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Notation "_ + _" was already used in scope nat_scope.
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Notation "_ - _" was already used in scope nat_scope.
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Notation "_ <= _" was already used in scope nat_scope.
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Notation "_ < _" was already used in scope nat_scope.
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Notation "_ >= _" was already used in scope nat_scope.
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Notation "_ > _" was already used in scope nat_scope.
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Notation "_ <= _ <= _" was already used in scope
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Notation "_ < _ <= _" was already used in scope
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Notation "_ <= _ < _" was already used in scope
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Notation "_ < _ < _" was already used in scope
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Notation "_ * _" was already used in scope nat_scope.
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(** We define a function converting a constant [c] to a function that
    ignores its argument and returns [c].  *)
Definition constant {X Y : Type} (c : Y) : (X -> Y) := fun _ => c.

(** We define a notation for the big concatenation operator.*)
  
 Reserved Notation "\cat_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
   format "'[' \cat_ ( m <= i < n ) '/ ' F ']'").

Notation "\cat_ ( m <= i < n ) F" :=
  (\big[cat/[::]]_(m <= i < n) F%N) : nat_scope.

Reserved Notation "\cat_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, P at level 41, i, m, n at level 50,
   format "'[' \cat_ ( m <= i < n | P ) '/ ' F ']'").

Notation "\cat_ ( m <= i < n | P ) F" :=
  (\big[cat/[::]]_(m <= i < n | P) F%N) : nat_scope.

Reserved Notation "\cat_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
   format "'[' \cat_ ( i < n ) '/ ' F ']'").

Notation "\cat_ ( i < n ) F" :=
  (\big[cat/[::]]_(i < n) F%N) : nat_scope.

Reserved Notation "\cat_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
   format "'[' \cat_ ( i < n | P ) '/ ' F ']'").

Notation "\cat_ ( i < n | P ) F" :=
  (\big[cat/[::]]_(i < n | P) F%N) : nat_scope.

Reserved Notation "\cat_ ( x <- xs | P ) F"
  (at level 41, F at level 41, x, xs at level 50,
   format "'[' \cat_ ( x <- xs | P ) '/ ' F ']'").

Notation "\cat_ ( x <- xs | P ) F" :=
  (\big[cat/[::]]_(x <- xs | P) F) : big_scope.

Reserved Notation "\cat_ ( x <- xs ) F"
  (at level 41, F at level 41, x, xs at level 50,
   format "'[' \cat_ ( x <- xs ) '/ ' F ']'").

Notation "\cat_ ( x <- xs ) F" :=
   (\big[cat/[::]]_(x <- xs) F) : big_scope.