feat: 3234 week 4

cs3234/labs/week-04_specifications-that-depend-on-other-specifications.v

@@ -33,7 +33,13 @@ Theorem O_is_left_neutral_for_recursive_addition :
     recursive_specification_of_addition add ->
     forall n : nat,
       add 0 n = n.
-Admitted.
+Proof.
+  intros add.
+  unfold recursive_specification_of_addition.
+  intros [S_add_O S_add_S].
+  exact (S_add_O).
+Qed.
+
 
 (* ********** *)
 
@@ -79,6 +85,7 @@ Proof.
 Admitted.
 
 
+
 Property O_is_left_absorbing_wrt_multiplication_alt :
   forall mul : nat -> nat -> nat,
     recursive_specification_of_multiplication mul ->
@@ -190,6 +197,19 @@ Theorem addition_is_commutative :
     forall n1 n2 : nat,
       add n1 n2 = add n2 n1.
 Proof.
+  intros add S_add.
+  assert (S_tmp := S_add).
+  unfold recursive_specification_of_addition in S_tmp.
+  destruct S_tmp as [S_add_O S_add_S].
+  intro x.
+  induction x as [| x' IHx'].
+  - intro y.
+    rewrite -> (S_add_O y).
+    rewrite -> (O_is_right_neutral_wrt_addition add S_add y).
+    reflexivity.
+  - intro y.
+    rewrite -> (S_add_S x' y).
+    rewrite -> (IHx' y).
 Admitted.
 
 Property SO_is_right_neutral_wrt_multiplication :
@@ -322,12 +342,60 @@ Proof.
     Check (multiplication_is_right_distributive_over_addition add S_add mul S_mul).
     rewrite -> (multiplication_is_right_distributive_over_addition add S_add mul S_mul (mul x' y) y z).
     reflexivity.
+    Show Proof.
 Qed.
 
 (* ********** *)
 
 (* Exercise 05 *)
 
+Check Nat.mul_0_r.
+
+Theorem addition_is_associative :
+  forall add : nat -> nat -> nat,
+    recursive_specification_of_addition add ->
+    forall x y z : nat,
+      add (add x y) z = add x (add y z).
+  Proof.
+    Admitted.
+
+Lemma commutativity_of_recursive_multiplication_S :
+  forall add : nat -> nat -> nat,
+    recursive_specification_of_addition add ->
+    forall mul : nat -> nat -> nat,
+      recursive_specification_of_multiplication mul ->
+      forall x y : nat,
+        mul x (S y) = add (mul x y) x.
+Proof.
+  intros add S_add mul S_mul.
+  assert (S_tmp := S_mul).
+  unfold recursive_specification_of_multiplication in S_tmp.
+  destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
+  clear S_tmp.
+  assert (S_tmp := S_add).
+  unfold recursive_specification_of_addition in S_tmp.
+  destruct S_tmp as [S_add_O S_add_S].
+  intro x.
+  induction x as [| x' IHx'].
+  - intro y.
+    rewrite -> (S_mul_O (S y)).
+    rewrite -> (S_mul_O y).
+    rewrite -> (S_add_O 0).
+    reflexivity.
+  - intro y.
+    rewrite -> (addition_is_commutative add S_add (mul (S x') y) (S x')).
+    rewrite -> (S_add_S x' (mul (S x') y)).
+    rewrite -> (S_mul_S x' y).
+    rewrite -> (S_mul_S x' (S y)).
+    rewrite -> (IHx' y).
+    rewrite -> (addition_is_commutative add S_add (add (mul x' y) x') (S y)).
+    rewrite -> (S_add_S y (add (mul x' y) x')).
+    rewrite -> (addition_is_commutative add S_add (add (mul x' y) x') (S y)).
+    rewrite -> (addition_is_associative )
+
+
+
+
 Property multiplication_is_commutative :
   forall add : nat -> nat -> nat,
     recursive_specification_of_addition add ->
@@ -344,7 +412,6 @@ Proof.
   assert (S_tmp := S_add).
   unfold recursive_specification_of_addition in S_tmp.
   destruct S_tmp as [S_add_O S_add_S].
-
   intro x.
   induction x as [| x' IHx'].
   - intro y.
@@ -352,12 +419,49 @@ Proof.
     rewrite -> (O_is_right_absorbing_wrt_multiplication add S_add mul S_mul y).
     reflexivity.
   - intro y.
+    (* rewrite -> (S_mul_S x' y). *)
+    (* rewrite ->  (addition_is_commutative add S_add (mul x' y) y). *)
+    (* rewrite -> (IHx' y). *)
+    (* rewrite <- (SO_is_right_neutral_wrt_multiplication add S_add mul S_mul x') at 1. *)
+    (* rewrite -> (multiplication_is_associative add S_add mul S_mul y x' 1). *)
+    (* rewrite <- (IHx' y). *)
+    (* rewrite <- (multiplication_is_associative add S_add mul S_mul x' y 1). *)
+    (* rewrite -> (SO_is_right_neutral_wrt_multiplication add S_add mul S_mul y). *)
     rewrite -> (S_mul_S x' y).
-    rewrite -> (IHx' y).
-    Check (multiplication_is_right_distributive_over_addition add S_add mul S_mul).
+    Check (add (mul x' y) y).
+
+    (* TODO FINISH THIS *)
+
+Admitted. (* <-- needed for later on *)
+
+Lemma S_mul_S_reversed :
+  forall add : nat -> nat -> nat,
+    recursive_specification_of_addition add ->
+    forall mul : nat -> nat -> nat,
+      recursive_specification_of_multiplication mul ->
+      forall x y : nat,
+        mul y (S x) = add (mul x y) y.
+Proof.
+  intros add S_add mul S_mul.
+  assert (S_tmp := S_mul).
+  unfold recursive_specification_of_multiplication in S_tmp.
+  destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
+  clear S_tmp.
+  assert (S_tmp := S_add).
+  unfold recursive_specification_of_addition in S_tmp.
+  destruct S_tmp as [S_add_O S_add_S].
+  intro x.
+  induction x as [| x' IHx'].
+  - intro y.
+    rewrite -> (S_mul_O y).
+    rewrite -> (SO_is_right_neutral_wrt_multiplication add S_add mul S_mul y).
+    rewrite -> (S_add_O y).
+    reflexivity.
+  - intro y.
+    Abort.
+
 
 
-Qed. (* <-- needed for later on *)
 
 (* ********** *)