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Proofgold Proof
pf
Apply df_dir__df_tail__df_plusf__df_mgm__df_sgrp__df_mnd__df_mhm__df_submnd__df_frmd__df_vrmd__df_grp__df_minusg__df_sbg__df_mulg__df_subg__df_nsg__df_eqg__df_ghm with
wceq
cmnd
(
crab
(
λ x0 .
wsbc
(
λ x1 .
wsbc
(
λ x2 .
wrex
(
λ x3 .
wral
(
λ x4 .
wa
(
wceq
(
co
(
cv
x3
)
(
cv
x4
)
(
cv
x2
)
)
(
cv
x4
)
)
(
wceq
(
co
(
cv
x4
)
(
cv
x3
)
(
cv
x2
)
)
(
cv
x4
)
)
)
(
λ x4 .
cv
x1
)
)
(
λ x3 .
cv
x1
)
)
(
cfv
(
cv
x0
)
cplusg
)
)
(
cfv
(
cv
x0
)
cbs
)
)
(
λ x0 .
csgrp
)
)
.
Assume H0:
wceq
cdir
(
cab
(
λ x0 .
wa
(
wa
(
wrel
(
cv
x0
)
)
(
wss
(
cres
cid
(
cuni
(
cuni
(
cv
x0
)
)
)
)
(
cv
x0
)
)
)
(
wa
(
wss
(
ccom
(
cv
x0
)
(
cv
x0
)
)
(
cv
x0
)
)
(
wss
(
cxp
(
cuni
(
cuni
(
cv
x0
)
)
)
(
cuni
(
cuni
(
cv
x0
)
)
)
)
(
ccom
(
ccnv
(
cv
x0
)
)
(
cv
x0
)
)
)
)
)
)
.
Assume H1:
wceq
ctail
(
cmpt
(
λ x0 .
cdir
)
(
λ x0 .
cmpt
(
λ x1 .
cuni
(
cuni
(
cv
x0
)
)
)
(
λ x1 .
cima
(
cv
x0
)
(
csn
(
cv
x1
)
)
)
)
)
.
Assume H2:
wceq
cplusf
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cbs
)
(
λ x1 x2 .
cfv
(
cv
x0
)
cbs
)
(
λ x1 x2 .
co
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
cplusg
)
)
)
)
.
Assume H3:
wceq
cmgm
(
cab
(
λ x0 .
wsbc
(
λ x1 .
wsbc
(
λ x2 .
wral
(
λ x3 .
wral
(
λ x4 .
wcel
(
co
(
cv
x3
)
(
cv
x4
)
(
cv
x2
)
)
(
cv
x1
)
)
(
λ x4 .
cv
x1
)
)
(
λ x3 .
cv
x1
)
)
(
cfv
(
cv
x0
)
cplusg
)
)
(
cfv
(
cv
x0
)
cbs
)
)
)
.
Assume H4:
wceq
csgrp
(
crab
(
λ x0 .
wsbc
(
λ x1 .
wsbc
(
λ x2 .
wral
(
λ x3 .
wral
(
λ x4 .
wral
(
λ x5 .
wceq
(
co
(
co
(
cv
x3
)
(
cv
x4
)
(
cv
x2
)
)
(
cv
x5
)
(
cv
x2
)
)
(
co
(
cv
x3
)
(
co
(
cv
x4
)
(
cv
x5
)
(
cv
x2
)
)
(
cv
x2
)
)
)
(
λ x5 .
cv
x1
)
)
(
λ x4 .
cv
x1
)
)
(
λ x3 .
cv
x1
)
)
(
cfv
(
cv
x0
)
cplusg
)
)
(
cfv
(
cv
x0
)
cbs
)
)
(
λ x0 .
cmgm
)
)
.
Assume H5:
wceq
cmnd
(
crab
(
λ x0 .
wsbc
(
λ x1 .
wsbc
(
λ x2 .
wrex
(
λ x3 .
wral
(
λ x4 .
wa
(
wceq
(
co
(
cv
x3
)
(
cv
...
)
...
)
...
)
...
)
...
)
...
)
...
)
...
)
...
)
.
...
■