Local Open Scope logic_base.
Local Open Scope syntax.
Class Adjointness (L: Language) {minL: MinimunLanguage L} (Gamma: ProofTheory L) (prodp funcp: expr -> expr -> expr): Prop := {
adjoint: forall x y z, |-- prodp x y --> z <-> |-- x --> (funcp y z)
}.
Class Commutativity (L: Language) {minL: MinimunLanguage L} (Gamma: ProofTheory L) (prodp: expr -> expr -> expr): Prop := {
prodp_comm_impp: forall x y, |-- prodp x y --> prodp y x
}.
Class Monotonicity (L: Language) {minL: MinimunLanguage L} (Gamma: ProofTheory L) (prodp: expr -> expr -> expr): Prop := {
prodp_mono: forall x1 y1 x2 y2, |-- x1 --> x2 -> |-- y1 --> y2 -> |-- prodp x1 y1 --> prodp x2 y2
}.
Class Associativity (L: Language) {minL: MinimunLanguage L} (Gamma: ProofTheory L) (prodp: expr -> expr -> expr): Prop := {
prodp_assoc1: forall x y z, |-- prodp x (prodp y z) --> prodp (prodp x y) z;
prodp_assoc2: forall x y z, |-- prodp (prodp x y) z --> prodp x (prodp y z)
}.
Class LeftUnit (L: Language) {minL: MinimunLanguage L} (Gamma: ProofTheory L) (e: expr) (prodp: expr -> expr -> expr): Prop := {
left_unit1: forall x, |-- prodp e x --> x;
left_unit2: forall x, |-- x --> prodp e x
}.
Class RightUnit (L: Language) {minL: MinimunLanguage L} (Gamma: ProofTheory L) (e: expr) (prodp: expr -> expr -> expr): Prop := {
right_unit1: forall x, |-- prodp x e --> x;
right_unit2: forall x, |-- x --> prodp x e
}.
Section AdjointTheorems.
Context {L: Language}
{minL: MinimunLanguage L}
{Gamma: ProofTheory L}
{minAX: MinimunAxiomatization L Gamma}
{prodp funcp: expr -> expr -> expr}
{Adj: Adjointness L Gamma prodp funcp}.

(* Specifications: *)
Lemma prodp_mono1: forall x1 x2 y, |-- x1 --> x2 -> |-- prodp x1 y --> prodp x2 y.
Lemma funcp_mono2: forall x y1 y2, |-- y1 --> y2 -> |-- funcp x y1 --> funcp x y2.
Lemma adjoint_modus_ponens:
forall x y, |-- prodp (funcp x y) x --> y.
Lemma adjoint_iter:
forall x xs y,
|-- fold_left prodp xs x --> y <-> |-- x --> (fold_right funcp y xs).