Add experimental formalization of (inefficient) solution.

This commit is contained in:
Danila Fedorin 2020-12-13 23:32:11 -08:00
parent 6ecae2b5bf
commit f53c65fb0d

110
day8.v
View File

@ -41,13 +41,17 @@ Module DayEight (Import M:Int).
Definition indices (n : nat) := VectorDef.t (fin n) n.
(* Change a jump to a nop, or a nop to a jump. *)
Definition replace (i : inst) : inst :=
Definition swap (i : inst) : inst :=
match i with
| (add, t) => (add, t)
| (nop, t) => (jmp, t)
| (jmp, t) => (nop, t)
end.
Inductive swappable : inst -> Prop :=
| swap_nop : forall t, swappable (nop, t)
| swap_jmp : forall t, swappable (jmp, t).
(* Compute the destination jump index, an integer. *)
Definition jump_t {n} (pc : fin n) (off : t) : t :=
M.add (nat_to_t (proj1_sig (to_nat pc))) off.
@ -96,20 +100,14 @@ Module DayEight (Import M:Int).
(* 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_add : forall inp pc' v acc t,
nth inp pc' = (add, t) ->
set_In pc' v ->
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_In pc' v ->
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_In pc' v ->
valid_jump_t pc' t = Some pc'' ->
step_noswap inp (weaken_one pc', v, acc) (pc'', set_remove Fin.eq_dec pc' v, acc).
Inductive step_noswap {n} : inst -> (fin n * t) -> (fin (S n) * t) -> Prop :=
| step_noswap_add : forall pc acc t,
step_noswap (add, t) (pc, acc) (FS pc, M.add acc t)
| step_noswap_nop : forall pc acc t,
step_noswap (nop, t) (pc, acc) (FS pc, acc)
| step_noswap_jmp : forall pc pc' acc t,
valid_jump_t pc t = Some pc' ->
step_noswap (jmp, t) (pc, acc) (pc', acc).
Inductive done {n} : input n -> state n -> Prop :=
| done_prog : forall inp v acc, done inp (nat_to_fin n, v, acc).
@ -121,8 +119,35 @@ Module DayEight (Import M:Int).
Inductive run_noswap {n} : input n -> state n -> state n -> Prop :=
| run_noswap_ok : forall inp st, done inp st -> run_noswap inp st st
| run_noswap_fail : forall inp st, stuck inp st -> run_noswap inp st st
| run_noswap_trans : forall inp st st' st'',
step_noswap inp st st' -> run_noswap inp st' st'' -> run_noswap inp st st''.
| run_noswap_trans : forall inp pc pc' v acc acc' st',
set_In pc v ->
step_noswap (nth inp pc) (pc, acc) (pc', acc') ->
run_noswap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
run_noswap inp (weaken_one pc, v, acc) st'.
Inductive run_swap {n} : input n -> state n -> state n -> Prop :=
| run_swap_normal : forall inp pc pc' v acc acc' st',
set_In pc v ->
~ swappable (nth inp pc) ->
step_noswap (nth inp pc) (pc, acc) (pc', acc') ->
run_swap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
run_swap inp (weaken_one pc, v, acc) st'
| run_swap_swapped_ok : forall inp pc pc' v acc acc' st',
set_In pc v ->
swappable (nth inp pc) ->
step_noswap (swap (nth inp pc)) (pc, acc) (pc', acc') ->
run_noswap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
done inp st' ->
run_swap inp (weaken_one pc, v, acc) st'
| run_swap_swapped_next : forall inp pc pc'w pc'n v acc acc'w acc'n st'w st'n,
set_In pc v ->
swappable (nth inp pc) ->
step_noswap (swap (nth inp pc)) (pc, acc) (pc'w, acc'w) ->
run_noswap inp (pc'w, set_remove Fin.eq_dec pc v, acc'w) st'w ->
stuck inp st'w ->
step_noswap (nth inp pc) (pc, acc) (pc'n, acc'n) ->
run_swap inp (pc'n, set_remove Fin.eq_dec pc v, acc'n) st'n ->
run_swap inp (weaken_one pc, v, acc) st'n.
Inductive valid_inst {n} : inst -> fin n -> Prop :=
| valid_inst_add : forall t f, valid_inst (add, t) f
@ -140,19 +165,16 @@ Module DayEight (Import M:Int).
Variable inp : input n.
Hypothesis Hv : valid_input inp.
(* If the current address, which is not the end of the array, is
present in the "allowed" set, the program can continue. *)
Lemma step_if_possible : forall pcs v acc,
set_In pcs v ->
exists pc' acc', step_noswap inp (weaken_one pcs, v, acc) (pc', set_remove Fin.eq_dec pcs v, acc').
Theorem valid_input_can_step : forall pc acc, exists pc' acc',
step_noswap (nth inp pc) (pc, acc) (pc', acc').
Proof.
intros pcs v acc Hin.
remember (nth inp pcs) as instr. destruct instr as [op t]. destruct op.
+ exists (FS pcs). exists (M.add acc t). apply step_noswap_add; auto.
+ exists (FS pcs). exists acc. apply step_noswap_nop with t; auto.
+ unfold valid_input in Hv. specialize (Hv pcs).
rewrite <- Heqinstr in Hv. inversion Hv; subst.
exists f'. exists acc. apply step_noswap_jmp with t; auto.
intros pc acc.
destruct (nth inp pc) eqn:Hop.
destruct o.
- exists (FS pc). exists (M.add acc t0). apply step_noswap_add.
- exists (FS pc). exists acc. eapply step_noswap_nop.
- specialize (Hv pc). rewrite Hop in Hv. inversion Hv; subst.
exists f'. exists acc. eapply step_noswap_jmp. apply H0.
Qed.
(* A program is either done, stuck (at an invalid/visited address), or can step. *)
@ -161,7 +183,7 @@ Module DayEight (Import M:Int).
(exists pcs, pc = weaken_one pcs /\
((~ set_In pcs v /\ stuck inp (pc, v, acc)) \/
(exists pc' acc', set_In pcs v /\
step_noswap inp (pc, v, acc) (pc', set_remove Fin.eq_dec pcs v, acc')))).
step_noswap (nth inp pcs) (pcs, acc) (pc', acc')))).
Proof.
intros pc v acc.
(* Have we reached the end? *)
@ -174,8 +196,8 @@ Module DayEight (Import M:Int).
destruct (set_In_dec Fin.eq_dec pcs v).
- (* It is. *)
right.
destruct (step_if_possible pcs v acc) as [pc' [acc' Hstep]]; auto.
exists pc'. exists acc'. split; auto.
destruct (valid_input_can_step pcs acc) as [pc' [acc' Hstep]].
exists pc'; exists acc'; auto.
- (* It is not. *)
left. split; auto. apply stuck_prog; auto.
Qed.
@ -186,30 +208,19 @@ Module DayEight (Import M:Int).
{ measure (length v) }:
(exists pc', run_noswap inp (pc, v, acc) pc') :=
match valid_input_progress pc v acc with
| or_introl (conj Heq Hdone) =>
inhabited_sig_to_exists
(inhabits
(@exist (state n)
(fun x => run_noswap inp (pc, v, acc) x) (pc, v, acc) (run_noswap_ok _ _ Hdone)))
| or_introl (conj Heq Hdone) => _
| or_intror (ex_intro _ pcs (conj Hw w)) =>
match w with
| or_introl (conj Hnin Hstuck) =>
inhabited_sig_to_exists
(inhabits
(@exist (state n)
(fun x => run_noswap inp (pc, v, acc) x) (pc, v, acc) (run_noswap_fail _ _ Hstuck)))
| or_introl (conj Hnin Hstuck) => _
| or_intror (ex_intro _ pc' (ex_intro _ acc' (conj Hin Hst))) =>
match valid_input_terminates pc' (set_remove Fin.eq_dec pcs v) acc' (set_remove_nodup Fin.eq_dec pcs Hnd) with
| ex_intro _ pc'' Hrun =>
inhabited_sig_to_exists
(inhabits
(@exist (state n)
(fun x => run_noswap inp (pc, v, acc) x) pc''
(run_noswap_trans _ _ (pc', set_remove Fin.eq_dec pcs v, acc') _ Hst Hrun)))
| ex_intro _ pc'' Hrun => _
end
end
end.
Obligation 1.
Obligation 1. eexists. apply run_noswap_ok. assumption. Qed.
Obligation 2. eexists. apply run_noswap_fail. assumption. Qed.
Obligation 3.
clear Heq_anonymous. clear valid_input_terminates. clear Hst.
induction v.
- inversion Hin.
@ -220,5 +231,6 @@ Module DayEight (Import M:Int).
specialize (IHv H2 H).
simpl. rewrite Heq_dec. simpl. lia.
Qed.
Obligation 4. eexists. eapply run_noswap_trans; auto. apply Hst. apply Hrun. Qed.
End ValidInput.
End DayEight.