Section OPERATION_PROPS.
Context {A : Type} `{SETOID : isSetoid A}.
Class isAssociative (f : A -> A -> A) : Prop :=
assoc x y z : f x (f y z) == f (f x y) z.
Class isCommutative (f : A -> A -> A) : Prop :=
comm x y : f x y == f y x.
Class isIdempotent (f : A -> A -> A) : Prop :=
idem x : f x x == x.
Class distributesOver (MUL : A -> A -> A) (ADD : A -> A -> A) : Prop :=
{ left_distr x y z
: MUL x (ADD y z) == ADD (MUL x y) (MUL x z)
; right_distr x y z
: MUL (ADD y z) x == ADD (MUL y x) (MUL z x)
}.
Class isIdentityElementOf (e : A) (f : A -> A -> A) : Prop :=
{ left_id x
: f e x == x
; right_id x
: f x e == x
}.
Class isInverseOperatorOf (INV : A -> A) (OP : A -> A -> A) {e : A} {IDENTITY : isIdentityElementOf e OP} : Prop :=
{ left_inv x
: OP (INV x) x == e
; right_inv x
: OP x (INV x) == e
}.
Class AbsorptionLaw (OP1 : A -> A -> A) (OP2 : A -> A -> A) : Prop :=
{ absorption_law1 x y
: OP1 x (OP2 x y) == x
; absorption_law2 x y
: OP2 x (OP1 x y) == x
}.
Class isAnnihilatorFor (a : A) (OP : A -> A -> A) : Prop :=
{ left_ann x
: OP a x == a
; right_ann x
: OP x a == a
}.
#[local]
Instance inverse_compatWith_eqProp (OP : A -> A -> A) (e : A) (INV : A -> A)
(COMPAT : eqPropCompatible2 OP)
(ASSOCIATIVE : isAssociative OP)
(IDENTITY : isIdentityElementOf e OP)
(INVERSE : isInverseOperatorOf INV OP)
: eqPropCompatible1 INV.
End OPERATION_PROPS.
Class isClosureOperator {A : Type} `{PROSET : isProset A} (cl : A -> A) : Prop :=
{ cl_op_extensive : forall x, x =< cl x
; cl_op_idempotent : forall x, cl (cl x) == cl x
; cl_op_monotonic :: isMonotonic1 cl
}.
Definition eqProp_cl {A : Type} `{SETOID : isSetoid A} (X : ensemble A) : ensemble A :=
fun z => exists x, z == x /\ x \in X.
#[global]
Instance eqProp_cl_isClosureOperator {A : Type} `{SETOID : isSetoid A}
: isClosureOperator eqProp_cl.
Definition isCompatibleWith_eqProp {A : Type} `{SETOID : isSetoid A} (P : A -> Prop) : Prop :=
forall x, P x -> forall y, x == y -> P y.

Lemma eqb_neq {A : Type} {hasEqDec : hasEqDec A} (x : A) (y : A)
: eqb x y = false <-> x <> y.