Flail around with this goddamn proof some more.
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day8.v
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day8.v
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@ -51,50 +51,12 @@ Module DayEight (Import M:Int).
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| FS f' => FS (weaken_one f')
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| FS f' => FS (weaken_one f')
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end.
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end.
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(* One modification: we really want to use 'allowed' addresses,
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a set that shrinks as the program continues, rather than 'visited'
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addresses, a set that increases as the program continues. *)
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Inductive step_noswap {n} : input n -> state n -> state n -> Prop :=
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| step_noswap_add : forall inp pc' v acc t,
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nth inp pc' = (add, t) ->
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set_mem Fin.eq_dec pc' v = true ->
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step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, M.add acc t)
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| step_noswap_nop : forall inp pc' v acc t,
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nth inp pc' = (nop, t) ->
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set_mem Fin.eq_dec pc' v = true ->
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step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, acc)
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| step_noswap_jmp : forall inp pc' pc'' v acc t,
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nth inp pc' = (jmp, t) ->
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set_mem Fin.eq_dec pc' v = true ->
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valid_jump_t pc' t = Some pc'' ->
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step_noswap inp (weaken_one pc', v, acc) (pc'', set_remove Fin.eq_dec pc' v, acc).
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Fixpoint nat_to_fin (n : nat) : fin (S n) :=
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Fixpoint nat_to_fin (n : nat) : fin (S n) :=
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match n with
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match n with
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| O => F1
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| O => F1
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| S n' => FS (nat_to_fin n')
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| S n' => FS (nat_to_fin n')
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end.
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end.
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Inductive run_noswap {n} : input n -> state n -> state n -> Prop :=
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| run_noswap_ok : forall inp v acc,
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run_noswap inp (nat_to_fin n, v, acc) (nat_to_fin n, v, acc)
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| run_noswap_fail : forall inp pc' v acc,
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set_mem Fin.eq_dec pc' v = false ->
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run_noswap inp (weaken_one pc', v, acc) (weaken_one pc', v, acc)
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| run_noswap_trans : forall inp st st' st'',
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step_noswap inp st st' -> run_noswap inp st' st'' -> run_noswap inp st st''.
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Inductive valid_inst {n} : inst -> fin n -> Prop :=
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| valid_inst_add : forall t f, valid_inst (add, t) f
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| valid_inst_nop : forall t f f',
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valid_jump_t f t = Some f' -> valid_inst (nop, t) f
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| valid_inst_jmp : forall t f f',
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valid_jump_t f t = Some f' -> valid_inst (jmp, t) f.
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(* An input is valid if all its instructions are valid. *)
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Definition valid_input {n} (inp : input n) : Prop := forall (pc : fin n),
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valid_inst (nth inp pc) pc.
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Lemma fin_big_or_small : forall {n} (f : fin (S n)),
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Lemma fin_big_or_small : forall {n} (f : fin (S n)),
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(f = nat_to_fin n) \/ (exists (f' : fin n), f = weaken_one f').
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(f = nat_to_fin n) \/ (exists (f' : fin n), f = weaken_one f').
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Proof.
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Proof.
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@ -114,6 +76,311 @@ Module DayEight (Import M:Int).
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exists (FS f'). simpl. rewrite Heq.
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exists (FS f'). simpl. rewrite Heq.
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reflexivity.
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reflexivity.
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Qed.
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Qed.
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Lemma weaken_one_inj : forall n (f1 f2 : fin n),
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(weaken_one f1 = weaken_one f2 -> f1 = f2).
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Proof.
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remember (fun {n} (a b : fin n) => weaken_one a = weaken_one b -> a = b) as P.
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(* Base case for rect2 *)
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assert (forall n, @P (S n) F1 F1).
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{rewrite HeqP. intros n Heq. reflexivity. }
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(* 'Impossible' cases for rect2. *)
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assert (forall {n} (f : fin n), P (S n) F1 (FS f)).
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{rewrite HeqP. intros n f Heq. simpl in Heq. inversion Heq. }
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assert (forall {n} (f : fin n), P (S n) (FS f) F1).
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{rewrite HeqP. intros n f Heq. simpl in Heq. inversion Heq. }
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(* Recursive case for rect2. *)
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assert (forall {n} (f g : fin n), P n f g -> P (S n) (FS f) (FS g)).
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{rewrite HeqP. intros n f g IH Heq.
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simpl in Heq. injection Heq as Heq'.
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apply inj_pair2_eq_dec in Heq'.
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- rewrite IH. reflexivity. assumption.
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- apply eq_nat_dec. }
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(* Actually apply recursion. *)
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(* This can't be _the_ way to do this. *)
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intros n.
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specialize (@Fin.rect2 P H H0 H1 H2 n) as Hind.
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rewrite HeqP in Hind. apply Hind.
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Qed.
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Lemma weaken_neq_to_fin : forall {n} (f : fin (S n)),
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nat_to_fin (S n) <> weaken_one f.
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Proof.
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apply Fin.rectS; intros n Heq.
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- inversion Heq.
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- intros IH. simpl. intros Heq'.
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injection Heq' as Hinj. apply inj_pair2_eq_dec in Hinj.
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+ simpl in IH. apply IH. apply Hinj.
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+ apply eq_nat_dec.
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Qed.
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(* One modification: we really want to use 'allowed' addresses,
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a set that shrinks as the program continues, rather than 'visited'
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addresses, a set that increases as the program continues. *)
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Inductive step_noswap {n} : input n -> state n -> state n -> Prop :=
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| step_noswap_add : forall inp pc' v acc t,
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nth inp pc' = (add, t) ->
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set_In pc' v ->
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step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, M.add acc t)
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| step_noswap_nop : forall inp pc' v acc t,
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nth inp pc' = (nop, t) ->
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set_In pc' v ->
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step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, acc)
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| step_noswap_jmp : forall inp pc' pc'' v acc t,
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nth inp pc' = (jmp, t) ->
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set_In pc' v ->
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valid_jump_t pc' t = Some pc'' ->
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step_noswap inp (weaken_one pc', v, acc) (pc'', set_remove Fin.eq_dec pc' v, acc).
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Inductive done {n} : input n -> state n -> Prop :=
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| done_prog : forall inp v acc, done inp (nat_to_fin n, v, acc).
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Inductive stuck {n} : input n -> state n -> Prop :=
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| stuck_prog : forall inp pc' v acc,
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~ set_In pc' v -> stuck inp (weaken_one pc', v, acc).
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Inductive run_noswap {n} : input n -> state n -> state n -> Prop :=
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| run_noswap_ok : forall inp st, done inp st -> run_noswap inp st st
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| run_noswap_fail : forall inp st, stuck inp st -> run_noswap inp st st
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| run_noswap_trans : forall inp st st' st'',
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step_noswap inp st st' -> run_noswap inp st' st'' -> run_noswap inp st st''.
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Inductive valid_inst {n} : inst -> fin n -> Prop :=
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| valid_inst_add : forall t f, valid_inst (add, t) f
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| valid_inst_nop : forall t f f',
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valid_jump_t f t = Some f' -> valid_inst (nop, t) f
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| valid_inst_jmp : forall t f f',
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valid_jump_t f t = Some f' -> valid_inst (jmp, t) f.
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(* An input is valid if all its instructions are valid. *)
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Definition valid_input {n} (inp : input n) : Prop := forall (pc : fin n),
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valid_inst (nth inp pc) pc.
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Section ValidInput.
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Variable n : nat.
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Variable inp : input n.
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Hypothesis Hv : valid_input inp.
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Lemma step_if_possible : forall pcs v acc,
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set_In pcs v ->
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exists pc' acc', step_noswap inp (weaken_one pcs, v, acc) (pc', set_remove Fin.eq_dec pcs v, acc').
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Proof.
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intros pcs v acc Hin.
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remember (nth inp pcs) as instr. destruct instr as [op t]. destruct op.
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+ exists (FS pcs). exists (M.add acc t). apply step_noswap_add; auto.
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+ exists (FS pcs). exists acc. apply step_noswap_nop with t; auto.
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+ unfold valid_input in Hv. specialize (Hv pcs).
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rewrite <- Heqinstr in Hv. inversion Hv; subst.
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exists f'. exists acc. apply step_noswap_jmp with t; auto.
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Qed.
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Theorem valid_input_progress : forall pc v acc,
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(pc = nat_to_fin n /\ done inp (pc, v, acc)) \/
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exists pcs, pc = weaken_one pcs /\
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((~ set_In pcs v /\ stuck inp (pc, v, acc)) \/
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(exists pc' acc', set_In pcs v /\ step_noswap inp (pc, v, acc) (pc', set_remove Fin.eq_dec pcs v, acc'))).
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Proof.
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intros pc v acc.
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(* Have we reached the end? *)
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destruct (fin_big_or_small pc).
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(* We're at the end, so we're done. *)
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left. rewrite H. split. reflexivity. apply done_prog.
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(* We're not at the end. Is the PC valid? *)
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right. destruct H as [pcs H]. exists pcs. rewrite H. split. reflexivity.
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destruct (set_In_dec Fin.eq_dec pcs v).
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- (* It is. *)
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right.
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destruct (step_if_possible pcs v acc) as [pc' [acc' Hstep]]; auto.
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exists pc'. exists acc'. split; auto.
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- (* It i not. *)
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left. split; auto. apply stuck_prog; auto.
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Qed.
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Theorem run_ok_or_fail : forall st st',
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run_noswap inp st st' ->
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(exists v acc, st' = (nat_to_fin n, v, acc) /\ done inp st') \/
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(exists pcs v acc, st' = (weaken_one pcs, v, acc) /\ stuck inp st').
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Proof.
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intros st st' Hr.
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induction Hr.
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- left. inversion H; subst. exists v. exists acc. auto.
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- right. inversion H; subst. exists pc'. exists v. exists acc. auto.
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- apply IHHr; auto.
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Qed.
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Theorem set_induction {A : Type}
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(Aeq_dec : forall (x y : A), { x = y } + { x <> y })
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(P : set A -> Prop) :
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P (@empty_set A) -> (forall a st, P (set_remove Aeq_dec a st) -> P st) ->
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forall st, P st.
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Proof. Admitted.
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(* Theorem add_terminates_later : forall pcs a v acc,
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List.NoDup v -> pcs <> a ->
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(forall pc' acc', exists st', run_noswap inp (pc', set_remove Fin.eq_dec a v, acc') st') ->
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exists st', run_noswap inp (weaken_one pcs, v, acc) st'.
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Proof.
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intros pcs a v acc Hnd Hneq He.
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assert (exists st', run_noswap inp (weaken_one pcs, set_remove Fin.eq_dec a v, acc) st').
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{ specialize (He (weaken_one pcs) acc). apply He. }
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destruct H as [st' Hr].
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inversion Hr; subst.
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- inversion H. destruct n. inversion pcs. apply weaken_neq_to_fin in H2 as [].
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- inversion H; subst. apply weaken_one_inj in H0. subst.
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eexists. eapply run_noswap_fail. apply stuck_prog.
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intros Hin. apply H2. apply set_remove_3; auto.
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-
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inversion H; subst; apply weaken_one_inj in H1; subst.
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+ eexists. eapply run_noswap_trans. apply step_noswap_add.
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* apply H6.
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* apply (@set_remove_1 _ Fin.eq_dec pcs a v H7).
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*
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destruct He as [st' Hr].
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dependent induction Hr.
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- inversion H. destruct n. inversion pcs. apply weaken_neq_to_fin in H2 as [].
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- inversion H; subst. apply weaken_one_inj in H0. subst.
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eexists. eapply run_noswap_fail. apply stuck_prog.
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intros Hin. apply H2. apply set_remove_3; auto.
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- destruct st' as [[pc' v'] a']. *)
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Lemma set_remove_comm : forall {A:Type} Aeq_dec a b (st : set A),
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set_remove Aeq_dec a (set_remove Aeq_dec b st) = set_remove Aeq_dec b (set_remove Aeq_dec a st).
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Admitted.
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Theorem remove_pc_safe : forall pc a v acc,
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List.NoDup v ->
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(exists st', run_noswap inp (pc, v, acc) st') ->
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exists st', run_noswap inp (pc, set_remove Fin.eq_dec a v, acc) st'.
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Proof.
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intros pc a v acc Hnd [st' He].
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dependent induction He.
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- inversion H; subst.
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eexists. apply run_noswap_ok. apply done_prog.
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- inversion H; subst.
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eexists. apply run_noswap_fail. apply stuck_prog.
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intros Contra. apply H2. eapply set_remove_1. apply Contra.
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- destruct st' as [[pc' v'] acc'].
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inversion H; subst; destruct (Fin.eq_dec pc'0 a); subst.
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+ eexists. apply run_noswap_fail. apply stuck_prog.
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intros Contra. apply set_remove_2 in Contra; auto.
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+ edestruct IHHe; auto. apply set_remove_nodup; auto.
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eexists. eapply run_noswap_trans.
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* apply step_noswap_add. apply H3. apply set_remove_iff; auto.
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* rewrite set_remove_comm. apply H0.
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+ eexists. apply run_noswap_fail. apply stuck_prog.
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intros Contra. apply set_remove_2 in Contra; auto.
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+ edestruct IHHe; auto. apply set_remove_nodup; auto.
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eexists. eapply run_noswap_trans.
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* apply step_noswap_nop with t0. apply H3. apply set_remove_iff; auto.
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* rewrite set_remove_comm. apply H0.
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+ eexists. apply run_noswap_fail. apply stuck_prog.
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intros Contra. apply set_remove_2 in Contra; auto.
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+ edestruct IHHe; auto. apply set_remove_nodup; auto.
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eexists. eapply run_noswap_trans.
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* apply step_noswap_jmp with t0. apply H5. apply set_remove_iff; auto.
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apply H9.
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* rewrite set_remove_comm. apply H0.
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Qed.
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Theorem valid_input_terminates : forall pc v acc,
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List.NoDup v -> exists st', run_noswap inp (pc, v, acc) st'.
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Proof.
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intros pc v acc.
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generalize dependent pc. generalize dependent acc.
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induction v as [|a v] using (@set_induction _ Fin.eq_dec); intros acc pc Hnd.
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- destruct (valid_input_progress pc (empty_set _) acc) as [|[pcs [Hpc []]]].
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+ (* The program is done at this PC. *)
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destruct H as [Hpc Hd].
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eexists. apply run_noswap_ok. auto.
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+ (* The PC is not done, so we must have failed. *)
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destruct H as [Hin Hst].
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eexists. apply run_noswap_fail. auto.
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+ (* The program can't possibly take a step if
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it's not done and the set of valid PCs is
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empty. This case is absurd. *)
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destruct H as [pc' [acc' [Hin Hstep]]].
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inversion Hstep; subst; inversion H6.
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- (* How did the program terminate? *)
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destruct (valid_input_progress pc (set_remove Fin.eq_dec a v) acc) as [|[pcs H]].
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+ (* We were done without a, so we're still done now. *)
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destruct H as [Hpc Hd]. rewrite Hpc.
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eexists. apply run_noswap_ok. apply done_prog.
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+ (* We were not done without a. *)
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destruct H as [Hw H].
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(* Is a equal to the current address? *)
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destruct (Fin.eq_dec pcs a) as [Heq_dec|Heq_dec].
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* (* a is equal to the current address. Now, we can resume,
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even though we couldn't have before. *)
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subst. destruct H.
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{ destruct (set_In_dec Fin.eq_dec a v).
|
||||||
|
- destruct (step_if_possible a v acc s) as [pc' [acc' Hstep]].
|
||||||
|
destruct (IHv acc' pc') as [st' Hr].
|
||||||
|
apply set_remove_nodup. auto.
|
||||||
|
exists st'. eapply run_noswap_trans. apply Hstep. apply Hr.
|
||||||
|
- eexists. apply run_noswap_fail. apply stuck_prog. assumption. }
|
||||||
|
{ destruct H as [pc' [acc' [Hf]]].
|
||||||
|
exfalso. apply set_remove_2 in Hf. apply Hf. auto. auto. }
|
||||||
|
* (* a is not equal to the current address.
|
||||||
|
This case is not straightforward. *)
|
||||||
|
subst. destruct H.
|
||||||
|
{ (* We were stuck without a, and since PC is not a, we're no less
|
||||||
|
stuck now. *)
|
||||||
|
destruct H. eexists. eapply run_noswap_fail. apply stuck_prog.
|
||||||
|
intros Hin. apply H. apply set_remove_iff; auto. }
|
||||||
|
{ (* We could move on without a. We can move on still. *)
|
||||||
|
destruct H as [pc' [acc' [Hin Hs]]].
|
||||||
|
destruct (step_if_possible pcs v acc) as [pc'' [acc'' Hs']].
|
||||||
|
- apply (set_remove_1 Fin.eq_dec pcs a v Hin).
|
||||||
|
- eexists. eapply run_noswap_trans. apply Hs'.
|
||||||
|
destruct (IHv acc'' pc'') as [st' Hr].
|
||||||
|
+ apply set_remove_nodup; auto.
|
||||||
|
+ specialize (IHv acc'' pc'') as IH.
|
||||||
|
Admitted.
|
||||||
|
|
||||||
|
(* specialize (IHv acc pc (set_remove_nodup Fin.eq_dec a Hnd)) as [st' Hr].
|
||||||
|
dependent induction Hr; subst.
|
||||||
|
{ inversion H0. destruct n. inversion pcs. apply weaken_neq_to_fin in H3 as []. }
|
||||||
|
{ destruct H.
|
||||||
|
- eexists. apply run_noswap_fail. apply stuck_prog.
|
||||||
|
intros Hin. specialize (set_remove_3 Fin.eq_dec v Hin Heq_dec) as Hin'.
|
||||||
|
destruct H. apply H. apply Hin'.
|
||||||
|
- destruct H as [pc' [acc' [Hin Hstep]]].
|
||||||
|
inversion H0; subst. apply weaken_one_inj in H. subst.
|
||||||
|
exfalso. apply H2. apply Hin. }
|
||||||
|
{ destruct H.
|
||||||
|
- destruct H as [Hin Hst].
|
||||||
|
inversion H0; subst; apply weaken_one_inj in H; subst;
|
||||||
|
exfalso; apply Hin; assumption.
|
||||||
|
- assert (weaken_one pcs = weaken_one pcs) as Hrefl by reflexivity.
|
||||||
|
specialize (IHHr Hv0 (weaken_one pcs) Hrefl acc v Hnd a (or_intror H) Heq_dec Hv).
|
||||||
|
apply IHHr.
|
||||||
|
|
||||||
|
(* We were broken without a, but now we could be working again. *)
|
||||||
|
destruct H as [Hin Hst]. rewrite Hpc. admit.
|
||||||
|
+ (* We could make a step without the a. We can still do so now. *)
|
||||||
|
|
||||||
|
|
||||||
|
destruct H as [pc' [acc' [Hin Hstep]]].
|
||||||
|
destruct (IHv acc pc) as [st' Hr]. inversion Hr.
|
||||||
|
* inversion H; subst.
|
||||||
|
destruct n. inversion pcs. apply weaken_neq_to_fin in H5 as [].
|
||||||
|
* inversion H; subst.
|
||||||
|
apply weaken_one_inj in H3. subst.
|
||||||
|
exfalso. apply H5. apply Hin.
|
||||||
|
* subst. apply set_remove_1 in Hin.
|
||||||
|
destruct (step_if_possible pcs v acc Hin) as [pc'' [acc'' Hstep']].
|
||||||
|
eexists. eapply run_noswap_trans.
|
||||||
|
apply Hstep'. specialize (IHv acc'' pc'') as [st''' Hr'].
|
||||||
|
|
||||||
|
|
||||||
|
destruct (fin_big_or_small pc).
|
||||||
|
(* If we're at the end, we're done. *)
|
||||||
|
eexists. rewrite H. eapply run_noswap_ok.
|
||||||
|
(* We're not at the end. *) *)
|
||||||
|
End ValidInput.
|
||||||
|
(*
|
||||||
|
|
||||||
Lemma set_add_idempotent : forall {A:Type}
|
Lemma set_add_idempotent : forall {A:Type}
|
||||||
(Aeq_dec : forall x y : A, { x = y } + { x <> y })
|
(Aeq_dec : forall x y : A, { x = y } + { x <> y })
|
||||||
|
|
@ -169,44 +436,6 @@ Module DayEight (Import M:Int).
|
||||||
forall (s : set A), P s.
|
forall (s : set A), P s.
|
||||||
Proof. Admitted.
|
Proof. Admitted.
|
||||||
|
|
||||||
Lemma weaken_one_inj : forall n (f1 f2 : fin n),
|
|
||||||
(weaken_one f1 = weaken_one f2 -> f1 = f2).
|
|
||||||
Proof.
|
|
||||||
remember (fun {n} (a b : fin n) => weaken_one a = weaken_one b -> a = b) as P.
|
|
||||||
(* Base case for rect2 *)
|
|
||||||
assert (forall n, @P (S n) F1 F1).
|
|
||||||
{rewrite HeqP. intros n Heq. reflexivity. }
|
|
||||||
(* 'Impossible' cases for rect2. *)
|
|
||||||
assert (forall {n} (f : fin n), P (S n) F1 (FS f)).
|
|
||||||
{rewrite HeqP. intros n f Heq. simpl in Heq. inversion Heq. }
|
|
||||||
assert (forall {n} (f : fin n), P (S n) (FS f) F1).
|
|
||||||
{rewrite HeqP. intros n f Heq. simpl in Heq. inversion Heq. }
|
|
||||||
(* Recursive case for rect2. *)
|
|
||||||
assert (forall {n} (f g : fin n), P n f g -> P (S n) (FS f) (FS g)).
|
|
||||||
{rewrite HeqP. intros n f g IH Heq.
|
|
||||||
simpl in Heq. injection Heq as Heq'.
|
|
||||||
apply inj_pair2_eq_dec in Heq'.
|
|
||||||
- rewrite IH. reflexivity. assumption.
|
|
||||||
- apply eq_nat_dec. }
|
|
||||||
|
|
||||||
(* Actually apply recursion. *)
|
|
||||||
(* This can't be _the_ way to do this. *)
|
|
||||||
intros n.
|
|
||||||
specialize (@Fin.rect2 P H H0 H1 H2 n) as Hind.
|
|
||||||
rewrite HeqP in Hind. apply Hind.
|
|
||||||
Qed.
|
|
||||||
|
|
||||||
Lemma weaken_neq_to_fin : forall {n} (f : fin (S n)),
|
|
||||||
nat_to_fin (S n) <> weaken_one f.
|
|
||||||
Proof.
|
|
||||||
apply Fin.rectS; intros n Heq.
|
|
||||||
- inversion Heq.
|
|
||||||
- intros IH. simpl. intros Heq'.
|
|
||||||
injection Heq' as Hinj. apply inj_pair2_eq_dec in Hinj.
|
|
||||||
+ simpl in IH. apply IH. apply Hinj.
|
|
||||||
+ apply eq_nat_dec.
|
|
||||||
Qed.
|
|
||||||
|
|
||||||
Lemma add_pc_safe_step : forall {n} (inp : input n) (pc : fin (S n)) i is acc st',
|
Lemma add_pc_safe_step : forall {n} (inp : input n) (pc : fin (S n)) i is acc st',
|
||||||
step_noswap inp (pc, is, acc) st' ->
|
step_noswap inp (pc, is, acc) st' ->
|
||||||
exists st'', step_noswap inp (pc, (set_add Fin.eq_dec i is), acc) st''.
|
exists st'', step_noswap inp (pc, (set_add Fin.eq_dec i is), acc) st''.
|
||||||
|
|
@ -413,5 +642,5 @@ Module DayEight (Import M:Int).
|
||||||
|
|
||||||
|
|
||||||
(* Stoppped here. *)
|
(* Stoppped here. *)
|
||||||
Admitted.
|
Admitted. *)
|
||||||
End DayEight.
|
End DayEight.
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue
Block a user