feat: 3234 week 4 wip
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(* week-04_specifications-that-depend-on-other-specifications.v *)
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(* LPP 2024 - CS3234 2023-2024, Sem2 *)
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(* Olivier Danvy <danvy@yale-nus.edu.sg> *)
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(* Version of 09 Feb 2024 *)
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(* Paraphernalia: *)
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Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.
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Require Import Arith Bool.
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(* ********** *)
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Definition recursive_specification_of_addition (add : nat -> nat -> nat) :=
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(forall y : nat,
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add O y = y)
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/\
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(forall x' y : nat,
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add (S x') y = S (add x' y)).
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Definition tail_recursive_specification_of_addition (add : nat -> nat -> nat) :=
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(forall y : nat,
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add O y = y)
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/\
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(forall x' y : nat,
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add (S x') y = add x' (S y)).
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(* ***** *)
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Theorem O_is_left_neutral_for_recursive_addition :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall n : nat,
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add 0 n = n.
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Admitted.
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(* ********** *)
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Definition recursive_specification_of_multiplication (mul : nat -> nat -> nat) :=
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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(forall y : nat,
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mul O y = 0)
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/\
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(forall x' y : nat,
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mul (S x') y = add (mul x' y) y).
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(* ********** *)
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Property O_is_left_absorbing_wrt_multiplication :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall y : nat,
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mul 0 y = 0.
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Proof.
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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Check (S_tmp add S_add).
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destruct (S_tmp add S_add) as [S_mul_O _].
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clear S_tmp.
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exact S_mul_O.
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Qed.
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Fixpoint add_v1 (i j : nat) : nat :=
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match i with
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O =>
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j
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| S i' =>
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S (add_v1 i' j)
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end.
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Theorem add_v1_satisfies_the_recursive_specification_of_addition :
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recursive_specification_of_addition add_v1.
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Proof.
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Admitted.
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Property O_is_left_absorbing_wrt_multiplication_alt :
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall y : nat,
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mul 0 y = 0.
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Proof.
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intros mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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Check (S_tmp add_v1).
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Check (S_tmp add_v1 add_v1_satisfies_the_recursive_specification_of_addition).
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destruct (S_tmp add_v1 add_v1_satisfies_the_recursive_specification_of_addition) as [S_mul_O _].
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clear S_tmp.
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exact S_mul_O.
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Qed.
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(* ********** *)
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Theorem O_is_left_neutral_wrt_addition :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall y : nat,
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add 0 y = y.
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Proof.
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intros add S_add.
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assert (S_tmp := S_add).
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unfold recursive_specification_of_addition in S_tmp.
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destruct S_tmp as [S_add_O _].
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exact S_add_O.
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Qed.
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Property SO_is_left_neutral_wrt_multiplication :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall y : nat,
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mul 1 y = y.
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Proof.
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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Check (S_tmp add S_add).
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destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
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clear S_tmp.
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intro y.
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Check (S_mul_S 0 y).
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rewrite -> (S_mul_S 0 y).
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rewrite -> (S_mul_O y).
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Check (O_is_left_neutral_wrt_addition add S_add y).
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exact (O_is_left_neutral_wrt_addition add S_add y).
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Qed.
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(* ********** *)
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(* Exercise 01 *)
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Theorem O_is_right_neutral_wrt_addition :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall y : nat,
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add y 0 = y.
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Proof.
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intros add S_add.
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assert (S_tmp := S_add).
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unfold recursive_specification_of_addition in S_tmp.
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destruct S_tmp as [S_add_O S_add_S].
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intro y.
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induction y as [| y' IHy'].
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- exact (S_add_O 0).
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- rewrite -> (S_add_S y' 0).
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rewrite -> IHy'.
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reflexivity.
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Qed.
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Property O_is_right_absorbing_wrt_multiplication :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall x : nat,
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mul x 0 = 0.
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Proof.
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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Check (S_tmp add S_add).
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destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
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clear S_tmp.
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intro x.
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induction x as [ | x' IHx'].
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- exact (S_mul_O 0).
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- rewrite -> (S_mul_S x' 0).
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rewrite -> IHx'.
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Check (O_is_right_neutral_wrt_addition add S_add).
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exact (O_is_right_neutral_wrt_addition add S_add 0).
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Qed.
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(* ********** *)
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(* Exercise 02 *)
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Theorem addition_is_commutative :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall n1 n2 : nat,
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add n1 n2 = add n2 n1.
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Proof.
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Admitted.
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Property SO_is_right_neutral_wrt_multiplication :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall x : nat,
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mul x 1 = x.
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Proof.
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(* My implementation *)
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
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clear S_tmp.
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intro x.
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induction x as [| x' IHx'].
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- exact (S_mul_O 1).
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- rewrite -> (S_mul_S x' 1).
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rewrite -> IHx'.
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assert (S_tmp := S_add).
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destruct S_tmp as [S_add_O S_add_S].
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Check (addition_is_commutative add S_add x' 1).
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rewrite -> (addition_is_commutative add S_add x' 1).
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rewrite -> (S_add_S 0 x').
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rewrite -> (S_add_O x').
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reflexivity.
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Restart.
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
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clear S_tmp.
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intro x.
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induction x as [ | x' IHx'].
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- exact (S_mul_O 1).
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- rewrite -> (S_mul_S x' 1).
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rewrite -> IHx'.
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Check (addition_is_commutative add S_add).
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Check (addition_is_commutative add S_add x' 1).
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rewrite -> (addition_is_commutative add S_add x' 1).
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assert (S_tmp := S_add).
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unfold recursive_specification_of_addition in S_tmp.
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destruct S_tmp as [S_add_O S_add_S].
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rewrite -> (S_add_S 0 x').
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rewrite -> (S_add_O x').
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reflexivity.
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Qed.
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(* ********** *)
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(* Exercise 03 *)
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Theorem associativity_of_recursive_addition :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall n1 n2 n3 : nat,
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add n1 (add n2 n3) = add (add n1 n2) n3.
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Proof.
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Admitted.
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Property multiplication_is_right_distributive_over_addition :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall x y z : nat,
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mul (add x y) z = add (mul x z) (mul y z).
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Proof.
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
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clear S_tmp.
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assert (S_tmp := S_add).
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unfold recursive_specification_of_addition in S_tmp.
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destruct S_tmp as [S_add_O S_add_S].
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intros x.
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induction x as [| x' IHx'].
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- intros y z.
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rewrite -> (S_add_O y).
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rewrite -> (S_mul_O z).
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rewrite -> (S_add_O (mul y z)).
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reflexivity.
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- intros y z.
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rewrite -> (S_add_S x' y).
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rewrite -> (S_mul_S (add x' y) z).
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rewrite -> (S_mul_S x' z).
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rewrite -> (IHx' y z).
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rewrite -> (addition_is_commutative add S_add (add (mul x' z) (mul y z)) z).
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rewrite -> (associativity_of_recursive_addition add S_add z (mul x' z) (mul y z)).
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rewrite -> (addition_is_commutative add S_add z (mul x' z)).
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reflexivity.
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Qed. (* <-- needed for later on *)
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(* ********** *)
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(* Exercise 04 *)
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Property multiplication_is_associative :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall x y z : nat,
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mul x (mul y z) = mul (mul x y) z.
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Proof.
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
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clear S_tmp.
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intros x.
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induction x as [| x' IHx'].
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- intros y z.
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rewrite -> (S_mul_O (mul y z)).
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rewrite -> (S_mul_O y).
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rewrite -> (S_mul_O z).
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reflexivity.
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- intros y z.
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rewrite -> (S_mul_S x' (mul y z)).
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rewrite -> (IHx' y z).
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rewrite -> (S_mul_S x' y).
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Check (multiplication_is_right_distributive_over_addition add S_add mul S_mul).
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rewrite -> (multiplication_is_right_distributive_over_addition add S_add mul S_mul (mul x' y) y z).
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reflexivity.
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Qed.
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(* ********** *)
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(* Exercise 05 *)
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Property multiplication_is_commutative :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall x y : nat,
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mul x y = mul y x.
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Proof.
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intros add S_add mul S_mul.
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assert (S_tmp := S_mul).
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unfold recursive_specification_of_multiplication in S_tmp.
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destruct (S_tmp add S_add) as [S_mul_O S_mul_S].
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clear S_tmp.
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assert (S_tmp := S_add).
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unfold recursive_specification_of_addition in S_tmp.
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destruct S_tmp as [S_add_O S_add_S].
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intro x.
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induction x as [| x' IHx'].
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- intro y.
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rewrite -> (S_mul_O y).
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rewrite -> (O_is_right_absorbing_wrt_multiplication add S_add mul S_mul y).
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reflexivity.
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- intro y.
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rewrite -> (S_mul_S x' y).
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rewrite -> (IHx' y).
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Check (multiplication_is_right_distributive_over_addition add S_add mul S_mul).
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Qed. (* <-- needed for later on *)
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(* ********** *)
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(* Exercise 06 *)
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Property multiplication_is_left_distributive_over_addition :
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forall add : nat -> nat -> nat,
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recursive_specification_of_addition add ->
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forall mul : nat -> nat -> nat,
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recursive_specification_of_multiplication mul ->
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forall x y z : nat,
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mul x (add y z) = add (mul x y) (mul x z).
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Proof.
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intros add S_add mul S_mul z x y.
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rewrite -> (multiplication_is_commutative add S_add mul S_mul z (add x y)).
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rewrite -> (multiplication_is_commutative add S_add mul S_mul z x).
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rewrite -> (multiplication_is_commutative add S_add mul S_mul z y).
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exact (multiplication_is_right_distributive_over_addition add S_add mul S_mul x y z).
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Qed.
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(* ********** *)
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(* The resident McCoys: *)
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Search (0 * _ = 0).
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(* Nat.mul_0_l : forall n : nat, 0 * n = 0 *)
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Check Nat.mul_0_r.
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(* Nat.mul_0_r : forall n : nat, n * 0 = 0 *)
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Search (1 * _ = _).
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(* Nat.mul_1_l: forall n : nat, 1 * n = n *)
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Check Nat.mul_1_r.
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(* Nat.mul_1_r : forall n : nat, n * 1 = n *)
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Search ((_ + _) * _ = _).
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(* Nat.mul_add_distr_r : forall n m p : nat, (n + m) * p = n * p + m * p
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*)
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Check Nat.mul_add_distr_l.
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(* Nat.mul_add_distr_l : forall n m p : nat, n * (m + p) = n * m + n * p *)
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Search ((_ * _) * _ = _).
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(* mult_assoc_reverse: forall n m p : nat, n * m * p = n * (m * p) *)
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Search (_ * (_ * _) = _).
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(* Nat.mul_assoc: forall n m p : nat, n * (m * p) = n * m * p *)
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Check Nat.mul_comm.
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(* Nat.mul_comm : forall n m : nat, n * m = m * n *)
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(* ********** *)
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(* end of week-04_specifications-that-depend-on-other-specifications.v *)
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