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dnegmod
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855d84d3
Commit
855d84d3
authored
Apr 07, 2017
by
litak
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855d84d3
...
...
@@ 65,8 +65,8 @@ Notation "{ Z + G } [ A" := (KIbox Z G A) (at level 79).
(
**
*
Equivalence
Proofs
for
the
Translations
*
)
(
**
In
this
section
we
prove
that
the
given
three
translations
result
in
equivalent
formulas
.
We
will
later
refer
to
this
fact
as
"the triangle"
,
because
one
can
view
it
as
a
triangle
of
equivalent
formula
s
.
*
)
(
**
In
this
section
we
prove
that
the
regular
translations
result
in
equivalent
formulas
.
In
some
parts
of
the
proof
script
below
,
an
old
name
for
such
translations
as
being
"in the triangle"
persisted
.
This
name
one
can
see
the
regular
translations
as
lying
in
a
triangle
given
by
the
refinement
order
on
monotone
modular
one
s
.
*
)
Lemma
eq_ggr_kur
:
forall
Z
f
,
KIbox
Z
[]
((
ggr
f
)
<<>>
(
kur
f
)).
Proof
.
...
...
@@ 206,15 +206,15 @@ Fixpoint dis_free_under_box (f : form) : Prop :=
(
**
*
Towards
Enveloped
Implications
Theorem
*
)
(
**
This
sections
defines
KClbox
,
which
is
simply
an
extension
of
KIbox
by
the
axiom
([
+!
p
>>
p
]),
and
introduces
the
notion
of
adequa
teness
.
A
translation
for
a
logic
([
+!
p
>>
p
]),
and
introduces
the
notion
of
adequa
cy
.
A
translation
for
a
logic
defined
by
given
axiom
is
adequate
if
the
translation
of
a
formula
holds
intuitionistically
iff
the
original
formula
holds
classically
.
We
show
that
adequa
teness
is
equivalent
for
the
translations
in
the
triangle
and
that
adequa
cy
is
equivalent
for
the
translations
in
the
triangle
and
that
a
system
extended
by
an
axiom
is
adequate
iff
the
translation
of
the
axioms
holds
intuitionistically
.
After
that
,
we
introduce
the
notion
of
envelopes
to
capture
the
properties
,
axioms
are
required
to
have
,
in
order
to
ensure
adequa
teness
of
the
resulting
system
.
axioms
are
required
to
have
,
in
order
to
ensure
adequa
cy
of
the
resulting
system
.
Since
the
notion
of
an
envelope
is
abstract
,
the
next
part
shows
examples
of
envelopes
.
The
last
part
explains
which
kind
of
envelopes
result
in
an
adequate
system
.
*
)
...
...
@@ 264,7 +264,7 @@ Proof.
Qed
.
(
*
Definition
and
equivalence
of
adequa
teness
*
)
(
*
Definition
and
equivalence
of
adequa
cy
*
)
Definition
adequate
(
f
:
form
>
form
)
(
Z
:
form
)
:=
forall
G
A
,
KIbox
Z
(
map
f
G
)
(
f
A
)
<>
KClbox
Z
G
A
.
...
...
@@ 461,7 +461,7 @@ Qed.
(
**
**
Definitions
of
pre
/
post
/
¬¬

envelopes
for
KIbox
*
)
(
**
Now
we
define
different
kinds
of
envelopes
.
They
are
used
to
describe
the
required
properties
for
adequa
teness
in
the
next
parts
.
describe
the
required
properties
for
adequa
cy
in
the
next
parts
.
Note
the
small
difference
between
¬¬

and
Kuroda

envelopes
.
*
)
...
...
@@ 494,8 +494,8 @@ Qed.
(
**
**
Envelope
Criteria
Lemma
*
)
(
**
Following
are
concrete
examples
of
envelopes
.
Our
first
attempt
at
proving
adequa
teness
was
done
with
axioms
being
either
disjunction
free
([
dis_free_under_box
])
and
[
shallow
]
or
completely
[
box_free
].
It
turned
out
that
we
could
prove
adequa
teness
for
an
even
less
restrictive
class
of
logics
using
envelopes
.
Still
those
stronger
requirements
are
relevant
for
practical
purpose
and
we
will
now
show
that
our
results
apply
to
them
as
well
.
(
**
Following
are
concrete
examples
of
envelopes
.
Our
first
attempt
at
proving
adequa
cy
was
done
with
axioms
being
either
disjunction
free
([
dis_free_under_box
])
and
[
shallow
]
or
completely
[
box_free
].
It
turned
out
that
we
could
prove
adequa
cy
for
an
even
less
restrictive
class
of
logics
using
envelopes
.
Still
those
stronger
requirements
are
relevant
for
practical
purpose
and
we
will
now
show
that
our
results
apply
to
them
as
well
.
*
)
Lemma
sub_vars_dneg_eq
:
...
...
@@ 611,7 +611,7 @@ Qed.
(
**
**
Enveloped
Implications
Theorem
*
)
(
**
Axioms
of
the
form
[(
B
>>
C
)],
where
B
is
a
post

envelope
and
C
a
pre

envelope
,
form
a
adequate
system
.
Kuroda

and
¬¬

envelopes
also
guarantee
adequa
teness
.
form
a
adequate
system
.
Kuroda

and
¬¬

envelopes
also
guarantee
adequa
cy
.
*
)
Theorem
thenvimpl_kol
:
...
...
@@ 674,7 +674,7 @@ Qed.
(
**
We
introduce
the
Glivenko

translation
,
which
is
simply
prefixing
formulas
by
double
negation
.
Assuming
[

[]

+!
A
>>
+!

[]

A
],
which
is
known
as
the
Kuroda
axiom
,
the
Glivenko

translation
becomes
equivalent
to
those
in
the
triangle
.
We
can
thus
deduct
a
adequateness
result
from
it
as
well
,
becomes
equivalent
to
those
in
the
triangle
.
We
can
thus
deduct
a
n
adequacy
result
from
it
as
well
,
of
course
still
assuming
the
Kuroda
axiom
.
Additionally
,
we
prove
that
Kuroda
axiom
follows
from
the
axiom
[
p
>>

[]

p
].
The
final
theorem
states
that
a
translation
from
the
triangle
or
glv
and
an
axiom
[
Z
]
are
adequate
if
the
...
...
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