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week-09_formalizing-two-by-two-matrices.v~
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week-09_formalizing-two-by-two-matrices.v~
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(* week-09_formalizing-two-by-two-matrices.v *)
(* LPP 2024 - CS3234 2023-2024, Sem2 *)
(* Olivier Danvy <danvy@yale-nus.edu.sg> *)
(* Version of Fri 22 Mar 2024 *)
(* ********** *)
Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.
Require Import Arith.
(* ********** *)
(*
In plain english,
A 2 by 2 matrix whose entries are all natural numbers
*)
Inductive m22 : Type := M22 : nat -> nat -> nat -> nat -> m22.
Property componential_equality_m22 :
forall x11 x12 x21 x22 y11 y12 y21 y22 : nat,
M22 x11 x12
x21 x22 =
M22 y11 y12
y21 y22
<->
x11 = y11 /\ x12 = y12 /\ x21 = y21 /\ x22 = y22.
Proof.
intros x11 x12 x21 x22 y11 y12 y21 y22.
split.
- intro H_tmp.
injection H_tmp as H11 H12 H21 H22.
rewrite -> H11.
rewrite -> H12.
rewrite -> H21.
rewrite -> H22.
split; [reflexivity | split; [reflexivity | split; [reflexivity | reflexivity]]].
- intros [H11 [H12 [H21 H22]]].
rewrite -> H11.
rewrite -> H12.
rewrite -> H21.
rewrite -> H22.
reflexivity.
Qed.
(* ***** *)
Definition zero_m22 := M22 0 0
0 0.
Definition add_m22 (x y : m22) : m22 :=
match (x, y) with
(M22 x11 x12
x21 x22,
M22 y11 y12
y21 y22) =>
M22 (x11 + y11) (x12 + y12)
(x21 + y21) (x22 + y22)
end.
Lemma add_m22_assoc :
forall x y z : m22,
add_m22 x (add_m22 y z) =
add_m22 (add_m22 x y) z.
Proof.
intros [x11 x12
x21 x22]
[y11 y12
y21 y22]
[z11 z12
z21 z22].
unfold add_m22.
rewrite ->4 Nat.add_assoc.
reflexivity.
Qed.
Lemma add_m22_0_l :
forall x : m22,
add_m22 zero_m22 x =
x.
Proof.
intros [x11 x12
x21 x22].
unfold add_m22, zero_m22.
rewrite ->4 Nat.add_0_l.
reflexivity.
Qed.
Lemma add_m22_0_r :
forall x : m22,
add_m22 x zero_m22 =
x.
Proof.
intros [x11 x12
x21 x22].
unfold add_m22, zero_m22.
rewrite ->4 Nat.add_0_r.
reflexivity.
Qed.
(* ********** *)
Inductive mm22 : Type := MM22 : m22 -> m22 -> m22 -> m22 -> mm22.
(* To get used to matrices, let us try proving
that matrix addition is commutative *)
Theorem add_m22_comm:
forall x y : m22,
add_m22 x y = add_m22 y x.
Proof.
intros [x11 x12 x21 x22] [y11 y12 y21 y22].
unfold add_m22.
rewrite -> (Nat.add_comm x11 y11).
rewrite -> (Nat.add_comm x12 y12).
rewrite -> (Nat.add_comm x21 y21).
rewrite -> (Nat.add_comm x22 y22).
reflexivity.
Qed.
(* The standard formula
"row by column" kids
learn *)
Definition mul_m22 (x y : m22) : m22 :=
match (x, y) with
(M22 x11 x12
x21 x22,
M22 y11 y12
y21 y22)
=> (M22 (x11*y11 + x12*y21) (x11*y12 + x12*y22)
(x21*y11 + x22*y21) (x21*y12 + x22*y22))
end.
Theorem mul_m22_not_comm:
exists x y : m22,
mul_m22 x y <> mul_m22 y x.
(* Cool question:
Suppose we have two matrices
whose entries are natural numbers
ranging from 0 to n.
What is the probability that the
two matrices satisfiy commutativity?
Delete before submission, it is just
a thought
*)
Proof.
exists (M22 1 2 3 4).
exists (M22 5 6 7 8).
compute.
intros H.
discriminate.
Qed.
Theorem mul_22_assoc:
forall x y z : m22,
mul_m22 x (mul_m22 y z) = mul_m22 (mul_m22 x y) z.
Proof.
intros [x11 x12 x21 x22]
[y11 y12 y21 y22]
[z11 z12 z21 z22].
unfold mul_m22.
rewrite -> componential_equality_m22.
split;
(* If anyone can clean this up,
please do so, this is definitely ugly *)
[rewrite ->2 Nat.mul_add_distr_l;
rewrite ->2 Nat.mul_add_distr_r;
rewrite ->4 Nat.mul_assoc;
rewrite -> (Nat.add_shuffle1);
reflexivity
| split;
[rewrite ->2 Nat.mul_add_distr_l;
rewrite ->2 Nat.mul_add_distr_r;
rewrite ->4 Nat.mul_assoc;
rewrite -> (Nat.add_shuffle1);
reflexivity
| split;
rewrite ->2 Nat.mul_add_distr_l;
rewrite ->2 Nat.mul_add_distr_r;
rewrite ->4 Nat.mul_assoc;
rewrite -> (Nat.add_shuffle1);
reflexivity
]].
Qed.
(* TODO: Write a nice and clean unit test *)
Compute mul_m22 (M22 1 1 1 1) (M22 0 0 0 0).
Compute mul_m22 (M22 1 0 1 0) (M22 1 1 1 1).
Compute mul_m22 (M22 2 2 0 3) (M22 1 2 3 4).
Compute mul_m22 (M22 18 2 3 123) (M22 1 2 3 4).
Definition id_m22 :=
M22 1 0 0 1.
Fixpoint expt_m22 (x : m22) (n : nat) :=
match n with
| 0 =>
id_m22
| S n' =>
mul_m22 x (expt_m22 x n')
end.
Lemma fold_unfold_expt_m22_0:
forall x : m22,
expt_m22 x 0 = id_m22.
Proof.
fold_unfold_tactic expt_m22.
Qed.
Lemma fold_unfold_expt_m22_S:
forall (x : m22) (n' : nat),
expt_m22 x (S n') = mul_m22 x (expt_m22 x n').
Proof.
fold_unfold_tactic expt_m22.
Qed.
(* TODO : Add proper unit tests and show
left and right identity *)
(* Now to show the cute property *)
Proposition cute_m22:
forall n : nat,
expt_m22 (M22 1 1 0 1) n = (M22 1 n 0 1).
Proof.
intros n.
induction n as [ | n' IHn'].
* rewrite -> fold_unfold_expt_m22_0.
unfold id_m22.
reflexivity.
* (* Just doing algebra on n *)
rewrite -> fold_unfold_expt_m22_S.
rewrite -> IHn'.
unfold mul_m22.
simpl (1 * 1 + 1 * 0).
simpl (0 * 1 + 1 * 0).
simpl (0 * n' + 1 * 1).
rewrite ->2 Nat.mul_1_l.
rewrite ->Nat.add_1_r.
reflexivity.
Qed.
Proposition sixteen_m22:
forall n : nat,
expt_m22 (M22 1 1 1 1) (S n) = (M22 (2^n) (2^n) (2^n) (2^n)).
Proof.
intros n.
induction n as [ | n' IHn'].
* compute.
reflexivity.
* rewrite -> fold_unfold_expt_m22_S.
rewrite -> IHn'.
unfold mul_m22.
rewrite -> (Nat.mul_1_l).
assert (H : 2 ^ n' + 2 ^ n' = 2 ^ (S n')).
{
rewrite <- (Nat.add_1_r n').
Search (_ ^ (_ + _)).
rewrite -> (Nat.pow_add_r).
simpl (2 ^ 1).
Search (_ + _ = _ * _ ).
rewrite <- (Nat.mul_1_r (2^n')) at 1.
rewrite -> mult_n_Sm.
reflexivity.
}
rewrite -> H.
reflexivity.
Qed.
(* Let's just hop straight to the general result *)
Search (_ ^ _ = (_ ^ _) ^ _).
Lemma expt_m22_mul_r:
forall x
Proposition pattern_in_the_propositons:
forall m n: nat,
expt_m22 (M22 (2^m) (2^m)
(2^m) (2^m))
(S n)
=
(M22 (2^(m*(S n) + n)) (2^(m*(S n) + n))
(2^(m*(S n) + n)) (2^(m*(S n) + n))).
Proof.
intros m.
induction m.
* exact sixteen_m22.
* intros n.
rewrite <- sixteen_m22.
rewrite <- sixteen_m22.
(* ********** *)
(* week-09_formalizing-two-by-two-matrices.v *)