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Proofgold Proof
pf
Apply df_ppi__df_mu__df_sgm__df_dchr__df_lgs__df_itv__df_lng__df_trkgc__df_trkgb__df_trkgcb__df_trkge__df_trkgld__df_trkg__df_cgrg__df_ismt__df_leg__df_hlg__df_mir with
wceq
ccgrg
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wa
(
wcel
(
cv
x1
)
(
co
(
cfv
(
cv
x0
)
cbs
)
cr
cpm
)
)
(
wcel
(
cv
x2
)
(
co
(
cfv
(
cv
x0
)
cbs
)
cr
cpm
)
)
)
(
wa
(
wceq
(
cdm
(
cv
x1
)
)
(
cdm
(
cv
x2
)
)
)
(
wral
(
λ x3 .
wral
(
λ x4 .
wceq
(
co
(
cfv
(
cv
x3
)
(
cv
x1
)
)
(
cfv
(
cv
x4
)
(
cv
x1
)
)
(
cfv
(
cv
x0
)
cds
)
)
(
co
(
cfv
(
cv
x3
)
(
cv
x2
)
)
(
cfv
(
cv
x4
)
(
cv
x2
)
)
(
cfv
(
cv
x0
)
cds
)
)
)
(
λ x4 .
cdm
(
cv
x1
)
)
)
(
λ x3 .
cdm
(
cv
x1
)
)
)
)
)
)
)
.
Assume H0:
wceq
cppi
(
cmpt
(
λ x0 .
cr
)
(
λ x0 .
cfv
(
cin
(
co
cc0
(
cv
x0
)
cicc
)
cprime
)
chash
)
)
.
Assume H1:
wceq
cmu
(
cmpt
(
λ x0 .
cn
)
(
λ x0 .
cif
(
wrex
(
λ x1 .
wbr
(
co
(
cv
x1
)
c2
cexp
)
(
cv
x0
)
cdvds
)
(
λ x1 .
cprime
)
)
cc0
(
co
(
cneg
c1
)
(
cfv
(
crab
(
λ x1 .
wbr
(
cv
x1
)
(
cv
x0
)
cdvds
)
(
λ x1 .
cprime
)
)
chash
)
cexp
)
)
)
.
Assume H2:
wceq
csgm
(
cmpt2
(
λ x0 x1 .
cc
)
(
λ x0 x1 .
cn
)
(
λ x0 x1 .
csu
(
crab
(
λ x2 .
wbr
(
cv
x2
)
(
cv
x1
)
cdvds
)
(
λ x2 .
cn
)
)
(
λ x2 .
co
(
cv
x2
)
(
cv
x0
)
ccxp
)
)
)
.
Assume H3:
wceq
cdchr
(
cmpt
(
λ x0 .
cn
)
(
λ x0 .
csb
(
cfv
(
cv
x0
)
czn
)
(
λ x1 .
csb
(
crab
(
λ x2 .
wss
(
cxp
(
cdif
(
cfv
(
cv
x1
)
cbs
)
(
cfv
(
cv
x1
)
cui
)
)
(
csn
cc0
)
)
(
cv
x2
)
)
(
λ x2 .
co
(
cfv
(
cv
x1
)
cmgp
)
(
cfv
ccnfld
cmgp
)
cmhm
)
)
(
λ x2 .
cpr
(
cop
(
cfv
cnx
cbs
)
(
cv
x2
)
)
(
cop
(
cfv
cnx
cplusg
)
(
cres
(
cof
cmul
)
(
cxp
(
cv
x2
)
(
cv
x2
)
)
)
)
)
)
)
)
.
Assume H4:
wceq
clgs
(
cmpt2
(
λ x0 x1 .
cz
)
(
λ x0 x1 .
cz
)
(
λ x0 x1 .
cif
(
wceq
(
cv
x1
)
cc0
)
(
cif
(
wceq
(
co
(
cv
x0
)
c2
cexp
)
c1
)
c1
cc0
)
(
co
(
cif
(
wa
(
wbr
(
cv
x1
)
...
...
)
...
)
...
...
)
...
...
)
)
)
.
...
■