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Notation "[ rel _ _ | _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope. [notation-overridden,parsing]
Require Export prosa.util.notation. Require Export prosa.util.rel. Require Export prosa.util.nat. Section SumsOverSequences. (** Consider any type [I] with a decidable equality ... *) Variable (I : eqType). (** ... and assume we are given a sequence ... *) Variable (r : seq I). (** ... and a predicate [P]. *) Variable (P : pred I). (** First, we will show some properties of the sum performed over a single function yielding natural numbers. *) Section SumOfOneFunction. (** Consider any function that yields natural numbers. *) Variable (F : I -> nat). (** We start showing that having every member of [r] equal to zero is equivalent to having the sum of all the elements of [r] equal to zero, and vice-versa. *)
I: eqType
r: seq I
P: pred I
F: I -> nat

(\sum_(i <- r | P i) F i == 0) = all (fun x : I => F x == 0) [seq x <- r | P x]
I: eqType
r: seq I
P: pred I
F: I -> nat

(\sum_(i <- r | P i) F i == 0) = all (fun x : I => F x == 0) [seq x <- r | P x]
I: eqType
r: seq I
P: pred I
F: I -> nat
x: I
r': seq I
IH: (\sum_(i <- r' | P i) F i == 0) = all (fun x : I => F x == 0) [seq x <- r' | P x]

((if P x then F x + \sum_(j <- r' | P j) F j else \sum_(j <- r' | P j) F j) == 0) = all (fun x : I => F x == 0) (if P x then x :: [seq x <- r' | P x] else [seq x <- r' | P x])
by case: ifP; rewrite ?addn_eq0 IH. Qed. (** In the same way, if at least one element of [r] is not zero, then the sum of all elements of [r] must be strictly positive, and vice-versa. *)
I: eqType
r: seq I
P: pred I
F: I -> nat

(0 < \sum_(i <- r | P i) F i) = has (fun x : I => 0 < F x) [seq x <- r | P x]
I: eqType
r: seq I
P: pred I
F: I -> nat

(0 < \sum_(i <- r | P i) F i) = has (fun x : I => 0 < F x) [seq x <- r | P x]
I: eqType
r: seq I
P: pred I
F: I -> nat

all (fun x : I => F x == 0) [seq x <- r | P x] = all (predC (fun x : I => 0 < F x)) [seq x <- r | P x]
by apply/eq_all => ?; rewrite /= lt0n negbK. Qed. (** Next, we show that if a number [a] is not contained in [r], then filtering or not filtering [a] when summing leads to the same result. *)
I: eqType
r: seq I
P: pred I
F: I -> nat
a: I

a \notin r -> \sum_(x <- r | P x && (x != a)) F x = \sum_(x <- r | P x) F x
I: eqType
r: seq I
P: pred I
F: I -> nat
a: I

a \notin r -> \sum_(x <- r | P x && (x != a)) F x = \sum_(x <- r | P x) F x
I: eqType
r: seq I
P: pred I
F: I -> nat
a: I
a_notin_r: a \notin r

\sum_(i <- r | [&& i \in r, P i & i != a]) F i = \sum_(i <- r | (i \in r) && P i) F i
I: eqType
r: seq I
P: pred I
F: I -> nat
a: I
a_notin_r: a \notin r
x: I
xinr: (x \in r) = true

P x && (x != a) = P x
I: eqType
r: seq I
P: pred I
F: I -> nat
a: I
a_notin_r: a \notin r
x: I
xinr: (x \in r) = true
xa: x = a

P x && ~~ true = P x
by move: xinr a_notin_r; rewrite xa => ->. Qed. (** We prove that if any element of [r] is bounded by constant [c], then the sum of the whole set is bounded by [c * size r]. *)
I: eqType
r: seq I
P: pred I
F: I -> nat
c: nat

(forall a : I, a \in r -> P a -> F a <= c) -> \sum_(j <- r | P j) F j <= c * size [seq j <- r | P j]
I: eqType
r: seq I
P: pred I
F: I -> nat
c: nat

(forall a : I, a \in r -> P a -> F a <= c) -> \sum_(j <- r | P j) F j <= c * size [seq j <- r | P j]
I: eqType
r: seq I
P: pred I
F: I -> nat
c: nat
Fa_le_c: forall a : I, a \in r -> P a -> F a <= c

\sum_(j <- r | P j) F j <= c * size [seq j <- r | P j]
I: eqType
r: seq I
P: pred I
F: I -> nat
c: nat
Fa_le_c: forall a : I, a \in r -> P a -> F a <= c

\sum_(j <- r | P j) F j <= \sum_(i <- r | P i) c
I: eqType
r: seq I
P: pred I
F: I -> nat
c: nat
Fa_le_c: forall a : I, a \in r -> P a -> F a <= c

\sum_(i <- r | (i \in r) && P i) F i <= \sum_(i <- r | (i \in r) && P i) c
apply: leq_sum => i /andP[ir Pi]; exact: Fa_le_c. Qed. (** Next, we show that the sum of the elements in [r] respecting [P] can be obtained by removing from the total sum over [r] the sum of the elements in [r] not respecting [P]. *)
I: eqType
r: seq I
P: pred I
F: I -> nat

\sum_(r <- r | P r) F r = \sum_(r <- r) F r - \sum_(r <- r | ~~ P r) F r
I: eqType
r: seq I
P: pred I
F: I -> nat

\sum_(r <- r | P r) F r = \sum_(r <- r) F r - \sum_(r <- r | ~~ P r) F r
by rewrite [X in X - _](bigID P)/= addnK. Qed. End SumOfOneFunction. (** In this section, we show some properties of the sum performed over two different functions. *) Section SumOfTwoFunctions. (** Consider three functions that yield natural numbers. *) Variable (E E1 E2 : I -> nat). (** Besides earlier introduced predicate [P], we add two additional predicates [P1] and [P2]. *) Variable (P1 P2 : pred I). (** Assume that [E2] dominates [E1] in all the points contained in the set [r] and respecting the predicate [P]. We prove that, if we sum both function over those points, then the sum of [E2] will dominate the sum of [E1]. *)
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I

(forall i : I, i \in r -> P i -> E1 i <= E2 i) -> \sum_(i <- r | P i) E1 i <= \sum_(i <- r | P i) E2 i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I

(forall i : I, i \in r -> P i -> E1 i <= E2 i) -> \sum_(i <- r | P i) E1 i <= \sum_(i <- r | P i) E2 i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I
le: forall i : I, i \in r -> P i -> E1 i <= E2 i

\sum_(i <- r | (i \in r) && P i) E1 i <= \sum_(i <- r | (i \in r) && P i) E2 i
apply: leq_sum => i /andP[]; exact: le. Qed. (** In the same way, if [E1] equals [E2] in all the points considered above, then also the two sums will be identical. *)
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I

(forall i : I, i \in r -> P i -> E1 i == E2 i) -> \sum_(i <- r | P i) E1 i == \sum_(i <- r | P i) E2 i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I

(forall i : I, i \in r -> P i -> E1 i == E2 i) -> \sum_(i <- r | P i) E1 i == \sum_(i <- r | P i) E2 i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I
eqE: forall i : I, i \in r -> P i -> E1 i == E2 i

\sum_(i <- [seq x <- r | P x]) E1 i = \sum_(i <- [seq x <- r | P x]) E2 i
apply: eq_big_seq => x; rewrite mem_filter => /andP[Px xr]; exact/eqP/eqE. Qed. (** Assume that [P1] implies [P2] in all the points contained in the set [r]. We prove that, if we sum both functions over those points, then the sum of [E] conditioned by [P2] will dominate the sum of [E] conditioned by [P1]. *)
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I

(forall i : I, i \in r -> P1 i -> P2 i) -> \sum_(i <- r | P1 i) E i <= \sum_(i <- r | P2 i) E i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I

(forall i : I, i \in r -> P1 i -> P2 i) -> \sum_(i <- r | P1 i) E i <= \sum_(i <- r | P2 i) E i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I
imp: forall i : I, i \in r -> P1 i -> P2 i

\sum_(i <- r | P1 i) E i <= \sum_(i <- r | P2 i && P1 i) E i + \sum_(i <- r | P2 i && ~~ P1 i) E i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I
imp: forall i : I, i \in r -> P1 i -> P2 i

\sum_(i <- r | (i \in r) && P1 i) E i <= \sum_(i <- r | [&& i \in r, P2 i & P1 i]) E i + \sum_(i <- r | P2 i && ~~ P1 i) E i
I: eqType
r: seq I
P: pred I
E, E1, E2: I -> nat
P1, P2: pred I
imp: forall i : I, i \in r -> P1 i -> P2 i
i: I

(i \in r) && P1 i = [&& i \in r, P2 i & P1 i]
by case ir: (i \in r); case P1i: (P1 i); rewrite ?andbF //= (imp i). Qed. End SumOfTwoFunctions. End SumsOverSequences. (** For technical (and temporary) reasons related to the proof style, the next two lemmas are stated outside of the section, but conceptually make use of a similar context. First, we observe that summing over a subset of a given sequence, if all summands are natural numbers, cannot yield a larger sum. *)
I: eqType
r, r': seq I
P: pred I
F: I -> nat

subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
I: eqType
r, r': seq I
P: pred I
F: I -> nat

subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
IH: forall r' : seq I, subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
r': seq I

subseq (x :: r) r' -> \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
x': I
r': seq I
IH': subseq (x :: r) r' -> \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i
IH: \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i

\sum_(i <- (x :: r) | P i) F i <= \sum_(i <- (x :: r') | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
IH: forall r' : seq I, subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
x': I
r': seq I
IH': \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i
\sum_(i <- (x :: r) | P i) F i <= \sum_(i <- (x' :: r') | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
x': I
r': seq I
IH': subseq (x :: r) r' -> \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i
IH: \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i

\sum_(i <- (x :: r) | P i) F i <= \sum_(i <- (x :: r') | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
IH: forall r' : seq I, subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
x': I
r': seq I
IH': \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i
\sum_(i <- (x :: r) | P i) F i <= \sum_(i <- (x' :: r') | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
IH: forall r' : seq I, subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
x': I
r': seq I
IH': \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i

\sum_(i <- (x :: r) | P i) F i <= \sum_(i <- (x' :: r') | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
IH: forall r' : seq I, subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
x': I
r': seq I
IH': \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i

\sum_(i <- (x :: r) | P i) F i <= \sum_(i <- (x' :: r') | P i) F i
I: eqType
P: pred I
F: I -> nat
x: I
r: seq I
IH: forall r' : seq I, subseq r r' -> \sum_(i <- r | P i) F i <= \sum_(i <- r' | P i) F i
x': I
r': seq I
IH': \sum_(i <- (x :: r) | P i) F i <= \sum_(i <- r' | P i) F i

\sum_(i <- (x :: r) | P i) F i <= F x' + \sum_(j <- r' | P j) F j
exact: leq_trans (leq_addl _ _). Qed. (** Second, we repeat the above observation that summing a superset of natural numbers cannot yield a lesser sum, but phrase the claim differently. Requiring the absence of duplicate in [r] is a simple way to guarantee that the set inclusion [r <= rs] implies the actually required multiset inclusion. *)
I: eqType
r: seq I
F: I -> nat
rs: seq I

uniq r -> {subset r <= rs} -> \sum_(i <- r) F i <= \sum_(i <- rs) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I

uniq r -> {subset r <= rs} -> \sum_(i <- r) F i <= \sum_(i <- rs) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

\sum_(i <- r) F i <= \sum_(i <- rs) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

\sum_(i <- r) F i <= \sum_(i <- rs | i \in r) F i + \sum_(i <- rs | i \notin r) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

\sum_(i <- r) F i <= \sum_(i <- rs | i \in r) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

\big[addn_comoid/0]_(i <- undup [seq x <- rs | x \in r]) F i <= \sum_(i <- rs | i \in r) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}
perm_eq r (undup [seq x <- rs | x \in r])
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

\big[addn_comoid/0]_(i <- undup [seq x <- rs | x \in r]) F i <= \sum_(i <- rs | i \in r) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}
perm_eq r (undup [seq x <- rs | x \in r])
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

\sum_(i <- undup rs | (i \in r) && true) F i <= \sum_(i <- rs | i \in r) F i
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}
perm_eq r (undup [seq x <- rs | x \in r])
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

perm_eq r (undup [seq x <- rs | x \in r])
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}

perm_eq r (undup [seq x <- rs | x \in r])
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}
x: I

(x \in r) = (x \in undup [seq x <- rs | x \in r])
I: eqType
r: seq I
F: I -> nat
rs: seq I
uniq_r: uniq r
sub_r_rs: {subset r <= rs}
x: I

(x \in r) = (x \in r) && (x \in rs)
by case xinr: (x \in r); rewrite // (sub_r_rs _ xinr). Qed. (** We continue establishing properties of sums over sequences, but start a new section here because some of the below proofs depend lemmas in the preceding section in their full generality. *) Section SumsOverSequences. (** Consider any type [I] with a decidable equality ... *) Variable (I : eqType). (** ... and assume we are given a sequence ... *) Variable (r : seq I). (** ... and a predicate [P]. *) Variable (P : pred I). (** Consider two functions that yield natural numbers. *) Variable (E1 E2 : I -> nat). (** First, as an auxiliary lemma, we observe that, if [E1 j] is less than [E2 j] for some element [j] involved in a summation (filtered by [P]), then the corresponding totals are not equal. *)
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
j: I

j \in r -> P j -> E1 j < E2 j -> (forall i : I, i \in r -> P i -> E1 i <= E2 i) -> \sum_(x <- r | P x) E1 x < \sum_(x <- r | P x) E2 x
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
j: I

j \in r -> P j -> E1 j < E2 j -> (forall i : I, i \in r -> P i -> E1 i <= E2 i) -> \sum_(x <- r | P x) E1 x < \sum_(x <- r | P x) E2 x
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j
le: forall i : I, i \in r -> P i -> E1 i <= E2 i

\sum_(x <- r | P x) E1 x < \sum_(x <- r | P x) E2 x
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j
le: forall i : I, i \in r -> P i -> E1 i <= E2 i

\sum_(y <- rem (T:=I) j r | P y) E1 y <= \sum_(y <- rem (T:=I) j r | P y) E2 y
apply: leq_sum_seq => i /mem_rem; exact: le. Qed. (** Next, we prove that if for any element i of a set [r] the following two statements hold (1) [E1 i] is less than or equal to [E2 i] and (2) the sum [E1 x_1, ..., E1 x_n] is equal to the sum of [E2 x_1, ..., E2 x_n], then [E1 x] is equal to [E2 x] for any element x of [xs]. *)
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat

(forall i : I, i \in r -> P i -> E1 i <= E2 i) -> (\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = all (fun x : I => E1 x == E2 x) [seq x <- r | P x]
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat

(forall i : I, i \in r -> P i -> E1 i <= E2 i) -> (\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = all (fun x : I => E1 x == E2 x) [seq x <- r | P x]
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = true

(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = true
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false

(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false

exists2 j : I, (j \in r) && P j & E1 j < E2 j
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j
(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r

~~ (P j ==> (E1 j == E2 j)) -> exists2 j : I, (j \in r) && P j & E1 j < E2 j
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j
(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
neq: E1 j != E2 j

exists2 j : I, (j \in r) && P j & E1 j < E2 j
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j
(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
neq: E1 j != E2 j

exists2 j : I, (j \in r) && P j & E1 j < E2 j
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j
(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j

(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
I: eqType
r: seq I
P: pred I
E1, E2: I -> nat
le: forall i : I, i \in r -> P i -> E1 i <= E2 i
aE: all (SimplPred (fun i : I => P i ==> (E1 i == E2 i))) r = false
j: I
jr: j \in r
Pj: P j
ltj: E1 j < E2 j

(\sum_(x <- r | P x) E1 x == \sum_(x <- r | P x) E2 x) = false
by apply/negbTE; rewrite neq_ltn (ltn_sum_leq_seq j). Qed. End SumsOverSequences. (** In this section, we prove a variety of properties of sums performed over ranges. *) Section SumsOverRanges. (** First, we prove that the sum of Δ ones is equal to Δ . *)

forall t Δ : nat, \sum_(t <= x < t + Δ) 1 = Δ

forall t Δ : nat, \sum_(t <= x < t + Δ) 1 = Δ
by move=> t Δ; rewrite big_const_nat iter_addn_0 mul1n addKn. Qed. (** Next, we show that a sum of natural numbers equals zero if and only if all terms are zero. *)
m, n: nat
F: nat -> nat

\sum_(m <= i < n) F i = 0 <-> (forall i : nat, m <= i < n -> F i = 0)
m, n: nat
F: nat -> nat

\sum_(m <= i < n) F i = 0 <-> (forall i : nat, m <= i < n -> F i = 0)
m, n: nat
F: nat -> nat

\sum_(m <= i < n) F i = 0 -> forall i : nat, m <= i < n -> F i = 0
m, n: nat
F: nat -> nat
(forall i : nat, m <= i < n -> F i = 0) -> \sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat

\sum_(m <= i < n) F i = 0 -> forall i : nat, m <= i < n -> F i = 0
m, n: nat
F: nat -> nat
(forall i : nat, m <= i < n -> F i = 0) -> \sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat

\sum_(i <- iota m (n - m)) F i == 0 -> forall i : nat, m <= i < n -> F i = 0
m, n: nat
F: nat -> nat
(forall i : nat, m <= i < n -> F i = 0) -> \sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat
ZERO: {in iota m (n - m), forall x : nat_eqType, F x == 0}
i: nat

m <= i < n -> F i = 0
m, n: nat
F: nat -> nat
(forall i : nat, m <= i < n -> F i = 0) -> \sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat
ZERO: {in iota m (n - m), forall x : nat_eqType, F x == 0}
i: nat
IN: i \in iota m (n - m)

F i = 0
m, n: nat
F: nat -> nat
(forall i : nat, m <= i < n -> F i = 0) -> \sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat

(forall i : nat, m <= i < n -> F i = 0) -> \sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat

(forall i : nat, m <= i < n -> F i = 0) -> \sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat
ZERO: forall i : nat, m <= i < n -> F i = 0

\sum_(m <= i < n) F i = 0
m, n: nat
F: nat -> nat
ZERO: forall i : nat, m <= i < n -> F i = 0

\sum_(m <= i < n) 0 = 0
exact: big1_eq. Qed. (** Moreover, the fact that the sum is smaller than the range of the summation implies the existence of a zero element. *)

forall (f : nat -> nat) (t Δ : nat), \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0

forall (f : nat -> nat) (t Δ : nat), \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
H: \sum_(t <= x < t + Δ.+1) f x < Δ.+1

exists x : nat, t <= x < t + Δ.+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
H: \sum_(t <= x < t + Δ.+1) f x < Δ.+1
EQ: f (t + Δ) = 0

exists x : nat, t <= x < t + Δ.+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
H: \sum_(t <= x < t + Δ.+1) f x < Δ.+1
n: nat
EQ: f (t + Δ) = n.+1
exists x : nat, t <= x < t + Δ.+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
H: \sum_(t <= x < t + Δ.+1) f x < Δ.+1
EQ: f (t + Δ) = 0

exists x : nat, t <= x < t + Δ.+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
H: \sum_(t <= x < t + Δ.+1) f x < Δ.+1
EQ: f (t + Δ) = 0

t <= t + Δ < t + Δ.+1
by apply/andP; split; [rewrite leq_addr | rewrite addnS ltnS].
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
H: \sum_(t <= x < t + Δ.+1) f x < Δ.+1
n: nat
EQ: f (t + Δ) = n.+1

exists x : nat, t <= x < t + Δ.+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
H: \sum_(t <= x < t + Δ.+1) f x < Δ.+1
n: nat
EQ: f (t + Δ) = n.+1

exists x : nat, t <= x < t + Δ.+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ

exists x : nat, t <= x < (t + Δ).+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ

\sum_(t <= x < t + Δ) f x < Δ
f: nat -> nat
t, Δ: nat
IHΔ: exists x : nat, t <= x < t + Δ /\ f x = 0
n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ
exists x : nat, t <= x < (t + Δ).+1 /\ f x = 0
f: nat -> nat
t, Δ: nat
IHΔ: \sum_(t <= x < t + Δ) f x < Δ -> exists x : nat, t <= x < t + Δ /\ f x = 0
n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ

\sum_(t <= x < t + Δ) f x < Δ
by apply leq_ltn_trans with (\sum_(t <= i < t + Δ) f i + n); first rewrite leq_addr.
f: nat -> nat
t, Δ: nat
IHΔ: exists x : nat, t <= x < t + Δ /\ f x = 0
n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ

exists x : nat, t <= x < (t + Δ).+1 /\ f x = 0
f: nat -> nat
t, Δ, n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ
z: nat
LE: t <= z
GE: z < t + Δ
ZERO: f z = 0

exists x : nat, t <= x < (t + Δ).+1 /\ f x = 0
f: nat -> nat
t, Δ, n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ
z: nat
LE: t <= z
GE: z < t + Δ
ZERO: f z = 0

t <= z < (t + Δ).+1
f: nat -> nat
t, Δ, n: nat
EQ: f (t + Δ) = n.+1
H: \sum_(t <= i < t + Δ) f i + n < Δ
z: nat
LE: t <= z
GE: z < t + Δ
ZERO: f z = 0

z < (t + Δ).+1
by rewrite ltnS ltnW. } Qed. (** Next, we prove that the summing over the difference of two functions is the same as summing over the two functions separately, and then taking the difference of the two sums. Since we are using natural numbers, we have to require that one function dominates the other in the summing range. *)
m, n: nat
F, G: nat -> nat

(forall i : nat, m <= i < n -> G i <= F i) -> \sum_(m <= i < n) (F i - G i) = \sum_(m <= i < n) F i - \sum_(m <= i < n) G i
m, n: nat
F, G: nat -> nat

(forall i : nat, m <= i < n -> G i <= F i) -> \sum_(m <= i < n) (F i - G i) = \sum_(m <= i < n) F i - \sum_(m <= i < n) G i
m, n: nat
F, G: nat -> nat
le: forall i : nat, m <= i < n -> G i <= F i

\sum_(m <= i < n) (F i - G i) = \sum_(m <= i < n) F i - \sum_(m <= i < n) G i
m, n: nat
F, G: nat -> nat
le: forall i : nat, m <= i < n -> G i <= F i

\sum_(m <= i < n | (m <= i < n) && true) (F i - G i) = \sum_(m <= i < n | (m <= i < n) && true) F i - \sum_(m <= i < n | (m <= i < n) && true) G i
rewrite sumnB// => i; rewrite andbT; exact: le. Qed. End SumsOverRanges. (** In this section, we show how it is possible to equate the result of two sums performed on two different functions and on different intervals, provided that the two functions match point-wise. *) Section SumOfTwoIntervals. (** Consider two equally-sized intervals <<[t1, t1+d)>> and <<[t2, t2+d)>>... *) Variable (t1 t2 : nat). Variable (d : nat). (** ...and two functions [F1] and [F2]. *) Variable (F1 F2 : nat -> nat). (** Assume that the two functions match point-wise with each other, with the points taken in their respective interval. *) Hypothesis equal_before_d: forall g, g < d -> F1 (t1 + g) = F2 (t2 + g). (** The then summations of [F1] over <<[t1, t1 + d)>> and [F2] over <<[t2, t2 + d)>> are equal. *)
t1, t2, d: nat
F1, F2: nat -> nat
equal_before_d: forall g : nat, g < d -> F1 (t1 + g) = F2 (t2 + g)

\sum_(t1 <= t < t1 + d) F1 t = \sum_(t2 <= t < t2 + d) F2 t
t1, t2, d: nat
F1, F2: nat -> nat
equal_before_d: forall g : nat, g < d -> F1 (t1 + g) = F2 (t2 + g)

\sum_(t1 <= t < t1 + d) F1 t = \sum_(t2 <= t < t2 + d) F2 t
t1, t2, d: nat
F1, F2: nat -> nat
n: nat
equal_before_d: forall g : nat, g < n.+1 -> F1 (t1 + g) = F2 (t2 + g)
IHn: (forall g : nat, g < n -> F1 (t1 + g) = F2 (t2 + g)) -> \sum_(t1 <= t < t1 + n) F1 t = \sum_(t2 <= t < t2 + n) F2 t

\sum_(t1 <= t < t1 + n.+1) F1 t = \sum_(t2 <= t < t2 + n.+1) F2 t
t1, t2, d: nat
F1, F2: nat -> nat
n: nat
equal_before_d: forall g : nat, g < n.+1 -> F1 (t1 + g) = F2 (t2 + g)
IHn: (forall g : nat, g < n -> F1 (t1 + g) = F2 (t2 + g)) -> \sum_(t1 <= t < t1 + n) F1 t = \sum_(t2 <= t < t2 + n) F2 t

addn_monoid (\big[addn_monoid/0]_(t1 <= i < t1 + n) F1 i) (F1 (t1 + n)) = addn_monoid (\big[addn_monoid/0]_(t2 <= i < t2 + n) F2 i) (F2 (t2 + n))
t1, t2, d: nat
F1, F2: nat -> nat
n: nat
equal_before_d: forall g : nat, g < n.+1 -> F1 (t1 + g) = F2 (t2 + g)
IHn: (forall g : nat, g < n -> F1 (t1 + g) = F2 (t2 + g)) -> \sum_(t1 <= t < t1 + n) F1 t = \sum_(t2 <= t < t2 + n) F2 t

\sum_(t2 <= t < t2 + n) F2 t + F1 (t1 + n) = \sum_(t2 <= i < t2 + n) F2 i + F2 (t2 + n)
by rewrite equal_before_d. Qed. End SumOfTwoIntervals. (** In this section, we relate the sum of items with the sum over partitions of those items. *) Section SumOverPartitions. (** Consider an item type [X] and a partition type [Y]. *) Variable X Y : eqType. (** [x_to_y] is the mapping from an item to the partition it is contained in. *) Variable x_to_y : X -> Y. (** Consider [f], a function from [X] to [nat]. *) Variable f : X -> nat. (** Consider an arbitrary predicate [P] on [X]. *) Variable P : pred X. (** Consider a sequence of items [xs] and a sequence of partitions [ys]. *) Variable xs : seq X. Variable ys : seq Y. (** We assume that any item in [xs] has its corresponding partition in the sequence of partitions [ys]. *) Hypothesis H_no_partition_missing : forall x, x \in xs -> x_to_y x \in ys. (** Consider the sum of [f x] over all [x] in a given partition [y]. *) Let sum_of_partition y := \sum_(x <- xs | P x && (x_to_y x == y)) f x. (** We prove that summation of [f x] over all [x] is less than or equal to the summation of [sum_of_partition] over all partitions. *)
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat

\sum_(x <- xs | P x) f x <= \sum_(y <- ys) sum_of_partition y
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat

\sum_(x <- xs | P x) f x <= \sum_(y <- ys) sum_of_partition y
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat

\sum_(x <- xs | P x) f x <= \sum_(y <- ys) \sum_(x <- xs | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> let sum_of_partition := fun y : Y => \sum_(x <- xs' | P x && (x_to_y x == y)) f x in \sum_(x <- xs' | P x) f x <= \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(x <- (x' :: xs') | P x) f x <= \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> let sum_of_partition := fun y : Y => \sum_(x <- xs' | P x && (x_to_y x == y)) f x in \sum_(x <- xs' | P x) f x <= \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j

\sum_(x <- (x' :: xs') | P x) f x <= \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> let sum_of_partition := fun y : Y => \sum_(x <- xs' | P x && (x_to_y x == y)) f x in \sum_(x <- xs' | P x) f x <= \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j

forall x : X, x \in xs' -> x_to_y x \in ys
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> let sum_of_partition := fun y : Y => \sum_(x <- xs' | P x && (x_to_y x == y)) f x in \sum_(x <- xs' | P x) f x <= \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
\sum_(x <- (x' :: xs') | P x) f x <= \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> let sum_of_partition := fun y : Y => \sum_(x <- xs' | P x && (x_to_y x == y)) f x in \sum_(x <- xs' | P x) f x <= \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j

forall x : X, x \in xs' -> x_to_y x \in ys
by move=> ??; apply H_no_partition_missing => //; rewrite in_cons; apply /orP; right.
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> let sum_of_partition := fun y : Y => \sum_(x <- xs' | P x && (x_to_y x == y)) f x in \sum_(x <- xs' | P x) f x <= \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys

\sum_(x <- (x' :: xs') | P x) f x <= \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> \sum_(x <- xs' | P x) f x <= \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j

\sum_(x <- (x' :: xs') | P x) f x <= \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> \sum_(x <- xs' | P x) f x <= \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j

(if P x' then f x' + \sum_(j <- xs' | P j) f j else \sum_(j <- xs' | P j) f j) <= (if P x' then \sum_(i <- ys | P x' && (x_to_y x' == i)) f x' + \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j else \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j)
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> \sum_(x <- xs' | P x) f x <= \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j

f x' + \sum_(j <- xs' | P j) f j <= \sum_(i <- ys | x_to_y x' == i) f x' + \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> \sum_(x <- xs' | P x) f x <= \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j

f x' <= \sum_(i <- ys | x_to_y x' == i) f x'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> \sum_(x <- xs' | P x) f x <= \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j

f x' <= f x' * count (eq_op (x_to_y x')) ys
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> \sum_(x <- xs' | P x) f x <= \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j

has (eq_op (x_to_y x')) ys
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
P_HOLDS: forall (i : Y) (j : X), true -> P j && (x_to_y j == i) -> P j
IN_ys: forall x : X, x \in xs' -> x_to_y x \in ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> \sum_(x <- xs' | P x) f x <= \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j

x_to_y x' \in ys
by apply H_no_partition_missing, mem_head. Qed. (** Consider a partition [y']. *) Variable y' : Y. (** We prove that the sum of items excluding all items from a partition [y'] is less-than-or-equal to the sum over all partitions except [y']. *)
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y

\sum_(x <- xs | P x && (x_to_y x != y')) f x <= \sum_(y <- ys | y != y') sum_of_partition y
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y

\sum_(x <- xs | P x && (x_to_y x != y')) f x <= \sum_(y <- ys | y != y') sum_of_partition y
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y

\sum_(x <- xs | P x && (x_to_y x != y')) f x <= \sum_(j <- xs | P j && (x_to_y j != y')) \sum_(i <- ys | [&& i != y', P j & x_to_y j == i]) f j
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y

\sum_(x <- xs | P x && (x_to_y x != y')) f x <= \sum_(j <- xs | P j && (x_to_y j != y')) \sum_(i <- ys | [&& i != y', P j & x_to_y j == i]) f j
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y

\sum_(i <- xs | [&& i \in xs, P i & x_to_y i != y']) f i <= \sum_(i <- xs | [&& i \in xs, P i & x_to_y i != y']) \sum_(i0 <- ys | [&& i0 != y', P i & x_to_y i == i0]) f i
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
x': X
ARRo: x' \in xs
Px': P x'
NEQ: x_to_y x' != y'

f x' <= \sum_(i <- ys | [&& i != y', P x' & x_to_y x' == i]) f x'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
x': X
ARRo: x' \in xs
Px': P x'
NEQ: x_to_y x' != y'

f x' <= (if [&& x_to_y x' != y', P x' & x_to_y x' == x_to_y x'] then f x' else 0) + \sum_(y <- rem (T:=Y) (x_to_y x') ys | [&& y != y', P x' & x_to_y x' == y]) f x'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
x': X
ARRo: x' \in xs
Px': P x'
NEQ: x_to_y x' != y'
x_to_y x' \in ys
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
x': X
ARRo: x' \in xs
Px': P x'
NEQ: x_to_y x' != y'

x_to_y x' \in ys
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y

forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y

forall (i : Y) (j : X), i != y' -> P j && (x_to_y j == i) -> P j && (x_to_y j != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y', y_of_x': Y
x': X
NEQ: ~ y_of_x' == y'
EQ1: P x'
EQ2: x_to_y x' = y_of_x'

P x' && (x_to_y x' != y')
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y', y_of_x': Y
x': X
NEQ: ~ y_of_x' == y'
EQ1: P x'
EQ2: x_to_y x' = y_of_x'
CONTR: x_to_y x' == y'

False
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y', y_of_x': Y
x': X
EQ2: x_to_y x' = y_of_x'
CONTR: x_to_y x' == y'

y_of_x' == y'
by rewrite -EQ2. Qed. (** In this section, we prove a stronger result about the equality between the sum over all items and the sum over all partitions of those items. *) Section Equality. (** In order to prove the stronger result of equality, we additionally assume that the sequences [xs] and [ys] are sets, i.e., that each element is contained at most once. *) Hypothesis H_xs_unique : uniq xs. Hypothesis H_ys_unique : uniq ys. (** We prove that summation of [f x] over all [x] is equal to the summation of [sum_of_partition] over all partitions. *)
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq xs
H_ys_unique: uniq ys

\sum_(x <- xs | P x) f x = \sum_(y <- ys) sum_of_partition y
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq xs
H_ys_unique: uniq ys

\sum_(x <- xs | P x) f x = \sum_(y <- ys) sum_of_partition y
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
H_no_partition_missing: forall x : X, x \in xs -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- xs | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq xs
H_ys_unique: uniq ys

\sum_(x <- xs | P x) f x = \sum_(y <- ys) \sum_(x <- xs | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> let sum_of_partition := fun y : Y => \sum_(x <- xs' | P x && (x_to_y x == y)) f x in uniq xs' -> \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(x <- (x' :: xs') | P x) f x = \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> uniq xs' -> \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

forall x : X, x \in xs' -> x_to_y x \in ys
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: uniq xs' -> \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
uniq xs'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
\sum_(x <- (x' :: xs') | P x) f x = \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: (forall x : X, x \in xs' -> x_to_y x \in ys) -> uniq xs' -> \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

forall x : X, x \in xs' -> x_to_y x \in ys
by move => ??; apply H_no_partition_missing; rewrite in_cons; apply /orP; right.
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: uniq xs' -> \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

uniq xs'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
\sum_(x <- (x' :: xs') | P x) f x = \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: uniq xs' -> \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

uniq xs'
by move: H_xs_unique; rewrite cons_uniq => /andP [??].
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(x <- (x' :: xs') | P x) f x = \sum_(y <- ys) \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(x <- (x' :: xs') | P x) f x = \sum_(j <- (x' :: xs') | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

(if P x' then f x' + \sum_(j <- xs' | P j) f j else \sum_(j <- xs' | P j) f j) = (if P x' then \sum_(i <- ys | P x' && (x_to_y x' == i)) f x' + \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j else \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j)
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

f x' + \sum_(j <- xs' | P j) f j = \sum_(i <- ys | true && (x_to_y x' == i)) f x' + \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(i <- ys | true && (x_to_y x' == i)) f x' = f x'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
f x' + \sum_(j <- xs' | P j) f j = f x' + \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(i <- ys | true && (x_to_y x' == i)) f x' = f x'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(i <- [seq x <- ys | x_to_y x' == x]) f x' = f x'
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

[seq i <- ys | x_to_y x' == i] = [:: x_to_y x']
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

[seq i <- ys | x_to_y x' == i] = [seq i <- ys | i == x_to_y x']
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x
[seq i <- ys | i == x_to_y x'] = [:: x_to_y x']
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

[seq i <- ys | x_to_y x' == i] = [seq i <- ys | i == x_to_y x']
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys

[seq i <- ys | x_to_y x' == i] = [seq i <- ys | i == x_to_y x']
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
y'': Y
ys': seq Y
H_ys_unique: uniq (y'' :: ys')
LE_TAILy: uniq ys' -> [seq i <- ys' | x_to_y x' == i] = [seq i <- ys' | i == x_to_y x']

[seq i <- y'' :: ys' | x_to_y x' == i] = [seq i <- y'' :: ys' | i == x_to_y x']
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
y'': Y
ys': seq Y
H_ys_unique: uniq (y'' :: ys')
LE_TAILy: [seq i <- ys' | x_to_y x' == i] = [seq i <- ys' | i == x_to_y x']

[seq i <- y'' :: ys' | x_to_y x' == i] = [seq i <- y'' :: ys' | i == x_to_y x']
by rewrite //= LE_TAILy //= eq_sym.
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

[seq i <- ys | i == x_to_y x'] = [:: x_to_y x']
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

x_to_y x' \in ys
by apply H_no_partition_missing; rewrite in_cons; apply /orP; left.
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

f x' + \sum_(j <- xs' | P j) f j = f x' + \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j
X, Y: eqType
x_to_y: X -> Y
f: X -> nat
P: pred X
xs: seq X
ys: seq Y
x': X
xs': seq X
H_no_partition_missing: forall x : X, x \in x' :: xs' -> x_to_y x \in ys
sum_of_partition:= fun y : Y => \sum_(x <- (x' :: xs') | P x && (x_to_y x == y)) f x: Y -> nat
y': Y
H_xs_unique: uniq (x' :: xs')
H_ys_unique: uniq ys
LE_TAIL: \sum_(x <- xs' | P x) f x = \sum_(y <- ys) \sum_(x <- xs' | P x && (x_to_y x == y)) f x

\sum_(j <- xs' | P j) f j = \sum_(j <- xs' | P j) \sum_(i <- ys | P j && (x_to_y j == i)) f j
by rewrite LE_TAIL (exchange_big_dep P) //=; move=> ??? /andP[??]. Qed. End Equality. End SumOverPartitions. (** We observe a trivial monotonicity-preserving property with regard to [leq]. *) Section Monotonicity. (** Consider any type of indices, any predicate, ... *) Variables (I : eqType) (P : pred I). (** ... and any function that maps each index to a [nat -> nat] function. *) Variable F : I -> nat -> nat. (** Consider any set of indices ... *) Variable r : seq I. (** ... and suppose that [F i] is monotonic with regard to [leq] for every index in [r]. *) Hypothesis H_mono : forall i, i \in r -> monotone leq (F i). (** Consider the sum of [F] over [r], for any given [x]. *) Let f x := \sum_(i <- r | P i) F i x. (** We note that this sum [f] is monotonic with regard to [leq] in [x]. *)
I: eqType
P: pred I
F: I -> nat -> nat
r: seq I
H_mono: forall i : I, i \in r -> monotone leq (F i)
f:= fun x : nat => \sum_(i <- r | P i) F i x: nat -> nat

monotone leq f
I: eqType
P: pred I
F: I -> nat -> nat
r: seq I
H_mono: forall i : I, i \in r -> monotone leq (F i)
f:= fun x : nat => \sum_(i <- r | P i) F i x: nat -> nat

monotone leq f
I: eqType
P: pred I
F: I -> nat -> nat
r: seq I
H_mono: forall i : I, i \in r -> monotone leq (F i)
f:= fun x : nat => \sum_(i <- r | P i) F i x: nat -> nat
x, y: nat
LEQ: x <= y

f x <= f y
I: eqType
P: pred I
F: I -> nat -> nat
r: seq I
H_mono: forall i : I, i \in r -> monotone leq (F i)
f:= fun x : nat => \sum_(i <- r | P i) F i x: nat -> nat
x, y: nat
LEQ: x <= y

\sum_(i <- r | (i \in r) && P i) F i x <= \sum_(i <- r | (i \in r) && P i) F i y
I: eqType
P: pred I
F: I -> nat -> nat
r: seq I
H_mono: forall i : I, i \in r -> monotone leq (F i)
f:= fun x : nat => \sum_(i <- r | P i) F i x: nat -> nat
x, y: nat
LEQ: x <= y
i: I
IN: i \in r

F i x <= F i y
by apply: H_mono. Qed. End Monotonicity.