Section Defix.
Variables (T : Type) (n : nat) (f : forall _ : T, T) (x : T).
Fixpoint loop (m : nat) : T :=
match m return T with
| O => x
| S i => f (loop i)
end.
Definition iter := loop n.
End Defix.
Parameter eq_bigl :
forall (R : Type) (idx : R) (op : forall (_ : R) (_ : R), R) (I : Type)
(r : list I) (P1 P2 : pred I) (F : forall _ : I, R) (_ : @eqfun bool I P1 P2),
@eq R (@bigop R I idx r (fun i : I => @BigBody R I i op (P1 i) (F i)))
(@bigop R I idx r (fun i : I => @BigBody R I i op (P2 i) (F i))).
Parameter eq_big :
forall (R : Type) (idx : R) (op : forall (_ : R) (_ : R), R) (I : Type)
(r : list I) (P1 P2 : pred I) (F1 F2 : forall _ : I, R) (_ : @eqfun bool I P1 P2)
(_ : forall (i : I) (_ : is_true (P1 i)), @eq R (F1 i) (F2 i)),
@eq R (@bigop R I idx r (fun i : I => @BigBody R I i op (P1 i) (F1 i)))
(@bigop R I idx r (fun i : I => @BigBody R I i op (P2 i) (F2 i))).
Parameter eq_bigr :
forall (R : Type) (idx : R) (op : forall (_ : R) (_ : R), R) (I : Type)
(r : list I) (P : pred I) (F1 F2 : forall _ : I, R)
(_ : forall (i : I) (_ : is_true (P i)), @eq R (F1 i) (F2 i)),
@eq R (@bigop R I idx r (fun i : I => @BigBody R I i op (P i) (F1 i)))
(@bigop R I idx r (fun i : I => @BigBody R I i op (P i) (F2 i))).
Parameter big_const_nat :
forall (R : Type) (idx : R) (op : forall (_ : R) (_ : R), R) (m n : nat) (x : R),
@eq R (@bigop R nat idx (index_iota m n) (fun i : nat => @BigBody R nat i op true x))
(@iter R (subn n m) (op x) idx).
Delimit Scope N_scope with num.
Delimit Scope nat_scope with N.
Reserved Notation "\sum_ ( m <= i < n | P ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \sum_ ( m  <=  i  <  n  |  P ) '/  '  F ']'").
Reserved Notation "\sum_ ( m <= i < n ) F"
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \sum_ ( m  <=  i  <  n ) '/  '  F ']'").
Local Notation "+%N" := addn (at level 0, only parsing).
Notation "\sum_ ( m <= i < n | P ) F" :=
(\big[+%N/0%N]_(m <= i < n | P%B) F%N) : (*nat_scope*) big_scope.
Notation "\sum_ ( m <= i < n ) F" :=
(\big[+%N/0%N]_(m <= i < n) F%N) : (*nat_scope*) big_scope.
Parameter iter_addn_0 : forall m n : nat, @eq nat (@iter nat n (addn m) O) (muln m n).
End Mocks.
(*****************************************************************************)

Lemma test_big_nested_1 (F G : nat -> nat) (m n : nat) :
\sum_(0 <= i < m) \sum_(0 <= j < n | odd (j * 1)) (i + j) =
\sum_(0 <= i < m) \sum_(0 <= j < n | odd j) (j + i).