Module Coqlib


This file collects a number of definitions and theorems that are used throughout the development. It complements the Coq standard library.

Require Export ZArith.
Require Export Znumtheory.
Require Export List.
Require Export Bool.

Useful tactics


Ltac inv H := inversion H; clear H; subst.

Ltac predSpec pred predspec x y :=
  generalize (predspec x y); case (pred x y); intro.

Ltac caseEq name :=
  generalize (refl_equal name); pattern name at -1 in |- *; case name.

Ltac destructEq name :=
  destruct name eqn:?.

Ltac decEq :=
  match goal with
  | [ |- _ = _ ] => f_equal
  | [ |- (?X ?A <> ?X ?B) ] =>
      cut (A <> B); [intro; congruence | try discriminate]
  end.

Ltac byContradiction :=
  cut False; [contradiction|idtac].

Ltac omegaContradiction :=
  cut False; [contradiction|omega].

Lemma modusponens: forall (P Q: Prop), P -> (P -> Q) -> Q.
Proof.
auto. Qed.

Ltac exploit x :=
    refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _ _) _)
 || refine (modusponens _ _ (x _ _ _) _)
 || refine (modusponens _ _ (x _ _) _)
 || refine (modusponens _ _ (x _) _).

Definitions and theorems over the type positive


Definition peq: forall (x y: positive), {x = y} + {x <> y} := Pos.eq_dec.
Global Opaque peq.

Lemma peq_true:
  forall (A: Type) (x: positive) (a b: A), (if peq x x then a else b) = a.
Proof.
  intros. case (peq x x); intros.
  auto.
  elim n; auto.
Qed.

Lemma peq_false:
  forall (A: Type) (x y: positive) (a b: A), x <> y -> (if peq x y then a else b) = b.
Proof.
  intros. case (peq x y); intros.
  elim H; auto.
  auto.
Qed.

Definition Plt: positive -> positive -> Prop := Pos.lt.

Lemma Plt_ne:
  forall (x y: positive), Plt x y -> x <> y.
Proof.
  unfold Plt; intros. red; intro. subst y. eelim Pos.lt_irrefl; eauto.
Qed.
Hint Resolve Plt_ne: coqlib.

Lemma Plt_trans:
  forall (x y z: positive), Plt x y -> Plt y z -> Plt x z.
Proof (Pos.lt_trans).

Lemma Plt_succ:
  forall (x: positive), Plt x (Psucc x).
Proof.
  unfold Plt; intros. apply Pos.lt_succ_r. apply Pos.le_refl.
Qed.
Hint Resolve Plt_succ: coqlib.

Lemma Plt_trans_succ:
  forall (x y: positive), Plt x y -> Plt x (Psucc y).
Proof.
  intros. apply Plt_trans with y. assumption. apply Plt_succ.
Qed.
Hint Resolve Plt_succ: coqlib.

Lemma Plt_succ_inv:
  forall (x y: positive), Plt x (Psucc y) -> Plt x y \/ x = y.
Proof.
  unfold Plt; intros. rewrite Pos.lt_succ_r in H.
  apply Pos.le_lteq; auto.
Qed.

Definition plt (x y: positive) : {Plt x y} + {~ Plt x y}.
Proof.
  unfold Plt, Pos.lt; intros. destruct (Pos.compare x y).
  - right; congruence.
  - left; auto.
  - right; congruence.
Defined.
Global Opaque plt.

Definition Ple: positive -> positive -> Prop := Pos.le.

Lemma Ple_refl: forall (p: positive), Ple p p.
Proof (Pos.le_refl).

Lemma Ple_trans: forall (p q r: positive), Ple p q -> Ple q r -> Ple p r.
Proof (Pos.le_trans).

Lemma Plt_Ple: forall (p q: positive), Plt p q -> Ple p q.
Proof (Pos.lt_le_incl).

Lemma Ple_succ: forall (p: positive), Ple p (Psucc p).
Proof.
  intros. apply Plt_Ple. apply Plt_succ.
Qed.

Lemma Plt_Ple_trans:
  forall (p q r: positive), Plt p q -> Ple q r -> Plt p r.
Proof (Pos.lt_le_trans).

Lemma Plt_strict: forall p, ~ Plt p p.
Proof (Pos.lt_irrefl).

Hint Resolve Ple_refl Plt_Ple Ple_succ Plt_strict: coqlib.

Ltac xomega := unfold Plt, Ple in *; zify; omega.
Ltac xomegaContradiction := exfalso; xomega.

Peano recursion over positive numbers.

Section POSITIVE_ITERATION.

Lemma Plt_wf: well_founded Plt.
Proof.
  apply well_founded_lt_compat with nat_of_P.
  intros. apply nat_of_P_lt_Lt_compare_morphism. exact H.
Qed.

Variable A: Type.
Variable v1: A.
Variable f: positive -> A -> A.

Lemma Ppred_Plt:
  forall x, x <> xH -> Plt (Ppred x) x.
Proof.
  intros. elim (Psucc_pred x); intro. contradiction.
  set (y := Ppred x) in *. rewrite <- H0. apply Plt_succ.
Qed.

Let iter (x: positive) (P: forall y, Plt y x -> A) : A :=
  match peq x xH with
  | left EQ => v1
  | right NOTEQ => f (Ppred x) (P (Ppred x) (Ppred_Plt x NOTEQ))
  end.

Definition positive_rec : positive -> A :=
  Fix Plt_wf (fun _ => A) iter.

Lemma unroll_positive_rec:
  forall x,
  positive_rec x = iter x (fun y _ => positive_rec y).
Proof.
  unfold positive_rec. apply (Fix_eq Plt_wf (fun _ => A) iter).
  intros. unfold iter. case (peq x 1); intro. auto. decEq. apply H.
Qed.

Lemma positive_rec_base:
  positive_rec 1%positive = v1.
Proof.
  rewrite unroll_positive_rec. unfold iter. case (peq 1 1); intro.
  auto. elim n; auto.
Qed.

Lemma positive_rec_succ:
  forall x, positive_rec (Psucc x) = f x (positive_rec x).
Proof.
  intro. rewrite unroll_positive_rec. unfold iter.
  case (peq (Psucc x) 1); intro.
  destruct x; simpl in e; discriminate.
  rewrite Ppred_succ. auto.
Qed.

Lemma positive_Peano_ind:
  forall (P: positive -> Prop),
  P xH ->
  (forall x, P x -> P (Psucc x)) ->
  forall x, P x.
Proof.
  intros.
  apply (well_founded_ind Plt_wf P).
  intros.
  case (peq x0 xH); intro.
  subst x0; auto.
  elim (Psucc_pred x0); intro. contradiction. rewrite <- H2.
  apply H0. apply H1. apply Ppred_Plt. auto.
Qed.

End POSITIVE_ITERATION.

Definitions and theorems over the type Z


Definition zeq: forall (x y: Z), {x = y} + {x <> y} := Z.eq_dec.

Lemma zeq_true:
  forall (A: Type) (x: Z) (a b: A), (if zeq x x then a else b) = a.
Proof.
  intros. case (zeq x x); intros.
  auto.
  elim n; auto.
Qed.

Lemma zeq_false:
  forall (A: Type) (x y: Z) (a b: A), x <> y -> (if zeq x y then a else b) = b.
Proof.
  intros. case (zeq x y); intros.
  elim H; auto.
  auto.
Qed.

Open Scope Z_scope.

Definition zlt: forall (x y: Z), {x < y} + {x >= y} := Z_lt_dec.

Lemma zlt_true:
  forall (A: Type) (x y: Z) (a b: A),
  x < y -> (if zlt x y then a else b) = a.
Proof.
  intros. case (zlt x y); intros.
  auto.
  omegaContradiction.
Qed.

Lemma zlt_false:
  forall (A: Type) (x y: Z) (a b: A),
  x >= y -> (if zlt x y then a else b) = b.
Proof.
  intros. case (zlt x y); intros.
  omegaContradiction.
  auto.
Qed.

Definition zle: forall (x y: Z), {x <= y} + {x > y} := Z_le_gt_dec.

Lemma zle_true:
  forall (A: Type) (x y: Z) (a b: A),
  x <= y -> (if zle x y then a else b) = a.
Proof.
  intros. case (zle x y); intros.
  auto.
  omegaContradiction.
Qed.

Lemma zle_false:
  forall (A: Type) (x y: Z) (a b: A),
  x > y -> (if zle x y then a else b) = b.
Proof.
  intros. case (zle x y); intros.
  omegaContradiction.
  auto.
Qed.

Properties of powers of two.

Lemma two_power_nat_O : two_power_nat O = 1.
Proof.
reflexivity. Qed.

Lemma two_power_nat_pos : forall n : nat, two_power_nat n > 0.
Proof.
  induction n. rewrite two_power_nat_O. omega.
  rewrite two_power_nat_S. omega.
Qed.

Lemma two_power_nat_two_p:
  forall x, two_power_nat x = two_p (Z_of_nat x).
Proof.
  induction x. auto.
  rewrite two_power_nat_S. rewrite inj_S. rewrite two_p_S. omega. omega.
Qed.

Lemma two_p_monotone:
  forall x y, 0 <= x <= y -> two_p x <= two_p y.
Proof.
  intros.
  replace (two_p x) with (two_p x * 1) by omega.
  replace y with (x + (y - x)) by omega.
  rewrite two_p_is_exp; try omega.
  apply Zmult_le_compat_l.
  assert (two_p (y - x) > 0). apply two_p_gt_ZERO. omega. omega.
  assert (two_p x > 0). apply two_p_gt_ZERO. omega. omega.
Qed.

Lemma two_p_monotone_strict:
  forall x y, 0 <= x < y -> two_p x < two_p y.
Proof.
  intros. assert (two_p x <= two_p (y - 1)). apply two_p_monotone; omega.
  assert (two_p (y - 1) > 0). apply two_p_gt_ZERO. omega.
  replace y with (Zsucc (y - 1)) by omega. rewrite two_p_S. omega. omega.
Qed.

Lemma two_p_strict:
  forall x, x >= 0 -> x < two_p x.
Proof.
  intros x0 GT. pattern x0. apply natlike_ind.
  simpl. omega.
  intros. rewrite two_p_S; auto. generalize (two_p_gt_ZERO x H). omega.
  omega.
Qed.

Lemma two_p_strict_2:
  forall x, x >= 0 -> 2 * x - 1 < two_p x.
Proof.
  intros. assert (x = 0 \/ x - 1 >= 0) by omega. destruct H0.
  subst. vm_compute. auto.
  replace (two_p x) with (2 * two_p (x - 1)).
  generalize (two_p_strict _ H0). omega.
  rewrite <- two_p_S. decEq. omega. omega.
Qed.

Properties of Zmin and Zmax

Lemma Zmin_spec:
  forall x y, Zmin x y = if zlt x y then x else y.
Proof.
  intros. case (zlt x y); unfold Zlt, Zge; intro z.
  unfold Zmin. rewrite z. auto.
  unfold Zmin. caseEq (x ?= y); intro.
  apply Zcompare_Eq_eq. auto.
  contradiction.
  reflexivity.
Qed.

Lemma Zmax_spec:
  forall x y, Zmax x y = if zlt y x then x else y.
Proof.
  intros. case (zlt y x); unfold Zlt, Zge; intro z.
  unfold Zmax. rewrite <- (Zcompare_antisym y x).
  rewrite z. simpl. auto.
  unfold Zmax. rewrite <- (Zcompare_antisym y x).
  caseEq (y ?= x); intro; simpl.
  symmetry. apply Zcompare_Eq_eq. auto.
  contradiction. reflexivity.
Qed.

Lemma Zmax_bound_l:
  forall x y z, x <= y -> x <= Zmax y z.
Proof.
  intros. generalize (Zmax1 y z). omega.
Qed.
Lemma Zmax_bound_r:
  forall x y z, x <= z -> x <= Zmax y z.
Proof.
  intros. generalize (Zmax2 y z). omega.
Qed.

Properties of Euclidean division and modulus.

Lemma Zdiv_small:
  forall x y, 0 <= x < y -> x / y = 0.
Proof.
  intros. assert (y > 0). omega.
  assert (forall a b,
    0 <= a < y ->
    0 <= y * b + a < y ->
    b = 0).
  intros.
  assert (b = 0 \/ b > 0 \/ (-b) > 0). omega.
  elim H3; intro.
  auto.
  elim H4; intro.
  assert (y * b >= y * 1). apply Zmult_ge_compat_l. omega. omega.
  omegaContradiction.
  assert (y * (-b) >= y * 1). apply Zmult_ge_compat_l. omega. omega.
  rewrite <- Zopp_mult_distr_r in H6. omegaContradiction.
  apply H1 with (x mod y).
  apply Z_mod_lt. auto.
  rewrite <- Z_div_mod_eq. auto. auto.
Qed.

Lemma Zmod_small:
  forall x y, 0 <= x < y -> x mod y = x.
Proof.
  intros. assert (y > 0). omega.
  generalize (Z_div_mod_eq x y H0).
  rewrite (Zdiv_small x y H). omega.
Qed.

Lemma Zmod_unique:
  forall x y a b,
  x = a * y + b -> 0 <= b < y -> x mod y = b.
Proof.
  intros. subst x. rewrite Zplus_comm.
  rewrite Z_mod_plus. apply Zmod_small. auto. omega.
Qed.

Lemma Zdiv_unique:
  forall x y a b,
  x = a * y + b -> 0 <= b < y -> x / y = a.
Proof.
  intros. subst x. rewrite Zplus_comm.
  rewrite Z_div_plus. rewrite (Zdiv_small b y H0). omega. omega.
Qed.

Lemma Zdiv_Zdiv:
  forall a b c,
  b > 0 -> c > 0 -> (a / b) / c = a / (b * c).
Proof.
  intros.
  generalize (Z_div_mod_eq a b H). generalize (Z_mod_lt a b H). intros.
  generalize (Z_div_mod_eq (a/b) c H0). generalize (Z_mod_lt (a/b) c H0). intros.
  set (q1 := a / b) in *. set (r1 := a mod b) in *.
  set (q2 := q1 / c) in *. set (r2 := q1 mod c) in *.
  symmetry. apply Zdiv_unique with (r2 * b + r1).
  rewrite H2. rewrite H4. ring.
  split.
  assert (0 <= r2 * b). apply Zmult_le_0_compat. omega. omega. omega.
  assert ((r2 + 1) * b <= c * b).
  apply Zmult_le_compat_r. omega. omega.
  replace ((r2 + 1) * b) with (r2 * b + b) in H5 by ring.
  replace (c * b) with (b * c) in H5 by ring.
  omega.
Qed.

Lemma Zmult_le_compat_l_neg :
  forall n m p:Z, n >= m -> p <= 0 -> p * n <= p * m.
Proof.
  intros.
  assert ((-p) * n >= (-p) * m). apply Zmult_ge_compat_l. auto. omega.
  replace (p * n) with (- ((-p) * n)) by ring.
  replace (p * m) with (- ((-p) * m)) by ring.
  omega.
Qed.

Lemma Zdiv_interval_1:
  forall lo hi a b,
  lo <= 0 -> hi > 0 -> b > 0 ->
  lo * b <= a < hi * b ->
  lo <= a/b < hi.
Proof.
  intros.
  generalize (Z_div_mod_eq a b H1). generalize (Z_mod_lt a b H1). intros.
  set (q := a/b) in *. set (r := a mod b) in *.
  split.
  assert (lo < (q + 1)).
  apply Zmult_lt_reg_r with b. omega.
  apply Zle_lt_trans with a. omega.
  replace ((q + 1) * b) with (b * q + b) by ring.
  omega.
  omega.
  apply Zmult_lt_reg_r with b. omega.
  replace (q * b) with (b * q) by ring.
  omega.
Qed.

Lemma Zdiv_interval_2:
  forall lo hi a b,
  lo <= a <= hi -> lo <= 0 -> hi >= 0 -> b > 0 ->
  lo <= a/b <= hi.
Proof.
  intros.
  assert (lo <= a / b < hi+1).
  apply Zdiv_interval_1. omega. omega. auto.
  assert (lo * b <= lo * 1). apply Zmult_le_compat_l_neg. omega. omega.
  replace (lo * 1) with lo in H3 by ring.
  assert ((hi + 1) * 1 <= (hi + 1) * b). apply Zmult_le_compat_l. omega. omega.
  replace ((hi + 1) * 1) with (hi + 1) in H4 by ring.
  omega.
  omega.
Qed.

Lemma Zmod_recombine:
  forall x a b,
  a > 0 -> b > 0 ->
  x mod (a * b) = ((x/b) mod a) * b + (x mod b).
Proof.
  intros.
  set (xb := x/b).
  apply Zmod_unique with (xb/a).
  generalize (Z_div_mod_eq x b H0); fold xb; intro EQ1.
  generalize (Z_div_mod_eq xb a H); intro EQ2.
  rewrite EQ2 in EQ1.
  eapply trans_eq. eexact EQ1. ring.
  generalize (Z_mod_lt x b H0). intro.
  generalize (Z_mod_lt xb a H). intro.
  assert (0 <= xb mod a * b <= a * b - b).
    split. apply Zmult_le_0_compat; omega.
    replace (a * b - b) with ((a - 1) * b) by ring.
    apply Zmult_le_compat; omega.
  omega.
Qed.

Properties of divisibility.

Lemma Zdivides_trans:
  forall x y z, (x | y) -> (y | z) -> (x | z).
Proof.
  intros x y z [a A] [b B]; subst. exists (a*b); ring.
Qed.

Definition Zdivide_dec:
  forall (p q: Z), p > 0 -> { (p|q) } + { ~(p|q) }.
Proof.
  intros. destruct (zeq (Zmod q p) 0).
  left. exists (q / p).
  transitivity (p * (q / p) + (q mod p)). apply Z_div_mod_eq; auto.
  transitivity (p * (q / p)). omega. ring.
  right; red; intros. elim n. apply Z_div_exact_1; auto.
  inv H0. rewrite Z_div_mult; auto. ring.
Defined.
Global Opaque Zdivide_dec.

Lemma Zdivide_interval:
  forall a b c,
  0 < c -> 0 <= a < b -> (c | a) -> (c | b) -> 0 <= a <= b - c.
Proof.
  intros. destruct H1 as [x EQ1]. destruct H2 as [y EQ2]. subst. destruct H0.
  split. omega. exploit Zmult_lt_reg_r; eauto. intros.
  replace (y * c - c) with ((y - 1) * c) by ring.
  apply Zmult_le_compat_r; omega.
Qed.

Conversion from Z to nat.

Definition nat_of_Z: Z -> nat := Z.to_nat.

Lemma nat_of_Z_of_nat:
  forall n, nat_of_Z (Z_of_nat n) = n.
Proof.
  exact Nat2Z.id.
Qed.

Lemma nat_of_Z_max:
  forall z, Z_of_nat (nat_of_Z z) = Zmax z 0.
Proof.
  intros. unfold Zmax. destruct z; simpl; auto.
  change (Z.of_nat (Z.to_nat (Zpos p)) = Zpos p).
  apply Z2Nat.id. compute; intuition congruence.
Qed.

Lemma nat_of_Z_eq:
  forall z, z >= 0 -> Z_of_nat (nat_of_Z z) = z.
Proof.
  unfold nat_of_Z; intros. apply Z2Nat.id. omega.
Qed.

Lemma nat_of_Z_neg:
  forall n, n <= 0 -> nat_of_Z n = O.
Proof.
  destruct n; unfold Zle; simpl; auto. congruence.
Qed.

Lemma nat_of_Z_plus:
  forall p q,
  p >= 0 -> q >= 0 ->
  nat_of_Z (p + q) = (nat_of_Z p + nat_of_Z q)%nat.
Proof.
  unfold nat_of_Z; intros. apply Z2Nat.inj_add; omega.
Qed.


Alignment: align n amount returns the smallest multiple of amount greater than or equal to n.

Definition align (n: Z) (amount: Z) :=
  ((n + amount - 1) / amount) * amount.

Lemma align_le: forall x y, y > 0 -> x <= align x y.
Proof.
  intros. unfold align.
  generalize (Z_div_mod_eq (x + y - 1) y H). intro.
  replace ((x + y - 1) / y * y)
     with ((x + y - 1) - (x + y - 1) mod y).
  generalize (Z_mod_lt (x + y - 1) y H). omega.
  rewrite Zmult_comm. omega.
Qed.

Lemma align_divides: forall x y, y > 0 -> (y | align x y).
Proof.
  intros. unfold align. apply Zdivide_factor_l.
Qed.

Definitions and theorems on the data types option, sum and list


Set Implicit Arguments.

Comparing option types.

Definition option_eq (A: Type) (eqA: forall (x y: A), {x=y} + {x<>y}):
  forall (x y: option A), {x=y} + {x<>y}.
Proof.
decide equality. Defined.
Global Opaque option_eq.

Mapping a function over an option type.

Definition option_map (A B: Type) (f: A -> B) (x: option A) : option B :=
  match x with
  | None => None
  | Some y => Some (f y)
  end.

Mapping a function over a sum type.

Definition sum_left_map (A B C: Type) (f: A -> B) (x: A + C) : B + C :=
  match x with
  | inl y => inl C (f y)
  | inr z => inr B z
  end.

Properties of List.nth (n-th element of a list).

Hint Resolve in_eq in_cons: coqlib.

Lemma nth_error_in:
  forall (A: Type) (n: nat) (l: list A) (x: A),
  List.nth_error l n = Some x -> In x l.
Proof.
  induction n; simpl.
   destruct l; intros.
    discriminate.
    injection H; intro; subst a. apply in_eq.
   destruct l; intros.
    discriminate.
    apply in_cons. auto.
Qed.
Hint Resolve nth_error_in: coqlib.

Lemma nth_error_nil:
  forall (A: Type) (idx: nat), nth_error (@nil A) idx = None.
Proof.
  induction idx; simpl; intros; reflexivity.
Qed.
Hint Resolve nth_error_nil: coqlib.

Compute the length of a list, with result in Z.

Fixpoint list_length_z_aux (A: Type) (l: list A) (acc: Z) {struct l}: Z :=
  match l with
  | nil => acc
  | hd :: tl => list_length_z_aux tl (Zsucc acc)
  end.

Remark list_length_z_aux_shift:
  forall (A: Type) (l: list A) n m,
  list_length_z_aux l n = list_length_z_aux l m + (n - m).
Proof.
  induction l; intros; simpl.
  omega.
  replace (n - m) with (Zsucc n - Zsucc m) by omega. auto.
Qed.

Definition list_length_z (A: Type) (l: list A) : Z :=
  list_length_z_aux l 0.

Lemma list_length_z_cons:
  forall (A: Type) (hd: A) (tl: list A),
  list_length_z (hd :: tl) = list_length_z tl + 1.
Proof.
  intros. unfold list_length_z. simpl.
  rewrite (list_length_z_aux_shift tl 1 0). omega.
Qed.

Lemma list_length_z_pos:
  forall (A: Type) (l: list A),
  list_length_z l >= 0.
Proof.
  induction l; simpl. unfold list_length_z; simpl. omega.
  rewrite list_length_z_cons. omega.
Qed.

Lemma list_length_z_map:
  forall (A B: Type) (f: A -> B) (l: list A),
  list_length_z (map f l) = list_length_z l.
Proof.
  induction l. reflexivity. simpl. repeat rewrite list_length_z_cons. congruence.
Qed.

Extract the n-th element of a list, as List.nth_error does, but the index n is of type Z.

Fixpoint list_nth_z (A: Type) (l: list A) (n: Z) {struct l}: option A :=
  match l with
  | nil => None
  | hd :: tl => if zeq n 0 then Some hd else list_nth_z tl (Zpred n)
  end.

Lemma list_nth_z_in:
  forall (A: Type) (l: list A) n x,
  list_nth_z l n = Some x -> In x l.
Proof.
  induction l; simpl; intros.
  congruence.
  destruct (zeq n 0). left; congruence. right; eauto.
Qed.

Lemma list_nth_z_map:
  forall (A B: Type) (f: A -> B) (l: list A) n,
  list_nth_z (List.map f l) n = option_map f (list_nth_z l n).
Proof.
  induction l; simpl; intros.
  auto.
  destruct (zeq n 0). auto. eauto.
Qed.

Lemma list_nth_z_range:
  forall (A: Type) (l: list A) n x,
  list_nth_z l n = Some x -> 0 <= n < list_length_z l.
Proof.
  induction l; simpl; intros.
  discriminate.
  rewrite list_length_z_cons. destruct (zeq n 0).
  generalize (list_length_z_pos l); omega.
  exploit IHl; eauto. unfold Zpred. omega.
Qed.

Properties of List.incl (list inclusion).

Lemma incl_cons_inv:
  forall (A: Type) (a: A) (b c: list A),
  incl (a :: b) c -> incl b c.
Proof.
  unfold incl; intros. apply H. apply in_cons. auto.
Qed.
Hint Resolve incl_cons_inv: coqlib.

Lemma incl_app_inv_l:
  forall (A: Type) (l1 l2 m: list A),
  incl (l1 ++ l2) m -> incl l1 m.
Proof.
  unfold incl; intros. apply H. apply in_or_app. left; assumption.
Qed.

Lemma incl_app_inv_r:
  forall (A: Type) (l1 l2 m: list A),
  incl (l1 ++ l2) m -> incl l2 m.
Proof.
  unfold incl; intros. apply H. apply in_or_app. right; assumption.
Qed.

Hint Resolve incl_tl incl_refl incl_app_inv_l incl_app_inv_r: coqlib.

Lemma incl_same_head:
  forall (A: Type) (x: A) (l1 l2: list A),
  incl l1 l2 -> incl (x::l1) (x::l2).
Proof.
  intros; red; simpl; intros. intuition.
Qed.

Properties of List.map (mapping a function over a list).

Lemma list_map_exten:
  forall (A B: Type) (f f': A -> B) (l: list A),
  (forall x, In x l -> f x = f' x) ->
  List.map f' l = List.map f l.
Proof.
  induction l; simpl; intros.
  reflexivity.
  rewrite <- H. rewrite IHl. reflexivity.
  intros. apply H. tauto.
  tauto.
Qed.

Lemma list_map_compose:
  forall (A B C: Type) (f: A -> B) (g: B -> C) (l: list A),
  List.map g (List.map f l) = List.map (fun x => g(f x)) l.
Proof.
  induction l; simpl. reflexivity. rewrite IHl; reflexivity.
Qed.

Lemma list_map_identity:
  forall (A: Type) (l: list A),
  List.map (fun (x:A) => x) l = l.
Proof.
  induction l; simpl; congruence.
Qed.

Lemma list_map_nth:
  forall (A B: Type) (f: A -> B) (l: list A) (n: nat),
  nth_error (List.map f l) n = option_map f (nth_error l n).
Proof.
  induction l; simpl; intros.
  repeat rewrite nth_error_nil. reflexivity.
  destruct n; simpl. reflexivity. auto.
Qed.

Lemma list_length_map:
  forall (A B: Type) (f: A -> B) (l: list A),
  List.length (List.map f l) = List.length l.
Proof.
  induction l; simpl; congruence.
Qed.

Lemma list_in_map_inv:
  forall (A B: Type) (f: A -> B) (l: list A) (y: B),
  In y (List.map f l) -> exists x:A, y = f x /\ In x l.
Proof.
  induction l; simpl; intros.
  contradiction.
  elim H; intro.
  exists a; intuition auto.
  generalize (IHl y H0). intros [x [EQ IN]].
  exists x; tauto.
Qed.

Lemma list_append_map:
  forall (A B: Type) (f: A -> B) (l1 l2: list A),
  List.map f (l1 ++ l2) = List.map f l1 ++ List.map f l2.
Proof.
  induction l1; simpl; intros.
  auto. rewrite IHl1. auto.
Qed.

Lemma list_append_map_inv:
  forall (A B: Type) (f: A -> B) (m1 m2: list B) (l: list A),
  List.map f l = m1 ++ m2 ->
  exists l1, exists l2, List.map f l1 = m1 /\ List.map f l2 = m2 /\ l = l1 ++ l2.
Proof.
  induction m1; simpl; intros.
  exists (@nil A); exists l; auto.
  destruct l; simpl in H; inv H.
  exploit IHm1; eauto. intros [l1 [l2 [P [Q R]]]]. subst l.
  exists (a0 :: l1); exists l2; intuition. simpl; congruence.
Qed.

Folding a function over a list

Section LIST_FOLD.

Variables A B: Type.
Variable f: A -> B -> B.

This is exactly List.fold_left from Coq's standard library, with f taking arguments in a different order.

Fixpoint list_fold_left (accu: B) (l: list A) : B :=
  match l with nil => accu | x :: l' => list_fold_left (f x accu) l' end.

This is exactly List.fold_right from Coq's standard library, except that it runs in constant stack space.

Definition list_fold_right (l: list A) (base: B) : B :=
  list_fold_left base (List.rev' l).

Remark list_fold_left_app:
  forall l1 l2 accu,
  list_fold_left accu (l1 ++ l2) = list_fold_left (list_fold_left accu l1) l2.
Proof.
  induction l1; simpl; intros.
  auto.
  rewrite IHl1. auto.
Qed.

Lemma list_fold_right_eq:
  forall l base,
  list_fold_right l base =
  match l with nil => base | x :: l' => f x (list_fold_right l' base) end.
Proof.
  unfold list_fold_right; intros.
  destruct l.
  auto.
  unfold rev'. rewrite <- ! rev_alt. simpl.
  rewrite list_fold_left_app. simpl. auto.
Qed.

Lemma list_fold_right_spec:
  forall l base, list_fold_right l base = List.fold_right f base l.
Proof.
  induction l; simpl; intros; rewrite list_fold_right_eq; congruence.
Qed.

End LIST_FOLD.

Properties of list membership.

Lemma in_cns:
  forall (A: Type) (x y: A) (l: list A), In x (y :: l) <-> y = x \/ In x l.
Proof.
  intros. simpl. tauto.
Qed.

Lemma in_app:
  forall (A: Type) (x: A) (l1 l2: list A), In x (l1 ++ l2) <-> In x l1 \/ In x l2.
Proof.
  intros. split; intro. apply in_app_or. auto. apply in_or_app. auto.
Qed.

Lemma list_in_insert:
  forall (A: Type) (x: A) (l1 l2: list A) (y: A),
  In x (l1 ++ l2) -> In x (l1 ++ y :: l2).
Proof.
  intros. apply in_or_app; simpl. elim (in_app_or _ _ _ H); intro; auto.
Qed.

list_disjoint l1 l2 holds iff l1 and l2 have no elements in common.

Definition list_disjoint (A: Type) (l1 l2: list A) : Prop :=
  forall (x y: A), In x l1 -> In y l2 -> x <> y.

Lemma list_disjoint_cons_l:
  forall (A: Type) (a: A) (l1 l2: list A),
  list_disjoint l1 l2 -> ~In a l2 -> list_disjoint (a :: l1) l2.
Proof.
  unfold list_disjoint; simpl; intros. destruct H1. congruence. apply H; auto.
Qed.

Lemma list_disjoint_cons_r:
  forall (A: Type) (a: A) (l1 l2: list A),
  list_disjoint l1 l2 -> ~In a l1 -> list_disjoint l1 (a :: l2).
Proof.
  unfold list_disjoint; simpl; intros. destruct H2. congruence. apply H; auto.
Qed.

Lemma list_disjoint_cons_left:
  forall (A: Type) (a: A) (l1 l2: list A),
  list_disjoint (a :: l1) l2 -> list_disjoint l1 l2.
Proof.
  unfold list_disjoint; simpl; intros. apply H; tauto.
Qed.

Lemma list_disjoint_cons_right:
  forall (A: Type) (a: A) (l1 l2: list A),
  list_disjoint l1 (a :: l2) -> list_disjoint l1 l2.
Proof.
  unfold list_disjoint; simpl; intros. apply H; tauto.
Qed.

Lemma list_disjoint_notin:
  forall (A: Type) (l1 l2: list A) (a: A),
  list_disjoint l1 l2 -> In a l1 -> ~(In a l2).
Proof.
  unfold list_disjoint; intros; red; intros.
  apply H with a a; auto.
Qed.

Lemma list_disjoint_sym:
  forall (A: Type) (l1 l2: list A),
  list_disjoint l1 l2 -> list_disjoint l2 l1.
Proof.
  unfold list_disjoint; intros.
  apply sym_not_equal. apply H; auto.
Qed.

Lemma list_disjoint_dec:
  forall (A: Type) (eqA_dec: forall (x y: A), {x=y} + {x<>y}) (l1 l2: list A),
  {list_disjoint l1 l2} + {~list_disjoint l1 l2}.
Proof.
  induction l1; intros.
  left; red; intros. elim H.
  case (In_dec eqA_dec a l2); intro.
  right; red; intro. apply (H a a); auto with coqlib.
  case (IHl1 l2); intro.
  left; red; intros. elim H; intro.
    red; intro; subst a y. contradiction.
    apply l; auto.
  right; red; intros. elim n0. eapply list_disjoint_cons_left; eauto.
Defined.

list_equiv l1 l2 holds iff the lists l1 and l2 contain the same elements.

Definition list_equiv (A : Type) (l1 l2: list A) : Prop :=
  forall x, In x l1 <-> In x l2.

list_norepet l holds iff the list l contains no repetitions, i.e. no element occurs twice.

Inductive list_norepet (A: Type) : list A -> Prop :=
  | list_norepet_nil:
      list_norepet nil
  | list_norepet_cons:
      forall hd tl,
      ~(In hd tl) -> list_norepet tl -> list_norepet (hd :: tl).

Lemma list_norepet_dec:
  forall (A: Type) (eqA_dec: forall (x y: A), {x=y} + {x<>y}) (l: list A),
  {list_norepet l} + {~list_norepet l}.
Proof.
  induction l.
  left; constructor.
  destruct IHl.
  case (In_dec eqA_dec a l); intro.
  right. red; intro. inversion H. contradiction.
  left. constructor; auto.
  right. red; intro. inversion H. contradiction.
Defined.

Lemma list_map_norepet:
  forall (A B: Type) (f: A -> B) (l: list A),
  list_norepet l ->
  (forall x y, In x l -> In y l -> x <> y -> f x <> f y) ->
  list_norepet (List.map f l).
Proof.
  induction 1; simpl; intros.
  constructor.
  constructor.
  red; intro. generalize (list_in_map_inv f _ _ H2).
  intros [x [EQ IN]]. generalize EQ. change (f hd <> f x).
  apply H1. tauto. tauto.
  red; intro; subst x. contradiction.
  apply IHlist_norepet. intros. apply H1. tauto. tauto. auto.
Qed.

Remark list_norepet_append_commut:
  forall (A: Type) (a b: list A),
  list_norepet (a ++ b) -> list_norepet (b ++ a).
Proof.
  intro A.
  assert (forall (x: A) (b: list A) (a: list A),
           list_norepet (a ++ b) -> ~(In x a) -> ~(In x b) ->
           list_norepet (a ++ x :: b)).
    induction a; simpl; intros.
    constructor; auto.
    inversion H. constructor. red; intro.
    elim (in_app_or _ _ _ H6); intro.
    elim H4. apply in_or_app. tauto.
    elim H7; intro. subst a. elim H0. left. auto.
    elim H4. apply in_or_app. tauto.
    auto.
  induction a; simpl; intros.
  rewrite <- app_nil_end. auto.
  inversion H0. apply H. auto.
  red; intro; elim H3. apply in_or_app. tauto.
  red; intro; elim H3. apply in_or_app. tauto.
Qed.

Lemma list_norepet_app:
  forall (A: Type) (l1 l2: list A),
  list_norepet (l1 ++ l2) <->
  list_norepet l1 /\ list_norepet l2 /\ list_disjoint l1 l2.
Proof.
  induction l1; simpl; intros; split; intros.
  intuition. constructor. red;simpl;auto.
  tauto.
  inversion H; subst. rewrite IHl1 in H3. rewrite in_app in H2.
  intuition.
  constructor; auto. red; intros. elim H2; intro. congruence. auto.
  destruct H as [B [C D]]. inversion B; subst.
  constructor. rewrite in_app. intuition. elim (D a a); auto. apply in_eq.
  rewrite IHl1. intuition. red; intros. apply D; auto. apply in_cons; auto.
Qed.

Lemma list_norepet_append:
  forall (A: Type) (l1 l2: list A),
  list_norepet l1 -> list_norepet l2 -> list_disjoint l1 l2 ->
  list_norepet (l1 ++ l2).
Proof.
  generalize list_norepet_app; firstorder.
Qed.

Lemma list_norepet_append_right:
  forall (A: Type) (l1 l2: list A),
  list_norepet (l1 ++ l2) -> list_norepet l2.
Proof.
  generalize list_norepet_app; firstorder.
Qed.

Lemma list_norepet_append_left:
  forall (A: Type) (l1 l2: list A),
  list_norepet (l1 ++ l2) -> list_norepet l1.
Proof.
  generalize list_norepet_app; firstorder.
Qed.

is_tail l1 l2 holds iff l2 is of the form l ++ l1 for some l.

Inductive is_tail (A: Type): list A -> list A -> Prop :=
  | is_tail_refl:
      forall c, is_tail c c
  | is_tail_cons:
      forall i c1 c2, is_tail c1 c2 -> is_tail c1 (i :: c2).

Lemma is_tail_in:
  forall (A: Type) (i: A) c1 c2, is_tail (i :: c1) c2 -> In i c2.
Proof.
  induction c2; simpl; intros.
  inversion H.
  inversion H. tauto. right; auto.
Qed.

Lemma is_tail_cons_left:
  forall (A: Type) (i: A) c1 c2, is_tail (i :: c1) c2 -> is_tail c1 c2.
Proof.
  induction c2; intros; inversion H.
  constructor. constructor. constructor. auto.
Qed.

Hint Resolve is_tail_refl is_tail_cons is_tail_in is_tail_cons_left: coqlib.

Lemma is_tail_incl:
  forall (A: Type) (l1 l2: list A), is_tail l1 l2 -> incl l1 l2.
Proof.
  induction 1; eauto with coqlib.
Qed.

Lemma is_tail_trans:
  forall (A: Type) (l1 l2: list A),
  is_tail l1 l2 -> forall (l3: list A), is_tail l2 l3 -> is_tail l1 l3.
Proof.
  induction 1; intros. auto. apply IHis_tail. eapply is_tail_cons_left; eauto.
Qed.

list_forall2 P [x1 ... xN] [y1 ... yM] holds iff N = M and P xi yi holds for all i.

Section FORALL2.

Variable A: Type.
Variable B: Type.
Variable P: A -> B -> Prop.

Inductive list_forall2: list A -> list B -> Prop :=
  | list_forall2_nil:
      list_forall2 nil nil
  | list_forall2_cons:
      forall a1 al b1 bl,
      P a1 b1 ->
      list_forall2 al bl ->
      list_forall2 (a1 :: al) (b1 :: bl).

Lemma list_forall2_app:
  forall a2 b2 a1 b1,
  list_forall2 a1 b1 -> list_forall2 a2 b2 ->
  list_forall2 (a1 ++ a2) (b1 ++ b2).
Proof.
  induction 1; intros; simpl. auto. constructor; auto.
Qed.

Lemma list_forall2_length:
  forall l1 l2,
  list_forall2 l1 l2 -> length l1 = length l2.
Proof.
  induction 1; simpl; congruence.
Qed.

End FORALL2.

Lemma list_forall2_imply:
  forall (A B: Type) (P1: A -> B -> Prop) (l1: list A) (l2: list B),
  list_forall2 P1 l1 l2 ->
  forall (P2: A -> B -> Prop),
  (forall v1 v2, In v1 l1 -> In v2 l2 -> P1 v1 v2 -> P2 v1 v2) ->
  list_forall2 P2 l1 l2.
Proof.
  induction 1; intros.
  constructor.
  constructor. auto with coqlib. apply IHlist_forall2; auto.
  intros. auto with coqlib.
Qed.

Dropping the first N elements of a list.

Fixpoint list_drop (A: Type) (n: nat) (x: list A) {struct n} : list A :=
  match n with
  | O => x
  | S n' => match x with nil => nil | hd :: tl => list_drop n' tl end
  end.

Lemma list_drop_incl:
  forall (A: Type) (x: A) n (l: list A), In x (list_drop n l) -> In x l.
Proof.
  induction n; simpl; intros. auto.
  destruct l; auto with coqlib.
Qed.

Lemma list_drop_norepet:
  forall (A: Type) n (l: list A), list_norepet l -> list_norepet (list_drop n l).
Proof.
  induction n; simpl; intros. auto.
  inv H. constructor. auto.
Qed.

Lemma list_map_drop:
  forall (A B: Type) (f: A -> B) n (l: list A),
  list_drop n (map f l) = map f (list_drop n l).
Proof.
  induction n; simpl; intros. auto.
  destruct l; simpl; auto.
Qed.

A list of n elements, all equal to x.

Fixpoint list_repeat {A: Type} (n: nat) (x: A) {struct n} :=
  match n with
  | O => nil
  | S m => x :: list_repeat m x
  end.

Lemma length_list_repeat:
  forall (A: Type) n (x: A), length (list_repeat n x) = n.
Proof.
  induction n; simpl; intros. auto. decEq; auto.
Qed.

Lemma in_list_repeat:
  forall (A: Type) n (x: A) y, In y (list_repeat n x) -> y = x.
Proof.
  induction n; simpl; intros. elim H. destruct H; auto.
Qed.

Definitions and theorems over boolean types


Definition proj_sumbool (P Q: Prop) (a: {P} + {Q}) : bool :=
  if a then true else false.

Implicit Arguments proj_sumbool [P Q].

Coercion proj_sumbool: sumbool >-> bool.

Lemma proj_sumbool_true:
  forall (P Q: Prop) (a: {P}+{Q}), proj_sumbool a = true -> P.
Proof.
  intros P Q a. destruct a; simpl. auto. congruence.
Qed.

Lemma proj_sumbool_is_true:
  forall (P: Prop) (a: {P}+{~P}), P -> proj_sumbool a = true.
Proof.
  intros. unfold proj_sumbool. destruct a. auto. contradiction.
Qed.

Ltac InvBooleans :=
  match goal with
  | [ H: _ && _ = true |- _ ] =>
      destruct (andb_prop _ _ H); clear H; InvBooleans
  | [ H: _ || _ = false |- _ ] =>
      destruct (orb_false_elim _ _ H); clear H; InvBooleans
  | [ H: proj_sumbool ?x = true |- _ ] =>
      generalize (proj_sumbool_true _ H); clear H; intro; InvBooleans
  | _ => idtac
  end.

Section DECIDABLE_EQUALITY.

Variable A: Type.
Variable dec_eq: forall (x y: A), {x=y} + {x<>y}.
Variable B: Type.

Lemma dec_eq_true:
  forall (x: A) (ifso ifnot: B),
  (if dec_eq x x then ifso else ifnot) = ifso.
Proof.
  intros. destruct (dec_eq x x). auto. congruence.
Qed.

Lemma dec_eq_false:
  forall (x y: A) (ifso ifnot: B),
  x <> y -> (if dec_eq x y then ifso else ifnot) = ifnot.
Proof.
  intros. destruct (dec_eq x y). congruence. auto.
Qed.

Lemma dec_eq_sym:
  forall (x y: A) (ifso ifnot: B),
  (if dec_eq x y then ifso else ifnot) =
  (if dec_eq y x then ifso else ifnot).
Proof.
  intros. destruct (dec_eq x y).
  subst y. rewrite dec_eq_true. auto.
  rewrite dec_eq_false; auto.
Qed.

End DECIDABLE_EQUALITY.

Section DECIDABLE_PREDICATE.

Variable P: Prop.
Variable dec: {P} + {~P}.
Variable A: Type.

Lemma pred_dec_true:
  forall (a b: A), P -> (if dec then a else b) = a.
Proof.
  intros. destruct dec. auto. contradiction.
Qed.

Lemma pred_dec_false:
  forall (a b: A), ~P -> (if dec then a else b) = b.
Proof.
  intros. destruct dec. contradiction. auto.
Qed.

End DECIDABLE_PREDICATE.

Well-founded orderings


Require Import Relations.

A non-dependent version of lexicographic ordering.

Section LEX_ORDER.

Variable A: Type.
Variable B: Type.
Variable ordA: A -> A -> Prop.
Variable ordB: B -> B -> Prop.

Inductive lex_ord: A*B -> A*B -> Prop :=
  | lex_ord_left: forall a1 b1 a2 b2,
      ordA a1 a2 -> lex_ord (a1,b1) (a2,b2)
  | lex_ord_right: forall a b1 b2,
      ordB b1 b2 -> lex_ord (a,b1) (a,b2).

Lemma wf_lex_ord:
  well_founded ordA -> well_founded ordB -> well_founded lex_ord.
Proof.
  intros Awf Bwf.
  assert (forall a, Acc ordA a -> forall b, Acc ordB b -> Acc lex_ord (a, b)).
    induction 1. induction 1. constructor; intros. inv H3.
    apply H0. auto. apply Bwf.
    apply H2; auto.
  red; intros. destruct a as [a b]. apply H; auto.
Qed.

Lemma transitive_lex_ord:
  transitive _ ordA -> transitive _ ordB -> transitive _ lex_ord.
Proof.
  intros trA trB; red; intros.
  inv H; inv H0.
  left; eapply trA; eauto.
  left; auto.
  left; auto.
  right; eapply trB; eauto.
Qed.

End LEX_ORDER.

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