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Definition
csect
:=
csect
Definition
cinv
:=
cinv
Definition
ccic
:=
ccic
Definition
ciso
:=
ciso
Definition
c0
:=
c0
Definition
csupp
:=
csupp
Definition
wex
:=
wex
Definition
wfn
:=
wfn
Definition
wrex
:=
wrex
Definition
cpw
:=
cpw
Definition
cresc
:=
cresc
Definition
cress
:=
cress
Definition
csts
:=
csts
Definition
csubc
:=
csubc
Definition
cab
:=
cab
Definition
chomf
:=
chomf
Definition
cssc
:=
cssc
Definition
wsbc
wsbc
:=
wsbc
Definition
w3a
:=
w3a
Definition
wf
:=
wf
Definition
cmap
:=
cmap
Definition
ccid
:=
ccid
Definition
cidfu
:=
cidfu
Definition
cid
:=
cid
Definition
ccofu
:=
ccofu
Definition
cresf
:=
cresf
Definition
cvv
:=
cvv
Definition
cdm
:=
cdm
Definition
cres
:=
cres
Definition
cful
:=
cful
Definition
crn
:=
crn
Definition
cfth
:=
cfth
Definition
copab
:=
copab
Definition
wa
:=
wa
Definition
wbr
:=
wbr
Definition
wfun
:=
wfun
Definition
ccnv
:=
ccnv
Definition
cixp
:=
cixp
Definition
cfuc
:=
cfuc
Definition
ctp
:=
ctp
Definition
cnx
:=
cnx
Definition
cxp
:=
cxp
Definition
cfunc
:=
cfunc
Definition
csb
:=
csb
Definition
c2nd
:=
c2nd
Definition
cmpt2
:=
cmpt2
Definition
cnat
:=
cnat
Definition
cop
:=
cop
Definition
cco
:=
cco
Definition
crab
:=
crab
Definition
wral
:=
wral
Definition
weu
:=
weu
Definition
wcel
:=
wcel
Definition
co
:=
co
Definition
chom
:=
chom
Definition
cbs
:=
cbs
Definition
czeroo
:=
czeroo
Definition
cmpt
:=
cmpt
Definition
ccat
:=
ccat
Definition
cin
:=
cin
Definition
cinito
:=
cinito
Definition
cfv
:=
cfv
Definition
cv
:=
cv
Definition
ctermo
:=
ctermo
Definition
wceq
wceq
:=
wceq
Definition
cdoma
:=
cdoma
Definition
ccom
:=
ccom
Definition
c1st
:=
c1st
Known
df_inv__df_iso__df_cic__df_ssc__df_resc__df_subc__df_func__df_idfu__df_cofu__df_resf__df_full__df_fth__df_nat__df_fuc__df_inito__df_termo__df_zeroo__df_doma
:
∀ x0 : ο .
(
wceq
cinv
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cmpt2
(
λ x2 x3 .
cfv
(
cv
x1
)
cbs
)
(
λ x2 x3 .
cfv
(
cv
x1
)
cbs
)
(
λ x2 x3 .
cin
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
csect
)
)
(
ccnv
(
co
(
cv
x3
)
(
cv
x2
)
(
cfv
(
cv
x1
)
csect
)
)
)
)
)
)
⟶
wceq
ciso
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
ccom
(
cmpt
(
λ x2 .
cvv
)
(
λ x2 .
cdm
(
cv
x2
)
)
)
(
cfv
(
cv
x1
)
cinv
)
)
)
⟶
wceq
ccic
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
co
(
cfv
(
cv
x1
)
ciso
)
c0
csupp
)
)
⟶
wceq
cssc
(
copab
(
λ x1 x2 .
wex
(
λ x3 .
wa
(
wfn
(
cv
x2
)
(
cxp
(
cv
x3
)
(
cv
x3
)
)
)
(
wrex
(
λ x4 .
wcel
(
cv
x1
)
(
cixp
(
λ x5 .
cxp
(
cv
x4
)
(
cv
x4
)
)
(
λ x5 .
cpw
(
cfv
(
cv
x5
)
(
cv
x2
)
)
)
)
)
(
λ x4 .
cpw
(
cv
x3
)
)
)
)
)
)
⟶
wceq
cresc
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
co
(
co
(
cv
x1
)
(
cdm
(
cdm
(
cv
x2
)
)
)
cress
)
(
cop
(
cfv
cnx
chom
)
(
cv
x2
)
)
csts
)
)
⟶
wceq
csubc
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cab
(
λ x2 .
wa
(
wbr
(
cv
x2
)
(
cfv
(
cv
x1
)
chomf
)
cssc
)
(
wsbc
(
λ x3 .
wral
(
λ x4 .
wa
(
wcel
(
cfv
(
cv
x4
)
(
cfv
(
cv
x1
)
ccid
)
)
(
co
(
cv
x4
)
(
cv
x4
)
(
cv
x2
)
)
)
(
wral
(
λ x5 .
wral
(
λ x6 .
wral
(
λ x7 .
wral
(
λ x8 .
wcel
(
co
(
cv
x8
)
(
cv
x7
)
(
co
(
cop
(
cv
x4
)
(
cv
x5
)
)
(
cv
x6
)
(
cfv
(
cv
x1
)
cco
)
)
)
(
co
(
cv
x4
)
(
cv
x6
)
(
cv
x2
)
)
)
(
λ x8 .
co
(
cv
x5
)
(
cv
x6
)
(
cv
x2
)
)
)
(
λ x7 .
co
(
cv
x4
)
(
cv
x5
)
(
cv
x2
)
)
)
(
λ x6 .
cv
x3
)
)
(
λ x5 .
cv
x3
)
)
)
(
λ x4 .
cv
x3
)
)
(
cdm
(
cdm
(
cv
x2
)
)
)
)
)
)
)
⟶
wceq
cfunc
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wsbc
(
λ x5 .
w3a
(
wf
(
cv
x5
)
(
cfv
(
cv
x2
)
cbs
)
(
cv
x3
)
)
(
wcel
(
cv
x4
)
(
cixp
(
λ x6 .
cxp
(
cv
x5
)
(
cv
x5
)
)
(
λ x6 .
co
(
co
(
cfv
(
cfv
(
cv
x6
)
c1st
)
(
cv
x3
)
)
(
cfv
(
cfv
(
cv
x6
)
c2nd
)
(
cv
x3
)
)
(
cfv
(
cv
x2
)
chom
)
)
(
cfv
(
cv
x6
)
(
cfv
(
cv
x1
)
chom
)
)
cmap
)
)
)
(
wral
(
λ x6 .
wa
(
wceq
(
cfv
(
cfv
(
cv
x6
)
(
cfv
(
cv
x1
)
ccid
)
)
(
co
(
cv
x6
)
(
cv
x6
)
(
cv
x4
)
)
)
(
cfv
(
cfv
(
cv
x6
)
(
cv
x3
)
)
(
cfv
(
cv
x2
)
ccid
)
)
)
(
wral
(
λ x7 .
wral
(
λ x8 .
wral
(
λ x9 .
wral
(
λ x10 .
wceq
(
cfv
(
co
(
cv
x10
)
(
cv
x9
)
(
co
(
cop
(
cv
x6
)
(
cv
x7
)
)
(
cv
x8
)
(
cfv
(
cv
x1
)
cco
)
)
)
(
co
(
cv
x6
)
(
cv
x8
)
(
cv
x4
)
)
)
(
co
(
cfv
(
cv
x10
)
(
co
(
cv
x7
)
(
cv
x8
)
(
cv
x4
)
)
)
(
cfv
(
cv
x9
)
(
co
(
cv
x6
)
(
cv
x7
)
(
cv
x4
)
)
)
(
co
(
cop
(
cfv
(
cv
x6
)
(
cv
x3
)
)
(
cfv
(
cv
x7
)
(
cv
x3
)
)
)
(
cfv
(
cv
x8
)
(
cv
x3
)
)
(
cfv
(
cv
x2
)
cco
)
)
)
)
(
λ x10 .
co
(
cv
x7
)
(
cv
x8
)
(
cfv
(
cv
x1
)
chom
)
)
)
(
λ x9 .
co
(
cv
x6
)
(
cv
x7
)
(
cfv
(
cv
x1
)
chom
)
)
)
(
λ x8 .
cv
x5
)
)
(
λ x7 .
cv
x5
)
)
)
(
λ x6 .
cv
x5
)
)
)
(
cfv
(
cv
x1
)
cbs
)
)
)
)
⟶
wceq
cidfu
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
csb
(
cfv
(
cv
x1
)
cbs
)
(
λ x2 .
cop
(
cres
cid
(
cv
x2
)
)
(
cmpt
(
λ x3 .
cxp
(
cv
x2
)
(
cv
x2
)
)
(
λ x3 .
cres
cid
(
cfv
(
cv
x3
)
(
cfv
(
cv
x1
)
chom
)
)
)
)
)
)
)
⟶
wceq
ccofu
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cop
(
ccom
(
cfv
(
cv
x1
)
c1st
)
(
cfv
(
cv
x2
)
c1st
)
)
(
cmpt2
(
λ x3 x4 .
cdm
(
cdm
(
cfv
(
cv
x2
)
c2nd
)
)
)
(
λ x3 x4 .
cdm
(
cdm
(
cfv
(
cv
x2
)
c2nd
)
)
)
(
λ x3 x4 .
ccom
(
co
(
cfv
(
cv
x3
)
(
cfv
(
cv
x2
)
c1st
)
)
(
cfv
(
cv
x4
)
(
cfv
(
cv
x2
)
c1st
)
)
(
cfv
(
cv
x1
)
c2nd
)
)
(
co
(
cv
x3
)
(
cv
x4
)
(
cfv
(
cv
x2
)
c2nd
)
)
)
)
)
)
⟶
wceq
cresf
(
cmpt2
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cvv
)
(
λ x1 x2 .
cop
(
cres
(
cfv
(
cv
x1
)
c1st
)
(
cdm
(
cdm
(
cv
x2
)
)
)
)
(
cmpt
(
λ x3 .
cdm
(
cv
x2
)
)
(
λ x3 .
cres
(
cfv
(
cv
x3
)
(
cfv
(
cv
x1
)
c2nd
)
)
(
cfv
(
cv
x3
)
(
cv
x2
)
)
)
)
)
)
⟶
wceq
cful
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wa
(
wbr
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
)
(
wral
(
λ x5 .
wral
(
λ x6 .
wceq
(
crn
(
co
(
cv
x5
)
(
cv
x6
)
(
cv
x4
)
)
)
(
co
(
cfv
(
cv
x5
)
(
cv
x3
)
)
(
cfv
(
cv
x6
)
(
cv
x3
)
)
(
cfv
(
cv
x2
)
chom
)
)
)
(
λ x6 .
cfv
(
cv
x1
)
cbs
)
)
(
λ x5 .
cfv
(
cv
x1
)
cbs
)
)
)
)
)
⟶
wceq
cfth
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
copab
(
λ x3 x4 .
wa
(
wbr
(
cv
x3
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
)
(
wral
(
λ x5 .
wral
(
λ x6 .
wfun
(
ccnv
(
co
(
cv
x5
)
(
cv
x6
)
(
cv
x4
)
)
)
)
(
λ x6 .
cfv
(
cv
x1
)
cbs
)
)
(
λ x5 .
cfv
(
cv
x1
)
cbs
)
)
)
)
)
⟶
wceq
cnat
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
cmpt2
(
λ x3 x4 .
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
(
λ x3 x4 .
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
(
λ x3 x4 .
csb
(
cfv
(
cv
x3
)
c1st
)
(
λ x5 .
csb
(
cfv
(
cv
x4
)
c1st
)
(
λ x6 .
crab
(
λ x7 .
wral
(
λ x8 .
wral
(
λ x9 .
wral
(
λ x10 .
wceq
(
co
(
cfv
(
cv
x9
)
(
cv
x7
)
)
(
cfv
(
cv
x10
)
(
co
(
cv
x8
)
(
cv
x9
)
(
cfv
(
cv
x3
)
c2nd
)
)
)
(
co
(
cop
(
cfv
(
cv
x8
)
(
cv
x5
)
)
(
cfv
(
cv
x9
)
(
cv
x5
)
)
)
(
cfv
(
cv
x9
)
(
cv
x6
)
)
(
cfv
(
cv
x2
)
cco
)
)
)
(
co
(
cfv
(
cv
x10
)
(
co
(
cv
x8
)
(
cv
x9
)
(
cfv
(
cv
x4
)
c2nd
)
)
)
(
cfv
(
cv
x8
)
(
cv
x7
)
)
(
co
(
cop
(
cfv
(
cv
x8
)
(
cv
x5
)
)
(
cfv
(
cv
x8
)
(
cv
x6
)
)
)
(
cfv
(
cv
x9
)
(
cv
x6
)
)
(
cfv
(
cv
x2
)
cco
)
)
)
)
(
λ x10 .
co
(
cv
x8
)
(
cv
x9
)
(
cfv
(
cv
x1
)
chom
)
)
)
(
λ x9 .
cfv
(
cv
x1
)
cbs
)
)
(
λ x8 .
cfv
(
cv
x1
)
cbs
)
)
(
λ x7 .
cixp
(
λ x8 .
cfv
(
cv
x1
)
cbs
)
(
λ x8 .
co
(
cfv
(
cv
x8
)
(
cv
x5
)
)
(
cfv
(
cv
x8
)
(
cv
x6
)
)
(
cfv
(
cv
x2
)
chom
)
)
)
)
)
)
)
)
⟶
wceq
cfuc
(
cmpt2
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ccat
)
(
λ x1 x2 .
ctp
(
cop
(
cfv
cnx
cbs
)
(
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
)
(
cop
(
cfv
cnx
chom
)
(
co
(
cv
x1
)
(
cv
x2
)
cnat
)
)
(
cop
(
cfv
cnx
cco
)
(
cmpt2
(
λ x3 x4 .
cxp
(
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
(
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
)
(
λ x3 x4 .
co
(
cv
x1
)
(
cv
x2
)
cfunc
)
(
λ x3 x4 .
csb
(
cfv
(
cv
x3
)
c1st
)
(
λ x5 .
csb
(
cfv
(
cv
x3
)
c2nd
)
(
λ x6 .
cmpt2
(
λ x7 x8 .
co
(
cv
x6
)
(
cv
x4
)
(
co
(
cv
x1
)
(
cv
x2
)
cnat
)
)
(
λ x7 x8 .
co
(
cv
x5
)
(
cv
x6
)
(
co
(
cv
x1
)
(
cv
x2
)
cnat
)
)
(
λ x7 x8 .
cmpt
(
λ x9 .
cfv
(
cv
x1
)
cbs
)
(
λ x9 .
co
(
cfv
(
cv
x9
)
(
cv
x7
)
)
(
cfv
(
cv
x9
)
(
cv
x8
)
)
(
co
(
cop
(
cfv
(
cv
x9
)
(
cfv
(
cv
x5
)
c1st
)
)
(
cfv
(
cv
x9
)
(
cfv
(
cv
x6
)
c1st
)
)
)
(
cfv
(
cv
x9
)
(
cfv
(
cv
x4
)
c1st
)
)
(
cfv
(
cv
x2
)
cco
)
)
)
)
)
)
)
)
)
)
)
⟶
wceq
cinito
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
crab
(
λ x2 .
wral
(
λ x3 .
weu
(
λ x4 .
wcel
(
cv
x4
)
(
co
(
cv
x2
)
(
cv
x3
)
(
cfv
(
cv
x1
)
chom
)
)
)
)
(
λ x3 .
cfv
(
cv
x1
)
cbs
)
)
(
λ x2 .
cfv
(
cv
x1
)
cbs
)
)
)
⟶
wceq
ctermo
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
crab
(
λ x2 .
wral
(
λ x3 .
weu
(
λ x4 .
wcel
(
cv
x4
)
(
co
(
cv
x3
)
(
cv
x2
)
(
cfv
(
cv
x1
)
chom
)
)
)
)
(
λ x3 .
cfv
(
cv
x1
)
cbs
)
)
(
λ x2 .
cfv
(
cv
x1
)
cbs
)
)
)
⟶
wceq
czeroo
(
cmpt
(
λ x1 .
ccat
)
(
λ x1 .
cin
(
cfv
(
cv
x1
)
cinito
)
(
cfv
(
cv
x1
)
ctermo
)
)
)
⟶
wceq
cdoma
(
ccom
c1st
c1st
)
⟶
x0
)
⟶
x0
Theorem
df_iso
:
wceq
ciso
(
cmpt
(
λ x0 .
ccat
)
(
λ x0 .
ccom
(
cmpt
(
λ x1 .
cvv
)
(
λ x1 .
cdm
(
cv
x1
)
)
)
(
cfv
(
cv
x0
)
cinv
)
)
)
(proof)
Theorem
df_subc
:
wceq
csubc
(
cmpt
(
λ x0 .
ccat
)
(
λ x0 .
cab
(
λ x1 .
wa
(
wbr
(
cv
x1
)
(
cfv
(
cv
x0
)
chomf
)
cssc
)
(
wsbc
(
λ x2 .
wral
(
λ x3 .
wa
(
wcel
(
cfv
(
cv
x3
)
(
cfv
(
cv
x0
)
ccid
)
)
(
co
(
cv
x3
)
(
cv
x3
)
(
cv
x1
)
)
)
(
wral
(
λ x4 .
wral
(
λ x5 .
wral
(
λ x6 .
wral
(
λ x7 .
wcel
(
co
(
cv
x7
)
(
cv
x6
)
(
co
(
cop
(
cv
x3
)
(
cv
x4
)
)
(
cv
x5
)
(
cfv
(
cv
x0
)
cco
)
)
)
(
co
(
cv
x3
)
(
cv
x5
)
(
cv
x1
)
)
)
(
λ x7 .
co
(
cv
x4
)
(
cv
x5
)
(
cv
x1
)
)
)
(
λ x6 .
co
(
cv
x3
)
(
cv
x4
)
(
cv
x1
)
)
)
(
λ x5 .
cv
x2
)
)
(
λ x4 .
cv
x2
)
)
)
(
λ x3 .
cv
x2
)
)
(
cdm
(
cdm
(
cv
x1
)
)
)
)
)
)
)
(proof)
Theorem
df_resf
:
wceq
cresf
(
cmpt2
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cvv
)
(
λ x0 x1 .
cop
(
cres
(
cfv
(
cv
x0
)
c1st
)
(
cdm
(
cdm
(
cv
x1
)
)
)
)
(
cmpt
(
λ x2 .
cdm
(
cv
x1
)
)
(
λ x2 .
cres
(
cfv
(
cv
x2
)
(
cfv
(
cv
x0
)
c2nd
)
)
(
cfv
(
cv
x2
)
(
cv
x1
)
)
)
)
)
)
(proof)
Theorem
df_fuc
:
wceq
cfuc
(
cmpt2
(
λ x0 x1 .
ccat
)
(
λ x0 x1 .
ccat
)
(
λ x0 x1 .
ctp
(
cop
(
cfv
cnx
cbs
)
(
co
(
cv
x0
)
(
cv
x1
)
cfunc
)
)
(
cop
(
cfv
cnx
chom
)
(
co
(
cv
x0
)
(
cv
x1
)
cnat
)
)
(
cop
(
cfv
cnx
cco
)
(
cmpt2
(
λ x2 x3 .
cxp
(
co
(
cv
x0
)
(
cv
x1
)
cfunc
)
(
co
(
cv
x0
)
(
cv
x1
)
cfunc
)
)
(
λ x2 x3 .
co
(
cv
x0
)
(
cv
x1
)
cfunc
)
(
λ x2 x3 .
csb
(
cfv
(
cv
x2
)
c1st
)
(
λ x4 .
csb
(
cfv
(
cv
x2
)
c2nd
)
(
λ x5 .
cmpt2
(
λ x6 x7 .
co
(
cv
x5
)
(
cv
x3
)
(
co
(
cv
x0
)
(
cv
x1
)
cnat
)
)
(
λ x6 x7 .
co
(
cv
x4
)
(
cv
x5
)
(
co
(
cv
x0
)
(
cv
x1
)
cnat
)
)
(
λ x6 x7 .
cmpt
(
λ x8 .
cfv
(
cv
x0
)
cbs
)
(
λ x8 .
co
(
cfv
(
cv
x8
)
(
cv
x6
)
)
(
cfv
(
cv
x8
)
(
cv
x7
)
)
(
co
(
cop
(
cfv
(
cv
x8
)
(
cfv
(
cv
x4
)
c1st
)
)
(
cfv
(
cv
x8
)
(
cfv
(
cv
x5
)
c1st
)
)
)
(
cfv
(
cv
x8
)
(
cfv
(
cv
x3
)
c1st
)
)
(
cfv
(
cv
x1
)
cco
)
)
)
)
)
)
)
)
)
)
)
(proof)
Theorem
df_doma
:
wceq
cdoma
(
ccom
c1st
c1st
)
(proof)
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