2020-12-13 23:32:11 -08:00

#### 237 lines 9.1 KiB Coq Raw Blame History

 ```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. ``` ```Require Import Coq.Program.Wf. ``` ```Require Import Lia. ``` ``` ``` ```Module DayEight (Import M:Int). ``` ``` (* We need to coerce natural numbers into integers to add them. *) ``` ``` Parameter nat_to_t : nat -> t. ``` ``` (* We need a way to convert integers back into finite sets. *) ``` ``` Parameter clamp : forall {n}, t -> option (Fin.t n). ``` ``` ``` ``` Definition fin := Fin.t. ``` ``` ``` ``` (* The opcode of our instructions. *) ``` ``` Inductive opcode : Type := ``` ``` | add ``` ``` | nop ``` ``` | jmp. ``` ``` ``` ``` (* The result of running a program is either the accumulator ``` ``` or an infinite loop error. In the latter case, we return the ``` ``` set of instructions that we tried. *) ``` ``` Inductive run_result {n : nat} : Type := ``` ``` | Ok : t -> run_result ``` ``` | Fail : set (fin n) -> run_result. ``` ``` ``` ``` (* A single program state .*) ``` ``` Definition state n : Type := (fin (S n) * set (fin n) * t). ``` ``` ``` ``` (* An instruction is a pair of an opcode and an argument. *) ``` ``` Definition inst : Type := (opcode * t). ``` ``` (* An input is a bounded list of instructions. *) ``` ``` Definition input (n : nat) := VectorDef.t inst n. ``` ``` (* 'indices' represents the list of instruction ``` ``` addresses, which are used for calculating jumps. *) ``` ``` Definition indices (n : nat) := VectorDef.t (fin n) n. ``` ``` ``` ``` (* Change a jump to a nop, or a nop to a jump. *) ``` ``` 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. ``` ``` ``` ``` (* Compute a destination index that's valid. ``` ``` 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). ``` ``` ``` ``` (* Cast a fin n to a fin (S n). *) ``` ``` Fixpoint weaken_one {n} (f : fin n) : fin (S n) := ``` ``` match f with ``` ``` | F1 => F1 ``` ``` | FS f' => FS (weaken_one f') ``` ``` end. ``` ``` ``` ``` (* Convert a nat to fin. *) ``` ``` Fixpoint nat_to_fin (n : nat) : fin (S n) := ``` ``` match n with ``` ``` | O => F1 ``` ``` | S n' => FS (nat_to_fin n') ``` ``` end. ``` ``` ``` ``` (* A finite natural is either its maximum value (aka nat_to_fin n), ``` ``` or it's not thatbig, which means it can be cast down to ``` ``` a fin (pred n). *) ``` ``` 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. ``` ``` ``` ``` (* 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} : 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). ``` ``` ``` ``` Inductive stuck {n} : input n -> state n -> Prop := ``` ``` | stuck_prog : forall inp pc' v acc, ``` ``` ~ set_In pc' v -> stuck inp (weaken_one pc', v, acc). ``` ``` ``` ``` 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 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 ``` ``` | valid_inst_nop : forall t f f', ``` ``` valid_jump_t f t = Some f' -> valid_inst (nop, t) f ``` ``` | valid_inst_jmp : forall t f f', ``` ``` valid_jump_t f t = Some f' -> valid_inst (jmp, t) f. ``` ``` ``` ``` (* An input is valid if all its instructions are valid. *) ``` ``` Definition valid_input {n} (inp : input n) : Prop := forall (pc : fin n), ``` ``` valid_inst (nth inp pc) pc. ``` ``` ``` ``` Section ValidInput. ``` ``` Variable n : nat. ``` ``` Variable inp : input n. ``` ``` Hypothesis Hv : valid_input inp. ``` ``` ``` ``` Theorem valid_input_can_step : forall pc acc, exists pc' acc', ``` ``` step_noswap (nth inp pc) (pc, acc) (pc', acc'). ``` ``` Proof. ``` ``` 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. *) ``` ``` Theorem valid_input_progress : forall pc v acc, ``` ``` (pc = nat_to_fin n /\ done inp (pc, v, acc)) \/ ``` ``` (exists pcs, pc = weaken_one pcs /\ ``` ``` ((~ set_In pcs v /\ stuck inp (pc, v, acc)) \/ ``` ``` (exists pc' acc', set_In pcs v /\ ``` ``` step_noswap (nth inp pcs) (pcs, acc) (pc', acc')))). ``` ``` Proof. ``` ``` intros pc v acc. ``` ``` (* Have we reached the end? *) ``` ``` destruct (fin_big_or_small pc). ``` ``` (* We're at the end, so we're done. *) ``` ``` left. rewrite H. split. reflexivity. apply done_prog. ``` ``` (* We're not at the end. *) ``` ``` right. destruct H as [pcs H]. exists pcs. rewrite H. split. reflexivity. ``` ``` (* We're not at the end. Is the PC valid? *) ``` ``` destruct (set_In_dec Fin.eq_dec pcs v). ``` ``` - (* It is. *) ``` ``` right. ``` ``` 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. ``` ``` ``` ``` (* A valid input always terminates, either by getting to the end of the program, ``` ``` or by looping and thus getting stuck. *) ``` ``` Program Fixpoint valid_input_terminates (pc : fin (S n)) (v : set (fin n)) (acc : t) (Hnd : List.NoDup v) ``` ``` { 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) => _ ``` ``` | or_intror (ex_intro _ pcs (conj Hw w)) => ``` ``` match w with ``` ``` | 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 => _ ``` ``` end ``` ``` end ``` ``` end. ``` ``` 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. ``` ``` - destruct (Fin.eq_dec pcs a) eqn:Heq_dec. ``` ``` + simpl. rewrite Heq_dec. lia. ``` ``` + inversion Hnd; subst. ``` ``` inversion Hin. subst. exfalso. apply n0. auto. ``` ``` 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. ```