complete basics
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133
Basics.v
133
Basics.v
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@ -282,17 +282,20 @@ Proof. simpl. reflexivity. Qed.
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model of the [orb] tests above.) The function should return [true]
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if either or both of its inputs are [false]. *)
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Definition nandb (b1:bool) (b2:bool) : bool
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(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
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Definition nandb (b1:bool) (b2:bool) : bool :=
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match b1, b2 with
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| true, true => false
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| _, _ => true
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end.
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Example test_nandb1: (nandb true false) = true.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_nandb2: (nandb false false) = true.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_nandb3: (nandb false true) = true.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_nandb4: (nandb true true) = false.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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(** [] *)
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(** **** Exercise: 1 star, standard (andb3)
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@ -301,17 +304,20 @@ Example test_nandb4: (nandb true true) = false.
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return [true] when all of its inputs are [true], and [false]
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otherwise. *)
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Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool
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(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
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Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool :=
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match b1, b2, b3 with
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| true, true, true => true
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| _, _, _ => false
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end.
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Example test_andb31: (andb3 true true true) = true.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_andb32: (andb3 false true true) = false.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_andb33: (andb3 true false true) = false.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_andb34: (andb3 true true false) = false.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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(** [] *)
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(* ================================================================= *)
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@ -698,13 +704,16 @@ Fixpoint exp (base power : nat) : nat :=
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Translate this into Coq. *)
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Fixpoint factorial (n:nat) : nat
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(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
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Fixpoint factorial (n:nat) : nat :=
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match n with
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| 0 => 1
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| S n => (S n) * factorial (n)
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end.
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Example test_factorial1: (factorial 3) = 6.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_factorial2: (factorial 5) = (mult 10 12).
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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(** [] *)
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(** Again, we can make numerical expressions easier to read and write
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@ -791,17 +800,17 @@ Proof. simpl. reflexivity. Qed.
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function. (It can be done with just one previously defined
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function, but you can use two if you need to.) *)
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Definition ltb (n m : nat) : bool
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(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
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Definition ltb (n m : nat) : bool :=
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1 <=? m - n.
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Notation "x <? y" := (ltb x y) (at level 70) : nat_scope.
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Example test_ltb1: (ltb 2 2) = false.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_ltb2: (ltb 2 4) = true.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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Example test_ltb3: (ltb 4 2) = false.
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(* FILL IN HERE *) Admitted.
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Proof. reflexivity. Qed.
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(** [] *)
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(* ################################################################# *)
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@ -953,7 +962,12 @@ Proof.
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Theorem plus_id_exercise : forall n m o : nat,
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n = m -> m = o -> n + m = m + o.
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Proof.
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(* FILL IN HERE *) Admitted.
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intros n m o.
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intros H J.
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rewrite -> H.
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rewrite -> J.
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reflexivity.
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Qed.
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(** [] *)
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(** The [Admitted] command tells Coq that we want to skip trying
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@ -985,7 +999,12 @@ Theorem mult_S_1 : forall n m : nat,
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m = S n ->
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m * (1 + n) = m * m.
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Proof.
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(* FILL IN HERE *) Admitted.
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intros n m.
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intros H.
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simpl.
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rewrite <- H.
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reflexivity.
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Qed.
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(* (N.b. This proof can actually be completed with tactics other than
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[rewrite], but please do use [rewrite] for the sake of the exercise.)
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@ -1213,15 +1232,21 @@ Qed.
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Theorem andb_true_elim2 : forall b c : bool,
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andb b c = true -> c = true.
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Proof.
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(* FILL IN HERE *) Admitted.
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(** [] *)
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intros [] [].
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- reflexivity.
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- intros H. destruct H. reflexivity.
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- reflexivity.
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- intros H. destruct H. reflexivity.
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Qed.
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(** **** Exercise: 1 star, standard (zero_nbeq_plus_1) *)
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Theorem zero_nbeq_plus_1 : forall n : nat,
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0 =? (n + 1) = false.
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Proof.
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(* FILL IN HERE *) Admitted.
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(** [] *)
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intros [].
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- reflexivity.
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- reflexivity.
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Qed.
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(* ================================================================= *)
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(** ** More on Notation (Optional) *)
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@ -1327,9 +1352,11 @@ Theorem identity_fn_applied_twice :
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(forall (x : bool), f x = x) ->
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forall (b : bool), f (f b) = b.
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Proof.
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(* FILL IN HERE *) Admitted.
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(** [] *)
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intros f x b.
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rewrite -> x.
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rewrite -> x.
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reflexivity.
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Qed.
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(** **** Exercise: 1 star, standard (negation_fn_applied_twice)
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@ -1360,7 +1387,12 @@ Theorem andb_eq_orb :
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(andb b c = orb b c) ->
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b = c.
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Proof.
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(* FILL IN HERE *) Admitted.
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intros [] [].
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- reflexivity.
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- simpl. intros H. rewrite -> H. reflexivity.
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- simpl. intros H. rewrite -> H. reflexivity.
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- reflexivity.
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Qed.
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(** [] *)
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@ -1399,11 +1431,19 @@ Inductive bin : Type :=
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for binary numbers, and a function [bin_to_nat] to convert
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binary numbers to unary numbers. *)
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Fixpoint incr (m:bin) : bin
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(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
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Fixpoint incr (m:bin) : bin :=
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match m with
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| Z => B Z
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| A n => B n
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| B n => A (incr n)
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end.
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Fixpoint bin_to_nat (m:bin) : nat
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(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
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Fixpoint bin_to_nat (m:bin) : nat :=
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match m with
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| Z => 0
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| A n => 2 * (bin_to_nat n)
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| B n => 1 + 2 * (bin_to_nat n)
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end.
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(** (b) Write five unit tests [test_bin_incr1], [test_bin_incr2], etc.
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for your increment and binary-to-unary functions. (A "unit
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@ -1413,10 +1453,31 @@ Fixpoint bin_to_nat (m:bin) : nat
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then converting it to unary should yield the same result as
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first converting it to unary and then incrementing. *)
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(* FILL IN HERE *)
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Example test_bin_incr1 : incr Z = B Z.
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Proof. reflexivity. Qed.
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Example test_bin_incr2 : incr (B Z) = A (B Z).
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Proof. reflexivity. Qed.
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Example test_bin_incr3 : incr (A (B Z)) = B (B Z).
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Proof. reflexivity. Qed.
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Example test_bin_incr4 : incr (B (B Z)) = A (A (B Z)).
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Proof. reflexivity. Qed.
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Example test_bin_incr5 : incr (A (A (B Z))) = B (A (B Z)).
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Proof. reflexivity. Qed.
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Example test_bin_to_nat0 : bin_to_nat Z = 0.
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Proof. reflexivity. Qed.
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Example test_bin_to_nat1 : bin_to_nat (B Z) = 1.
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Proof. reflexivity. Qed.
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Example test_bin_to_nat2 : bin_to_nat (A (B Z)) = 2.
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Proof. reflexivity. Qed.
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Example test_bin_to_nat3 : bin_to_nat (B (B Z)) = 3.
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Proof. reflexivity. Qed.
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Example test_bin_to_nat4 : bin_to_nat (A (A (B Z))) = 4.
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Proof. reflexivity. Qed.
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(* Do not modify the following line: *)
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Definition manual_grade_for_binary : option (nat*string) := None.
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(** [] *)
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(* Wed Jan 9 12:02:44 EST 2019 *)
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