Add results about permutations (#42)

* added bunch of results to permutations

* everything

* Proof repair; added decidability of permutations

* name change

* final cleanup

* fixes to small files

* modulus

* new file yay

* dirty work

* delay progress

* final move

* fix build

* opam

---------

Co-authored-by: William Spencer <wjbs@uchicago.edu>

Author: lczielinski

Bits.v

@@ -1,4 +1,4 @@
-Require Import VectorStates Measurement.
+Require Import Bits VectorStates Measurement.
 
 (** This file describes some theory of discrete probability distributions.
   Its main feature is 'apply_u', a function to describe the output distribution 

DiscreteProb.v

@@ -0,0 +1,98 @@
+Require Import Prelim.
+
+Lemma div_mod_inj {a b} (c :nat) : c > 0 ->
+  (a mod c) = (b mod c) /\ (a / c) = (b / c) -> a = b.
+Proof.
+  intros Hc.
+  intros [Hmod Hdiv].
+  rewrite (Nat.div_mod_eq a c).
+  rewrite (Nat.div_mod_eq b c).
+  rewrite Hmod, Hdiv.
+  easy.
+Qed.
+
+Lemma mod_add_n_r : forall m n, 
+    (m + n) mod n = m mod n.
+Proof.
+    intros m n.
+    replace (m + n)%nat with (m + 1 * n)%nat by lia.
+    destruct n.
+    - cbn; easy.
+    - rewrite Nat.mod_add;
+        lia.
+Qed.
+
+Lemma mod_eq_sub : forall m n,
+    m mod n = (m - n * (m / n))%nat.
+Proof.
+    intros m n.
+    destruct n.
+    - cbn; lia.
+    - assert (H: (S n <> 0)%nat) by easy.
+        pose proof (Nat.div_mod m (S n) H) as Heq.
+        lia.
+Qed.
+
+Lemma mod_of_scale : forall m n q, 
+    (n * q <= m < n * S q)%nat -> m mod n = (m - q * n)%nat.
+Proof.
+    intros m n q [Hmq HmSq].
+    rewrite mod_eq_sub.
+    replace (m/n)%nat with q; [lia|].
+    apply Nat.le_antisymm.
+    - apply Nat.div_le_lower_bound; lia. 
+    - epose proof (Nat.div_lt_upper_bound m n (S q) _ _).
+        lia.
+        Unshelve.
+        all: lia.
+Qed.
+
+Lemma mod_n_to_2n : forall m n, 
+    (n <= m < 2 * n)%nat -> m mod n = (m - n)%nat.
+Proof.
+    intros.
+    epose proof (mod_of_scale m n 1 _).
+    lia.
+    Unshelve.
+    lia.
+Qed.
+
+Lemma mod_n_to_n_plus_n : forall m n, 
+    (n <= m < n + n)%nat -> m mod n = (m - n)%nat.
+Proof.
+    intros.
+    apply mod_n_to_2n; lia.
+Qed.
+
+Ltac simplify_mods_of a b :=
+    first [
+        rewrite (Nat.mod_small a b) in * by lia
+    | rewrite (mod_n_to_2n a b) in * by lia
+    ].
+
+Ltac solve_simple_mod_eqns :=
+  let __fail_if_has_mods a :=
+    match a with
+    | context[_ mod _] => fail 1
+    | _ => idtac
+    end
+  in
+    match goal with
+    | |- context[if _ then _ else _] => fail 1 "Cannot solve equation with if"
+    | _ =>
+        repeat first [
+      easy
+      |	lia
+        |	match goal with 
+            | |- context[?a mod ?b] => __fail_if_has_mods a; __fail_if_has_mods b; 
+                    simplify_mods_of a b
+            | H: context[?a mod ?b] |- _ => __fail_if_has_mods a; __fail_if_has_mods b; 
+                    simplify_mods_of a b
+            end 
+        | match goal with
+            | |- context[?a mod ?b] => (* idtac a b; *) bdestruct (a <? b);
+                    [rewrite (Nat.mod_small a b) by lia 
+                    | try rewrite (mod_n_to_2n a b) by lia]
+            end
+        ]
+    end.