Documentation
Pdl
.
Beth
Search
return to top
source
Imports
Init
Pdl.Interpolation
Imported by
Formula
.
impDef
Formula
.
expDef
beth
Beth Definability (Corollary 7.5)
#
source
def
Formula
.
impDef
(
φ
:
Formula
)
(
p
:
ℕ
)
:
Prop
Equations
φ
.
impDef
p
=
∀ (
p0
p1
:
ℕ
),
repl_in_F
p
(
·
p0
)
φ
⋀
repl_in_F
p
(
·
p1
)
φ
⊨
~
(
·
p0
⋀
~
·
p1
)
⋀
~
(
·
p1
⋀
~
·
p0
)
Instances For
source
def
Formula
.
expDef
(
ψ
:
Formula
)
(
p
:
ℕ
)
(
φ
:
Formula
)
:
Prop
Equations
ψ
.
expDef
p
φ
=
(
ψ
.
voc
⊆
φ
.
voc
\
{
Sum.inl
p
}
∧
φ
⊨
~
(
·
p
⋀
~
ψ
)
⋀
~
(
ψ
⋀
~
·
p
))
Instances For
source
theorem
beth
{
p
:
ℕ
}
(
φ
:
Formula
)
(
h
:
φ
.
impDef
p
)
:
∃ (
ψ
:
Formula
),
ψ
.
expDef
p
φ
For any implicit definition there exists an explicit one.