478 lines
8.9 KiB
Coq
478 lines
8.9 KiB
Coq
From LF Require Export Basics.
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Theorem add_0_r : forall n : nat,
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n + 0 = n.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem minus_n_n : forall n,
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n - n = 0.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem mul_0_r : forall n : nat,
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n * 0 = 0.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem plus_n_Sm : forall n m : nat,
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S (n + m) = n + (S m).
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- intros m.
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simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem add_comm : forall n m : nat,
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n + m = m + n.
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Proof.
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induction n as [ | n' IHn'].
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- intros m.
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rewrite add_0_r.
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reflexivity.
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- intros m.
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simpl.
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rewrite plus_n_Sm.
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rewrite IHn'.
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simpl.
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rewrite plus_n_Sm.
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reflexivity.
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Qed.
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Theorem sn_eq_sm_n_eq_m : forall n m : nat,
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S n = S m -> n = m.
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Proof.
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intros n m H.
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inversion H.
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reflexivity.
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Qed.
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Theorem n_eq_m_sn_eq_sm : forall n m : nat,
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n = m -> S n = S m.
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Proof.
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intros n m H.
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rewrite H.
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reflexivity.
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Qed.
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Theorem add_assoc : forall n m p : nat,
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n + (m + p) = (n + m) + p.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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intros m p.
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apply n_eq_m_sn_eq_sm.
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rewrite IHn'.
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reflexivity.
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Qed.
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Fixpoint double (n : nat) :=
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match n with
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| O => O
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| S n' => S (S (double n'))
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end.
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Lemma double_plus : forall n, double n = n + n.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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apply n_eq_m_sn_eq_sm.
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rewrite IHn'.
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apply plus_n_Sm.
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Qed.
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Theorem eqb_refl : forall n : nat,
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(n =? n) = true.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem even_S : forall n : nat,
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even (S n) = negb (even n).
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- rewrite IHn'.
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rewrite negb_involutive.
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reflexivity.
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Qed.
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Theorem mult_0_plus' : forall n m : nat,
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(n + 0 + 0) * m = n * m.
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Proof.
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intros n m.
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assert (H : n + 0 + 0 = n).
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- rewrite add_comm.
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simpl.
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rewrite add_comm.
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reflexivity.
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- rewrite H.
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reflexivity.
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Qed.
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Lemma n_zero_zero : forall n : nat,
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n * 0 = 0.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem add_shuffle3 : forall n m p : nat,
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n + (m + p) = m + (n + p).
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Proof.
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intros n m p.
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rewrite add_assoc.
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assert (H: n + m = m + n).
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{ rewrite add_comm. reflexivity. }
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rewrite H.
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rewrite add_assoc.
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reflexivity.
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Qed.
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Lemma shuffle_4 : forall (a b c d : nat),
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(a + b) + (c + d) = (a + c) + (b + d).
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Proof.
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intros a b c d.
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assert (H: a + b + (c + d) = a + ((b + c) + d)).
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{ rewrite add_assoc. rewrite add_assoc. rewrite add_assoc. reflexivity. }
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rewrite H.
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assert (H1: b + c = c + b).
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{ rewrite add_comm. reflexivity. }
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rewrite H1.
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rewrite add_assoc.
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rewrite add_assoc.
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rewrite add_assoc.
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reflexivity.
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Qed.
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Theorem mult_plus_distr_l : forall (n m k : nat),
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n * (m + k) = n * m + n * k.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- intros m k.
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simpl.
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rewrite IHn'.
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rewrite shuffle_4.
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reflexivity.
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Qed.
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Theorem n_one_n : forall n : nat,
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n * 1 = n.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem mul_comm : forall m n : nat,
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m * n = n * m.
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Proof.
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induction n as [ | n' IHn'].
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- rewrite n_zero_zero.
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reflexivity.
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- simpl.
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assert (H: S n' = n' + 1).
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{ rewrite add_comm. reflexivity. }
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rewrite H.
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rewrite mult_plus_distr_l.
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rewrite n_one_n.
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rewrite IHn'.
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rewrite add_comm.
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reflexivity.
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Qed.
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Theorem plus_leb_compact_l : forall n m p : nat,
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n <=? m = true -> (p + n) <=? (p + m) = true.
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Proof.
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induction p as [ | p' IHp'].
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- auto.
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- intros H.
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simpl.
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apply IHp'.
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rewrite H.
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reflexivity.
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Qed.
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Theorem leb_refl : forall n : nat,
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(n <=? n) = true.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem zero_neqb_S : forall n : nat,
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0 =? (S n) = false.
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Proof.
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reflexivity.
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Qed.
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Theorem andb_false_r : forall b : bool,
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andb b false = false.
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Proof.
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intros.
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rewrite andb_commutative.
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reflexivity.
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Qed.
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Theorem S_neqb_0 : forall n : nat,
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(S n) =? 0 = false.
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Proof.
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reflexivity.
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Qed.
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Theorem mult_1_l : forall n : nat, 1 * n = n.
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Proof.
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intros n.
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simpl.
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rewrite add_0_r.
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reflexivity.
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Qed.
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Theorem all3_spec : forall b c : bool,
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b && c || (negb b || negb c) = true.
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Proof.
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intros [] [].
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- reflexivity.
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- reflexivity.
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- reflexivity.
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- reflexivity.
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Qed.
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Theorem mult_plus_distr_r : forall n m p : nat,
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(n + m) * p = (n * p) + (m * p).
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Proof.
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induction p as [ | p' IHp'].
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- rewrite n_zero_zero.
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rewrite n_zero_zero.
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rewrite n_zero_zero.
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reflexivity.
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- rewrite mul_comm.
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simpl.
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rewrite mul_comm.
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rewrite IHp'.
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assert (H: forall n', n' * S p' = S p' * n').
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{ intros. rewrite mul_comm. reflexivity. }
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rewrite H.
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rewrite H.
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simpl.
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rewrite shuffle_4.
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assert (H': forall n', p' * n' = n' * p').
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{ intros. rewrite mul_comm. reflexivity. }
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rewrite H'.
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rewrite H'.
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reflexivity.
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Qed.
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Theorem mult_assoc : forall n m p : nat,
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n * (m * p) = (n * m) * p.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- intros.
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simpl.
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rewrite mult_plus_distr_r.
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rewrite IHn'.
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reflexivity.
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Qed.
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Theorem add_shuffle3' : forall n m p : nat,
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n + (m + p) = m + (n + p).
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Proof.
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intros.
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rewrite add_assoc.
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rewrite add_assoc.
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replace (n + m) with (m + n).
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- reflexivity.
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- rewrite add_comm.
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reflexivity.
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Qed.
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Inductive bin : Type :=
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| Z
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| B0 (n : bin)
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| B1 (n : bin).
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Fixpoint incr (m : bin) : bin :=
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match m with
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| Z => B1 Z
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| B0 b => B1 b
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| B1 b => B0 (incr b)
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end.
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Fixpoint bin_to_nat (m : bin) : nat :=
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match m with
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| Z => O
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| B0 b => 2 * bin_to_nat b
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| B1 b => 1 + 2 * bin_to_nat b
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end.
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Theorem sn_np1 : forall n : nat,
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S n = n + 1.
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Proof.
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intros.
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rewrite add_comm.
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reflexivity.
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Qed.
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Theorem add_both_r : forall n m k : nat,
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n = m -> n + k = m + k.
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Proof.
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intros n m k H.
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rewrite H.
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reflexivity.
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Qed.
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Theorem add_cancel_r : forall n m k : nat,
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n + k = m + k -> n = m.
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Proof.
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induction k as [ | k' IHk'].
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- rewrite add_0_r.
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rewrite add_0_r.
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intro H.
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rewrite H.
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reflexivity.
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- simpl.
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rewrite <- plus_n_Sm.
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rewrite <- plus_n_Sm.
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intro H.
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inversion H.
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apply IHk'.
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rewrite H1.
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reflexivity.
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Qed.
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Theorem bin_to_nat_pres_incr : forall b : bin,
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bin_to_nat (incr b) = S (bin_to_nat b).
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Proof.
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induction b as [ | b0 | b1 IHb].
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- reflexivity.
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- reflexivity.
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- simpl.
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rewrite add_0_r.
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rewrite IHb.
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rewrite add_0_r.
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rewrite sn_np1.
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assert (H: S (S (bin_to_nat b1 + bin_to_nat b1)) = S (bin_to_nat b1 + bin_to_nat b1) + 1).
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{ rewrite sn_np1. reflexivity. }
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rewrite H.
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assert (H1 : S (bin_to_nat b1 + bin_to_nat b1) + 1 = bin_to_nat b1 + bin_to_nat b1 + 1 + 1).
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{ rewrite sn_np1. reflexivity. }
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rewrite H1.
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rewrite shuffle_4.
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rewrite add_assoc.
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reflexivity.
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Qed.
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Fixpoint nat_to_bin (n : nat) : bin :=
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match n with
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| O => Z
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| S n' => incr (nat_to_bin n')
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end.
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Theorem nat_bin_nat : forall n : nat,
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bin_to_nat (nat_to_bin n) = n.
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite bin_to_nat_pres_incr.
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rewrite IHn'.
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reflexivity.
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Qed.
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Lemma double_incr : forall n : nat,
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double (S n) = S (S (double n)).
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Proof.
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intros [ | n' ].
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- reflexivity.
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- simpl.
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reflexivity.
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Qed.
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Definition double_bin (b : bin) : bin :=
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match b with
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| Z => Z
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| b' => B0 b'
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end.
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Lemma double_incr_bin : forall b,
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double_bin (incr b) = incr (incr (double_bin b)).
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Proof.
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intros [ | b0 | b1 ] ; reflexivity.
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Qed.
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Fixpoint normalize (b : bin) : bin :=
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match b with
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| B0 Z => Z
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| Z => Z
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| B0 b' => double_bin (normalize b')
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| B1 b' => B1 (normalize b')
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end.
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Theorem double_natural : forall n : nat,
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nat_to_bin (double n) = double_bin (nat_to_bin n).
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Proof.
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induction n as [ | n' IHn'].
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- reflexivity.
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- simpl.
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rewrite double_incr_bin.
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rewrite IHn'.
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reflexivity.
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Qed.
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Lemma incr_double_B1 : forall b,
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incr (double_bin b) = B1 b.
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Proof.
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induction b as [ | b0 IHb0 | b1 IHb1] ; reflexivity.
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Qed.
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Theorem bin_nat_bin : forall b, nat_to_bin (bin_to_nat b) = normalize b.
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induction b as [ | b0 IHb0 | b1 IHb1].
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- reflexivity.
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- simpl.
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rewrite add_0_r.
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rewrite <- double_plus.
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rewrite double_natural.
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rewrite IHb0.
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destruct b0 as [ | b0' | b1'] ; reflexivity.
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- simpl.
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rewrite add_0_r.
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rewrite <- double_plus.
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rewrite double_natural.
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rewrite IHb1.
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rewrite incr_double_B1.
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reflexivity.
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Qed.
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