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Imp.v
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Imp.v
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Require Export SfLib.
Module AExp.
Inductive aexp : Type :=
| ANum : nat -> aexp
| APlus : aexp -> aexp -> aexp
| AMinus : aexp -> aexp -> aexp
| AMult : aexp -> aexp -> aexp.
Inductive bexp : Type :=
| BTrue : bexp
| BFalse : bexp
| BEq : aexp -> aexp -> bexp
| BLe : aexp -> aexp -> bexp
| BNot : bexp -> bexp
| BAnd : bexp -> bexp -> bexp.
Fixpoint aeval (e: aexp) : nat :=
match e with
| ANum n => n
| APlus a1 a2 => (aeval a1) + (aeval a2)
| AMinus a1 a2 => (aeval a1) - (aeval a2)
| AMult a1 a2 => (aeval a1) * (aeval a2)
end.
Example test_aeval1:
aeval (APlus (ANum 2) (ANum 2)) = 4.
Proof. reflexivity. Qed.
Fixpoint beval (e: bexp) : bool :=
match e with
| BTrue => true
| BFalse => false
| BEq a1 a2 => beq_nat (aeval a1) (aeval a2)
| BLe a1 a2 => ble_nat (aeval a1) (aeval a2)
| BNot b1 => negb (beval b1)
| BAnd b1 b2 => andb (beval b1) (beval b2)
end.
Fixpoint optimize_0plus (a:aexp) : aexp :=
match a with
| ANum n => ANum n
| APlus (ANum 0) e2 => optimize_0plus e2
| APlus e1 e2 => APlus (optimize_0plus e1) (optimize_0plus e2)
| AMinus e1 e2 => AMinus (optimize_0plus e1) (optimize_0plus e2)
| AMult e1 e2 => AMult (optimize_0plus e1) (optimize_0plus e2)
end.
Example test_optimize_0plus:
optimize_0plus (APlus (ANum 2)
(APlus (ANum 0)
(APlus (ANum 0) (ANum 1)))) =
APlus (ANum 2) (ANum 1).
Proof. reflexivity. Qed.
Theorem optimize_0plus_sound: forall e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e. induction e.
Case "ANum". reflexivity.
Case "APlus". destruct e1.
SCase "e1 = ANum n". destruct n.
SSCase "n = 0". simpl. apply IHe2.
SSCase "n <> 0". simpl. rewrite IHe2. reflexivity.
SCase "e1 = APlus e1_1 e1_2".
simpl. simpl in IHe1. rewrite IHe1. rewrite IHe2. reflexivity.
SCase "e1 = AMinus e1_1 e1_2".
simpl. simpl in IHe1. rewrite IHe1. rewrite IHe2. reflexivity.
SCase "e1 = AMult e1_1 e1_2".
simpl. simpl in IHe1. rewrite IHe1. rewrite IHe2. reflexivity.
Case "AMinus".
simpl. rewrite IHe1. rewrite IHe2. reflexivity.
Case "AMult".
simpl. rewrite IHe1. rewrite IHe2. reflexivity. Qed.
Theorem optimize_0plus_sound': forall e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e.
induction e;
try (simpl; rewrite IHe1; rewrite IHe2; reflexivity).
Case "ANum". reflexivity.
Case "APlus".
destruct e1;
try (simpl; simpl in IHe1; rewrite IHe1; rewrite IHe2; reflexivity).
SCase "e1 = ANum n". destruct n;
simpl; rewrite IHe2; reflexivity. Qed.
Theorem optimize_0plus_sound'': forall e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e.
induction e;
try (simpl; rewrite IHe1; rewrite IHe2; reflexivity);
try reflexivity.
Case "APlus".
destruct e1;
try (simpl; simpl in IHe1; rewrite IHe1; rewrite IHe2; reflexivity).
SCase "e1 = ANum n". destruct n;
simpl; rewrite IHe2; reflexivity. Qed.
Tactic Notation "simpl_and_try" tactic(c) :=
simpl;
try c.
Tactic Notation "aexp_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "ANum" | Case_aux c "APlus"
| Case_aux c "AMinus" | Case_aux c "AMult" ].
Theorem optimize_0plus_sound''': forall e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e.
aexp_cases (induction e) Case;
try (simpl; rewrite IHe1; rewrite IHe2; reflexivity);
try reflexivity.
Case "APlus".
aexp_cases (destruct e1) SCase;
try (simpl; simpl in IHe1; rewrite IHe1; rewrite IHe2; reflexivity).
SCase "ANum". destruct n;
simpl; rewrite IHe2; reflexivity. Qed.
Fixpoint optimize_0plus_all (b:bexp) : bexp :=
match b with
| BTrue => BTrue
| BFalse => BFalse
| BEq a1 a2 => BEq (optimize_0plus a1) (optimize_0plus a2)
| BLe a1 a2 => BLe (optimize_0plus a1) (optimize_0plus a2)
| BNot b1 => BNot b1
| BAnd b1 b2 => BAnd b1 b2
end.
Tactic Notation "bexp_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "BTrue" | Case_aux c "BFalse"
| Case_aux c "BEq" | Case_aux c "BLe" | Case_aux c "BNot"
| Case_aux c "BAnd" ].
Theorem optimize_0plus_all_sound: forall (e:bexp),
beval (optimize_0plus_all e) = beval e.
Proof.
intros e.
bexp_cases (destruct e) Case;
try (simpl; rewrite optimize_0plus_sound; rewrite optimize_0plus_sound);
try reflexivity.
Qed.
Fixpoint optimize_and (b: bexp): bexp :=
match b with
| BTrue => BTrue
| BFalse => BFalse
| BEq a1 a2 => BEq a1 a2
| BLe a1 a2 => BLe a1 a2
| BNot a1 => BNot (optimize_and a1)
| BAnd BFalse a2 => BFalse
| BAnd a1 a2 => BAnd (optimize_and a1) (optimize_and a2)
end.
Theorem optimize_and_sound: forall (e:bexp),
beval (optimize_and e) = beval e.
Proof.
intros e.
bexp_cases (induction e) Case;
try (simpl; rewrite IHe);
try reflexivity.
Case "BAnd". simpl.
bexp_cases (destruct e1) SCase;
try reflexivity;
try (simpl; rewrite IHe2; reflexivity);
try (simpl; simpl in IHe1; rewrite IHe2; rewrite IHe1; reflexivity).
Qed.
Module aevalR_first_try.
Inductive aevalR : aexp -> nat -> Prop :=
| E_Anum : forall (n : nat),
aevalR (ANum n) n
| E_APlus : forall (e1 e2 : aexp) (n1 n2 : nat),
aevalR e1 n1 -> aevalR e2 n2 -> aevalR (APlus e1 e2) (n1 + n2)
| E_AMinus: forall (e1 e2 : aexp) (n1 n2 : nat),
aevalR e1 n1 -> aevalR e2 n2 -> aevalR (AMinus e1 e2) (n1 - n2)
| E_AMult : forall (e1 e2 : aexp) (n1 n2 : nat),
aevalR e1 n1 -> aevalR e2 n2 -> aevalR (AMult e1 e2) (n1 * n2).
Notation "e '==>' n" := (aevalR e n) (at level 40).
End aevalR_first_try.
Reserved Notation "e '==>' n" (at level 40).
Inductive aevalR : aexp -> nat -> Prop :=
| E_ANum : forall (n : nat), (ANum n) ==> n
| E_APlus : forall (e1 e2 : aexp) (n1 n2 : nat),
(e1 ==> n1) -> (e2 ==> n2) -> (APlus e1 e2) ==> (n1 + n2)
| E_AMinus : forall (e1 e2 : aexp) (n1 n2 : nat),
(e1 ==> n1) -> (e2 ==> n2) -> (AMinus e1 e2) ==> (n1 - n2)
| E_AMult : forall (e1 e2 : aexp) (n1 n2 : nat),
(e1 ==> n1) -> (e2 ==> n2) -> (AMult e1 e2) ==> (n1 * n2)
where "e '==>' n" := (aevalR e n).
Inductive bevalR : bexp -> bool -> Prop :=
| E_BTrue : bevalR (BTrue) true
| E_BFalse : bevalR (BFalse) false
| E_BEq : forall (e1 e2: aexp) (n1 n2 : nat),
aevalR e1 n1 -> aevalR e2 n2 -> bevalR (BEq e1 e2) (beq_nat n1 n2)
| E_BLe : forall (e1 e2 : aexp) (n1 n2 : nat),
aevalR e1 n1 -> aevalR e2 n2 -> bevalR (BLe e1 e2) (ble_nat n1 n2)
| E_BNot : forall (e1 : bexp) (b : bool),
bevalR e1 b -> bevalR (BNot e1) (negb b)
| E_BAnd : forall (e1 e2 : bexp) (b1 b2 : bool),
bevalR e1 b1 -> bevalR e2 b2 -> bevalR (BAnd e1 e2) (andb b1 b2).
Tactic Notation "aevalR_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "E_Anum" | Case_aux c "E_APlus"
| Case_aux c "E_AMinus" | Case_aux c "E_AMult" ].
Theorem aeval_iff_aevalR : forall a n,
(a ==> n) <-> aeval a = n.
Proof.
split.
Case "->".
intros H; induction H; subst; reflexivity;
Case "<-".
generalize dependent n.
induction a; simpl; intros; subst; constructor;
try apply IHa1; try apply IHa2; reflexivity.
Qed.
Tactic Notation "bevalR_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "E_BTrue" | Case_aux c "E_BFalse"
| Case_aux c "E_BEq" | Case_aux c "E_BLe" | Case_aux c "E_BNot"
| Case_aux c "E_BAnd" ].
Theorem beval_iff_bevalR : forall a b,
bevalR a b <-> beval a = b.
Proof.
split.
intros H.
induction H.
simpl in |- *.
reflexivity.
simpl in |- *.
reflexivity.
simpl in |- *.
apply aeval_iff_aevalR in H.
apply aeval_iff_aevalR in H0.
subst.
reflexivity.
simpl in |- *.
apply aeval_iff_aevalR in H0.
apply aeval_iff_aevalR in H.
subst.
reflexivity.
simpl in |- *.
rewrite IHbevalR in |- *.
reflexivity.
simpl in |- *.
rewrite IHbevalR1 in |- *.
rewrite IHbevalR2 in |- *.
reflexivity.
generalize dependent b.
induction a.
intros b H. subst. constructor.
intros b H.
subst.
constructor.
intros b H.
rewrite <- H in |- *.
constructor.
simpl in H.
remember (aeval a)as A in *.
assert (aeval a = A).
subst.
reflexivity.
apply aeval_iff_aevalR in H0.
apply H0.
remember (aeval a0)as A in *.
assert (aeval a0 = A).
subst.
reflexivity.
apply aeval_iff_aevalR in H0.
apply H0.
intros b.
intros H.
simpl in H.
rewrite <- H in |- *.
simpl in |- *.
constructor.
remember (aeval a)as A in *.
assert (aeval a = A).
subst.
reflexivity.
apply aeval_iff_aevalR in H0.
apply H0.
remember (aeval a0)as A in *.
assert (aeval a0 = A).
subst.
reflexivity.
apply aeval_iff_aevalR in H0.
apply H0.
intros b.
intros H.
simpl in |- *.
simpl in H.
rewrite <- H in |- *.
constructor.
apply IHa.
reflexivity.
intros b.
simpl in |- *.
intros H.
rewrite <- H in |- *.
constructor.
apply IHa1.
reflexivity.
apply IHa2.
reflexivity.
Qed.
End AExp.
Example silly_presburger_formula : forall m n o p,
m + n <= n + o /\ o + 3 = p + 3 ->
m <= p.
Proof.
intros. omega.
Qed.
Module Id.
Inductive id : Type :=
Id : nat -> id.
Definition beq_id id1 id2 :=
match (id1, id2) with
(Id n1, Id n2) => beq_nat n1 n2
end.
Theorem beq_id_refl : forall i,
true = beq_id i i.
Proof.
intros. destruct i.
apply beq_nat_refl. Qed.
Theorem beq_id_eq : forall i1 i2,
true = beq_id i1 i2 -> i1 = i2.
Proof.
intros i1 i2.
destruct i1.
destruct i2.
unfold beq_id in |- *.
intros H.
apply beq_nat_eq in H.
subst.
reflexivity.
Qed.
Theorem beq_id_false_not_eq : forall i1 i2,
beq_id i1 i2 = false -> i1 <> i2.
Proof.
intros i1 i2.
destruct i1.
destruct i2.
unfold beq_id in |- *.
intros H.
intros C.
unfold beq_id in |- *.
assert (false = beq_nat n n0).
subst.
rewrite <- H in |- *.
reflexivity.
apply beq_false_not_eq in H0.
apply H0.
subst.
inversion C.
reflexivity.
Qed.
Theorem not_eq_beq_id_false : forall i1 i2,
i1 <> i2 -> beq_id i1 i2 = false.
Proof.
intros i1 i2.
destruct i1. destruct i2.
unfold beq_id in |- *.
intros H.
apply not_eq_beq_false.
intros C.
apply H.
subst.
reflexivity.
Qed.
Theorem beq_nat_sym : forall (n m:nat), forall (b: bool),
beq_nat n m = b -> beq_nat m n = b.
Proof.
intros n.
induction n as [| n'].
Case "n = O".
intros m b eq1.
induction m.
SCase "m = 0".
apply eq1.
SCase "m = S m'".
apply eq1.
Case "n = S n'".
induction m.
SCase "m = 0".
intros b eq1.
apply eq1.
SCase "m = S m'".
intros b eq1.
apply IHn'.
apply eq1.
Qed.
Theorem beq_id_sym : forall i1 i2,
beq_id i1 i2 = beq_id i2 i1.
Proof.
intros i1 i2.
destruct i1. destruct i2.
unfold beq_id.
apply beq_nat_sym.
reflexivity.
Qed.
End Id.
Definition state := id -> nat.
Definition empty_state : state := fun _ => 0.
Definition update (st : state) (V:id) (n:nat) : state :=
fun V' => if beq_id V V' then n else st V'.
Theorem update_eq : forall n V st,
(update st V n) V = n.
Proof.
intros n V st.
unfold update.
rewrite <- beq_id_refl.
reflexivity.
Qed.
Theorem update_neq : forall V2 V1 n st,
beq_id V2 V1 = false ->
(update st V2 n) V1 = (st V1).
Proof.
intros V2 V1 n st.
intros H.
unfold update.
rewrite H.
reflexivity.
Qed.
Theorem update_example : forall (n:nat),
(update empty_state (Id 2) n) (Id 3) = 0.
Proof.
unfold update.
intros n.
simpl.
unfold empty_state.
reflexivity.
Qed.
Theorem update_shadow : forall x1 x2 k1 k2 (f:state),
(update (update f k2 x1) k2 x2) k1 = (update f k2 x2) k1.
Proof.
intros x1 x2 k1 k2 f.
unfold update in |- *.
destruct (beq_id k2 k1);
reflexivity.
Qed.
Theorem update_same : forall x1 k1 k2 (f : state),
f k1 = x1 ->
(update f k1 x1) k2 = f k2.
Proof.
unfold update in |- *.
intros x1 k1 k2 f.
remember (beq_id k1 k2)as A in *.
destruct A.
apply beq_id_eq in HeqA.
subst.
intros H.
subst.
reflexivity.
intros H.
reflexivity.
Qed.
Theorem update_permute : forall x1 x2 k1 k2 k3 f,
beq_id k2 k1 = false ->
(update (update f k2 x1) k1 x2) k3 = (update (update f k1 x2) k2 x1) k3.
Proof.
intros x1 x2 k1 k2 k3 f.
intros H.
unfold update in |- *.
remember (beq_id k1 k3)as A1 in *.
destruct A1.
apply beq_id_eq in HeqA1.
rewrite HeqA1 in H.
destruct (beq_id k2 k3).
inversion H.
reflexivity.
remember (beq_id k2 k3)as B in *.
destruct B;
try reflexivity.
Qed.
Inductive aexp : Type :=
| ANum : nat -> aexp
| AId : id -> aexp (* <----- NEW *)
| APlus : aexp -> aexp -> aexp
| AMinus : aexp -> aexp -> aexp
| AMult : aexp -> aexp -> aexp.
Tactic Notation "aexp_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "ANum" | Case_aux c "AId" | Case_aux c "APlus"
| Case_aux c "AMinus" | Case_aux c "AMult" ].
Definition X : id := Id 0.
Definition Y : id := Id 1.
Definition Z : id := Id 2.
Inductive bexp : Type :=
| BTrue : bexp
| BFalse : bexp
| BEq : aexp -> aexp -> bexp
| BLe : aexp -> aexp -> bexp
| BNot : bexp -> bexp
| BAnd : bexp -> bexp -> bexp.
Tactic Notation "bexp_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "BTrue" | Case_aux c "BFalse" | Case_aux c "BEq"
| Case_aux c "BLe" | Case_aux c "BNot" | Case_aux c "BAnd" ].
Fixpoint aeval (st : state) (e : aexp) : nat :=
match e with
| ANum n => n
| AId i => st i (* <----- NEW *)
| APlus a1 a2 => (aeval st a1) + (aeval st a2)
| AMinus a1 a2 => (aeval st a1) - (aeval st a2)
| AMult a1 a2 => (aeval st a1) * (aeval st a2)
end.
Fixpoint beval (st : state) (e : bexp) : bool :=
match e with
| BTrue => true
| BFalse => false
| BEq a1 a2 => beq_nat (aeval st a1) (aeval st a2)
| BLe a1 a2 => ble_nat (aeval st a1) (aeval st a2)
| BNot b1 => negb (beval st b1)
| BAnd b1 b2 => andb (beval st b1) (beval st b2)
end.
Example aexp1 :
aeval (update empty_state X 5)
(APlus (ANum 3) (AMult (AId X) (ANum 2)))
= 13.
Proof. reflexivity. Qed.
Example bexp1 :
beval (update empty_state X 5)
(BAnd BTrue (BNot (BLe (AId X) (ANum 4))))
= true.
Proof. reflexivity. Qed.
Inductive com : Type :=
| CSkip : com
| CAss : id -> aexp -> com
| CSeq : com -> com -> com
| CIf : bexp -> com -> com -> com
| CWhile : bexp -> com -> com.
Tactic Notation "com_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "SKIP" | Case_aux c "::=" | Case_aux c ";"
| Case_aux c "IFB" | Case_aux c "WHILE" ].
Notation "'SKIP'" :=
CSkip.
Notation "l '::=' a" :=
(CAss l a) (at level 60).
Notation "c1 ; c2" :=
(CSeq c1 c2) (at level 80, right associativity).
Notation "'WHILE' b 'DO' c 'END'" :=
(CWhile b c) (at level 80, right associativity).
Notation "'IFB' e1 'THEN' e2 'ELSE' e3 'FI'" :=
(CIf e1 e2 e3) (at level 80, right associativity).
Definition plus2 : com :=
X ::= (APlus (AId X) (ANum 2)).
Definition XtimesYinZ : com :=
Z ::= (AMult (AId X) (AId Y)).
Definition subtract_slowly_body : com :=
Z ::= AMinus (AId Z) (ANum 1);
X ::= AMinus (AId X) (ANum 1).
Definition subtract_slowly : com :=
WHILE BNot (BEq (AId X) (ANum 0)) DO
subtract_slowly_body
END.
Definition subtract_3_from_5_slowly : com :=
X ::= ANum 3 ;
Z ::= ANum 5 ;
subtract_slowly.
Definition loop : com :=
WHILE BTrue DO
SKIP
END.
Definition fact_body : com :=
Y ::= AMult (AId Y) (AId Z) ;
Z ::= AMinus (AId Z) (ANum 1).
Definition fact_loop : com :=
WHILE BNot (BEq (AId Z) (ANum 0)) DO
fact_body
END.
Definition fact_com : com :=
Z ::= AId X ;
Y ::= ANum 1 ;
fact_loop.
Fixpoint ceval_step1 (st:state) (c:com) : state :=
match c with
| SKIP => st
| l ::= a1 =>
update st l (aeval st a1)
| c1 ; c2 =>
let st' := ceval_step1 st c1 in
ceval_step1 st' c2
| IFB b THEN c1 ELSE c2 FI =>
if (beval st b) then ceval_step1 st c1 else ceval_step1 st c2
| WHILE b1 DO c1 END =>
st (* bogus *)
end.
Fixpoint ceval_step2 (st: state) (c:com) (i:nat) : state :=
match i with
| O => empty_state
| S i' =>
match c with
| SKIP => st
| l ::= a1 =>
update st l (aeval st a1)
| c1 ; c2 => let st' := ceval_step2 st c1 i' in
ceval_step2 st' c2 i'
| IFB b THEN c1 ELSE c2 FI =>
if (beval st b) then ceval_step2 st c1 i' else ceval_step2 st c2 i'
| WHILE b1 DO c1 END =>
if (beval st b1)
then let st' := ceval_step2 st c1 i' in
ceval_step2 st' c i'
else st
end
end.
Fixpoint ceval_step3 (st: state) (c:com) (i:nat) : option state :=
match i with
| O => None
| S i' =>
match c with
| SKIP => Some st
| l ::= a1 =>
Some (update st l (aeval st a1))
| c1 ; c2 => match (ceval_step3 st c1 i') with
| Some st' => ceval_step3 st' c2 i'
| None => None
end
| IFB b THEN c1 ELSE c2 FI =>
if (beval st b) then ceval_step3 st c1 i' else ceval_step3 st c2 i'
| WHILE b1 DO c1 END =>
if (beval st b1)
then match (ceval_step3 st c1 i') with
| Some st' => ceval_step3 st' c i'
| None => None
end
else Some st
end
end.
Definition bind_option {X Y : Type} (xo : option X) (f : X -> option Y) : option Y :=
match xo with
| None => None
| Some x => f x
end.
Fixpoint ceval_step (st: state) (c:com) (i:nat) : option state :=
match i with
| O => None
| S i' =>
match c with
| SKIP => Some st
| l ::= a1 =>
Some (update st l (aeval st a1))
| c1 ; c2 => bind_option (ceval_step st c1 i')
(fun st' => ceval_step st' c2 i')
| IFB b THEN c1 ELSE c2 FI =>
if (beval st b) then ceval_step st c1 i' else ceval_step st c2 i'
| WHILE b1 DO c1 END =>
if (beval st b1)
then bind_option (ceval_step st c1 i')
(fun st' => ceval_step st' c i')
else Some st
end
end.
Definition test_ceval (st:state) (c:com) :=
match ceval_step st c 500 with
| None => None
| Some st => Some (st X, st Y, st Z)
end.
Definition pup_to_n: com :=
(Y ::= (ANum 0) ;
WHILE (BLe (ANum 1) (AId X)) DO
Y ::= (APlus (AId X) (AId Y)) ;
X ::= (AMinus (AId X) (ANum 1))
END) .
Example pup_to_n_1 :
test_ceval (update empty_state X 5) pup_to_n
= Some (0, 15, 0).
Proof. reflexivity. Qed.
Definition ceval_even : com :=
WHILE (BLe (ANum 2) (AId X)) DO
X ::= (AMinus (AId X) (ANum 2))
END ;
Z ::= (AId X).
Example ceval_even_test1 :
test_ceval (update empty_state X 20) ceval_even
= Some (0, 0, 0).
Proof. reflexivity. Qed.
Example ceval_even_test2 :
test_ceval (update empty_state X 21) ceval_even
= Some (1, 0, 1).
Proof. reflexivity. Qed.
Example ceval_even_test3 :
test_ceval (update empty_state X 2) ceval_even
= Some (0, 0, 0).
Proof. reflexivity. Qed.
Example ceval_even_test4 :
test_ceval (update empty_state X 1) ceval_even
= Some (1, 0, 1).
Proof. reflexivity. Qed.
Example ceval_even_test5 :
test_ceval (update empty_state X 0) ceval_even
= Some (0, 0, 0).
Proof. reflexivity. Qed.
Reserved Notation "cl '/' st '==>' st'" (at level 40, st at level 39).
Inductive ceval : com -> state -> state -> Prop :=
| E_Skip : forall st,
SKIP / st ==> st
| E_Ass : forall st al n l,
aeval st al = n ->
(l ::= al) / st ==> (update st l n)
| E_Seq : forall c1 c2 st st' st'',
c1 / st ==> st' ->
c2 / st' ==> st'' ->
(c1 ; c2) / st ==> st''
| E_IfTrue : forall st st' b1 c1 c2,
beval st b1 = true ->
c1 / st ==> st' ->
(IFB b1 THEN c1 ELSE c2 FI) / st ==> st'
| E_IfFalse : forall st st' b1 c1 c2,
beval st b1 = false ->
c2 / st ==> st' ->
(IFB b1 THEN c1 ELSE c2 FI) / st ==> st'
| E_WhileEnd : forall b1 st c1,
beval st b1 = false ->
(WHILE b1 DO c1 END) / st ==> st
| E_WhileLoop : forall st st' st'' b1 c1,
beval st b1 = true ->
c1 / st ==> st' ->
(WHILE b1 DO c1 END) / st' ==> st'' ->
(WHILE b1 DO c1 END) / st ==> st''
where "c1 '/' st '==>' st'" := (ceval c1 st st').
Tactic Notation "ceval_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "E_Skip" | Case_aux c "E_Ass" | Case_aux c "E_Seq"
| Case_aux c "E_IfTrue" | Case_aux c "E_IfFalse"
| Case_aux c "E_WhileEnd" | Case_aux c "E_WhileLoop" ].
Example ceval_example1 :
(X ::= ANum 2;
IFB BLe (AId X) (ANum 1)
THEN Y ::= ANum 3
ELSE Z ::= ANum 4
FI)
/ empty_state ==> (update (update empty_state X 2) Z 4).
Proof.
(* We must supply the intermediate state *)
apply E_Seq with (update empty_state X 2).
Case "assignment command".
apply E_Ass. reflexivity.
Case "if command".
apply E_IfFalse.
reflexivity.
apply E_Ass. reflexivity. Qed.
Example ceval_example2:
(X ::= ANum 0; Y ::= ANum 1; Z ::= ANum 2) / empty_state ==>
(update (update (update empty_state X 0) Y 1) Z 2).
Proof.
apply E_Seq with (update empty_state X 0);
try (apply E_Ass; reflexivity).
apply E_Seq with (update (update empty_state X 0) Y 1);
try (apply E_Ass; reflexivity).
Qed.
Theorem ceval_step__ceval : forall c st st',
(exists i, ceval_step st c i = Some st') ->
c / st ==> st'.
Proof.
intros c st st' H.
inversion H as (i, E).
clear H.
generalize dependent st'.
generalize dependent st.
generalize dependent c.
induction i as [| i'].
intros c st st' H.
inversion H.
intros c st st' H.
com_cases (destruct c) SCase; simpl in H; inversion H; subst; clear H.
apply E_Skip.
apply E_Ass.
reflexivity.
remember (ceval_step st c1 i')as r1 in *.
destruct r1.
apply E_Seq with s.
apply IHi'.
rewrite Heqr1 in |- *.
reflexivity.
apply IHi'.
simpl in H1.
assumption.
inversion H1.
remember (beval st b)as r in *.
destruct r.
apply E_IfTrue.
rewrite Heqr in |- *.
reflexivity.
apply IHi'.
assumption.
apply E_IfFalse.
rewrite Heqr in |- *.
reflexivity.
apply IHi'.
assumption.
remember (beval st b)as r in *.
destruct r.
remember (ceval_step st c i')as r1 in *.
destruct r1.
apply E_WhileLoop with s.
rewrite Heqr in |- *.
reflexivity.
apply IHi'.
rewrite Heqr1 in |- *.
reflexivity.
apply IHi'.
simpl in H1.
assumption.
inversion H1.
inversion H1.
apply E_WhileEnd.
rewrite Heqr in |- *.
subst.
reflexivity.
Qed.
Theorem ceval_step_more: forall i1 i2 st st' c,
i1 <= i2 -> ceval_step st c i1 = Some st' ->
ceval_step st c i2 = Some st'.
Proof.
induction i1 as [| i1']; intros i2 st st' c Hle Hceval.
inversion Hceval.
destruct i2 as [| i2'].
inversion Hle.
assert (i1' <= i2') as Hle' by complete omega.
com_cases (destruct c) SCase.
SCase "SKIP".
simpl in Hceval.
inversion Hceval.
reflexivity.
SCase "::=".
simpl in Hceval.
inversion Hceval.
reflexivity.
SCase ";".
simpl in Hceval.
simpl in |- *.
remember (ceval_step st c1 i1')as st1'o in *.
destruct st1'o.
SSCase "st1'o = Some".
symmetry in Heqst1'o.
apply (IHi1' i2') in Heqst1'o; try assumption.
rewrite Heqst1'o in |- *.
simpl in |- *.
simpl in Hceval.
apply (IHi1' i2') in Hceval; try assumption.
inversion Hceval.
SCase "IFB".
simpl in Hceval.
simpl in |- *.
remember (beval st b)as bval in *.
destruct bval; apply (IHi1' i2') in Hceval; assumption.
SCase "WHILE".
simpl in Hceval.
simpl in |- *.
destruct (beval st b); try assumption.
remember (ceval_step st c i1')as st1'o in *.
destruct st1'o.
SSCase "st1'o = Some".
symmetry in Heqst1'o.
apply (IHi1' i2') in Heqst1'o; try assumption.
rewrite Heqst1'o in |- *.
simpl in |- *.
simpl in Hceval.
apply (IHi1' i2') in Hceval; try assumption.
SSCase "i1'o = None".
simpl in Hceval.
inversion Hceval.
Qed.
Theorem ceval__ceval_step : forall c st st',
c / st ==> st' ->
exists i, ceval_step st c i = Some st'.
Proof.
intros c st st' Hce.
ceval_cases (induction Hce) Case.
exists 1%nat.
simpl in |- *.