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Mess around some more with proof for day 8.

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Danila Fedorin 2020-12-12 01:44:16 -08:00
parent 8459219e60
commit 51a679ec63
1 changed files with 333 additions and 14 deletions
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day8.v
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@ -2,6 +2,9 @@ Require Import Coq.ZArith.Int.
Require Import Coq.Lists.ListSet.
Require Import Coq.Vectors.VectorDef.
Require Import Coq.Vectors.Fin.
Require Import Coq.Program.Equality.
Require Import Coq.Logic.Eqdep_dec.
Require Import Coq.Arith.Peano_dec.
Module DayEight (Import M:Int).
(* We need to coerce natural numbers into integers to add them. *)
@ -42,27 +45,29 @@ Module DayEight (Import M:Int).
Not all inputs are valid, so this may fail. *)
Definition valid_jump_t {n} (pc : fin n) (off : t) : option (fin (S n)) := @clamp (S n) (jump_t pc off).
Definition weaken_one {n} (f : fin n) : fin (S n).
Proof.
apply (@cast (n + 1)).
+ apply L. apply f.
+ rewrite <- plus_n_Sm. rewrite <- plus_n_O. reflexivity.
Defined.
Fixpoint weaken_one {n} (f : fin n) : fin (S n) :=
match f with
| F1 => F1
| FS f' => FS (weaken_one f')
end.
(* One modification: we really want to use 'allowed' addresses,
a set that shrinks as the program continues, rather than 'visited'
addresses, a set that increases as the program continues. *)
Inductive step_noswap {n} : input n -> state n -> state n -> Prop :=
| step_noswap_acc : forall inp pc' v acc t,
| step_noswap_add : forall inp pc' v acc t,
nth inp pc' = (add, t) ->
~ set_mem Fin.eq_dec pc' v = true ->
step_noswap inp (weaken_one pc', v, acc) (FS pc', set_add Fin.eq_dec pc' v, M.add acc t)
set_mem Fin.eq_dec pc' v = true ->
step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, M.add acc t)
| step_noswap_nop : forall inp pc' v acc t,
nth inp pc' = (nop, t) ->
~ set_mem Fin.eq_dec pc' v = true ->
step_noswap inp (weaken_one pc', v, acc) (FS pc', set_add Fin.eq_dec pc' v, acc)
set_mem Fin.eq_dec pc' v = true ->
step_noswap inp (weaken_one pc', v, acc) (FS pc', set_remove Fin.eq_dec pc' v, acc)
| step_noswap_jmp : forall inp pc' pc'' v acc t,
nth inp pc' = (jmp, t) ->
~ set_mem Fin.eq_dec pc' v = true ->
set_mem Fin.eq_dec pc' v = true ->
valid_jump_t pc' t = Some pc'' ->
step_noswap inp (weaken_one pc', v, acc) (pc'', set_add Fin.eq_dec pc' v, acc).
step_noswap inp (weaken_one pc', v, acc) (pc'', set_remove Fin.eq_dec pc' v, acc).
Fixpoint nat_to_fin (n : nat) : fin (S n) :=
match n with
@ -74,7 +79,7 @@ Module DayEight (Import M:Int).
| run_noswap_ok : forall inp v acc,
run_noswap inp (nat_to_fin n, v, acc) (nat_to_fin n, v, acc)
| run_noswap_fail : forall inp pc' v acc,
set_mem Fin.eq_dec pc' v = true ->
set_mem Fin.eq_dec pc' v = false ->
run_noswap inp (weaken_one pc', v, acc) (weaken_one pc', v, acc)
| run_noswap_trans : forall inp st st' st'',
step_noswap inp st st' -> run_noswap inp st' st'' -> run_noswap inp st st''.
@ -90,9 +95,323 @@ Module DayEight (Import M:Int).
Definition valid_input {n} (inp : input n) : Prop := forall (pc : fin n),
valid_inst (nth inp pc) pc.
Lemma fin_big_or_small : forall {n} (f : fin (S n)),
(f = nat_to_fin n) \/ (exists (f' : fin n), f = weaken_one f').
Proof.
(* Hey, looks like the creator of Fin provided
us with nice inductive principles. Using Coq's
default `induction` breaks here.
Merci, Pierre! *)
apply Fin.rectS.
- intros n. destruct n.
+ left. reflexivity.
+ right. exists F1. auto.
- intros n p IH.
destruct IH.
+ left. rewrite H. reflexivity.
+ right. destruct H as [f' Heq].
exists (FS f'). simpl. rewrite Heq.
reflexivity.
Qed.
Lemma set_add_idempotent : forall {A:Type}
(Aeq_dec : forall x y : A, { x = y } + { x <> y })
(a : A) (s : set A), set_mem Aeq_dec a s = true -> set_add Aeq_dec a s = s.
Proof.
intros A Aeq_dec a s Hin.
induction s.
- inversion Hin.
- simpl. simpl in Hin.
destruct (Aeq_dec a a0).
+ reflexivity.
+ simpl. rewrite IHs; auto.
Qed.
Theorem set_add_append : forall {A:Type}
(Aeq_dec : forall x y : A, {x = y } + { x <> y })
(a : A) (s : set A), set_mem Aeq_dec a s = false ->
set_add Aeq_dec a s = List.app s (List.cons a List.nil).
Proof.
induction s.
- reflexivity.
- intros Hnm. simpl in Hnm.
destruct (Aeq_dec a a0) eqn:Heq_dec.
+ inversion Hnm.
+ simpl. rewrite Heq_dec. rewrite IHs.
reflexivity. assumption.
Qed.
Lemma list_append_or_nil : forall {A:Type} (l : list A),
l = List.nil \/ exists l' a, l = List.app l' (List.cons a List.nil).
Proof.
induction l.
- left. reflexivity.
- right. destruct IHl.
+ exists List.nil. exists a.
rewrite H. reflexivity.
+ destruct H as [l' [a' H]].
exists (List.cons a l'). exists a'.
rewrite H. reflexivity.
Qed.
Theorem list_append_induction : forall {A:Type}
(P : list A -> Prop),
P List.nil -> (forall (a : A) (l : list A), P l -> P (List.app l (List.cons a (List.nil)))) ->
forall l, P l.
Proof. Admitted.
Theorem set_induction : forall {A:Type}
(Aeq_dec : forall x y : A, { x = y } + {x <> y })
(P : set A -> Prop),
P (@empty_set A) -> (forall (a : A) (s' : set A), P s' -> P (set_add Aeq_dec a s')) ->
forall (s : set A), P s.
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',
step_noswap inp (pc, is, acc) st' ->
exists st'', step_noswap inp (pc, (set_add Fin.eq_dec i is), acc) st''.
Proof.
intros n inp pc' i is acc st' Hstep.
inversion Hstep.
- eexists. apply step_noswap_add. apply H4.
apply set_mem_correct2. apply set_add_intro1.
apply set_mem_correct1 with Fin.eq_dec. assumption.
- eexists. eapply step_noswap_nop. apply H4.
apply set_mem_correct2. apply set_add_intro1.
apply set_mem_correct1 with Fin.eq_dec. assumption.
- eexists. eapply step_noswap_jmp. apply H3.
apply set_mem_correct2. apply set_add_intro1.
apply set_mem_correct1 with Fin.eq_dec. assumption.
apply H6.
Qed.
Lemma remove_pc_safe_run : forall {n} (inp : input n) i pc v acc st',
run_noswap inp (pc, set_add Fin.eq_dec i v, acc) st' ->
exists st'', run_noswap inp (pc, v, acc) st''.
Proof.
intros n inp i pc v acc st' Hr.
dependent induction Hr.
- eexists. eapply run_noswap_ok.
- eexists. eapply run_noswap_fail.
apply set_mem_complete1 in H.
apply set_mem_complete2.
intros Hin. apply H. apply set_add_intro. right. apply Hin.
- inversion H; subst; destruct (set_mem Fin.eq_dec pc' v) eqn:Hm.
Admitted.
Lemma add_pc_safe_run : forall {n} (inp : input n) i pc v acc st',
run_noswap inp (pc, v, acc) st' ->
exists st'', run_noswap inp (pc, (set_add Fin.eq_dec i v), acc) st''.
Proof.
intros n inp i pc v acc st' Hr.
destruct (set_mem Fin.eq_dec i v) eqn:Hm.
(* If i is already in the set, nothing changes. *)
rewrite set_add_idempotent.
exists st'. assumption. assumption.
(* Otherwise, the behavior might have changed.. *)
destruct (fin_big_or_small pc).
- (* If we're done, we're done no matter what. *)
eexists. rewrite H. eapply run_noswap_ok.
- (* The PC points somewhere inside. We tried (and maybe failed)
to execute and instruction. The challenging part
is that adding i may change the outcome from 'fail' to 'ok' *)
destruct H as [pc' Heq].
generalize dependent st'.
induction v using (@set_induction (fin n) Fin.eq_dec);
intros st' Hr.
+ (* Our set of valid states is nearly empty. One step,
and it runs dry. *)
simpl. destruct (Fin.eq_dec pc' i) eqn:Heq_dec.
* (* The PC is the one allowed state. *)
remember (nth inp pc') as h. destruct h. destruct o.
{ (* Addition. *)
destruct (fin_big_or_small (FS pc')).
- (* The additional step puts as at the end. *)
eexists. eapply run_noswap_trans.
+ rewrite Heq. apply step_noswap_add.
symmetry. apply Heqh. simpl. rewrite Heq_dec. reflexivity.
+ rewrite H. apply run_noswap_ok.
- (* The additional step puts us somewhere else. *)
destruct H as [f' H].
eexists. eapply run_noswap_trans.
+ rewrite Heq. apply step_noswap_add.
symmetry. apply Heqh. simpl. rewrite Heq_dec. reflexivity.
+ rewrite H. apply run_noswap_fail.
simpl. rewrite Heq_dec. reflexivity. }
{ (* No-op *) admit. }
{ (* Jump*) admit. }
* (* The PC is not. We're done. *)
eexists. rewrite Heq. eapply run_noswap_fail.
simpl. rewrite Heq_dec. reflexivity.
+ destruct (set_mem Fin.eq_dec a v) eqn:Hm'.
* unfold fin. rewrite (set_add_idempotent Fin.eq_dec a).
{ apply step_noswap_nop.
- symmetry. apply Heqh.
- simpl. rewrite Heq_dec. reflexivity. }
(*
dependent induction Hr; subst.
+ (* We can't be in the OK state, since we already covered
that earlier. *)
destruct n. inversion pc'.
apply weaken_neq_to_fin in Heq as [].
+ apply weaken_one_inj in Heq as Hs. subst.
destruct (Fin.eq_dec pc'0 i) eqn:Heq_dec.
* admit.
* eexists. eapply run_noswap_fail.
assert (~set_In pc'0 v).
{ apply (set_mem_complete1 Fin.eq_dec). assumption. }
assert (~set_In pc'0 (List.cons i List.nil)).
{ simpl. intros [Heq'|[]]. apply n0. auto. }
assert (~set_In pc'0 (set_union Fin.eq_dec v (List.cons i List.nil))).
{ intros Hin. apply set_union_iff in Hin as [Hf|Hf].
- apply H0. apply Hf.
- apply H1. apply Hf. }
simpl in H2. apply set_mem_complete2. assumption.
+ apply (add_pc_safe_step _ _ i) in H as [st''' Hr'].
eexists. eapply run_noswap_trans.
apply Hr'. destruct st' as [[pc'' v''] acc''].
specialize (IHHr i pc'' v'' acc'').*)
(* intros n inp i pc v.
generalize dependent i.
generalize dependent pc.
induction v; intros pc i acc st Hr.
- inversion Hr; subst.
+ eexists. apply run_noswap_ok.
+ destruct (Fin.eq_dec pc' i) eqn:Heq_dec.
* admit.
* eexists. apply run_noswap_fail.
simpl. rewrite Heq_dec. reflexivity.
+ inversion H; subst; simpl in H7; inversion H7.
- inversion Hr; subst.
+ eexists. apply run_noswap_ok.
+ destruct (Fin.eq_dec pc' i) eqn:Heq_dec.
* admit.
* eexists. apply run_noswap_fail.
simpl. rewrite Heq_dec. simpl in H4.
apply H4.
+
destruct (nth inp pc') as [op t]. *)
Admitted.
Theorem valid_input_terminates : forall n (inp : input n) st,
valid_input inp -> exists st', run_noswap inp st st'.
Proof.
intros n inp st.
destruct st as [[pc is] acc].
generalize dependent inp.
generalize dependent pc.
generalize dependent acc.
induction is using (@set_induction (fin n) Fin.eq_dec); intros acc pc inp Hv;
(* The PC may point past the end of the
array, or it may not. *)
destruct (fin_big_or_small pc);
(* No matter what, if it's past the end
of the array, we're done, *)
try (eexists (pc, _, acc); rewrite H; apply run_noswap_ok).
- (* It's not past the end of the array,
and the 'allowed' list is empty.
Evaluation fails. *)
destruct H as [f' Heq].
exists (pc, Datatypes.nil, acc).
rewrite Heq. apply run_noswap_fail. reflexivity.
- (* We're not past the end of the array. However,
adding a new valid index still guarantees
evaluation terminates. *)
specialize (IHis acc pc inp Hv) as [st' Hr].
apply add_pc_safe_run with st'. assumption.
Qed.
(*
(* It's not past the end of the array,
and we're in the inductive case on is. *)
destruct H as [pc' Heq].
destruct (Fin.eq_dec pc' a) eqn:Heq_dec.
+ (* This PC is allowed. *)
(* That must mean we have a non-empty list. *)
remember (nth inp pc') as h. destruct h as [op t].
(* Unfortunately, we can't do eexists at the top
level, since that will mean the final state
has to be the same for every op. *)
destruct op.
(* Addition. *)
{ destruct (IHis (M.add acc t) (FS pc') inp Hv) as [st' Htrans].
eexists. eapply run_noswap_trans.
rewrite Heq. apply step_noswap_add.
- symmetry. apply Heqh.
- simpl. rewrite Heq_dec. reflexivity.
- simpl. rewrite Heq_dec. apply Htrans. }
(* No-ops *)
{ destruct (IHis acc (FS pc') inp Hv) as [st' Htrans].
eexists. eapply run_noswap_trans.
rewrite Heq. apply step_noswap_nop with t.
- symmetry. apply Heqh.
- simpl. rewrite Heq_dec. reflexivity.
- simpl. rewrite Heq_dec. apply Htrans. }
(* Jump. *)
{ (* A little more interesting. We need to know that the jump is valid. *)
assert (Hv' : valid_inst (jmp, t) pc').
{ specialize (Hv pc'). rewrite <- Heqh in Hv. assumption. }
inversion Hv'.
(* Now, proceed as usual. *)
destruct (IHis acc f' inp Hv) as [st' Htrans].
eexists. eapply run_noswap_trans.
rewrite Heq. apply step_noswap_jmp with t.
- symmetry. apply Heqh.
- simpl. rewrite Heq_dec. reflexivity.
- apply H0.
- simpl. rewrite Heq_dec. apply Htrans. }
+ (* The top PC is not allowed. *)
specialize (IHis acc pc inp Hv) as [st' Hr].
apply add_pc_safe_run with st'. assumption. *)
Qed.
(* Stoppped here. *)
Admitted.
End DayEight.