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doc published by
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Known
df_trls__df_trlson__df_pths__df_spths__df_pthson__df_spthson__df_clwlks__df_crcts__df_cycls__df_wwlks__df_wwlksn__df_wwlksnon__df_wspthsn__df_wspthsnon__df_clwwlk__df_clwwlkn__df_clwwlknOLD__df_clwwlknon
:
∀ x0 : ο .
(
wceq
ctrls
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
copab
(
λ x2 x3 .
wa
(
wbr
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
cwlks
)
)
(
wfun
(
ccnv
(
cv
x2
)
)
)
)
)
)
⟶
wceq
ctrlson
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
copab
(
λ x4 x5 .
wa
(
wbr
(
cv
x4
)
(
cv
x5
)
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
cwlkson
)
)
)
(
wbr
(
cv
x4
)
(
cv
x5
)
(
cfv
(
cv
x1
)
ctrls
)
)
)
)
)
)
⟶
wceq
cpths
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
copab
(
λ x2 x3 .
w3a
(
wbr
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
ctrls
)
)
(
wfun
(
ccnv
(
cres
(
cv
x3
)
(
co
c1
(
cfv
(
cv
x2
)
chash
)
cfzo
)
)
)
)
(
wceq
(
cin
(
cima
(
cv
x3
)
(
cpr
cc0
(
cfv
(
cv
x2
)
chash
)
)
)
(
cima
(
cv
x3
)
(
co
c1
(
cfv
(
cv
x2
)
chash
)
cfzo
)
)
)
c0
)
)
)
)
⟶
wceq
cspths
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
copab
(
λ x2 x3 .
wa
(
wbr
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
ctrls
)
)
(
wfun
(
ccnv
(
cv
x3
)
)
)
)
)
)
⟶
wceq
cpthson
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
copab
(
λ x4 x5 .
wa
(
wbr
(
cv
x4
)
(
cv
x5
)
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
ctrlson
)
)
)
(
wbr
(
cv
x4
)
(
cv
x5
)
(
cfv
(
cv
x1
)
cpths
)
)
)
)
)
)
⟶
wceq
cspthson
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
copab
(
λ x4 x5 .
wa
(
wbr
(
cv
x4
)
(
cv
x5
)
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
ctrlson
)
)
)
(
wbr
(
cv
x4
)
(
cv
x5
)
(
cfv
(
cv
x1
)
cspths
)
)
)
)
)
)
⟶
wceq
cclwlks
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
copab
(
λ x2 x3 .
wa
(
wbr
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
cwlks
)
)
(
wceq
(
cfv
cc0
(
cv
x3
)
)
(
cfv
(
cfv
(
cv
x2
)
chash
)
(
cv
x3
)
)
)
)
)
)
⟶
wceq
ccrcts
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
copab
(
λ x2 x3 .
wa
(
wbr
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
ctrls
)
)
(
wceq
(
cfv
cc0
(
cv
x3
)
)
(
cfv
(
cfv
(
cv
x2
)
chash
)
(
cv
x3
)
)
)
)
)
)
⟶
wceq
ccycls
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
copab
(
λ x2 x3 .
wa
(
wbr
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
cpths
)
)
(
wceq
(
cfv
cc0
(
cv
x3
)
)
(
cfv
(
cfv
(
cv
x2
)
chash
)
(
cv
x3
)
)
)
)
)
)
⟶
wceq
cwwlks
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
crab
(
λ x2 .
wa
(
wne
(
cv
x2
)
c0
)
(
wral
(
λ x3 .
wcel
(
cpr
(
cfv
(
cv
x3
)
(
cv
x2
)
)
(
cfv
(
co
(
cv
x3
)
c1
caddc
)
(
cv
x2
)
)
)
(
cfv
(
cv
x1
)
cedg
)
)
(
λ x3 .
co
cc0
(
co
(
cfv
(
cv
x2
)
chash
)
c1
cmin
)
cfzo
)
)
)
(
λ x2 .
cword
(
cfv
(
cv
x1
)
cvtx
)
)
)
)
⟶
wceq
cwwlksn
(
cmpt2
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
crab
(
λ x3 .
wceq
(
cfv
(
cv
x3
)
chash
)
(
co
(
cv
x1
)
c1
caddc
)
)
(
λ x3 .
cfv
(
cv
x2
)
cwwlks
)
)
)
⟶
wceq
cwwlksnon
(
cmpt2
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cmpt2
(
λ x3 x4 .
cfv
(
cv
x2
)
cvtx
)
(
λ x3 x4 .
cfv
(
cv
x2
)
cvtx
)
(
λ x3 x4 .
crab
(
λ x5 .
wa
(
wceq
(
cfv
cc0
(
cv
x5
)
)
(
cv
x3
)
)
(
wceq
(
cfv
(
cv
x1
)
(
cv
x5
)
)
(
cv
x4
)
)
)
(
λ x5 .
co
(
cv
x1
)
(
cv
x2
)
cwwlksn
)
)
)
)
⟶
wceq
cwwspthsn
(
cmpt2
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
crab
(
λ x3 .
wex
(
λ x4 .
wbr
(
cv
x4
)
(
cv
x3
)
(
cfv
(
cv
x2
)
cspths
)
)
)
(
λ x3 .
co
(
cv
x1
)
(
cv
x2
)
cwwlksn
)
)
)
⟶
wceq
cwwspthsnon
(
cmpt2
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cmpt2
(
λ x3 x4 .
cfv
(
cv
x2
)
cvtx
)
(
λ x3 x4 .
cfv
(
cv
x2
)
cvtx
)
(
λ x3 x4 .
crab
(
λ x5 .
wex
(
λ x6 .
wbr
(
cv
x6
)
(
cv
x5
)
(
co
(
cv
x3
)
(
cv
x4
)
(
cfv
(
cv
x2
)
cspthson
)
)
)
)
(
λ x5 .
co
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
cwwlksnon
)
)
)
)
)
⟶
wceq
cclwwlk
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
crab
(
λ x2 .
w3a
(
wne
(
cv
x2
)
c0
)
(
wral
(
λ x3 .
wcel
(
cpr
(
cfv
(
cv
x3
)
(
cv
x2
)
)
(
cfv
(
co
(
cv
x3
)
c1
caddc
)
(
cv
x2
)
)
)
(
cfv
(
cv
x1
)
cedg
)
)
(
λ x3 .
co
cc0
(
co
(
cfv
(
cv
x2
)
chash
)
c1
cmin
)
cfzo
)
)
(
wcel
(
cpr
(
cfv
(
cv
x2
)
clsw
)
(
cfv
cc0
(
cv
x2
)
)
)
(
cfv
(
cv
x1
)
cedg
)
)
)
(
λ x2 .
cword
(
cfv
(
cv
x1
)
cvtx
)
)
)
)
⟶
wceq
cclwwlkn
(
cmpt2
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
crab
(
λ x3 .
wceq
(
cfv
(
cv
x3
)
chash
)
(
cv
x1
)
)
(
λ x3 .
cfv
(
cv
x2
)
cclwwlk
)
)
)
⟶
wceq
cclwwlknold
(
cmpt2
(
λ x1 x2 .
cn
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
crab
(
λ x3 .
wceq
(
cfv
(
cv
x3
)
chash
)
(
cv
x1
)
)
(
λ x3 .
cfv
(
cv
x2
)
cclwwlk
)
)
)
⟶
wceq
cclwwlknon
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
cn0
)
(
λ x2 x3 .
crab
(
λ x4 .
wceq
(
cfv
cc0
(
cv
x4
)
)
(
cv
x2
)
)
(
λ x4 .
co
(
cv
x3
)
(
cv
x1
)
cclwwlkn
)
)
)
)
⟶
x0
)
⟶
x0
Theorem
df_trls
:
wceq
ctrls
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
cwlks
)
)
(
wfun
(
ccnv
(
cv
x1
)
)
)
)
)
)
(proof)
Theorem
df_trlson
:
wceq
ctrlson
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wa
(
wbr
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
cwlkson
)
)
)
(
wbr
(
cv
x3
)
(
cv
x4
)
(
cfv
(
cv
x0
)
ctrls
)
)
)
)
)
)
(proof)
Theorem
df_pths
:
wceq
cpths
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
w3a
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrls
)
)
(
wfun
(
ccnv
(
cres
(
cv
x2
)
(
co
c1
(
cfv
(
cv
x1
)
chash
)
cfzo
)
)
)
)
(
wceq
(
cin
(
cima
(
cv
x2
)
(
cpr
cc0
(
cfv
(
cv
x1
)
chash
)
)
)
(
cima
(
cv
x2
)
(
co
c1
(
cfv
(
cv
x1
)
chash
)
cfzo
)
)
)
c0
)
)
)
)
(proof)
Theorem
df_spths
:
wceq
cspths
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrls
)
)
(
wfun
(
ccnv
(
cv
x2
)
)
)
)
)
)
(proof)
Theorem
df_pthson
:
wceq
cpthson
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wa
(
wbr
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrlson
)
)
)
(
wbr
(
cv
x3
)
(
cv
x4
)
(
cfv
(
cv
x0
)
cpths
)
)
)
)
)
)
(proof)
Theorem
df_spthson
:
wceq
cspthson
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wa
(
wbr
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrlson
)
)
)
(
wbr
(
cv
x3
)
(
cv
x4
)
(
cfv
(
cv
x0
)
cspths
)
)
)
)
)
)
(proof)
Theorem
df_clwlks
:
wceq
cclwlks
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
cwlks
)
)
(
wceq
(
cfv
cc0
(
cv
x2
)
)
(
cfv
(
cfv
(
cv
x1
)
chash
)
(
cv
x2
)
)
)
)
)
)
(proof)
Theorem
df_crcts
:
wceq
ccrcts
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
ctrls
)
)
(
wceq
(
cfv
cc0
(
cv
x2
)
)
(
cfv
(
cfv
(
cv
x1
)
chash
)
(
cv
x2
)
)
)
)
)
)
(proof)
Theorem
df_cycls
:
wceq
ccycls
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
copab
(
λ x1 x2 .
wa
(
wbr
(
cv
x1
)
(
cv
x2
)
(
cfv
(
cv
x0
)
cpths
)
)
(
wceq
(
cfv
cc0
(
cv
x2
)
)
(
cfv
(
cfv
(
cv
x1
)
chash
)
(
cv
x2
)
)
)
)
)
)
(proof)
Theorem
df_wwlks
:
wceq
cwwlks
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
crab
(
λ x1 .
wa
(
wne
(
cv
x1
)
c0
)
(
wral
(
λ x2 .
wcel
(
cpr
(
cfv
(
cv
x2
)
(
cv
x1
)
)
(
cfv
(
co
(
cv
x2
)
c1
caddc
)
(
cv
x1
)
)
)
(
cfv
(
cv
x0
)
cedg
)
)
(
λ x2 .
co
cc0
(
co
(
cfv
(
cv
x1
)
chash
)
c1
cmin
)
cfzo
)
)
)
(
λ x1 .
cword
(
cfv
(
cv
x0
)
cvtx
)
)
)
)
(proof)
Theorem
df_wwlksn
:
wceq
cwwlksn
(
cmpt2
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
crab
(
λ x2 .
wceq
(
cfv
(
cv
x2
)
chash
)
(
co
(
cv
x0
)
c1
caddc
)
)
(
λ x2 .
cfv
(
cv
x1
)
cwwlks
)
)
)
(proof)
Theorem
df_wwlksnon
:
wceq
cwwlksnon
(
cmpt2
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
crab
(
λ x4 .
wa
(
wceq
(
cfv
cc0
(
cv
x4
)
)
(
cv
x2
)
)
(
wceq
(
cfv
(
cv
x0
)
(
cv
x4
)
)
(
cv
x3
)
)
)
(
λ x4 .
co
(
cv
x0
)
(
cv
x1
)
cwwlksn
)
)
)
)
(proof)
Theorem
df_wspthsn
:
wceq
cwwspthsn
(
cmpt2
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
crab
(
λ x2 .
wex
(
λ x3 .
wbr
(
cv
x3
)
(
cv
x2
)
(
cfv
(
cv
x1
)
cspths
)
)
)
(
λ x2 .
co
(
cv
x0
)
(
cv
x1
)
cwwlksn
)
)
)
(proof)
Theorem
df_wspthsnon
:
wceq
cwwspthsnon
(
cmpt2
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
cfv
(
cv
x1
)
cvtx
)
(
λ x2 x3 .
crab
(
λ x4 .
wex
(
λ x5 .
wbr
(
cv
x5
)
(
cv
x4
)
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
cspthson
)
)
)
)
(
λ x4 .
co
(
cv
x2
)
(
cv
x3
)
(
co
(
cv
x0
)
(
cv
x1
)
cwwlksnon
)
)
)
)
)
(proof)
Theorem
df_clwwlk
:
wceq
cclwwlk
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
crab
(
λ x1 .
w3a
(
wne
(
cv
x1
)
c0
)
(
wral
(
λ x2 .
wcel
(
cpr
(
cfv
(
cv
x2
)
(
cv
x1
)
)
(
cfv
(
co
(
cv
x2
)
c1
caddc
)
(
cv
x1
)
)
)
(
cfv
(
cv
x0
)
cedg
)
)
(
λ x2 .
co
cc0
(
co
(
cfv
(
cv
x1
)
chash
)
c1
cmin
)
cfzo
)
)
(
wcel
(
cpr
(
cfv
(
cv
x1
)
clsw
)
(
cfv
cc0
(
cv
x1
)
)
)
(
cfv
(
cv
x0
)
cedg
)
)
)
(
λ x1 .
cword
(
cfv
(
cv
x0
)
cvtx
)
)
)
)
(proof)
Theorem
df_clwwlkn
:
wceq
cclwwlkn
(
cmpt2
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
crab
(
λ x2 .
wceq
(
cfv
(
cv
x2
)
chash
)
(
cv
x0
)
)
(
λ x2 .
cfv
(
cv
x1
)
cclwwlk
)
)
)
(proof)
Theorem
df_clwwlknOLD
:
wceq
cclwwlknold
(
cmpt2
(
λ x0 x1 .
cn
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
crab
(
λ x2 .
wceq
(
cfv
(
cv
x2
)
chash
)
(
cv
x0
)
)
(
λ x2 .
cfv
(
cv
x1
)
cclwwlk
)
)
)
(proof)
Theorem
df_clwwlknon
:
wceq
cclwwlknon
(
cmpt
(
λ x0 .
cvv
)
(
λ x0 .
cmpt2
(
λ x1 x2 .
cfv
(
cv
x0
)
cvtx
)
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
crab
(
λ x3 .
wceq
(
cfv
cc0
(
cv
x3
)
)
(
cv
x1
)
)
(
λ x3 .
co
(
cv
x2
)
(
cv
x0
)
cclwwlkn
)
)
)
)
(proof)