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255 lines
9.7 KiB
255 lines
9.7 KiB
Require Import Coq.ZArith.Int.
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Require Import Coq.Lists.ListSet.
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Require Import Coq.Vectors.VectorDef.
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Require Import Coq.Vectors.Fin.
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Require Import Coq.Program.Equality.
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Require Import Coq.Logic.Eqdep_dec.
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Require Import Coq.Arith.Peano_dec.
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Require Import Coq.Program.Wf.
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Require Import Lia.
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Module DayEight (Import M:Int).
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(* We need to coerce natural numbers into integers to add them. *)
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Parameter nat_to_t : nat -> t.
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(* We need a way to convert integers back into finite sets. *)
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Parameter clamp : forall {n}, t -> option (Fin.t n).
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Definition fin := Fin.t.
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(* The opcode of our instructions. *)
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Inductive opcode : Type :=
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| add
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| nop
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| jmp.
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(* The result of running a program is either the accumulator
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or an infinite loop error. In the latter case, we return the
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set of instructions that we tried. *)
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Inductive run_result {n : nat} : Type :=
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| Ok : t -> run_result
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| Fail : set (fin n) -> run_result.
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(* A single program state .*)
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Definition state n : Type := (fin (S n) * set (fin n) * t).
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(* An instruction is a pair of an opcode and an argument. *)
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Definition inst : Type := (opcode * t).
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(* An input is a bounded list of instructions. *)
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Definition input (n : nat) := VectorDef.t inst n.
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(* 'indices' represents the list of instruction
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addresses, which are used for calculating jumps. *)
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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 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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(* Compute a destination index that's valid.
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Not all inputs are valid, so this may fail. *)
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Definition valid_jump_t {n} (pc : fin n) (off : t) : option (fin (S n)) := @clamp (S n) (jump_t pc off).
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(* Cast a fin n to a fin (S n). *)
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Fixpoint weaken_one {n} (f : fin n) : fin (S n) :=
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match f with
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| F1 => F1
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| FS f' => FS (weaken_one f')
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end.
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(* Convert a nat to fin. *)
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Fixpoint nat_to_fin (n : nat) : fin (S n) :=
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match n with
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| O => F1
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| S n' => FS (nat_to_fin n')
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end.
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(* A finite natural is either its maximum value (aka nat_to_fin n),
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or it's not thatbig, which means it can be cast down to
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a fin (pred 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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Proof.
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(* Hey, looks like the creator of Fin provided
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us with nice inductive principles. Using Coq's
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default `induction` breaks here.
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Merci, Pierre! *)
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apply Fin.rectS.
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- intros n. destruct n.
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+ left. reflexivity.
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+ right. exists F1. auto.
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- intros n p IH.
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destruct IH.
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+ left. rewrite H. reflexivity.
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+ right. destruct H as [f' Heq].
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exists (FS f'). simpl. rewrite Heq.
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reflexivity.
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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} : 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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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 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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| 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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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 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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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 /\
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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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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. *)
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right. destruct H as [pcs H]. exists pcs. rewrite H. split. reflexivity.
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(* We're not at the end. Is the PC valid? *)
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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 (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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Theorem list_length_induction {X : Type} (P : list X -> Prop) :
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(forall l, (forall l', length l' < length l -> P l') -> P l) ->
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forall l, P l.
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Proof.
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intros Hrec.
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assert (forall (l l' : list X), length l' <= length l -> P l').
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{ induction l; intros l' Hlen; apply Hrec; intros l'0 Hlen0.
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- simpl in Hlen. lia.
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- apply IHl. simpl in Hlen. lia. }
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intros l. apply H with l. lia.
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Qed.
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Theorem set_remove_length : forall (f : fin n) (s : set (fin n)),
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set_In f s ->
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length (set_remove Fin.eq_dec f s) < length s.
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Proof.
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intros f s Hin.
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induction s.
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- inversion Hin.
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- simpl. destruct (Fin.eq_dec f a) eqn:Heq.
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+ unfold lt. apply le_n. (* Why couldn't lia get this one? *)
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+ inversion Hin; subst. exfalso. apply n0. auto.
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apply IHs in H. simpl. lia.
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Qed.
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Theorem valid_input_terminates : forall (pc : fin (S n)) (v : set (fin n)) (acc : t),
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(exists pc', run_noswap inp (pc, v, acc) pc').
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Proof.
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intros pc v. generalize dependent pc.
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induction v using list_length_induction.
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intros pc acc.
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destruct (valid_input_progress pc l acc) as [[_ Hd]|[pc' [Hw [[_ Hst]|[pc'' [acc'' [Hin Hst]]]]]]].
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- (* We're done. *)
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eexists. apply run_noswap_ok. assumption.
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- (* We're stuck. *)
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eexists. apply run_noswap_fail. assumption.
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- (* We can make a step. This will remove our current PC from the valid list, *)
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edestruct (H (set_remove Fin.eq_dec pc' l)).
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(* Since the PC must be in the list, removing it makes the list smaller. *)
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apply (set_remove_length _ _ Hin).
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(* Without the current PC, our valid set shrinks.
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Since this is the inductive step, we have assumed
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that programs with smaller sets of valid PCs always
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terminate. Thus, after we make the step, we're done. *)
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exists x. subst. eapply run_noswap_trans.
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+ auto.
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+ apply Hst.
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+ apply H0.
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Qed.
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End ValidInput.
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End DayEight.
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