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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
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 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
(
wceq
(
cv
x0
)
cc0
)
cc0
(
cif
(
wbr
(
cv
x0
)
cc0
clt
)
(
cneg
c1
)
c1
)
)
)
.
Assume H5:
wceq
ccj
(
cmpt
(
λ x0 .
cc
)
(
λ x0 .
crio
(
λ x1 .
wa
(
wcel
(
co
(
cv
x0
)
(
cv
x1
)
caddc
)
cr
)
(
wcel
(
co
ci
(
co
(
cv
x0
)
(
cv
x1
)
cmin
)
cmul
)
cr
)
)
(
λ x1 .
cc
)
)
)
.
Assume H6:
wceq
cre
(
cmpt
(
λ x0 .
cc
)
(
λ x0 .
co
(
co
(
cv
x0
)
(
cfv
(
cv
x0
)
ccj
)
caddc
)
c2
cdiv
)
)
.
Assume H7:
wceq
cim
(
cmpt
(
λ x0 .
cc
)
(
λ x0 .
cfv
(
co
(
cv
x0
)
ci
cdiv
)
cre
)
)
.
Assume H8:
wceq
csqrt
(
cmpt
(
λ x0 .
cc
)
(
λ x0 .
crio
(
λ x1 .
w3a
(
wceq
...
...
)
...
...
)
...
)
)
.
...
■