some induction + lists

Induction.v

@@ -197,23 +197,38 @@ Proof.
 Theorem mult_0_r : forall n:nat,
   n * 0 = 0.
 Proof.
-  (* FILL IN HERE *) Admitted.
+  induction n.
+  - reflexivity.
+  - simpl. rewrite -> IHn. reflexivity.
+Qed.
 
 Theorem plus_n_Sm : forall n m : nat,
   S (n + m) = n + (S m).
 Proof.
-  (* FILL IN HERE *) Admitted.
+  induction n. induction m.
+  - reflexivity.
+  - reflexivity.
+  - intros m. simpl. rewrite -> IHn. reflexivity.
+Qed.
 
 Theorem plus_comm : forall n m : nat,
   n + m = m + n.
 Proof.
-  (* FILL IN HERE *) Admitted.
+  induction n. induction m.
+  - reflexivity.
+  - simpl. rewrite <- IHm. reflexivity.
+  - intros m. simpl. rewrite <- plus_n_Sm. rewrite -> IHn. reflexivity.
+Qed.
 
 Theorem plus_assoc : forall n m p : nat,
   n + (m + p) = (n + m) + p.
 Proof.
-  (* FILL IN HERE *) Admitted.
+  induction n. induction m. induction p.
-(** [] *)
+  - reflexivity.
+  - reflexivity.
+  - intros p. reflexivity.
+  - intros m p. simpl. rewrite <- IHn. reflexivity.
+Qed.
 
 (** **** Exercise: 2 stars, standard (double_plus)  
 
@@ -229,8 +244,10 @@ Fixpoint double (n:nat) :=
 
 Lemma double_plus : forall n, double n = n + n .
 Proof.
-  (* FILL IN HERE *) Admitted.
+  induction n.
-(** [] *)
+  - reflexivity.
+  - simpl. rewrite -> IHn. rewrite <- plus_n_Sm. reflexivity.
+Qed.
 
 (** **** Exercise: 2 stars, standard, optional (evenb_S)  
 
@@ -244,8 +261,9 @@ Proof.
 Theorem evenb_S : forall n : nat,
   evenb (S n) = negb (evenb n).
 Proof.
-  (* FILL IN HERE *) Admitted.
+  induction n.
-(** [] *)
+  - reflexivity.
+  - simpl.
 
 (** **** Exercise: 1 star, standard (destruct_induction)  
 
@@ -256,7 +274,7 @@ Proof.
 *)
 
 (* Do not modify the following line: *)
-Definition manual_grade_for_destruct_induction : option (nat*string) := None.
+Definition manual_grade_for_destruct_induction : option (nat*string) := None..
 (** [] *)
 
 (* ################################################################# *)

Lists.v

@@ -134,15 +134,19 @@ Proof.
 Theorem snd_fst_is_swap : forall (p : natprod),
   (snd p, fst p) = swap_pair p.
 Proof.
-  (* FILL IN HERE *) Admitted.
+  intros p.
-(** [] *)
+  destruct p as [n m].
+  reflexivity.
+Qed.
 
 (** **** Exercise: 1 star, standard, optional (fst_swap_is_snd)  *)
 Theorem fst_swap_is_snd : forall (p : natprod),
   fst (swap_pair p) = snd p.
 Proof.
-  (* FILL IN HERE *) Admitted.
+  intros p.
-(** [] *)
+  destruct p as [n m].
+  reflexivity.
+Qed.
 
 (* ################################################################# *)
 (** * Lists of Numbers *)
@@ -290,34 +294,41 @@ Proof. reflexivity.  Qed.
     [countoddmembers] below. Have a look at the tests to understand
     what these functions should do. *)
 
-Fixpoint nonzeros (l:natlist) : natlist
+Fixpoint nonzeros (l:natlist) : natlist :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  match l with
+  | nil => nil
+  | 0 :: t => nonzeros t
+  | n :: t => n :: nonzeros t
+  end.
 
 Example test_nonzeros:
   nonzeros [0;1;0;2;3;0;0] = [1;2;3].
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
-Fixpoint oddmembers (l:natlist) : natlist
+Fixpoint oddmembers (l:natlist) : natlist :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  match l with
+  | nil => nil
+  | n :: t => if oddb n then n :: oddmembers t else oddmembers t
+  end.
 
 Example test_oddmembers:
   oddmembers [0;1;0;2;3;0;0] = [1;3].
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
-Definition countoddmembers (l:natlist) : nat
+Definition countoddmembers (l:natlist) : nat :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  length (oddmembers l).
 
 Example test_countoddmembers1:
   countoddmembers [1;0;3;1;4;5] = 4.
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_countoddmembers2:
   countoddmembers [0;2;4] = 0.
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_countoddmembers3:
   countoddmembers nil = 0.
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 (** [] *)
 
 (** **** Exercise: 3 stars, advanced (alternate)  
@@ -334,24 +345,29 @@ Example test_countoddmembers3:
     of both lists at the same time.  One possible solution involves
     defining a new kind of pairs, but this is not the only way.)  *)
 
-Fixpoint alternate (l1 l2 : natlist) : natlist
+Fixpoint alternate (l1 l2 : natlist) : natlist :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  match l1, l2 with
+  | nil, nil => nil
+  | _ :: _, nil => l1
+  | nil, _ :: _ => l2
+  | h1 :: t1, h2 :: t2 => h1 :: h2 :: alternate t1 t2
+  end.
 
 Example test_alternate1:
   alternate [1;2;3] [4;5;6] = [1;4;2;5;3;6].
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_alternate2:
   alternate [1] [4;5;6] = [1;4;5;6].
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_alternate3:
   alternate [1;2;3] [4] = [1;4;2;3].
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_alternate4:
   alternate [] [20;30] = [20;30].
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 (** [] *)
 
 (* ----------------------------------------------------------------- *)
@@ -368,15 +384,18 @@ Definition bag := natlist.
     Complete the following definitions for the functions
     [count], [sum], [add], and [member] for bags. *)
 
-Fixpoint count (v:nat) (s:bag) : nat
+Fixpoint count (v:nat) (s:bag) : nat :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  match s with
+  | nil => 0
+  | h :: t => if v =? h then 1 + count v t else count v t
+  end.
 
 (** All these proofs can be done just by [reflexivity]. *)
 
 Example test_count1:              count 1 [1;2;3;1;4;1] = 3.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 Example test_count2:              count 6 [1;2;3;1;4;1] = 0.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 (** Multiset [sum] is similar to set [union]: [sum a b] contains all
     the elements of [a] and of [b].  (Mathematicians usually define
@@ -390,29 +409,28 @@ Example test_count2:              count 6 [1;2;3;1;4;1] = 0.
     think about whether [sum] can be implemented in another way --
     perhaps by using functions that have already been defined.  *)
 
-Definition sum : bag -> bag -> bag
+Definition sum : bag -> bag -> bag :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  app.
 
 Example test_sum1:              count 1 (sum [1;2;3] [1;4;1]) = 3.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
-Definition add (v:nat) (s:bag) : bag
+Definition add (v:nat) (s:bag) : bag :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  v :: s.
 
 Example test_add1:                count 1 (add 1 [1;4;1]) = 3.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 Example test_add2:                count 5 (add 1 [1;4;1]) = 0.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
-Definition member (v:nat) (s:bag) : bool
+Definition member (v:nat) (s:bag) : bool :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  negb ((count v s) =? 0).
 
 Example test_member1:             member 1 [1;4;1] = true.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_member2:             member 2 [1;4;1] = false.
-(* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
-(** [] *)
 
 (** **** Exercise: 3 stars, standard, optional (bag_more_functions)  
 
@@ -424,44 +442,53 @@ Example test_member2:             member 2 [1;4;1] = false.
     to fill in the definition of [remove_one] for a later
     exercise.) *)
 
-Fixpoint remove_one (v:nat) (s:bag) : bag
+Fixpoint remove_one (v:nat) (s:bag) : bag :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  match s with
+  | nil => nil
+  | h :: t => if eqb v h then t else h :: remove_one v t
+  end.
 
 Example test_remove_one1:
   count 5 (remove_one 5 [2;1;5;4;1]) = 0.
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_remove_one2:
   count 5 (remove_one 5 [2;1;4;1]) = 0.
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_remove_one3:
   count 4 (remove_one 5 [2;1;4;5;1;4]) = 2.
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
 Example test_remove_one4:
   count 5 (remove_one 5 [2;1;5;4;5;1;4]) = 1.
-  (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
-Fixpoint remove_all (v:nat) (s:bag) : bag
+Fixpoint remove_all (v:nat) (s:bag) : bag :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  match s with
+  | nil => nil
+  | h :: t => if v =? h then remove_all v t else h :: remove_all v t
+  end.
 
 Example test_remove_all1:  count 5 (remove_all 5 [2;1;5;4;1]) = 0.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 Example test_remove_all2:  count 5 (remove_all 5 [2;1;4;1]) = 0.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 Example test_remove_all3:  count 4 (remove_all 5 [2;1;4;5;1;4]) = 2.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 Example test_remove_all4:  count 5 (remove_all 5 [2;1;5;4;5;1;4;5;1;4]) = 0.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 
-Fixpoint subset (s1:bag) (s2:bag) : bool
+Fixpoint subset (s1:bag) (s2:bag) : bool :=
-  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
+  match s1 with
+  | nil => true
+  | h :: t => if count h s2 =? 0 then false else subset t (remove_one h s2)
+  end.
 
 Example test_subset1:              subset [1;2] [2;1;4;1] = true.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 Example test_subset2:              subset [1;2;2] [2;1;4;1] = false.
- (* FILL IN HERE *) Admitted.
+Proof. reflexivity. Qed.
 (** [] *)
 
 (** **** Exercise: 2 stars, standard, recommended (bag_theorem)  

_CoqProject

@@ -0,0 +1 @@
+-Q . LF