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Proofgold Proof
pf
Apply df_rtrcl__df_relexp__df_rtrclrec__df_shft__df_sgn__df_cj__df_re__df_im__df_sqrt__df_abs__df_limsup__df_clim__df_rlim__df_o1__df_lo1__df_sum__df_prod__df_risefac with
∀ x0 x1 :
ι →
ι → ο
.
∀ x2 .
wceq
(
cprod
(
λ x3 .
x0
x3
)
(
λ x3 .
x1
x3
)
)
(
cio
(
λ x3 .
wo
(
wrex
(
λ x4 .
w3a
(
wss
(
x0
x2
)
(
cfv
(
cv
x4
)
cuz
)
)
(
wrex
(
λ x5 .
wex
(
λ x6 .
wa
(
wne
(
cv
x6
)
cc0
)
(
wbr
(
cseq
cmul
(
cmpt
(
λ x7 .
cz
)
(
λ x7 .
cif
(
wcel
(
cv
x7
)
(
x0
x7
)
)
(
x1
x7
)
c1
)
)
(
cv
x5
)
)
(
cv
x6
)
cli
)
)
)
(
λ x5 .
cfv
(
cv
x4
)
cuz
)
)
(
wbr
(
cseq
cmul
(
cmpt
(
λ x5 .
cz
)
(
λ x5 .
cif
(
wcel
(
cv
x5
)
(
x0
x5
)
)
(
x1
x5
)
c1
)
)
(
cv
x4
)
)
(
cv
x3
)
cli
)
)
(
λ x4 .
cz
)
)
(
wrex
(
λ x4 .
wex
(
λ x5 .
wa
(
wf1o
(
co
c1
(
cv
x4
)
cfz
)
(
x0
x2
)
(
cv
x5
)
)
(
wceq
(
cv
x3
)
(
cfv
(
cv
x4
)
(
cseq
cmul
(
cmpt
(
λ x6 .
cn
)
(
λ x6 .
csb
(
cfv
(
cv
x6
)
(
cv
x5
)
)
(
λ x7 .
x1
x7
)
)
)
c1
)
)
)
)
)
(
λ x4 .
cn
)
)
)
)
.
Assume H0:
wceq
crtcl
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cint
(
cab
(
λ x1 .
w3a
(
wss
(
cres
cid
(
cun
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
(
cv
x1
)
)
(
wss
(
cv
x0
)
(
cv
x1
)
)
(
wss
(
ccom
(
cv
x1
)
(
cv
x1
)
)
(
cv
x1
)
)
)
)
)
)
.
Assume H1:
wceq
crelexp
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cif
(
wceq
(
cv
x1
)
cc0
)
(
cres
cid
(
cun
(
cdm
(
cv
x0
)
)
(
crn
(
cv
x0
)
)
)
)
(
cfv
(
cv
x1
)
(
cseq
(
cmpt2
(
λ x2 x3 .
cvv
)
(
λ x2 x3 .
cvv
)
(
λ x2 x3 .
ccom
(
cv
x2
)
(
cv
x0
)
)
)
(
cmpt
(
λ x2 .
cvv
)
(
λ x2 .
cv
x0
)
)
c1
)
)
)
)
.
Assume H2:
wceq
crtrcl
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
ciun
(
λ x1 .
cn0
)
(
λ x1 .
co
(
cv
x0
)
(
cv
x1
)
crelexp
)
)
)
.
Assume H3:
wceq
cshi
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cc
)
(
λ x0 x1 .
copab
(
λ x2 x3 .
wa
(
wcel
(
cv
x2
)
cc
)
(
wbr
(
co
(
cv
x2
)
(
cv
x1
)
cmin
)
(
cv
x3
)
(
cv
x0
)
)
)
)
)
.
Assume H4:
wceq
csgn
(
cmpt
(
λ x0 .
cxr
)
(
λ x0 .
cif
...
...
...
)
)
.
...
■