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Proofgold Proof
pf
Apply df_msub__df_mvh__df_mpst__df_msr__df_msta__df_mfs__df_mcls__df_mpps__df_mthm__df_m0s__df_msa__df_mwgfs__df_msyn__df_mtree__df_mst__df_msax__df_mufs__df_muv with
wceq
cmwgfs
(
crab
(
λ x0 .
∀ x1 x2 x3 .
wa
(
wcel
(
cotp
(
cv
x1
)
(
cv
x2
)
(
cv
x3
)
)
(
cfv
(
cv
x0
)
cmax
)
)
(
wcel
(
cfv
(
cv
x3
)
c1st
)
(
cfv
(
cv
x0
)
cmvt
)
)
⟶
wrex
(
λ x4 .
wcel
(
cv
x3
)
(
cima
(
cv
x4
)
(
cfv
(
cv
x0
)
cmsa
)
)
)
(
λ x4 .
crn
(
cfv
(
cv
x0
)
cmsub
)
)
)
(
λ x0 .
cmfs
)
)
.
Assume H0:
wceq
cmsub
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt
(
λ x1 .
co
(
cfv
(
cv
x0
)
cmrex
)
(
cfv
(
cv
x0
)
cmvar
)
cpm
)
(
λ x1 .
cmpt
(
λ x2 .
cfv
(
cv
x0
)
cmex
)
(
λ x2 .
cop
(
cfv
(
cv
x2
)
c1st
)
(
cfv
(
cfv
(
cv
x2
)
c2nd
)
(
cfv
(
cv
x1
)
(
cfv
(
cv
x0
)
cmrsub
)
)
)
)
)
)
)
.
Assume H1:
wceq
cmvh
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt
(
λ x1 .
cfv
(
cv
x0
)
cmvar
)
(
λ x1 .
cop
(
cfv
(
cv
x1
)
(
cfv
(
cv
x0
)
cmty
)
)
(
cs1
(
cv
x1
)
)
)
)
)
.
Assume H2:
wceq
cmpst
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cxp
(
cxp
(
crab
(
λ x1 .
wceq
(
ccnv
(
cv
x1
)
)
(
cv
x1
)
)
(
λ x1 .
cpw
(
cfv
(
cv
x0
)
cmdv
)
)
)
(
cin
(
cpw
(
cfv
(
cv
x0
)
cmex
)
)
cfn
)
)
(
cfv
(
cv
x0
)
cmex
)
)
)
.
Assume H3:
wceq
cmsr
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt
(
λ x1 .
cfv
(
cv
x0
)
cmpst
)
(
λ x1 .
csb
(
cfv
(
cfv
(
cv
x1
)
c1st
)
c2nd
)
(
λ x2 .
csb
(
cfv
(
cv
x1
)
c2nd
)
(
λ x3 .
cotp
(
cin
(
cfv
(
cfv
(
cv
x1
)
c1st
)
c1st
)
(
csb
(
cuni
(
cima
(
cfv
(
cv
x0
)
cmvrs
)
(
cun
(
cv
x2
)
(
csn
(
cv
x3
)
)
)
)
)
(
λ x4 .
cxp
(
cv
x4
)
(
cv
x4
)
)
)
)
(
cv
x2
)
(
cv
x3
)
)
)
)
)
)
.
Assume H4:
wceq
cmsta
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
crn
(
cfv
(
cv
x0
)
cmsr
)
)
)
.
Assume H5:
wceq
cmfs
(
cab
(
λ x0 .
wa
(
wa
(
wceq
(
cin
(
cfv
(
cv
x0
)
cmcn
)
(
cfv
(
cv
x0
)
cmvar
)
)
c0
)
(
wf
(
cfv
(
cv
x0
)
cmvar
)
(
cfv
(
cv
x0
)
cmtc
)
(
cfv
(
cv
x0
)
cmty
)
)
)
(
wa
(
wss
...
...
)
...
)
)
)
.
...
■