(* Copyright 2006, 2008 Andrew W. Appel, Robert Dockins and Xavier Leroy * This file is part of the List-machine Benchmark. * The List-machine Benchmark is licensed under GPLv3; see LICENSE. *)
(*** Machine specification ***)
(** Abstract syntax of machine code **)
Definition var: Set := positive.
Definition var_eq: forall (v1 v2: var), {v1=v2} + {v1<>v2} := peq.
Definition V (n: nat) : var := P_of_succ_nat n.
Definition label: Set := positive.
Definition label_eq: forall (l1 l2: label), {l1=l2} + {l1<>l2} := peq.
Definition L (n: nat) : label := P_of_succ_nat n.
Inductive instr: Set :=
| instr_jump: var -> instr
| instr_getlabel: label -> var -> instr
| instr_branch_if_nil: var -> label -> instr
| instr_fetch_field: var -> nat -> var -> instr
| instr_cons: var -> var -> var -> instr
| instr_halt: instr
| instr_seq: instr -> instr -> instr.
Infix ";;" := instr_seq (at level 60, right associativity).
Definition program := map instr.
(** Run-time values and environments **)
Inductive value: Set :=
| value_label: label -> value
| value_cons: value -> value -> value.
Definition store := map value.
Definition store_empty : store := empty value.
(** Execution of machine code **)
Inductive step: program -> store -> instr -> store -> instr -> Prop :=
| step_seq: forall p r i1 i2 i3,
(*----------------------------------------------------*)
step p r ((i1 ;; i2) ;; i3) r (i1 ;; i2 ;; i3)
| step_fetch_field_0: forall p r v1 v2 i a0 a1,
r#v1 = Some (value_cons a0 a1) ->
(*----------------------------------------------------*)
step p r (instr_fetch_field v1 0 v2 ;; i) (r#v2 <- a0) i
| step_fetch_field_1: forall p r v1 v2 i a0 a1,
r#v1 = Some (value_cons a0 a1) ->
(*----------------------------------------------------*)
step p r (instr_fetch_field v1 1 v2 ;; i) (r#v2 <- a1) i
| step_cons: forall p r v1 v2 v3 i a1 a2,
r#v1 = Some a1 ->
r#v2 = Some a2 ->
(*----------------------------------------------------*)
step p r (instr_cons v1 v2 v3 ;; i) (r#v3 <- (value_cons a1 a2)) i
| step_branch_not_taken: forall p r v l i a1 a2,
r#v = Some (value_cons a1 a2) ->
(*----------------------------------------------------*)
step p r (instr_branch_if_nil v l ;; i) r i
| step_branch_taken: forall p r v l i i',
r#v = Some (value_label (L 0)) ->
p#l = Some i' ->
(*----------------------------------------------------*)
step p r (instr_branch_if_nil v l ;; i) r i'
| step_jump: forall p r v l i',
r#v = Some (value_label l) ->
p#l = Some i' ->
(*----------------------------------------------------*)
step p r (instr_jump v) r i'
| step_getlabel: forall p r l v i,
(*----------------------------------------------------*)
step p r (instr_getlabel l v ;; i) (r#v <- (value_label l)) i.
(* Safety property *)
Definition entry_label := L 0.
Definition start_state (p: program) (s: store) (i: instr) :=
p#entry_label = Some i.
Inductive stepstar (p: program) : store -> instr -> store -> instr ->  Prop :=
| stepstar_O: forall s i, stepstar p s i s i
| stepstar_S: forall s i s' i' s'' i'',
step p s i s' i' -> stepstar p s' i' s'' i'' -> stepstar p s i s'' i''.
Inductive step_or_halt: program -> store -> instr -> Prop :=
| step_or_halt_step: forall p r i r' i',
step p r i r' i' -> step_or_halt p r i
| step_or_halt_halt: forall p r,
step_or_halt p r instr_halt.
Definition safe (p: program) (s: store) (i: instr) :=
forall s' i', stepstar p s i s' i' -> step_or_halt p s' i'.
Definition safe_prog (p: program) := forall s i, start_state p s i -> safe p s i.
Inductive stepN (p: program) : nat -> store -> instr -> store -> instr ->  Prop :=
| stepN_O: forall s i, stepN p O s i s i
| stepN_S: forall n s i s' i' s'' i'',
step p s i s' i' -> stepN p n s' i' s'' i'' -> stepN p (S n) s i s'' i''.
intros p s i s' i' H; induction H.
exists 0; apply stepN_O.
destruct IHstepstar as [n Hn].
exists (S n).
eapply stepN_S.
apply H.
apply Hn.
Qed.
intros p n s i s' i' H; induction H.
apply stepstar_O.
eapply stepstar_S.
apply H.
apply IHstepN.
Qed.
intros p s i s1 s2 i1 i2 H1 H2.
inversion H1; subst; inversion H2; subst;
intuition congruence.
Qed.
intros p n s s1 s2 s' i i1 i2 i' H1 H2 H3.
generalize (step_deterministic _ _ _ _ _ _ _ H1 H2).
intuition; subst; auto.
Qed.

Lemma step_triangle : forall p n1 n2 s s1 s2 i i1 i2,
n1 <= n2 ->
stepN p n1 s i s1 i1 ->
stepN p n2 s i s2 i2 ->
stepN p (n2 - n1) s1 i1 s2 i2.