Section ProofTheoryPatterns.
Context {L: Language}
{minL: MinimumLanguage L}
{pL: PropositionalLanguage L}
{Gamma: Provable L}
{minAX: MinimumAxiomatization L Gamma}
{ipAX: IntuitionisticPropositionalLogic L Gamma}.

Lemma Build_Associativity': forall {prodp: expr -> expr -> expr},
(forall x y z: expr, |-- prodp (prodp x y) z <--> prodp x (prodp y z)) ->
Associativity L Gamma prodp.