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Prop.v
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Prop.v
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(*
This file is part of the verified smart contract project of SECBIT Labs.
Copyright 2018 SECBIT Labs
This program is free software: you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public License
as published by the Free Software Foundation, either version 3 of
the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
*)
Require Export Lists.List.
Require Import Model.
Require Import Spec.
Require Export Arith.
(*
High-level properties implied by the ERC20 spec in Spec.v.
1) Fixed total supply: in any step of any execution, the sum of all balances
always equal to totalSupply:
Theorem Property_totalSupply_fixed :
forall env0 env msg ml C E C' E',
create env0 msg C E
-> env_step env0 env
-> run env C ml C' E'
-> Sum (st_balances (w_st C')) (st_totalSupply (w_st C')).
*)
(* Definition of sum of mapping *)
Inductive Sum : (@tmap address value) -> value -> Prop :=
| Sum_emp : Sum tmap_emp 0
| Sum_add : forall m v a' v',
Sum m v
-> m a' = 0
-> Sum (m $+ {a' <- v'}) (v + v')
| Sum_del : forall m v a',
Sum m v
-> Sum (m $+ {a' <- 0}) (v - (m a')).
Lemma address_dec : forall (a1 a2: address),
{a1 = a2} + {a1 <> a2}.
Proof.
intros.
remember (beq a1 a2) as Ha.
assert (beq a1 a2 = Ha). auto.
destruct Ha.
beq_elimH H. left. apply H.
right.
simplbeq.
trivial.
Qed.
Lemma Sum_dec2 : forall m t a,
Sum m t
-> Sum (m $+ {a <- -= m a}) (t - m a).
Proof.
unfold a2v_upd_dec.
intros.
assert (Ht : minus_with_underflow (m a) (m a) = 0).
assert (Ht1 : 0 = (m a) - (m a)).
auto with arith.
rewrite Ht1.
apply minus_safe; auto.
rewrite Ht.
apply Sum_del; trivial.
Qed.
Fixpoint sum (m: @tmap address value) (al: list address) : value :=
match al with
| nil => 0
| cons a al' => (m a) + sum m al'
end.
Open Scope list_scope.
Section List.
Context `{A: Type}.
Context `{BEq A}.
Fixpoint list_in (a: A) (al: list A) : bool :=
match al with
| nil => false
| cons a' al' => if beq a a' then true
else list_in a al'
end.
Fixpoint no_repeat (al: list A) : bool :=
match al with
| nil => true
| cons a' al' => andb (negb (list_in a' al')) (no_repeat al')
end.
End List.
Opaque beq.
Lemma sum_emp : forall al,
sum $0 al = 0.
Proof.
intros.
induction al.
simpl. trivial.
simpl. apply IHal.
Qed.
Lemma sum_add_cons : forall (al : list address) m (a: address),
list_in a al = false
-> no_repeat al = true
-> m a + sum m al = sum m (a :: al).
Proof.
induction al.
intros m a Hin Hnr.
simpl.
trivial.
intros m a' Hin' Hnr'.
assert (Hnin : list_in a' al = false).
simpl in Hin'.
decbeq a' a.
trivial.
substH IHal with (IHal m a' Hnin).
simpl.
simpl in IHal.
omega.
Qed.
Lemma sum_del_none : forall al m a,
list_in a al = false
-> no_repeat al = true
-> sum (m $+ {a <- 0}) al = sum m al.
Proof.
induction al.
intros m a Hin Hnr.
simpl.
trivial.
intros m a' Hin' Hnr.
simpl in Hin'.
simpl.
decbeq a a'; tmap_simpl.
simpl in Hnr.
desb Hnr as [Hnr1 Hnr2].
rewrite (IHal m a' Hin' Hnr2).
trivial.
Qed.
Lemma sum_del_any : forall al m a,
list_in a al = true
-> no_repeat al = true
-> m a + sum (m $+ {a <- 0}) al = sum m al.
Proof.
induction al.
intros m a Hin Hnr.
simpl in Hin.
discriminate.
intros m a' Hin' Hnr.
simpl in Hin'.
destruct (beq_dec a a').
simplbeq.
simpl.
simpl in Hnr.
desb Hnr as [Hnr1 Hnr2].
simpltm.
assert (a = a').
beq_elimH H.
trivial.
subst a'.
simplb.
rewrite sum_del_none; trivial.
simpl.
simplbeq.
tmap_simpl.
simpl in Hnr.
desb Hnr as [Hnr1 Hnr2].
rewrite <- (IHal m a' Hin' Hnr2).
omega.
Qed.
Lemma minus_minus: forall t a b,
t - a - b = t - (a + b).
Proof.
intros.
omega.
Qed.
Lemma sum_add_not_in : forall al m a v,
list_in a al = false
-> no_repeat al = true
-> sum (m $+ {a <- v}) al = sum m al.
Proof.
induction al.
intros m a v Hin Hnr.
simpl.
trivial.
intros m a' v' Hin' Hnr.
simpl in Hin'.
simpl in Hnr.
desb Hnr as [Hnr1 Hnr2].
simplb.
decbeq a a'; tmap_simpl.
simpl.
simpltm.
Qed.
Lemma sum_add_in : forall al m a v,
list_in a al = true
-> no_repeat al = true
-> m a = 0
-> sum (m $+ {a <- v}) al = sum m al + v.
Proof.
induction al.
intros m a v Hin Hnr Hma.
simpl in Hin.
discriminate.
intros m a' v' Hin' Hnr Hma.
simpl in Hin'.
simpl in Hnr.
desb Hnr as [Hnr1 Hnr2].
decbeq a a'; simpl; simpltm.
simplb.
beq_elimH Hb.
subst a'.
rewrite sum_add_not_in; auto.
rewrite Hma; simpl; trivial.
auto with arith.
simplb.
rewrite (IHal m a' v' Hin' Hnr2 Hma).
auto with arith.
Qed.
Lemma Sum_ge_strong : forall m t,
Sum m t
-> forall a al,
list_in a al = false
-> no_repeat al = true
-> t >= m a + sum m al.
Proof.
intros m t H.
induction H.
- intros a al Hal Hnr.
simpltm.
rewrite sum_emp.
auto with arith.
- intros a al Hal Hnr.
decbeq a a'.
simpltm.
substH IHSum with (IHSum a al Hal Hnr).
beq_elimH Hb.
subst a'.
rewrite H0 in IHSum.
rewrite (sum_add_not_in _ _ _ _ Hal Hnr).
omega.
simpltm.
substH IHSum with (IHSum a al Hal Hnr).
assert (Hx: list_in a' al = true \/ list_in a' al = false).
destruct (list_in a' al); [left | right]; trivial.
destruct Hx as [Hx | Hx].
rewrite (sum_add_in _ _ _ _ Hx Hnr H0).
omega.
rewrite (sum_add_not_in _ _ _ _ Hx Hnr).
omega.
- intros a al' Hnin Hnr.
decbeq a a'.
tmap_simpl.
beq_elimH Hb.
subst a'.
rewrite (sum_add_not_in _ _ _ _ Hnin Hnr).
substH IHSum with (IHSum a al' Hnin Hnr).
omega.
assert (Hx: list_in a' al' = true \/ list_in a' al' = false).
destruct (list_in a' al'); [left | right]; trivial.
destruct Hx as [Hx | Hx].
tmap_simpl.
assert (Hy:=sum_del_any al' m a' Hx Hnr).
substH IHSum with (IHSum a al' Hnin Hnr).
rewrite <- Hy in IHSum.
assert (Hxx: forall a b c d,
a >= b + (c + d)
-> a - c >= b + d).
clear.
intros.
omega.
apply Hxx; trivial.
tmap_simpl.
rewrite sum_del_none; auto.
assert (Hy: v >= m a + sum m (a' :: al')).
apply IHSum; trivial.
simpl.
simplbeq.
trivial.
simpl.
rewrite Hx.
simpl.
trivial.
simpl in Hy.
omega.
Qed.
Lemma Sum_ge : forall m a t,
Sum m t
-> t >= m a.
Proof.
intros.
assert (Hx:= Sum_ge_strong _ _ H a nil).
simpl in Hx.
substH Hx with (Hx (eq_refl _) (eq_refl _)).
omega.
Qed.
Lemma Sum_ge_2 : forall m a a' t,
Sum m t
-> beq a a' = false
-> t >= m a + m a'.
Proof.
intros.
assert (Hx:= Sum_ge_strong _ _ H a (a'::nil)).
simpl in Hx.
rewrite H0 in Hx.
assert (m a + m a' =m a + (m a' + 0)). omega.
rewrite <- H1 in Hx.
apply Hx.
trivial. trivial.
Qed.
Lemma Sum_sig :
forall m a t,
m = $0 $+ { a <- t }
-> Sum m t.
Proof.
intros m a t Hm.
rewrite Hm.
assert (t = 0 + t).
auto with arith.
rewrite H at 2.
constructor 2.
constructor 1.
simpltm.
trivial.
Qed.
Ltac arith_rewrite t :=
let H := fresh "Harith" in
match t with
| ?x = ?y => assert (H: t); [auto with arith; try omega | rewrite H; clear H]
end.
Lemma Sum_dec : forall m t a (v: value),
Sum m t
-> m a >= v
-> Sum (m $+ {a <- -= v}) (t - v).
Proof.
unfold a2v_upd_dec.
intros m t a v H.
generalize dependent v.
generalize dependent a.
induction H.
+ intros.
simpl in H.
assert (v = 0).
unfold value in * .
omega.
simpl.
assert ($0 $+ {a <- (0:value)} = $0).
simpl.
apply tmap_extensionality.
intro a'.
decbeq a a'.
rewrite (tmap_get_upd_eq2 $0 a a' (0:value)); auto.
tmap_simpl.
assert(Ht: minus_with_underflow 0 v = 0 - v).
apply minus_safe; trivial.
rewrite Ht.
rewrite H0.
assert(Hm: 0 - 0 = 0);auto.
rewrite Hm.
rewrite H1.
constructor 1.
+ intros a v2 H1.
decbeq a a'.
tmap_simpl.
beq_elimH Hb.
subst a'.
assert(Ht: minus_with_underflow v' v2 = v' - v2).
apply minus_safe; trivial.
rewrite Ht.
arith_rewrite (v + v' - v2 = v + (v' - v2)).
constructor 2; trivial.
simpltm.
rewrite (tmap_upd_upd_ne); simplbeq; auto.
assert (Hx : v + v' - v2 = v - v2 + v').
assert (v >= m a). apply Sum_ge. apply H.
assert (v >= v2). omega.
unfold value in *. omega.
rewrite Hx.
constructor; trivial.
apply IHSum; trivial.
simpltm.
+ intros a v0 H1.
decbeq a a'.
simpltm.
assert (v0 = 0).
unfold value in *. omega.
subst v0.
simpl.
simpltm.
beq_elimH Hb.
subst a'.
assert (Hx: v - m a - 0 = v - m a).
unfold value in *. omega.
rewrite Hx.
constructor; trivial.
simpltm.
rewrite (tmap_upd_upd_ne); simplbeq; auto.
assert (Hx: v - m a' - v0 = v - v0 - m a').
unfold value in *. omega.
rewrite Hx.
assert (Hm: m a' = (m $+ {a <- minus_with_underflow (m a) v0}) a').
simpltm.
rewrite Hm.
constructor 3.
apply IHSum; trivial.
Qed.
Lemma Sum_inc : forall m t a (v: value),
Sum m t
-> m a <= MAX_UINT256 - v
-> Sum (m $+ {a <- += v}) (t + v).
Proof.
unfold a2v_upd_inc.
intros m t a v H Hlt.
generalize dependent v.
generalize dependent a.
induction H.
+ intros.
simpltm.
assert (plus_with_overflow TMap.zero v = v).
apply plus_safe_lhs0; auto.
rewrite H.
constructor 2; auto; try constructor.
+ intros a v2 Hlt.
decbeq a a'.
tmap_simpl.
beq_elimH Hb.
subst a'.
arith_rewrite (v + v' + v2 = v + (v' + v2)).
rewrite (plus_safe_lt v' v2); trivial.
constructor 2; trivial.
simpltm.
substH IHSum with (IHSum a v2).
rewrite (tmap_upd_upd_ne); simplbeq; auto.
arith_rewrite (v + v' + v2 = v + v2 + v').
constructor 2; auto.
simpltm.
+ intros a v0.
decbeq a a'.
beq_elimH Hb.
subst a'.
assert (plus_with_overflow ((m $+ {a <- 0}) a) v0 = v0).
rewrite tmap_get_upd_eq.
rewrite plus_safe_lhs0; trivial.
rewrite H0.
constructor 2.
constructor 3.
trivial.
simpltm.
simpltm.
rewrite (tmap_upd_upd_ne); simplbeq; auto.
assert (Hx: v - m a' + v0 = v + v0 - m a').
assert (Hy: v >= m a').
apply (Sum_ge m a' v); trivial.
omega.
rewrite Hx.
assert (m a' = m $+ {a <- plus_with_overflow (m a) v0} a').
simpltm.
rewrite H0.
constructor 3.
apply IHSum; trivial.
Qed.
Lemma a2v_upd_inc_zero : forall m a,
m $+ {a <- += 0} = m.
Proof.
unfold a2v_upd_inc.
intros.
rewrite(plus_safe_rhs0 (m a) 0); trivial.
assert (Hx : m a + 0 = m a).
unfold value in *. omega.
rewrite Hx.
tmap_simpl.
Qed.
Lemma a2v_upd_dec_zero : forall m a,
m $+ {a <- -= 0} = m.
Proof.
unfold a2v_upd_dec.
intros.
rewrite(minus_safe (m a) 0).
assert (Hx : m a - 0 = m a).
unfold value in *. omega.
rewrite Hx.
tmap_simpl.
omega.
Qed.
Lemma test : forall a b,
a >= b -> a = a - b + b.
Proof.
intros.
omega.
Qed.
Lemma Sum_transfer : forall m t a1 a2 v m',
Sum m t
-> m a1 >= v
-> m a2 <= MAX_UINT256 - v
-> m' = m $+{a1 <- -= v} $+{a2 <- += v}
-> Sum m' t.
Proof.
intros.
decbeq a1 a2.
+ beq_elimH Hb.
subst a2.
rewrite H2.
assert (Ht: t = t - v + v).
assert (Ht1 : t >= m a1).
apply Sum_ge; trivial.
assert (Ht2 : t >= v).
omega.
clear - Ht2.
unfold value in t.
eapply test; eauto.
rewrite Ht.
eapply Sum_inc; eauto.
eapply Sum_dec; eauto.
unfold a2v_upd_dec.
rewrite (tmap_get_upd_eq m a1 _).
rewrite(minus_safe (m a1) v); auto.
omega.
+ remember (m a1) as Ha1.
remember (m a2) as Ha2.
rewrite HeqHa1 in H0.
destruct Ha1.
assert (Hv : v = 0).
rewrite <- HeqHa1 in H0.
auto with arith.
subst v.
rewrite a2v_upd_dec_zero in H2.
rewrite a2v_upd_inc_zero in H2.
rewrite H2; trivial.
assert (Hx := Sum_ge _ a1 _ H).
assert (Ht : t = t - v - (m $+ {a1 <- -= v} a2) + (m $+ {a1 <- -= v} a2) + v).
unfold a2v_upd_dec.
simpltm.
assert (Ht2 : t >= v).
omega.
assert (Ht3: t >= m a1 + m a2).
apply Sum_ge_2; trivial.
rewrite minus_minus.
arith_rewrite (t - (v + m a2) + m a2 + v = t - (v + m a2) + (v + m a2)).
rewrite <- test; trivial.
omega.
rewrite Ht.
rewrite H2.
unfold a2v_upd_dec in * .
unfold a2v_upd_inc in * .
apply Sum_inc.
simpltm.
assert (Ht3: t >= m a1 + m a2).
apply Sum_ge_2; trivial.
arith_rewrite (t - v - m a2 + m a2 = t - v).
apply Sum_dec; trivial.
simpltm.
omega.
Qed.
Definition assert_genesis_event (e: event) (E: eventlist) : Prop :=
match E with
| nil => False
| cons e' E => e = e'
end.
Lemma assert_genesis_event_app : forall e E E',
assert_genesis_event e E
-> assert_genesis_event e (E ++ E').
Proof.
intros.
destruct E.
+ simpl in H. inversion H.
+ simpl in H. auto.
Qed.
(* Invariant *)
Definition INV (env: env) (S: state) (E: eventlist) : Prop :=
let blncs := st_balances S in
(* balances not overflow *)
(forall a, blncs a <= MAX_UINT256) /\
(* totalSupply preserves *)
exists total,
total = st_totalSupply S
/\ Sum blncs total
/\ exists creator, exists name, exists decimals, exists sym,
assert_genesis_event (ev_EIP20 creator total name decimals sym) E.
(* step evaluation maintains invariant *)
Theorem step_INV: forall this env msg S E env' S' E',
step env (mk_contract this S) msg (mk_contract this S') E'
-> env_step env env'
-> INV env S E
-> INV env' S' (E ++ E').
Proof.
intros this env msg S E env' S' E'.
intros Hstep Henv' HI.
destruct HI as [Hblncs [total [Htotal [Htv [creator [name [decimals [sym Hassert]]]]]]]].
inversion_clear Hstep.
(* case: transfer *)
- unfold funcspec_transfer in H1.
subst spec preP evP postP.
simpl in *.
subst msg.
simpl in H5.
destruct H5 as [[Hx1a Hx1b] [Hx2 [Hx3 [Hx4 [Hx5 [Hx6 [Hx7 Hx8]]]]]]].
destruct Hx1b.
+ (* msg.sender == to *)
subst sender.
rewrite <- (a2v_dec_inc_id _ _ _ (Hblncs to) Hx1a) in Hx7.
rewrite <- Hx7 in *.
split; auto.
exists total.
repeat split; subst; auto.
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
+ (* msg.sender != to *)
destruct H0 as [Hsender Hof].
split.
* (* no overflow *)
rewrite Hx7.
intros.
destruct (beq_dec a to).
{
(* a == to *)
rewrite Nat.eqb_eq in H0.
subst a.
apply neq_beq_false in Hsender.
unfold a2v_upd_inc, a2v_upd_dec.
rewrite (tmap_upd_upd_ne _ _ _ _ _ Hsender).
rewrite (tmap_get_upd_ne _ _ _ _ Hsender).
rewrite (tmap_get_upd_eq _ _ _).
rewrite (tmap_get_upd_ne _ _ _ _ Hsender).
rewrite (plus_safe_lt _ _ Hof).
generalize(Hblncs sender). intros. omega.
}
{
(* a <> to *)
apply beq_sym in H0.
unfold a2v_upd_inc, a2v_upd_dec.
rewrite (tmap_get_upd_ne _ _ _ _ H0).
destruct (beq_dec a sender).
- (* a == sender *)
rewrite Nat.eqb_eq in H1.
subst a.
rewrite (tmap_get_upd_eq _ _ _).
rewrite (minus_safe _ _ Hx1a).
generalize (Hblncs sender). intros. omega.
- (* a <> sender *)
apply beq_sym in H1.
rewrite (tmap_get_upd_ne _ _ _ _ H1).
generalize (Hblncs a). intros. omega.
}
* (* totalSupply preserves *)
exists total.
rewrite <- Htotal in *.
repeat split; auto.
{
apply (Sum_transfer (st_balances S) total
sender to v (st_balances S'));
auto with arith.
}
{
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
}
(* case: transferFrom-1 *)
- unfold funcspec_transferFrom_1 in H1.
subst spec.
simpl in *.
destruct H2 as [[Hx1a [Hx1b [Hx1c Hx1d]]] [Hx2 [Hx3 [Hx4 [Hx5 [Hx6 [Hx7 Hx8]]]]]]].
destruct Hx1b.
+ (* from = to *)
subst from.
rewrite <- (a2v_dec_inc_id _ _ _ (Hblncs to) Hx1a) in Hx7.
rewrite <- Hx7 in *.
split; auto.
exists total.
repeat split; subst; auto.
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
+ (* from <> to *)
destruct H1 as [Hfrom Hof].
split.
* (* no overflow *)
rewrite Hx7.
intros.
destruct (beq_dec a to).
{
(* a == to *)
rewrite Nat.eqb_eq in H1.
subst a.
apply neq_beq_false in Hfrom.
unfold a2v_upd_inc, a2v_upd_dec.
rewrite (tmap_upd_upd_ne _ _ _ _ _ Hfrom).
rewrite (tmap_get_upd_ne _ _ _ _ Hfrom).
rewrite (tmap_get_upd_eq _ _ _).
rewrite (tmap_get_upd_ne _ _ _ _ Hfrom).
rewrite (plus_safe_lt _ _ Hof).
generalize(Hblncs from). intros. omega.
}
{
(* a <> to *)
apply beq_sym in H1.
unfold a2v_upd_inc, a2v_upd_dec.
rewrite (tmap_get_upd_ne _ _ _ _ H1).
destruct (beq_dec a from).
- (* a == from *)
rewrite Nat.eqb_eq in H2.
subst a.
rewrite (tmap_get_upd_eq _ _ _).
rewrite (minus_safe _ _ Hx1a).
generalize (Hblncs from). intros. omega.
- (* a <> sender *)
apply beq_sym in H2.
rewrite (tmap_get_upd_ne _ _ _ _ H2).
generalize (Hblncs a). intros. omega.
}
* (* totalSupply preserves *)
exists total.
rewrite <- Htotal in *.
repeat split; auto.
{
apply (Sum_transfer (st_balances S) total
from to v (st_balances S'));
auto with arith.
}
{
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
}
(* case: transferFrom-2 *)
- unfold funcspec_transferFrom_2 in H1.
subst spec.
simpl in *.
destruct H2 as [[Hx1a [Hx1b [Hx1c Hx1d]]] [Hx2 [Hx3 [Hx4 [Hx5 [Hx6 [Hx7 Hx8]]]]]]].
destruct Hx1b.
+ (* from = to *)
subst from.
rewrite <- (a2v_dec_inc_id _ _ _ (Hblncs to) Hx1a) in Hx7.
rewrite <- Hx7 in *.
split; auto.
exists total.
repeat split; subst; auto.
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
+ (* from <> to *)
destruct H1 as [Hfrom Hof].
split.
* (* no overflow *)
rewrite Hx7.
intros.
destruct (beq_dec a to).
{
(* a == to *)
rewrite Nat.eqb_eq in H1.
subst a.
apply neq_beq_false in Hfrom.
unfold a2v_upd_inc, a2v_upd_dec.
rewrite (tmap_upd_upd_ne _ _ _ _ _ Hfrom).
rewrite (tmap_get_upd_ne _ _ _ _ Hfrom).
rewrite (tmap_get_upd_eq _ _ _).
rewrite (tmap_get_upd_ne _ _ _ _ Hfrom).
rewrite (plus_safe_lt _ _ Hof).
generalize(Hblncs from). intros. omega.
}
{
(* a <> to *)
apply beq_sym in H1.
unfold a2v_upd_inc, a2v_upd_dec.
rewrite (tmap_get_upd_ne _ _ _ _ H1).
destruct (beq_dec a from).
- (* a == from *)
rewrite Nat.eqb_eq in H2.
subst a.
rewrite (tmap_get_upd_eq _ _ _).
rewrite (minus_safe _ _ Hx1a).
generalize (Hblncs from). intros. omega.
- (* a <> sender *)
apply beq_sym in H2.
rewrite (tmap_get_upd_ne _ _ _ _ H2).
generalize (Hblncs a). intros. omega.
}
* (* totalSupply preserves *)
exists total.
rewrite <- Htotal in *.
repeat split; auto.
{
apply (Sum_transfer (st_balances S) total
from to v (st_balances S'));
auto with arith.
}
{
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
}
(* case: balanceOf *)
- unfold funcspec_balanceOf in H2.
subst spec.
simpl in *.
destruct H2 as [Hx1 [Hx2 Hx3]].
subst S.
split; auto.
exists total.
simpl.
repeat split; auto.
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
(* case: approve *)
- unfold funcspec_approve in H1.
subst spec.
simpl in *.
destruct H2 as [Hx1 [Hx2 [Hx3 [Hx4 [Hx5 [Hx6 [Hx7 Hx8]]]]]]].
rewrite <- Hx7 in *.
split; auto.
exists total.
rewrite Hx3 in *.
rewrite Hx7 in *.
repeat split; auto.
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
(* case: allowance *)
- unfold funcspec_allowance in H1.
subst spec.
simpl in *.
destruct H2 as [Hx1 [Hx2 Hx3]].
subst S'.
split; auto.
exists total.
simpl.
repeat split; auto.
exists creator. exists name. exists decimals. exists sym.
apply assert_genesis_event_app; auto.
Qed.
(* create evaluation maintains invariant *)
Theorem create_INV : forall env0 env msg C E,
create env0 msg C E
-> env_step env0 env
-> INV env (w_st C) E.
Proof.
intros.
inversion_clear H.
subst spec preP evP postP; simpl.
unfold funcspec_EIP20 in H7.
simpl in H7.
destruct H7 as [Hx1 [Hx2 [Hx3 [Hx4 [Hx5 [Hx6 Hx7]]]]]].
unfold INV.
split.
- (* no overflow initially *)
subst.
rewrite Hx6. clear Hx6. simpl.
intros a.
destruct (beq_dec sender a).
+ (* a = sender *)
apply Nat.eqb_eq in H. subst a.
rewrite (tmap_get_upd_eq _ _ _).
auto.
+ (* a <> sender *)
rewrite (tmap_get_upd_ne _ _ _ _ H).
rewrite (tmap_emp_zero _).
unfold TMap.zero.
unfold value_Range.
omega.
- (* totalSupply preserves *)
exists ia.
repeat split; auto.
+ apply Sum_sig in Hx6.
trivial.
+ exists sender. exists name. exists dec. exists sym.
unfold assert_genesis_event.
rewrite Hx1.
rewrite H1.
simpl.
trivial.
Qed.
Lemma step_contract_address_constant : forall env C msg C' E',
step env C msg C' E'
-> w_a C = w_a C'.
Proof.
intros.
destruct C as [a S].
destruct C' as [a' S'].
induction H; simpl; auto; intuition.
Qed.
Lemma steps_INV: forall ml env C E,
INV env (w_st C) E
-> forall env' C' E', steps env C ml env' C' E'
-> INV env' (w_st C') (E ++ E').
Proof.
induction ml.
- (* nil *)
intros.
inversion_clear H0.
destruct H2.
subst.
rewrite app_nil_r.
trivial.
- (* a :: ml *)
intros.
inversion_clear H0.
rename x into envx.
rename a into msg.
destruct H1 as [Cx [Ex [Ey [H1 [H2 [H3 H4]]]]]].
subst E'.
assert (Hx : INV envx (w_st Cx) (E ++ Ex)).
{
assert (w_a C = w_a Cx).
{
apply step_contract_address_constant with env msg Ex. apply H1.
}
destruct C as [C_a C_st].
destruct Cx as [Cx_a Cx_st].
simpl. simpl in H. simpl in H0. generalize H. generalize H4.
apply step_INV with C_a msg.
subst Cx_a. apply H1.
}
substH IHml with (IHml envx Cx (E ++ Ex) Hx).
rewrite app_assoc.
apply IHml; trivial.
Qed.
Lemma INV_implies_totalSupply_fixed :
forall env S E,
INV env S E
-> Sum (st_balances S) (st_totalSupply S).
Proof.
intros env S E HI.
unfold INV in HI.
destruct HI as [_ [total [Ht [HT HI]]]].
rewrite Ht in HT.
trivial.
Qed.
(* Prop #1: total supply is equal to sum of balances *)
Theorem Property_totalSupply_equal_to_sum_balances :
forall env0 env msg ml C E C' E',
create env0 msg C E
-> env_step env0 env
-> run env C ml C' E'
-> Sum (st_balances (w_st C')) (st_totalSupply (w_st C')).
Proof.
intros env0 env msg il C E C' E' Hc Hs Hr.
unfold run in Hr.
destruct Hr as [env' Hsteps].
apply INV_implies_totalSupply_fixed with env' (E++E').
substH Hc with (create_INV _ _ _ _ _ Hc Hs).
eapply steps_INV; eauto.
Qed.
(* Prop #2: total supply is fixed with transfer *)
Theorem Property_totalSupply_fixed_transfer:
forall env C C' E' msg to v spec preP evP postP,