Add experimental formalization of (inefficient) solution.
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day8.v
110
day8.v
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@ -41,13 +41,17 @@ Module DayEight (Import M:Int).
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Definition indices (n : nat) := VectorDef.t (fin n) n.
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(* Change a jump to a nop, or a nop to a jump. *)
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Definition replace (i : inst) : inst :=
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Definition swap (i : inst) : inst :=
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match i with
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| (add, t) => (add, t)
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| (nop, t) => (jmp, t)
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| (jmp, t) => (nop, t)
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end.
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Inductive swappable : inst -> Prop :=
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| swap_nop : forall t, swappable (nop, t)
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| swap_jmp : forall t, swappable (jmp, t).
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(* Compute the destination jump index, an integer. *)
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Definition jump_t {n} (pc : fin n) (off : t) : t :=
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M.add (nat_to_t (proj1_sig (to_nat pc))) off.
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@ -96,20 +100,14 @@ Module DayEight (Import M:Int).
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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 step_noswap {n} : inst -> (fin n * t) -> (fin (S n) * t) -> Prop :=
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| step_noswap_add : forall pc acc t,
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step_noswap (add, t) (pc, acc) (FS pc, M.add acc t)
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| step_noswap_nop : forall pc acc t,
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step_noswap (nop, t) (pc, acc) (FS pc, acc)
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| step_noswap_jmp : forall pc pc' acc t,
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valid_jump_t pc t = Some pc' ->
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step_noswap (jmp, t) (pc, acc) (pc', 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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@ -121,8 +119,35 @@ Module DayEight (Import M:Int).
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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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| run_noswap_trans : forall inp pc pc' v acc acc' st',
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set_In pc v ->
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step_noswap (nth inp pc) (pc, acc) (pc', acc') ->
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run_noswap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
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run_noswap inp (weaken_one pc, v, acc) st'.
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Inductive run_swap {n} : input n -> state n -> state n -> Prop :=
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| run_swap_normal : forall inp pc pc' v acc acc' st',
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set_In pc v ->
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~ swappable (nth inp pc) ->
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step_noswap (nth inp pc) (pc, acc) (pc', acc') ->
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run_swap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
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run_swap inp (weaken_one pc, v, acc) st'
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| run_swap_swapped_ok : forall inp pc pc' v acc acc' st',
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set_In pc v ->
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swappable (nth inp pc) ->
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step_noswap (swap (nth inp pc)) (pc, acc) (pc', acc') ->
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run_noswap inp (pc', set_remove Fin.eq_dec pc v, acc') st' ->
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done inp st' ->
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run_swap inp (weaken_one pc, v, acc) st'
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| run_swap_swapped_next : forall inp pc pc'w pc'n v acc acc'w acc'n st'w st'n,
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set_In pc v ->
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swappable (nth inp pc) ->
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step_noswap (swap (nth inp pc)) (pc, acc) (pc'w, acc'w) ->
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run_noswap inp (pc'w, set_remove Fin.eq_dec pc v, acc'w) st'w ->
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stuck inp st'w ->
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step_noswap (nth inp pc) (pc, acc) (pc'n, acc'n) ->
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run_swap inp (pc'n, set_remove Fin.eq_dec pc v, acc'n) st'n ->
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run_swap inp (weaken_one pc, v, acc) st'n.
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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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@ -140,19 +165,16 @@ Module DayEight (Import M:Int).
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Variable inp : input n.
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Hypothesis Hv : valid_input inp.
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(* If the current address, which is not the end of the array, is
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present in the "allowed" set, the program can continue. *)
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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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Theorem valid_input_can_step : forall pc acc, exists pc' acc',
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step_noswap (nth inp pc) (pc, acc) (pc', 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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intros pc acc.
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destruct (nth inp pc) eqn:Hop.
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destruct o.
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- exists (FS pc). exists (M.add acc t0). apply step_noswap_add.
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- exists (FS pc). exists acc. eapply step_noswap_nop.
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- specialize (Hv pc). rewrite Hop in Hv. inversion Hv; subst.
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exists f'. exists acc. eapply step_noswap_jmp. apply H0.
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Qed.
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(* A program is either done, stuck (at an invalid/visited address), or can step. *)
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@ -161,7 +183,7 @@ Module DayEight (Import M:Int).
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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 /\
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step_noswap inp (pc, v, acc) (pc', set_remove Fin.eq_dec pcs v, acc')))).
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step_noswap (nth inp pcs) (pcs, acc) (pc', 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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@ -174,8 +196,8 @@ Module DayEight (Import M:Int).
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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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destruct (valid_input_can_step pcs acc) as [pc' [acc' Hstep]].
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exists pc'; exists acc'; auto.
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- (* It is not. *)
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left. split; auto. apply stuck_prog; auto.
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Qed.
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@ -186,30 +208,19 @@ Module DayEight (Import M:Int).
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{ measure (length v) }:
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(exists pc', run_noswap inp (pc, v, acc) pc') :=
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match valid_input_progress pc v acc with
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| or_introl (conj Heq Hdone) =>
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inhabited_sig_to_exists
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(inhabits
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(@exist (state n)
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(fun x => run_noswap inp (pc, v, acc) x) (pc, v, acc) (run_noswap_ok _ _ Hdone)))
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| or_introl (conj Heq Hdone) => _
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| or_intror (ex_intro _ pcs (conj Hw w)) =>
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match w with
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| or_introl (conj Hnin Hstuck) =>
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inhabited_sig_to_exists
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(inhabits
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(@exist (state n)
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(fun x => run_noswap inp (pc, v, acc) x) (pc, v, acc) (run_noswap_fail _ _ Hstuck)))
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| or_introl (conj Hnin Hstuck) => _
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| or_intror (ex_intro _ pc' (ex_intro _ acc' (conj Hin Hst))) =>
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match valid_input_terminates pc' (set_remove Fin.eq_dec pcs v) acc' (set_remove_nodup Fin.eq_dec pcs Hnd) with
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| ex_intro _ pc'' Hrun =>
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inhabited_sig_to_exists
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(inhabits
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(@exist (state n)
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(fun x => run_noswap inp (pc, v, acc) x) pc''
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(run_noswap_trans _ _ (pc', set_remove Fin.eq_dec pcs v, acc') _ Hst Hrun)))
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| ex_intro _ pc'' Hrun => _
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end
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end
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end.
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Obligation 1.
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Obligation 1. eexists. apply run_noswap_ok. assumption. Qed.
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Obligation 2. eexists. apply run_noswap_fail. assumption. Qed.
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Obligation 3.
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clear Heq_anonymous. clear valid_input_terminates. clear Hst.
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induction v.
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- inversion Hin.
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@ -220,5 +231,6 @@ Module DayEight (Import M:Int).
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specialize (IHv H2 H).
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simpl. rewrite Heq_dec. simpl. lia.
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Qed.
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Obligation 4. eexists. eapply run_noswap_trans; auto. apply Hst. apply Hrun. Qed.
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End ValidInput.
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End DayEight.
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