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concordium
smartcontractinteractions
Commits
de0c8b0f
Commit
de0c8b0f
authored
Apr 01, 2019
by
Jakob Botsch Nielsen
Browse files
Formalize and prove some circulation results
parent
59b60edd
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src/Circulation.v
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de0c8b0f
(
*
In
this
file
we
prove
various
results
about
the
circulation
of
coins
in
any
chain
implementing
a
chain
type
.
More
specifically
,
we
show
that
the
circulation
does
not
change
during
execution
of
blocks
.
This
is
proven
under
the
(
implicit
)
assumption
that
the
address
space
is
finite
.
*
)
From
Coq
Require
Import
List
Permutation
.
From
SmartContracts
Require
Import
Automation
Blockchain
Extras
.
Import
ListNotations
.
Section
Circulation
.
Context
{
Chain
:
ChainType
}
.
(
*
First
,
we
must
define
the
circulation
.
We
will
define
it
as
a
finite
sum
of
the
balance
over
all
valid
addresses
.
This
is
where
the
assumption
about
finite
address
space
comes
into
play
.
With
some
work
this
result
could
be
improved
to
assume
only
that
the
set
of
addresses
which
has
seen
transactions
is
finite
.
*
)
Definition
IsSetOfAllAddresses
(
addrs
:
list
Address
)
:
Prop
:=
NoDup
addrs
/
\
forall
a
,
valid_address
a
>
In
a
addrs
.
Context
{
addrs
:
list
Address
}
(
all_addrs
:
IsSetOfAllAddresses
addrs
).
Definition
circulation
(
chain
:
Chain
)
:=
sumZ
(
account_balance
chain
)
addrs
.
(
*
We
then
prove
that
over
any
single
step
,
the
circulation
is
preserved
.
The
idea
behind
this
proof
is
that
all_addrs
contain
from
and
to
so
we
can
move
them
to
the
beginning
of
the
sum
and
it
easily
follows
that
the
sum
of
their
balances
is
the
same
as
before
.
For
the
rest
of
the
list
the
total
balance
will
then
not
be
affected
which
follows
by
induction
.
*
)
Lemma
transfer_reorganize
{
pre
post
:
Environment
}
{
act
:
InternalAction
}
(
transfer
:
Transfer
act
pre
post
)
:
exists
suf
,
Permutation
([
iact_from
act
;
iact_to
act
]
++
suf
)
addrs
.
Proof
.
pose
proof
(
valid_address_from
transfer
).
pose
proof
(
valid_address_to
transfer
).
pose
proof
(
from_neq_to
transfer
).
unfold
IsSetOfAllAddresses
in
*
.
apply
(
NoDup_incl_reorganize
_
([
iact_from
act
;
iact_to
act
]));
repeat
constructor
;
unfold
incl
;
prove
.
Qed
.
Lemma
act_circulation_unchanged
{
pre
post
:
Environment
}
{
act
:
InternalAction
}
(
transfer
:
Transfer
act
pre
post
)
:
circulation
post
=
circulation
pre
.
Proof
.
unfold
circulation
.
destruct
(
transfer_reorganize
transfer
)
as
[
suf
perm
].
apply
Permutation_sym
in
perm
.
rewrite
2
!
(
sumZ_permutation
perm
);
prove
.
rewrite
(
account_balance_from_pre_post
transfer
).
rewrite
(
account_balance_to_pre_post
transfer
).
enough
(
sumZ
(
account_balance
pre
)
suf
=
sumZ
(
account_balance
post
)
suf
)
by
prove
.
destruct
all_addrs
as
[
addrs_set
all_addrs_in
].
pose
proof
(
Permutation_NoDup
perm
addrs_set
)
as
perm_set
.
assert
(
from_not_in_suf
:
~
In
(
iact_from
act
)
suf
).
{
apply
(
in_NoDup_app
_
[
iact_from
act
;
iact_to
act
]
_
);
prove
.
}
assert
(
to_not_in_suf
:
~
In
(
iact_to
act
)
suf
).
{
apply
(
in_NoDup_app
_
[
iact_from
act
;
iact_to
act
]
_
);
prove
.
}
clear
perm
perm_set
.
pose
proof
(
account_balance_unrelated_pre_post
transfer
).
induction
suf
as
[

x
xs
IH
];
prove
.
Qed
.
Hint
Resolve
act_circulation_unchanged
.
Lemma
step_circulation_unchanged
{
pre
post
:
Environment
}
{
x
y
:
list
InternalAction
}
(
step
:
ChainStep
pre
post
x
y
)
:
circulation
post
=
circulation
pre
.
Proof
.
destruct
step
;
eauto
.
Qed
.
Hint
Resolve
step_circulation_unchanged
.
(
*
Finally
,
we
get
the
result
over
block
traces
by
a
simple
induction
.
*
)
Lemma
block_trace_circulation_unchanged
{
pre
post
:
Environment
}
{
x
y
:
list
InternalAction
}
(
trace
:
BlockTrace
pre
post
x
y
)
:
circulation
post
=
circulation
pre
.
Proof
.
induction
trace
;
prove
;
eauto
.
Qed
.
End
Circulation
.
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