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Proofgold Proof
pf
Apply df_hash__df_word__df_lsw__df_concat__df_s1__df_substr__df_splice__df_reverse__df_reps__df_csh__df_s2__df_s3__df_s4__df_s5__df_s6__df_s7__df_s8__df_trcl with
wceq
cconcat
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cmpt
(
λ x2 .
co
cc0
(
co
(
cfv
(
cv
x0
)
chash
)
(
cfv
(
cv
x1
)
chash
)
caddc
)
cfzo
)
(
λ x2 .
cif
(
wcel
(
cv
x2
)
(
co
cc0
(
cfv
(
cv
x0
)
chash
)
cfzo
)
)
(
cfv
(
cv
x2
)
(
cv
x0
)
)
(
cfv
(
co
(
cv
x2
)
(
cfv
(
cv
x0
)
chash
)
cmin
)
(
cv
x1
)
)
)
)
)
.
Assume H0:
wceq
chash
(
cun
(
ccom
(
cres
(
crdg
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
co
(
cv
x0
)
c1
caddc
)
)
cc0
)
com
)
ccrd
)
(
cxp
(
cdif
cvv
cfn
)
(
csn
cpnf
)
)
)
.
Assume H1:
∀ x0 :
ι → ο
.
wceq
(
cword
x0
)
(
cab
(
λ x1 .
wrex
(
λ x2 .
wf
(
co
cc0
(
cv
x2
)
cfzo
)
x0
(
cv
x1
)
)
(
λ x2 .
cn0
)
)
)
.
Assume H2:
wceq
clsw
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cfv
(
co
(
cfv
(
cv
x0
)
chash
)
c1
cmin
)
(
cv
x0
)
)
)
.
Assume H3:
wceq
cconcat
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cmpt
(
λ x2 .
co
cc0
(
co
(
cfv
(
cv
x0
)
chash
)
(
cfv
(
cv
x1
)
chash
)
caddc
)
cfzo
)
(
λ x2 .
cif
(
wcel
(
cv
x2
)
(
co
cc0
(
cfv
(
cv
x0
)
chash
)
cfzo
)
)
(
cfv
(
cv
x2
)
(
cv
x0
)
)
(
cfv
(
co
(
cv
x2
)
(
cfv
(
cv
x0
)
chash
)
cmin
)
(
cv
x1
)
)
)
)
)
.
Assume H4:
∀ x0 :
ι → ο
.
wceq
(
cs1
x0
)
(
csn
(
cop
cc0
(
cfv
x0
cid
)
)
)
.
Assume H5:
wceq
csubstr
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cxp
cz
cz
)
(
λ x0 x1 .
cif
(
wss
(
co
(
cfv
(
cv
x1
)
c1st
)
(
cfv
(
cv
x1
)
c2nd
)
cfzo
)
(
cdm
(
cv
x0
)
)
)
(
cmpt
(
λ x2 .
co
cc0
(
co
(
cfv
(
cv
x1
)
c2nd
)
(
cfv
(
cv
x1
)
c1st
)
cmin
)
cfzo
)
(
λ x2 .
cfv
(
co
(
cv
x2
)
(
cfv
(
cv
x1
)
c1st
)
caddc
)
(
cv
x0
)
)
)
c0
)
)
.
Assume H6:
wceq
csplice
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
co
(
co
(
co
(
cv
x0
)
(
cop
cc0
(
cfv
(
cfv
(
cv
x1
)
c1st
)
c1st
)
)
csubstr
)
(
cfv
...
...
)
...
)
...
...
)
)
.
...
■