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doc published by
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Known
df_fallfac__df_bpoly__df_ef__df_e__df_sin__df_cos__df_tan__df_pi__df_dvds__df_bits__df_sad__df_smu__df_gcd__df_lcm__df_lcmf__df_prm__df_numer__df_denom
:
∀ x0 : ο .
(
wceq
cfallfac
(
cmpt2
(
λ x1 x2 .
cc
)
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
cprod
(
λ x3 .
co
cc0
(
co
(
cv
x2
)
c1
cmin
)
cfz
)
(
λ x3 .
co
(
cv
x1
)
(
cv
x3
)
cmin
)
)
)
⟶
wceq
cbp
(
cmpt2
(
λ x1 x2 .
cn0
)
(
λ x1 x2 .
cc
)
(
λ x1 x2 .
cfv
(
cv
x1
)
(
cwrecs
cn0
clt
(
cmpt
(
λ x3 .
cvv
)
(
λ x3 .
csb
(
cfv
(
cdm
(
cv
x3
)
)
chash
)
(
λ x4 .
co
(
co
(
cv
x2
)
(
cv
x4
)
cexp
)
(
csu
(
cdm
(
cv
x3
)
)
(
λ x5 .
co
(
co
(
cv
x4
)
(
cv
x5
)
cbc
)
(
co
(
cfv
(
cv
x5
)
(
cv
x3
)
)
(
co
(
co
(
cv
x4
)
(
cv
x5
)
cmin
)
c1
caddc
)
cdiv
)
cmul
)
)
cmin
)
)
)
)
)
)
⟶
wceq
ce
(
cmpt
(
λ x1 .
cc
)
(
λ x1 .
csu
cn0
(
λ x2 .
co
(
co
(
cv
x1
)
(
cv
x2
)
cexp
)
(
cfv
(
cv
x2
)
cfa
)
cdiv
)
)
)
⟶
wceq
ceu
(
cfv
c1
ce
)
⟶
wceq
csin
(
cmpt
(
λ x1 .
cc
)
(
λ x1 .
co
(
co
(
cfv
(
co
ci
(
cv
x1
)
cmul
)
ce
)
(
cfv
(
co
(
cneg
ci
)
(
cv
x1
)
cmul
)
ce
)
cmin
)
(
co
c2
ci
cmul
)
cdiv
)
)
⟶
wceq
ccos
(
cmpt
(
λ x1 .
cc
)
(
λ x1 .
co
(
co
(
cfv
(
co
ci
(
cv
x1
)
cmul
)
ce
)
(
cfv
(
co
(
cneg
ci
)
(
cv
x1
)
cmul
)
ce
)
caddc
)
c2
cdiv
)
)
⟶
wceq
ctan
(
cmpt
(
λ x1 .
cima
(
ccnv
ccos
)
(
cdif
cc
(
csn
cc0
)
)
)
(
λ x1 .
co
(
cfv
(
cv
x1
)
csin
)
(
cfv
(
cv
x1
)
ccos
)
cdiv
)
)
⟶
wceq
cpi
(
cinf
(
cin
crp
(
cima
(
ccnv
csin
)
(
csn
cc0
)
)
)
cr
clt
)
⟶
wceq
cdvds
(
copab
(
λ x1 x2 .
wa
(
wa
(
wcel
(
cv
x1
)
cz
)
(
wcel
(
cv
x2
)
cz
)
)
(
wrex
(
λ x3 .
wceq
(
co
(
cv
x3
)
(
cv
x1
)
cmul
)
(
cv
x2
)
)
(
λ x3 .
cz
)
)
)
)
⟶
wceq
cbits
(
cmpt
(
λ x1 .
cz
)
(
λ x1 .
crab
(
λ x2 .
wn
(
wbr
c2
(
cfv
(
co
(
cv
x1
)
(
co
c2
(
cv
x2
)
cexp
)
cdiv
)
cfl
)
cdvds
)
)
(
λ x2 .
cn0
)
)
)
⟶
wceq
csad
(
cmpt2
(
λ x1 x2 .
cpw
cn0
)
(
λ x1 x2 .
cpw
cn0
)
(
λ x1 x2 .
crab
(
λ x3 .
whad
(
wcel
(
cv
x3
)
(
cv
x1
)
)
(
wcel
(
cv
x3
)
(
cv
x2
)
)
(
wcel
c0
(
cfv
(
cv
x3
)
(
cseq
(
cmpt2
(
λ x4 x5 .
c2o
)
(
λ x4 x5 .
cn0
)
(
λ x4 x5 .
cif
(
wcad
(
wcel
(
cv
x5
)
(
cv
x1
)
)
(
wcel
(
cv
x5
)
(
cv
x2
)
)
(
wcel
c0
(
cv
x4
)
)
)
c1o
c0
)
)
(
cmpt
(
λ x4 .
cn0
)
(
λ x4 .
cif
(
wceq
(
cv
x4
)
cc0
)
c0
(
co
(
cv
x4
)
c1
cmin
)
)
)
cc0
)
)
)
)
(
λ x3 .
cn0
)
)
)
⟶
wceq
csmu
(
cmpt2
(
λ x1 x2 .
cpw
cn0
)
(
λ x1 x2 .
cpw
cn0
)
(
λ x1 x2 .
crab
(
λ x3 .
wcel
(
cv
x3
)
(
cfv
(
co
(
cv
x3
)
c1
caddc
)
(
cseq
(
cmpt2
(
λ x4 x5 .
cpw
cn0
)
(
λ x4 x5 .
cn0
)
(
λ x4 x5 .
co
(
cv
x4
)
(
crab
(
λ x6 .
wa
(
wcel
(
cv
x5
)
(
cv
x1
)
)
(
wcel
(
co
(
cv
x6
)
(
cv
x5
)
cmin
)
(
cv
x2
)
)
)
(
λ x6 .
cn0
)
)
csad
)
)
(
cmpt
(
λ x4 .
cn0
)
(
λ x4 .
cif
(
wceq
(
cv
x4
)
cc0
)
c0
(
co
(
cv
x4
)
c1
cmin
)
)
)
cc0
)
)
)
(
λ x3 .
cn0
)
)
)
⟶
wceq
cgcd
(
cmpt2
(
λ x1 x2 .
cz
)
(
λ x1 x2 .
cz
)
(
λ x1 x2 .
cif
(
wa
(
wceq
(
cv
x1
)
cc0
)
(
wceq
(
cv
x2
)
cc0
)
)
cc0
(
csup
(
crab
(
λ x3 .
wa
(
wbr
(
cv
x3
)
(
cv
x1
)
cdvds
)
(
wbr
(
cv
x3
)
(
cv
x2
)
cdvds
)
)
(
λ x3 .
cz
)
)
cr
clt
)
)
)
⟶
wceq
clcm
(
cmpt2
(
λ x1 x2 .
cz
)
(
λ x1 x2 .
cz
)
(
λ x1 x2 .
cif
(
wo
(
wceq
(
cv
x1
)
cc0
)
(
wceq
(
cv
x2
)
cc0
)
)
cc0
(
cinf
(
crab
(
λ x3 .
wa
(
wbr
(
cv
x1
)
(
cv
x3
)
cdvds
)
(
wbr
(
cv
x2
)
(
cv
x3
)
cdvds
)
)
(
λ x3 .
cn
)
)
cr
clt
)
)
)
⟶
wceq
clcmf
(
cmpt
(
λ x1 .
cpw
cz
)
(
λ x1 .
cif
(
wcel
cc0
(
cv
x1
)
)
cc0
(
cinf
(
crab
(
λ x2 .
wral
(
λ x3 .
wbr
(
cv
x3
)
(
cv
x2
)
cdvds
)
(
λ x3 .
cv
x1
)
)
(
λ x2 .
cn
)
)
cr
clt
)
)
)
⟶
wceq
cprime
(
crab
(
λ x1 .
wbr
(
crab
(
λ x2 .
wbr
(
cv
x2
)
(
cv
x1
)
cdvds
)
(
λ x2 .
cn
)
)
c2o
cen
)
(
λ x1 .
cn
)
)
⟶
wceq
cnumer
(
cmpt
(
λ x1 .
cq
)
(
λ x1 .
cfv
(
crio
(
λ x2 .
wa
(
wceq
(
co
(
cfv
(
cv
x2
)
c1st
)
(
cfv
(
cv
x2
)
c2nd
)
cgcd
)
c1
)
(
wceq
(
cv
x1
)
(
co
(
cfv
(
cv
x2
)
c1st
)
(
cfv
(
cv
x2
)
c2nd
)
cdiv
)
)
)
(
λ x2 .
cxp
cz
cn
)
)
c1st
)
)
⟶
wceq
cdenom
(
cmpt
(
λ x1 .
cq
)
(
λ x1 .
cfv
(
crio
(
λ x2 .
wa
(
wceq
(
co
(
cfv
(
cv
x2
)
c1st
)
(
cfv
(
cv
x2
)
c2nd
)
cgcd
)
c1
)
(
wceq
(
cv
x1
)
(
co
(
cfv
(
cv
x2
)
c1st
)
(
cfv
(
cv
x2
)
c2nd
)
cdiv
)
)
)
(
λ x2 .
cxp
cz
cn
)
)
c2nd
)
)
⟶
x0
)
⟶
x0
Theorem
df_fallfac
:
wceq
cfallfac
(
cmpt2
(
λ x0 x1 .
cc
)
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cprod
(
λ x2 .
co
cc0
(
co
(
cv
x1
)
c1
cmin
)
cfz
)
(
λ x2 .
co
(
cv
x0
)
(
cv
x2
)
cmin
)
)
)
(proof)
Theorem
df_bpoly
:
wceq
cbp
(
cmpt2
(
λ x0 x1 .
cn0
)
(
λ x0 x1 .
cc
)
(
λ x0 x1 .
cfv
(
cv
x0
)
(
cwrecs
cn0
clt
(
cmpt
(
λ x2 .
cvv
)
(
λ x2 .
csb
(
cfv
(
cdm
(
cv
x2
)
)
chash
)
(
λ x3 .
co
(
co
(
cv
x1
)
(
cv
x3
)
cexp
)
(
csu
(
cdm
(
cv
x2
)
)
(
λ x4 .
co
(
co
(
cv
x3
)
(
cv
x4
)
cbc
)
(
co
(
cfv
(
cv
x4
)
(
cv
x2
)
)
(
co
(
co
(
cv
x3
)
(
cv
x4
)
cmin
)
c1
caddc
)
cdiv
)
cmul
)
)
cmin
)
)
)
)
)
)
(proof)
Theorem
df_ef
:
wceq
ce
(
cmpt
(
λ x0 .
cc
)
(
λ x0 .
csu
cn0
(
λ x1 .
co
(
co
(
cv
x0
)
(
cv
x1
)
cexp
)
(
cfv
(
cv
x1
)
cfa
)
cdiv
)
)
)
(proof)
Theorem
df_e
:
wceq
ceu
(
cfv
c1
ce
)
(proof)
Theorem
df_sin
:
wceq
csin
(
cmpt
(
λ x0 .
cc
)
(
λ x0 .
co
(
co
(
cfv
(
co
ci
(
cv
x0
)
cmul
)
ce
)
(
cfv
(
co
(
cneg
ci
)
(
cv
x0
)
cmul
)
ce
)
cmin
)
(
co
c2
ci
cmul
)
cdiv
)
)
(proof)
Theorem
df_cos
:
wceq
ccos
(
cmpt
(
λ x0 .
cc
)
(
λ x0 .
co
(
co
(
cfv
(
co
ci
(
cv
x0
)
cmul
)
ce
)
(
cfv
(
co
(
cneg
ci
)
(
cv
x0
)
cmul
)
ce
)
caddc
)
c2
cdiv
)
)
(proof)
Theorem
df_tan
:
wceq
ctan
(
cmpt
(
λ x0 .
cima
(
ccnv
ccos
)
(
cdif
cc
(
csn
cc0
)
)
)
(
λ x0 .
co
(
cfv
(
cv
x0
)
csin
)
(
cfv
(
cv
x0
)
ccos
)
cdiv
)
)
(proof)
Theorem
df_pi
:
wceq
cpi
(
cinf
(
cin
crp
(
cima
(
ccnv
csin
)
(
csn
cc0
)
)
)
cr
clt
)
(proof)
Theorem
df_dvds
:
wceq
cdvds
(
copab
(
λ x0 x1 .
wa
(
wa
(
wcel
(
cv
x0
)
cz
)
(
wcel
(
cv
x1
)
cz
)
)
(
wrex
(
λ x2 .
wceq
(
co
(
cv
x2
)
(
cv
x0
)
cmul
)
(
cv
x1
)
)
(
λ x2 .
cz
)
)
)
)
(proof)
Theorem
df_bits
:
wceq
cbits
(
cmpt
(
λ x0 .
cz
)
(
λ x0 .
crab
(
λ x1 .
wn
(
wbr
c2
(
cfv
(
co
(
cv
x0
)
(
co
c2
(
cv
x1
)
cexp
)
cdiv
)
cfl
)
cdvds
)
)
(
λ x1 .
cn0
)
)
)
(proof)
Theorem
df_sad
:
wceq
csad
(
cmpt2
(
λ x0 x1 .
cpw
cn0
)
(
λ x0 x1 .
cpw
cn0
)
(
λ x0 x1 .
crab
(
λ x2 .
whad
(
wcel
(
cv
x2
)
(
cv
x0
)
)
(
wcel
(
cv
x2
)
(
cv
x1
)
)
(
wcel
c0
(
cfv
(
cv
x2
)
(
cseq
(
cmpt2
(
λ x3 x4 .
c2o
)
(
λ x3 x4 .
cn0
)
(
λ x3 x4 .
cif
(
wcad
(
wcel
(
cv
x4
)
(
cv
x0
)
)
(
wcel
(
cv
x4
)
(
cv
x1
)
)
(
wcel
c0
(
cv
x3
)
)
)
c1o
c0
)
)
(
cmpt
(
λ x3 .
cn0
)
(
λ x3 .
cif
(
wceq
(
cv
x3
)
cc0
)
c0
(
co
(
cv
x3
)
c1
cmin
)
)
)
cc0
)
)
)
)
(
λ x2 .
cn0
)
)
)
(proof)
Theorem
df_smu
:
wceq
csmu
(
cmpt2
(
λ x0 x1 .
cpw
cn0
)
(
λ x0 x1 .
cpw
cn0
)
(
λ x0 x1 .
crab
(
λ x2 .
wcel
(
cv
x2
)
(
cfv
(
co
(
cv
x2
)
c1
caddc
)
(
cseq
(
cmpt2
(
λ x3 x4 .
cpw
cn0
)
(
λ x3 x4 .
cn0
)
(
λ x3 x4 .
co
(
cv
x3
)
(
crab
(
λ x5 .
wa
(
wcel
(
cv
x4
)
(
cv
x0
)
)
(
wcel
(
co
(
cv
x5
)
(
cv
x4
)
cmin
)
(
cv
x1
)
)
)
(
λ x5 .
cn0
)
)
csad
)
)
(
cmpt
(
λ x3 .
cn0
)
(
λ x3 .
cif
(
wceq
(
cv
x3
)
cc0
)
c0
(
co
(
cv
x3
)
c1
cmin
)
)
)
cc0
)
)
)
(
λ x2 .
cn0
)
)
)
(proof)
Theorem
df_gcd
:
wceq
cgcd
(
cmpt2
(
λ x0 x1 .
cz
)
(
λ x0 x1 .
cz
)
(
λ x0 x1 .
cif
(
wa
(
wceq
(
cv
x0
)
cc0
)
(
wceq
(
cv
x1
)
cc0
)
)
cc0
(
csup
(
crab
(
λ x2 .
wa
(
wbr
(
cv
x2
)
(
cv
x0
)
cdvds
)
(
wbr
(
cv
x2
)
(
cv
x1
)
cdvds
)
)
(
λ x2 .
cz
)
)
cr
clt
)
)
)
(proof)
Theorem
df_lcm
:
wceq
clcm
(
cmpt2
(
λ x0 x1 .
cz
)
(
λ x0 x1 .
cz
)
(
λ x0 x1 .
cif
(
wo
(
wceq
(
cv
x0
)
cc0
)
(
wceq
(
cv
x1
)
cc0
)
)
cc0
(
cinf
(
crab
(
λ x2 .
wa
(
wbr
(
cv
x0
)
(
cv
x2
)
cdvds
)
(
wbr
(
cv
x1
)
(
cv
x2
)
cdvds
)
)
(
λ x2 .
cn
)
)
cr
clt
)
)
)
(proof)
Theorem
df_lcmf
:
wceq
clcmf
(
cmpt
(
λ x0 .
cpw
cz
)
(
λ x0 .
cif
(
wcel
cc0
(
cv
x0
)
)
cc0
(
cinf
(
crab
(
λ x1 .
wral
(
λ x2 .
wbr
(
cv
x2
)
(
cv
x1
)
cdvds
)
(
λ x2 .
cv
x0
)
)
(
λ x1 .
cn
)
)
cr
clt
)
)
)
(proof)
Theorem
df_prm
:
wceq
cprime
(
crab
(
λ x0 .
wbr
(
crab
(
λ x1 .
wbr
(
cv
x1
)
(
cv
x0
)
cdvds
)
(
λ x1 .
cn
)
)
c2o
cen
)
(
λ x0 .
cn
)
)
(proof)
Theorem
df_numer
:
wceq
cnumer
(
cmpt
(
λ x0 .
cq
)
(
λ x0 .
cfv
(
crio
(
λ x1 .
wa
(
wceq
(
co
(
cfv
(
cv
x1
)
c1st
)
(
cfv
(
cv
x1
)
c2nd
)
cgcd
)
c1
)
(
wceq
(
cv
x0
)
(
co
(
cfv
(
cv
x1
)
c1st
)
(
cfv
(
cv
x1
)
c2nd
)
cdiv
)
)
)
(
λ x1 .
cxp
cz
cn
)
)
c1st
)
)
(proof)
Theorem
df_denom
:
wceq
cdenom
(
cmpt
(
λ x0 .
cq
)
(
λ x0 .
cfv
(
crio
(
λ x1 .
wa
(
wceq
(
co
(
cfv
(
cv
x1
)
c1st
)
(
cfv
(
cv
x1
)
c2nd
)
cgcd
)
c1
)
(
wceq
(
cv
x0
)
(
co
(
cfv
(
cv
x1
)
c1st
)
(
cfv
(
cv
x1
)
c2nd
)
cdiv
)
)
)
(
λ x1 .
cxp
cz
cn
)
)
c2nd
)
)
(proof)