feat: sem7

cs3234/labs/week-11_induction-principles.v

@@ -247,6 +247,28 @@ Proof.
   induction n as [ | [ | n''] IHn'].
 Abort.
 
+Lemma nat_ind2 :
+  forall P : nat -> Prop,
+    P 0 ->
+    P 1 ->
+    (forall k : nat, P k -> P (S k) -> P (S (S k))) ->
+    forall n : nat, P n.
+Proof.
+  intros P H_P0 H_P1 H_PSS [ | [ | n'' ] ].
+  - exact H_P0.
+  - exact H_P1.
+  - assert (both : forall m : nat, P m /\ P(S m)).
+    { intro m.
+      induction m as [ | m' [ IHm'  IHSm' ] ].
+      - exact (conj H_P0 H_P1).
+      - split.
+        + exact IHSm'.
+        + exact (H_PSS m' IHm' IHSm' ).
+    }
+    destruct (both n'') as [Pn'' PSn''].
+    exact (H_PSS n'' Pn'' PSn'').
+Qed.
+
 (* Thus equipped, the following theorem is proved pretty directly: *)
 
 Theorem there_is_at_most_one_fibonacci_function :
@@ -262,7 +284,18 @@ Proof.
          [H_fib2_0 [H_fib2_1 H_fib2_SS]]
          n.
   induction n as [ | | n'' IHn'' IHSn''] using nat_ind2.
-Abort.
+   - rewrite -> H_fib1_0.
+     rewrite -> H_fib2_0.
+     reflexivity.
+   - rewrite -> H_fib1_1.
+     rewrite -> H_fib2_1.
+     reflexivity.
+  - rewrite -> (H_fib1_SS n'').
+    rewrite -> (H_fib2_SS n'').
+    rewrite -> IHn''.
+    rewrite -> IHSn''.
+    reflexivity.
+Qed.
 
 (* ***** *)