpt2
(
λ x3 x4 .
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
(
λ x3 x4 .
cfv
(
cv
x1
)
cbs
)
(
λ x3 x4 .
cfv
(
cv
x4
)
(
cfv
(
cv
x3
)
c1st
)
)
)
(
cmpt2
(
λ x3 x4 .
cxp
(
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
(
cfv
(
cv
x1
)
cbs
)
)
(
λ x3 x4 .
cxp
(
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
(
cfv
(
cv
x1
)
cbs
)
)
(
λ x3 x4 .
csb
(
cfv
(
cv
x3
)
c1st
)
(
λ x5 .
csb
(
cfv
(
cv
x4
)
c1st
)
(
λ x6 .
cmpt2
(
λ x7 x8 .
co
(
cv
x5
)
(
cv
x6
)
(
co
(
cv
x1
)
(
cv
x2
)
cnat
)
)
(
λ x7 x8 .
co
(
cfv
(
cv
x3
)
c2nd
)
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x1
)
chom
)
)
(
λ x7 x8 .
co
(
cfv
(
cfv
(
cv
x4
)
c2nd
)
(
cv
x7
)
)
(
cfv
(
cv
x8
)
(
co
(
cfv
(
cv
x3
)
c2nd
)
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x5
)
c2nd
)
)
)
(
co
(
cop
(
cfv
(
cfv
(
cv
x3
)
c2nd
)
(
cfv
(
cv
x5
)
c1st
)
)
(
cfv
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x5
)
c1st
)
)
)
(
cfv
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x6
)
c1st
)
)
(
cfv
(
cv
x2
)
cco
)
)
)
)
)
)
)
)
)
⟶
wceq
ccurf
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
csb
(
cfv
(
cv
x1
)
c1st
)
(
λ x3 .
csb
(
cfv
(
cv
x1
)
c2nd
)
(
λ x4 .
cop
(
cmpt
(
λ x5 .
cfv
(
cv
x3
)
cbs
)
(
λ x5 .
cop
(
cmpt
(
λ x6 .
cfv
(
cv
x4
)
cbs
)
(
λ x6 .
co
(
cv
x5
)
(
cv
x6
)
(
cfv
(
cv
x2
)
c1st
)
)
)
(
cmpt2
(
λ x6 x7 .
cfv
(
cv
x4
)
cbs
)
(
λ x6 x7 .
cfv
(
cv
x4
)
cbs
)
(
λ x6 x7 .
cmpt
(
λ x8 .
co
(
cv
x6
)
(
cv
x7
)
(
cfv
(
cv
x4
)
chom
)
)
(
λ x8 .
co
(
cfv
(
cv
x5
)
(
cfv
(
cv
x3
)
ccid
)
)
(
cv
x8
)
(
co
(
cop
(
cv
x5
)
(
cv
x6
)
)
(
cop
(
cv
x5
)
(
cv
x7
)
)
(
cfv
(
cv
x2
)
c2nd
)
)
)
)
)
)
)
(
cmpt2
(
λ x5 x6 .
cfv
(
cv
x3
)
cbs
)
(
λ x5 x6 .
cfv
(
cv
x3
)
cbs
)
(
λ x5 x6 .
cmpt
(
λ x7 .
co
(
cv
x5
)
(
cv
x6
)
(
cfv
(
cv
x3
)
chom
)
)
(
λ x7 .
cmpt
(
λ x8 .
cfv
(
cv
x4
)
cbs
)
(
λ x8 .
co
(
cv
x7
)
(
cfv
(
cv
x8
)
(
cfv
(
cv
x4
)
ccid
)
)
(
co
(
cop
(
cv
x5
)
(
cv
x8
)
)
(
cop
(
cv
x6
)
(
cv
x8
)
)
(
cfv
(
cv
x2
)
c2nd
)
)
)
)
)
)
)
)
)
)
⟶
wceq
cuncf
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
co
(
co
(
cfv
c1
(
cv
x1
)
)
(
cfv
c2
(
cv
x1
)
)
cevlf
)
(
co
(
co
(
cv
x2
)
(
co
(
cfv
cc0
(
cv
x1
)
)
(
cfv
c1
(
cv
x1
)
)
c1stf
)
ccofu
)
(
co
(
cfv
cc0
(
cv
x1
)
)
(
cfv
c1
(
cv
x1
)
)
c2ndf
)
cprf
)
ccofu
)
)
⟶
wceq
cdiag
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
co
(
cop
(
cv
x1
)
(
cv
x2
)
)
(
co
(
cv
x1
)
(
cv
x2
)
c1stf
)
ccurf
)
)
⟶
wceq
chof
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cop
(
cfv
(
cv
x1
)
chomf
)
(
csb
(
cfv
(
cv
x1
)
cbs
)
(
λ x2 .
cmpt2
(
λ x3 x4 .
cxp
(
cv
x2
)
(
cv
x2
)
)
(
λ x3 x4 .
cxp
(
cv
x2
)
(
cv
x2
)
)
(
λ x3 x4 .
cmpt2
(
λ x5 x6 .
co
(
cfv
(
cv
x4
)
c1st
)
(
cfv
(
cv
x3
)
c1st
)
(
cfv
(
cv
x1
)
chom
)
)
(
λ x5 x6 .
co
(
cfv
(
cv
x3
)
c2nd
)
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x1
)
chom
)
)
(
λ x5 x6 .
cmpt
(
λ x7 .
cfv
(
cv
x3
)
(
cfv
(
cv
x1
)
chom
)
)
(
λ x7 .
co
(
co
(
cv
x6
)
(
cv
x7
)
(
co
(
cv
x3
)
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x1
)
cco
)
)
)
(
cv
x5
)
(
co
(
cop
(
cfv
(
cv
x4
)
c1st
)
(
cfv
(
cv
x3
)
c1st
)
)
(
cfv
(
cv
x4
)
c2nd
)
(
cfv
(
cv
x1
)
cco
)
)
)
)
)
)
)
)
)
⟶
wceq
cyon
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
co
(
cop
(
cv
x1
)
(
cfv
(
cv
x1
)
coppc
)
)
(
cfv
(
cfv
(
cv
x1
)
coppc
)
chof
)
ccurf
)
)
⟶
x0
)
⟶
x0
type
prop
theory
SetMM
name
df_coda__df_homa__df_arw__df_ida__df_coa__df_setc__df_catc__df_estrc__df_xpc__df_1stf__df_2ndf__df_prf__df_evlf__df_curf__df_uncf__df_diag__df_hof__df_yon
proof
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