Section CommutativeAxioms.
Variable (T : Type) (zero one : T) (mul add : T -> T -> T) (inv : T -> T).
Hypothesis mulC : commutative mul.
(* Legacy mathcomp scripts have been relying on the fact that enum A and      *)
(* filter A (index_enum T) are convertible. This is likely to change in the   *)
(* next mathcomp release when enum, pick, subset and card are generalised to  *)
(* predicates with finite support in a choiceType - in fact the two will only *)
(* be equal up to permutation in this new theory.                             *)
(*  It is therefore advisable to stop relying on this, and use the new        *)
(* facilities provided in this library: lemmas big_enumP, big_enum, big_image *)
(* and such. Users wishing to test compliance should change the Defined in    *)
(* index_enum_key to Qed, and comment out the filter_index_enum compatibility *)
(* definition below (or Import Deprecation.Reject).                           *)
Fact index_enum_key : unit. Proof. split. Defined. (* Qed. *)
Definition index_enum (T : finType) :=
locked_with index_enum_key (Finite.enum T).
Hypotheses (fM : {morph f : x y / op x y}) (f_id : f idx = idx).
End SeqExtension.

Lemma big_nat_recl n m F : m <= n ->
\big[op/idx]_(m <= i < n.+1) F i =
op (F m) (\big[op/idx]_(m <= i < n) F i.+1).