(* The record of operations *)
Structure fparams: Type := {
K:> Set;               (* the scalar type *)
v0: K;                (* 0 *)
v1: K;                (* 1 *)
eqK: K -> K -> bool;  (* = as a boolean function *)
oppK: K -> K;         (* - unary *)
addK : K -> K -> K;   (* + *)
multK : K -> K -> K;  (* * *)
invK : K -> K         (* 1/x *)
}.
(* Recover the usual mathematical notations *)
Delimit Scope field_scope with f.
Notation "x ?= y" := (eqK _ x y) (at level 70) : field_scope.
Notation "x + y" := (addK _ x y) : field_scope.
Notation "- x" := (oppK _ x) : field_scope.
Notation "x ^-1" := (invK _ x) (at level 25) : field_scope.
Notation "x * y" := (multK _ x y) : field_scope.
Notation "0" := (v0 _) : field_scope.
Notation "1" := (v1 _) : field_scope.
Section ParamsProp.
Variable p: fparams.
Open Scope field_scope.
(* Properties associated with a params *)
Structure fparamsProp: Type := {
eqK_dec: forall x y: p, if x ?= y then x = y else x <> y;
(* Boolean equality *)
addK_assoc: forall x y z: p, (x + y) + z = x + (y + z);
(* Associativity for + *)
addK_com: forall x y: p, x + y = y + x;
(* Commutativity for + *)
addK0l: forall x: p, 0 + x = x;
(* Left neutral  for + *)
oppKr: forall x: p, x + - x = 0;
(* Right opposite  for + *)
multK_assoc: forall x y z: p, (x * y) * z = x * (y * z);
(* Associativity for * *)
multK_com: forall x y: p, x * y = y * x;
(* Commutativity for * *)
multK1l: forall x: p, 1 * x = x;
(* Left neutral element for * *)
add_multKl: forall x y z: p, (x + y) * z = x * z + y * z;
(* Left distributivity of + over * *)
one_diff_zero: 1 <> 0 :> p;
(* Inverse *)
invKl: forall x: p, x <> 0 -> x * x^-1 = 1
}.
Fixpoint expK  (k: p) (n:nat) {struct n}: p :=
match n with O => 1 | (S n1) => k * expK k n1 end.
Notation "x ^ k" := (expK x k) : field_scope.
(* Derived properties  *)
Variable Hp: fparamsProp.
Implicit Types x y z: p.

Lemma expKm1_sub m n: m <= n -> (-(1))^ (n - m) = (-(1))^ (n + m).