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DiscreteProb.v
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Require Import VectorStates Measurement.
(** This file describes some theory of discrete probability distributions.
Its main feature is 'apply_u', a function to describe the output distribution
of running a quantum circuit. *)
Definition Cmod2 (c : C) : R := fst c ^ 2 + snd c ^ 2.
Lemma Cmod2_ge_0 : forall c, 0 <= Cmod2 c.
Proof.
intros. simpl. field_simplify. apply Rplus_le_le_0_compat; apply pow2_ge_0.
Qed.
Lemma Cmod2_Cmod_sqr : forall c, (Cmod2 c = (Cmod c)^2)%R.
Proof.
intros. unfold Cmod2, Cmod. rewrite R_sqrt.pow2_sqrt. lra.
simpl. nra.
Qed.
(* ========================================== *)
(** * Definition of probability distribution **)
(* ========================================== *)
(** We represent a (discrete) probability distribution over (0,n)
using a length n list of real numbers. We support sampling from this
distribution using the 'sample' function. *)
Definition sum_over_list (l : list R) := big_sum (fun i => nth i l 0) (length l).
Definition distribution (l : list R) :=
Forall (fun x => 0 <= x) l /\ sum_over_list l = 1.
Lemma sum_over_list_nil : sum_over_list [] = 0.
Proof. unfold sum_over_list. simpl. reflexivity. Qed.
Lemma sum_over_list_cons : forall x l,
sum_over_list (x :: l) = (x + sum_over_list l)%R.
Proof.
intros x l.
unfold sum_over_list.
simpl length.
rewrite big_sum_shift.
simpl nth.
reflexivity.
Qed.
Lemma sum_over_list_append : forall l1 l2,
sum_over_list (l1 ++ l2) = (sum_over_list l1 + sum_over_list l2)%R.
Proof.
intros l1 l2.
induction l1.
unfold sum_over_list.
simpl. lra.
simpl.
rewrite 2 sum_over_list_cons, IHl1.
lra.
Qed.
Lemma sum_over_list_geq_0 : forall l,
Forall (fun x => 0 <= x) l ->
0 <= sum_over_list l.
Proof.
induction l; intros.
- unfold sum_over_list. simpl. lra.
- inversion H; subst. specialize (IHl H3).
rewrite sum_over_list_cons. lra.
Qed.
(* ================================ *)
(** * Sample from a distribution **)
(* ================================ *)
(** Choose an element from the distribution based on random number r ∈ (0,1).
Example: Say that the input list is l = (.2, .3, .4, .1) (which might correspond
to the probabilities of measuring the outcomes 00, 01, 10, 11). Then this
function will return:
- 0 for r ∈ (0, .2)
- 1 for r ∈ (.2, .5)
- 2 for r ∈ (.5, .9)
- 3 for r ∈ (.9, 1)
The probability of getting a particular outcome is the size of the intervals of
r values that produce that outcome. (See r_interval below.) *)
Fixpoint sample (l : list R) (r : R) : nat :=
match l with
| nil => 0 (* error case *)
| x :: l' => if Rlt_le_dec r x then 0 else S (sample l' (r-x))
end.
Lemma sample_ub : forall l r, (sample l r <= length l)%nat.
Proof.
induction l; intros. easy.
simpl. specialize (IHl (r - a)%R). destruct (Rlt_le_dec r a); lia.
Qed.
Lemma sample_ub_lt : forall l r,
0 <= r < sum_over_list l ->
(sample l r < length l)%nat.
Proof.
induction l; intros. unfold sum_over_list in H. simpl in H. lra.
simpl. destruct (Rlt_le_dec r a). lia.
apply lt_n_S. apply IHl. rewrite sum_over_list_cons in H. lra.
Qed.
Lemma sample_lb : forall l r, (0 <= sample l r)%nat.
Proof.
induction l; intros. easy.
simpl. specialize (IHl (r - a)%R). destruct (Rlt_le_dec r a); lia.
Qed.
Lemma sample_max : forall l r,
Forall (fun x => 0 <= x) l ->
sum_over_list l <= r ->
sample l r = length l.
Proof.
induction l; intros. easy.
simpl. rewrite sum_over_list_cons in H0.
inversion H; subst.
specialize (sum_over_list_geq_0 l H4) as G.
assert (sum_over_list l <= r - a)%R by lra.
specialize (IHl (r - a)%R H4 H1) as T. rewrite T.
destruct (Rlt_le_dec r a). lra. lia.
Qed.
Lemma sample_append_l : forall l1 l2 r,
0 <= r ->
r < sum_over_list l1 ->
sample (l1 ++ l2) r = sample l1 r.
Proof.
induction l1; intros.
- rewrite sum_over_list_nil in H0. lra.
- rewrite sum_over_list_cons in H0. simpl.
destruct (Rlt_le_dec r a).
reflexivity.
rewrite IHl1.
reflexivity. lra. lra.
Qed.
Lemma sample_append_r : forall l1 l2 r,
Forall (fun x => 0 <= x) l1 ->
Forall (fun x => 0 <= x) l2 ->
sum_over_list l1 <= r ->
(sample (l1 ++ l2) r = (length l1) + sample l2 (r - sum_over_list l1))%nat.
Proof.
induction l1; intros.
- simpl. f_equal. rewrite sum_over_list_nil. lra.
- inversion H; subst.
specialize (sum_over_list_geq_0 l1 H5) as G.
rewrite sum_over_list_cons in *.
simpl.
destruct (Rlt_le_dec r a); try lra.
f_equal. rewrite IHl1; try easy; try lra.
f_equal. f_equal. lra.
Qed.
Lemma sample_repeat_lb : forall m l r,
0 <= r ->
(m <= sample (repeat 0%R m ++ l) r)%nat.
Proof.
induction m; intros.
lia. simpl.
destruct (Rlt_le_dec r 0). lra.
specialize (IHm l r H).
replace (r - 0)%R with r by lra.
lia.
Qed.
(* ========================================================================= *)
(** * Probability that a distribution satisfies a predicate (pr_outcome_sum) *)
(* ========================================================================= *)
(** Intuitively, the probability that an element satisfies boolean predicate
f is the sum over all element for which f holds. *)
Definition pr_outcome_sum (l : list R) (f : nat -> bool) : R :=
big_sum (fun i => if f i then nth i l 0 else 0) (length l).
Lemma pr_outcome_sum_extend : forall x l f,
pr_outcome_sum (x :: l) f
= if f O
then (x + pr_outcome_sum l (fun y => f (S y)))%R
else pr_outcome_sum l (fun y => f (S y)).
Proof.
intros x l f.
unfold pr_outcome_sum.
simpl length.
rewrite big_sum_shift.
destruct (f O); simpl.
reflexivity.
lra.
Qed.
Lemma pr_outcome_sum_append : forall l1 l2 f,
pr_outcome_sum (l1 ++ l2) f
= (pr_outcome_sum l1 f + pr_outcome_sum l2 (fun x => f (length l1 + x)%nat))%R.
Proof.
intros l1 l2.
induction l1; intro f.
unfold pr_outcome_sum.
simpl.
lra.
simpl.
rewrite 2 pr_outcome_sum_extend.
rewrite IHl1.
destruct (f O); lra.
Qed.
Lemma pr_outcome_sum_repeat_false : forall n f,
pr_outcome_sum (repeat 0 n) f = 0.
Proof.
intros n f.
unfold pr_outcome_sum.
destruct n as [| n]; trivial.
rewrite big_sum_0_bounded; try easy.
intros x Hx.
destruct (f x); trivial.
destruct x; trivial.
rewrite repeat_length in Hx.
simpl.
replace n with (x + (S (n - x - 1)))%nat by lia.
rewrite <- repeat_combine.
simpl.
rewrite <- repeat_length with (n:=x) (x:=0) at 1.
rewrite nth_middle.
trivial.
Qed.
Definition pr_outcome_sum_extend' :
forall l f a,
(pr_outcome_sum (a :: l) f = (if (f O) then a else 0) + pr_outcome_sum l (fun i => f (S i)))%R.
Proof.
intros.
rewrite pr_outcome_sum_extend.
destruct (f O).
reflexivity. lra.
Qed.
Lemma pr_outcome_sum_replace_f : forall l f1 f2,
(forall x, (x < length l)%nat -> f1 x = f2 x) ->
pr_outcome_sum l f1 = pr_outcome_sum l f2.
Proof.
intros l f1 f2 H.
unfold pr_outcome_sum.
apply big_sum_eq_bounded.
intros.
rewrite H; auto.
Qed.
Lemma pr_outcome_sum_false : forall l f,
(forall i, (i < length l)%nat -> f i = false) ->
pr_outcome_sum l f = 0.
Proof.
induction l; intros f Hf.
reflexivity.
rewrite pr_outcome_sum_extend.
rewrite Hf.
apply IHl.
intros i Hi.
apply Hf.
simpl. lia.
simpl. lia.
Qed.
Lemma pr_outcome_sum_true : forall l f,
(forall i, (i < length l)%nat -> f i = true) ->
pr_outcome_sum l f = sum_over_list l.
Proof.
induction l; intros f Hf.
reflexivity.
rewrite pr_outcome_sum_extend.
rewrite Hf.
rewrite IHl.
rewrite sum_over_list_cons.
reflexivity.
intros i Hi.
apply Hf.
simpl. lia.
simpl. lia.
Qed.
Lemma pr_outcome_sum_negb : forall l f,
pr_outcome_sum l f = (sum_over_list l - pr_outcome_sum l (fun x => negb (f x)))%R.
Proof.
induction l; intro f.
unfold pr_outcome_sum, sum_over_list.
simpl. lra.
rewrite 2 pr_outcome_sum_extend.
rewrite sum_over_list_cons.
rewrite IHl.
destruct (f O); simpl; lra.
Qed.
Lemma pr_outcome_sum_orb : forall l f1 f2,
Forall (fun x => 0 <= x) l ->
pr_outcome_sum l f1 <= pr_outcome_sum l (fun rnd => f1 rnd || f2 rnd).
Proof.
intros l f1 f2 Hl.
gen f1 f2.
induction l; intros f1 f2.
unfold pr_outcome_sum.
simpl. lra.
rewrite 2 pr_outcome_sum_extend.
inversion Hl; subst.
destruct (f1 O); simpl.
apply Rplus_le_compat_l.
apply IHl; auto.
destruct (f2 O); simpl.
rewrite <- (Rplus_0_l (pr_outcome_sum l _)).
apply Rplus_le_compat; auto.
apply IHl; auto.
Qed.
Lemma pr_outcome_sum_implies : forall l f1 f2,
Forall (fun x => 0 <= x) l ->
(forall x, f1 x = true -> f2 x = true) ->
(pr_outcome_sum l f1 <= pr_outcome_sum l f2)%R.
Proof.
intros l f1 f2 Hl.
gen f1 f2.
induction l; intros f1 f2 H.
unfold pr_outcome_sum.
simpl. lra.
rewrite 2 pr_outcome_sum_extend.
inversion Hl; subst.
destruct (f1 O) eqn:f1O.
apply H in f1O.
rewrite f1O.
apply Rplus_le_compat_l.
auto.
destruct (f2 O).
rewrite <- (Rplus_0_l (pr_outcome_sum _ _)).
apply Rplus_le_compat; auto.
auto.
Qed.
Lemma pr_outcome_sum_ge_0 :
forall l f, Forall (fun x => 0 <= x) l -> 0 <= pr_outcome_sum l f.
Proof.
induction l; intros.
- unfold pr_outcome_sum. simpl. lra.
- inversion H; subst. unfold pr_outcome_sum.
replace (length (a :: l)) with (S (length l)) by easy.
rewrite big_sum_shift. simpl.
specialize (IHl (fun x => f (S x)) H3).
unfold pr_outcome_sum in IHl.
destruct (f O); lra.
Qed.
(* ================================================================== *)
(** * Probability that a distribution satisfies a predicate (pr_P) **)
(* ================================================================== *)
(** Mathematically, the probability that an element satisifes a (not necessarily
boolean) predicate is the size of the range of r-values for which the element
returned from 'sample' satisfies the predicate. *)
Inductive interval_sum (P : R -> Prop) (rl rr : R) : R -> Prop :=
| SingleInterval : forall r1 r2, rl <= r1 <= r2 /\ r2 <= rr ->
(forall r, r1 < r < r2 -> P r) ->
(forall r, rl < r < r1 -> ~ P r) ->
(forall r, r2 < r < rr -> ~ P r) ->
interval_sum P rl rr (r2 - r1)%R
(* We could add [~ P rm] to this case to guarantee unique intervals *)
| CombineIntervals : forall rm r1 r2, rl <= rm <= rr ->
interval_sum P rl rm r1 ->
interval_sum P rm rr r2 ->
interval_sum P rl rr (r1 + r2).
Lemma interval_sum_shift :
forall P rl rr r a,
interval_sum P rl rr r ->
interval_sum (fun x => P (x - a)%R) (rl + a)%R (rr + a)%R r.
Proof.
intros.
induction H.
- replace (r2 - r1)%R with ((r2 + a) - (r1 + a))%R by lra.
constructor; intros; try lra.
apply H0. lra.
apply H1. lra.
apply H2. lra.
- apply CombineIntervals with (rm := (rm + a)%R); try lra; try easy.
Qed.
Lemma interval_sum_same :
forall P1 P2 rl rr r,
interval_sum P1 rl rr r ->
(forall x, rl <= x < rr -> (P1 x <-> P2 x)) ->
interval_sum P2 rl rr r.
Proof.
intros.
induction H.
- constructor; intros; try lra.
apply H0. lra. apply H1. lra.
intro. apply H0 in H5. assert (~ P1 r). apply H2. lra. easy. lra.
intro. apply H0 in H5. assert (~ P1 r). apply H3. lra. easy. lra.
- apply CombineIntervals with (rm := rm); try lra.
apply IHinterval_sum1. intros. apply H0. lra.
apply IHinterval_sum2. intros. apply H0. lra.
Qed.
Lemma interval_sum_shift_alt :
forall P rl rr r a,
interval_sum (fun x => P (x + a)%R) (rl - a)%R (rr - a)%R r ->
interval_sum P rl rr r.
Proof.
intros.
replace rl with ((rl - a) + a)%R by lra.
replace rr with ((rr - a) + a)%R by lra.
remember (fun x => P (x + a)%R) as P'.
eapply interval_sum_same.
apply interval_sum_shift. apply H.
split; intros. subst. replace (x - a + a)%R with x in H1 by lra. easy.
subst. replace (x - a + a)%R with x by lra. easy.
Qed.
Lemma interval_sum_gt_0 : forall P rl rr r, interval_sum P rl rr r -> r >= 0.
Proof. intros. induction H; lra. Qed.
Lemma interval_sum_break :
forall P rl rm rr r,
interval_sum P rl rr r ->
rl <= rm <= rr ->
exists r1 r2 : R, interval_sum P rl rm r1 /\ interval_sum P rm rr r2 /\ (r = r1 + r2)%R.
Proof.
intros.
induction H.
- intros.
destruct (Rle_lt_dec rm r1); [| destruct (Rlt_le_dec rm r2)].
+ exists (rm - rm)%R, (r2 - r1)%R.
repeat split; try lra.
* apply SingleInterval; intros; auto; try lra.
apply H2; lra.
* apply SingleInterval; intros; auto; try lra.
apply H2; lra.
+ exists (rm - r1)%R, (r2 - rm)%R.
repeat split; try lra.
* apply SingleInterval; intros; auto; try lra.
apply H1; lra.
* apply SingleInterval; intros; auto; try lra.
apply H1; lra.
+ exists (r2 - r1)%R, (rm - rm)%R.
repeat split; try lra.
* apply SingleInterval; intros; auto; try lra.
apply H3; lra.
* apply SingleInterval; intros; auto; try lra.
apply H3; lra.
- intros.
destruct (Rtotal_order rm0 rm) as [L1 | [L2 | L3]].
+ clear IHinterval_sum1.
destruct IHinterval_sum2 as [x1 [x2 [S1 [S2 E]]]]. lra.
subst.
exists (r1 + x1)%R, x2.
repeat split; auto; try lra.
apply CombineIntervals with rm0; auto; lra.
+ subst. eauto.
+ clear IHinterval_sum2.
destruct IHinterval_sum1 as [x1 [x2 [S1 [S2 E]]]]. lra.
subst.
exists x1, (x2 + r2)%R.
repeat split; auto; try lra.
apply CombineIntervals with rm0; auto; lra.
Qed.
(* TODO: clean up this proof *)
Lemma interval_sum_unique : forall P rl rr r1 r2,
interval_sum P rl rr r1 ->
interval_sum P rl rr r2 ->
r1 = r2.
Proof.
intros.
(* r3 - r0 = r4 - r1, H 1234 5678 *)
(* P : R -> Prop
rl, rr, r0, r3 : R
H : interval_sum P rl rr (r3 - r0)
r1, r4 : R
H0 : interval_sum P rl rr (r4 - r1)
H1 : rl <= r0 <= r3 /\ r3 <= rr
H2 : forall r : R, r0 < r < r3 -> P r
H3 : forall r : R, rl < r < r0 -> ~ P r
H4 : forall r : R, r3 < r < rr -> ~ P r
H5 : rl <= r1 <= r4 /\ r4 <= rr
H6 : forall r : R, r1 < r < r4 -> P r
H7 : forall r : R, rl < r < r1 -> ~ P r
H8 : forall r : R, r4 < r < rr -> ~ P r
============================
(r3 - r0)%R = (r4 - r1)%R
*)
gen r2.
induction H; intros.
gen r1 r2.
induction H3; intros.
rename r2 into r4.
rename H2 into H8. rename H1 into H7. rename H6 into tmp6. rename H0 into H6. rename H5 into tmp5. rename H into H5. rename H4 into H1. rename tmp6 into H4. rename tmp5 into H2.
{
destruct (total_order_T r0 r3) as [[L03 | E03] | G03]; try lra.
2: {
destruct (total_order_T r1 r4) as [[L14 | E14] | G14]; try lra.
remember ((r1 + r4) / 2)%R as r14.
assert (r1 < r14 < r4)%R by lra.
destruct (total_order_T r0 r14) as [[L | E] | G]; try lra.
+ assert (P r14) by (apply H6; lra).
assert (~ P r14) by (apply H4; lra).
easy.
+ remember ((r14 + r4) / 2)%R as r144.
assert (r14 < r144 < r4)%R by lra.
assert (P r144) by (apply H6; lra).
assert (~ P r144) by (apply H4; lra).
easy.
+ assert (P r14) by (apply H6; lra).
assert (~ P r14) by (apply H3; lra).
easy.
}
destruct (total_order_T r0 r1) as [[L01 | E01] | G01].
- destruct (total_order_T r1 r3) as [[L13 | E13] | G13].
+ remember ((r0 + r1) / 2)%R as r01.
assert (r0 < r01 < r1)%R by lra.
assert (P r01) by (apply H2; lra).
assert (~ P r01) by (apply H7; lra).
easy.
+ remember ((r0 + r1) / 2)%R as r01.
assert (r0 < r01 < r1)%R by lra.
assert (P r01) by (apply H2; lra).
assert (~ P r01) by (apply H7; lra).
easy.
+ remember ((r0 + r3) / 2)%R as r03.
assert (r0 < r03 < r3)%R by lra.
assert (P r03) by (apply H2; lra).
assert (~ P r03) by (apply H7; lra).
easy.
- destruct (total_order_T r3 r4) as [[L34 | E34] | G34]; try lra.
+ remember ((r3 + r4) / 2)%R as r34.
assert (r3 < r34 < r4)%R by lra.
assert (P r34) by (apply H6; lra).
assert (~ P r34) by (apply H4; lra).
easy.
+ remember ((r3 + r4) / 2)%R as r43.
assert (r4 < r43 < r3)%R by lra.
assert (P r43) by (apply H2; lra).
assert (~ P r43) by (apply H8; lra).
easy.
- destruct (total_order_T r0 r4) as [[L04 | E04] | G04].
+ remember ((r0 + r1) / 2)%R as r10.
assert (r1 < r10 < r0)%R by lra.
assert (P r10) by (apply H6; lra).
assert (~ P r10) by (apply H3; lra).
easy.
+ remember ((r0 + r1) / 2)%R as r10.
assert (r1 < r10 < r0)%R by lra.
assert (P r10) by (apply H6; lra).
assert (~ P r10) by (apply H3; lra).
easy.
+ remember ((r0 + r3) / 2)%R as r03.
assert (r0 < r03 < r3)%R by lra.
assert (P r03) by (apply H2; lra).
assert (~ P r03) by (apply H8; lra).
easy.
}
- destruct (Rlt_le_dec r0 rm) as [LT0m | LE0m]; try lra;
destruct (Rlt_le_dec r3 rm) as [LT3m | LE3m]; try lra.
+ assert (r3 - r0 = r1)%R.
{ apply IHinterval_sum1; try intros. apply H1; lra. lra. apply H2; lra. apply H3; lra.
}
assert (rr - rr = r2)%R.
{ apply IHinterval_sum2; try intros. apply H3; lra. lra. apply H2; lra. apply H3; lra.
}
lra.
+ assert (rm - r0 = r1)%R.
{ apply IHinterval_sum1; try intros. apply H1; lra. lra. apply H2; lra. apply H3; lra.
}
assert (r3 - rm = r2)%R.
{ apply IHinterval_sum2; try intros. apply H3; lra. lra. apply H2; lra. apply H3; lra.
}
lra.
+ assert (rm - rm = r1)%R.
{ apply IHinterval_sum1; try intros. apply H1; lra. lra. apply H2; lra. apply H3; lra.
}
assert (r3 - r0 = r2)%R.
{ apply IHinterval_sum2; try intros. apply H1; lra. lra. apply H2; lra. apply H3; lra.
}
lra.
- inversion H2.
+ destruct (Rlt_le_dec r3 rm) as [LT3m | LE3m];
destruct (Rlt_le_dec r4 rm) as [LT4m | LE4m]; try lra.
* assert (r1 = r4 - r3)%R.
{ apply IHinterval_sum1.
constructor; try intros. lra.
apply H4; lra. apply H5; lra. apply H6; lra.
}
assert (r2 = rr - rr)%R.
{ apply IHinterval_sum2.
constructor; try intros. lra.
apply H4; lra. apply H6; lra. apply H6; lra.
}
lra.
* assert (r1 = rm - r3)%R.
{ apply IHinterval_sum1.
constructor; try intros. lra.
apply H4; lra. apply H5; lra. apply H6; lra.
}
assert (r2 = r4 - rm)%R.
{ apply IHinterval_sum2.
constructor; try intros. lra.
apply H4; lra. apply H6; lra. apply H6; lra.
}
lra.
* assert (r1 = rm - rm)%R.
{ apply IHinterval_sum1.
constructor; try intros. lra.
apply H4; lra. apply H5; lra. apply H6; lra.
}
assert (r2 = r4 - r3)%R.
{ apply IHinterval_sum2.
constructor; try intros. lra.
apply H4; lra. apply H5; lra. apply H6; lra.
}
lra.
+ destruct (Rlt_le_dec rm rm0) as [LTmm | LEmm].
* destruct (interval_sum_break P rl rm rm0 _ H4) as [l1 [l2 [G1 [G2 G3]]]]. lra.
assert (r1 = l1) by (apply IHinterval_sum1; easy).
assert (r2 = l2 + r4)%R.
{ apply IHinterval_sum2. apply CombineIntervals with (rm := rm0); try lra; try easy.
}
lra.
* destruct (interval_sum_break P rm0 rm rr _ H5) as [l1 [l2 [G1 [G2 G3]]]]. lra.
assert (r1 = r3 + l1)%R.
{ apply IHinterval_sum1. apply CombineIntervals with (rm := rm0); try lra; try easy.
}
assert (r2 = l2) by (apply IHinterval_sum2; easy).
lra.
Qed.
(** Mathematical measure of P on the interval (0,1) *)
Definition pr_P P r := interval_sum P 0%R 1%R r.
Lemma pr_P_same :
forall P1 P2 r,
(forall rnd, 0 <= rnd < 1 -> P1 rnd <-> P2 rnd) ->
pr_P P1 r ->
pr_P P2 r.
Proof.
unfold pr_P. intros.
apply interval_sum_same with (P1 := P1); assumption.
Qed.
Lemma pr_outcome_sum_eq_aux' : forall (l : list R) (f : nat -> bool) r,
Forall (fun x => 0 <= x) l ->
sum_over_list l = r ->
interval_sum (fun rnd => f (sample l rnd) = true) 0 r (pr_outcome_sum l f).
Proof.
induction l; intros.
- unfold sum_over_list in *.
simpl in *. subst. unfold pr_outcome_sum. simpl.
replace 0 with (0 - 0)%R by lra. constructor; try intros; try lra.
- rewrite sum_over_list_cons in H0.
remember (fun i => f (S i)) as sf.
assert (interval_sum (fun rnd : R => sf (sample l rnd) = true) 0 (r - a)%R (pr_outcome_sum l sf)).
{ apply IHl. inversion H; easy. lra.
}
rewrite pr_outcome_sum_extend'.
assert (interval_sum (fun rnd : R => f (sample (a :: l) rnd) = true) 0 a (if f O then a else 0)).
{ destruct (f O) eqn:E.
- replace a with (a - 0)%R by lra.
constructor;
try (inversion H; lra);
try (intros; simpl; destruct (Rlt_le_dec r0 (a - 0)); try easy; try lra).
- replace 0 with (a - a)%R by lra.
constructor;
try (inversion H; lra);
try (intros; simpl; destruct (Rlt_le_dec r0 a); try easy; try lra).
rewrite E. easy.
}
assert (interval_sum (fun rnd : R => f (sample (a :: l) rnd) = true) a r (pr_outcome_sum l sf)).
{ (*remember (fun rnd : R => f (sample (a :: l) rnd) = true) as P.*)
apply interval_sum_shift_alt with (a := a).
replace (a - a)%R with 0 by lra.
eapply interval_sum_same. apply H1.
split; intros.
- simpl. destruct (Rlt_le_dec (x + a) a); try lra. subst. replace (x + a - a)%R with x by lra. easy.
- subst. simpl in H4. destruct (Rlt_le_dec (x + a) a) in H4; try lra. replace (x + a - a)%R with x in H4 by lra. easy.
}
rewrite Heqsf in H3.
apply CombineIntervals with (rm := a); try easy.
inversion H.
assert (0 <= sum_over_list l) by (apply sum_over_list_geq_0; easy).
lra.
Qed.
(** The pr_outcome_sum and pr_P definitions of probability are consistent. *)
Lemma pr_outcome_sum_eq_aux : forall (l : list R) (f : nat -> bool),
distribution l ->
pr_P (fun rnd => f (sample l rnd) = true) (pr_outcome_sum l f).
Proof.
intros. destruct H as [H H0]. unfold pr_P.
apply pr_outcome_sum_eq_aux'; easy.
Qed.
Lemma pr_outcome_sum_leq_exists : forall l f r,
distribution l ->
pr_outcome_sum l f <= r ->
exists r0, (0 <= r0 <= r)%R /\ pr_P (fun rnd => f (sample l rnd) = true) r0.
Proof.
intros l f r HlHr.
exists (pr_outcome_sum l f).
split; auto.
split. apply pr_outcome_sum_ge_0. apply HlHr. auto.
apply pr_outcome_sum_eq_aux; auto.
Qed.
Lemma pr_P_unique : forall P r1 r2,
pr_P P r1 ->
pr_P P r2 ->
r1 = r2.
Proof.
intros.
apply interval_sum_unique with (P:=P) (rl:=R0) (rr:=R1); auto.
Qed.
(* Alternative definition that requires uniqueness. *)
Lemma pr_outcome_sum_eq : forall f l r,
distribution l ->
pr_outcome_sum l f = r <-> pr_P (fun rnd => f (sample l rnd) = true) r.
Proof.
split; intros.
- rewrite <- H0.
apply pr_outcome_sum_eq_aux.
easy.
- eapply pr_P_unique.
apply pr_outcome_sum_eq_aux; trivial.
easy.
Qed.
(* ======================================================= *)
(** * Distribution created by running a quantum program **)
(* ======================================================= *)
(** Given our definition of sample, we can define a function to apply a
quantum program and return the result of measuring all qubits.
rnd is a random input in (0,1). *)
Definition apply_u {dim} (u : Square (2 ^ dim)) : list R :=
let v := u × basis_vector (2^dim) 0 in
map Cmod2 (vec_to_list v).
Lemma pos_Cmod2_list :
forall l, Forall (fun x => 0 <= x) (map Cmod2 l).
Proof.
induction l; intros.
- simpl. constructor.
- simpl. constructor. apply Cmod2_ge_0. apply IHl.
Qed.
Local Opaque big_sum.
Lemma sum_over_list_Cmod2_vec_to_list' : forall d x l,
sum_over_list (map Cmod2 (@vec_to_list' x d l)) =
big_sum (fun i : nat => (Cmod (l (i + x - d)%nat O) ^ 2)%R) d.
Proof.
induction d; intros.
- unfold sum_over_list. reflexivity.
- simpl. rewrite sum_over_list_cons. rewrite IHd. simpl.
rewrite big_sum_shift.
replace (0 + x - S d)%nat with (x - S d)%nat by lia.
rewrite Cmod2_Cmod_sqr. simpl.
f_equal.
Qed.
Local Transparent big_sum.
Lemma sum_over_list_Cmod2_vec_to_list :
forall d (l : Vector d),
sum_over_list (map Cmod2 (vec_to_list l)) =
big_sum (fun i : nat => (Cmod (l i O) ^ 2)%R) d.
Proof.
intros. unfold vec_to_list.
rewrite sum_over_list_Cmod2_vec_to_list'.
apply big_sum_eq_bounded. intros.
replace (x + d - d)%nat with x by lia.
reflexivity.
Qed.
Lemma distribution_apply_u : forall {dim} u,
WF_Unitary u ->
distribution (@apply_u dim u).
Proof.
intros. unfold apply_u. split.
- apply pos_Cmod2_list.
- rewrite sum_over_list_Cmod2_vec_to_list.
rewrite <- rewrite_norm.
unfold inner_product.
rewrite Mmult_adjoint.
rewrite <- Mmult_assoc.
rewrite Mmult_assoc with (A := (basis_vector (2 ^ dim) 0) †).
destruct H as [_ H].
rewrite H.
restore_dims. rewrite Mmult_1_r.
rewrite basis_vector_product_eq. reflexivity.
apply pow_positive. lia.
apply WF_adjoint. apply basis_vector_WF.
apply pow_positive. lia.
Qed.
Lemma length_apply_u : forall n (u : Square (2 ^ n)),
length (apply_u u) = (2 ^ n)%nat.
Proof.
intros n u.
unfold apply_u.
rewrite map_length.
rewrite vec_to_list_length.
reflexivity.
Qed.
Lemma nth_apply_u_probability_of_outcome : forall n (u : Square (2 ^ n)) x,
(x < 2 ^ n)%nat ->
WF_Matrix u ->
nth x (apply_u u) 0
= probability_of_outcome
(basis_vector (2^n) x)
(u × basis_vector (2^n) 0).
Proof.
intros n u x Hx WFu.
unfold apply_u, probability_of_outcome.
rewrite nth_indep with (d':=Cmod2 0).
rewrite map_nth.
remember (u × basis_vector (2 ^ n) 0) as ψ.
rewrite nth_vec_to_list by assumption.
rewrite (basis_vector_decomp ψ) at 2.
unfold inner_product.
rewrite Mmult_Msum_distr_l, <- Cmod2_Cmod_sqr.
apply f_equal.
symmetry.
erewrite big_sum_unique.
2 : { exists x. split. assumption.
split.
rewrite Mscale_mult_dist_r.
rewrite basis_vector_product_eq.
reflexivity. assumption.
intros y Hy Hxy.
rewrite Mscale_mult_dist_r.
rewrite basis_vector_product_neq by auto.
lma. }
unfold scale, I.
bdestruct_all; try lia; lca.
subst.
apply WF_mult.
assumption.
apply basis_vector_WF.
apply pow_positive. lia.
rewrite map_length.
rewrite vec_to_list_length.
assumption.
Qed.
(* ========================== *)
(** * Uniform distribution **)
(* ========================== *)
(** Uniform sampling in the range (lower, upper) *)
Definition uniform (lower upper : nat) : list R :=
repeat 0 lower ++ repeat (1/ INR (upper - lower))%R (upper - lower).
Lemma repeat_gt0 : forall m r, 0 <= r -> Forall (fun x => 0 <= x) (repeat r m).
Proof.
induction m; intros. simpl. constructor.
simpl. constructor. easy. apply IHm. easy.
Qed.
Lemma sum_over_list_repeat : forall m x, (sum_over_list (repeat x m) = INR m * x)%R.
Proof.
induction m; intros.
- simpl. unfold sum_over_list. simpl. lra.
- simpl. rewrite sum_over_list_cons. rewrite IHm.
destruct m; simpl; lra.
Qed.
Lemma sample_uniform : forall l u r,
(l < u)%nat -> 0 <= r < 1 -> (l <= sample (uniform l u) r < u)%nat.
Proof.
intros. split.
- unfold uniform. apply sample_repeat_lb. easy.
- unfold uniform. rewrite sample_append_r. rewrite repeat_length.
assert (T: (forall a b c, a < c -> b < c - a -> a + b < c)%nat) by (intros; lia).
apply T. easy.
replace (u - l)%nat
with (length (repeat (1 / INR (u - l))%R (u - l))) at 3 by apply repeat_length.
apply sample_ub_lt.
replace l with (length (repeat 0 l)) at 1 2 by apply repeat_length.
repeat rewrite sum_over_list_repeat.
replace (INR (u - l) * (1 / INR (u - l)))%R
with (INR (u - l) * / INR (u - l))%R by lra.
rewrite Rinv_r. lra.
apply not_0_INR. lia.
apply repeat_gt0; lra.
apply repeat_gt0. unfold Rdiv. rewrite Rmult_1_l.
apply Rlt_le. apply Rinv_0_lt_compat. apply lt_0_INR. lia.
rewrite sum_over_list_repeat. lra.
Qed.
Lemma distribution_uniform : forall l r,
(l < r)%nat ->
distribution (uniform l r).
Proof.
intros. split; unfold uniform.
- apply Forall_app. split; apply repeat_gt0. lra.
unfold Rdiv.
assert (0 < / INR (r - l)).
{ apply Rinv_0_lt_compat. apply lt_0_INR. lia. }
lra.
- rewrite sum_over_list_append.
do 2 rewrite sum_over_list_repeat.
unfold Rdiv.
replace (INR l * 0 + INR (r - l) * (1 * / INR (r - l)))%R
with (INR (r - l) * / INR (r - l))%R by lra.
rewrite <- Rinv_r_sym. easy.
assert (0 < INR (r - l)) by (apply lt_0_INR; lia).
lra.
Qed.
Lemma length_uniform : forall l r, (l <= r)%nat -> (length (uniform l r) = r)%nat.
Proof.
intros. unfold uniform. rewrite app_length, repeat_length, repeat_length. lia.
Qed.
(* ======================= *)
(** * Joint distribution **)
(* ======================= *)
Fixpoint scale r l :=
match l with
| nil => nil
| h :: t => (r * h)%R :: scale r t
end.
(** Combine distributions l1 and l2, where l2 may depend on the value of l1 *)
Fixpoint join' l1 l2 n :=
match n with
| O => nil
| S n' => join' l1 l2 n' ++ scale (nth n' l1 0) (l2 n')
end.
Definition join l1 l2 := join' l1 l2 (length l1).
(** Given a nat consisting of (n+m) bits, extract the first n or last m.
Example application: when sampling from (join l1 l2) where |l1|=n and
|l2|=m, you can use fst and snd to split the result. *)
(* TODO: come up with better names *)
Definition fst (m x : nat) := (x / 2 ^ m)%nat.
Definition snd (m x : nat) := (x mod 2 ^ m)%nat.
Lemma fst_0 : forall m, fst m 0 = O.
Proof.
intros. unfold fst. apply Nat.div_0_l. apply Nat.pow_nonzero. lia.
Qed.
Lemma fst_plus : forall m x, fst m (2 ^ m + x) = S (fst m x).
Proof.
intros.
unfold fst.
rewrite <- (Nat.mul_1_l (2 ^ m)) at 1.
rewrite Nat.div_add_l.
lia.
apply Nat.pow_nonzero. lia.
Qed.
Lemma fst_small : forall m x, (x < 2 ^ m)%nat -> fst m x = O.
Proof. intros. unfold fst. apply Nat.div_small. auto. Qed.
Lemma snd_0 : forall m, snd m 0 = O.
Proof.
intros. unfold snd. apply Nat.mod_0_l. apply Nat.pow_nonzero. lia.
Qed.
Lemma snd_small : forall m x, (x < 2 ^ m)%nat -> snd m x = x.
Proof. intros. unfold snd. apply Nat.mod_small. auto. Qed.
Lemma snd_plus : forall m x, snd m (2 ^ m + x) = snd m x.
Proof.
intros.
unfold snd.
rewrite Nat.add_comm.
rewrite <- (Nat.mul_1_l (2 ^ m)) at 1.
apply Nat.mod_add.
apply Nat.pow_nonzero. lia.
Qed.
Lemma simplify_fst : forall n x y,
(y < 2 ^ n)%nat ->
fst n (x * 2 ^ n + y) = x.
Proof.
intros n x y Hy.
unfold fst.
rewrite Nat.div_add_l.
rewrite Nat.div_small by assumption.
lia.
apply Nat.pow_nonzero.
lia.
Qed.