Decorating proofs

We consider proofs in minimal logic, written in natural deduction style. The only rules are introduction and elimination for implication and the universal quantifier. The logical connectives ∃,∧ are seen as special cases of inductively defined predicates, and hence are defined by the introduction and elimination schemes ∀x(A→ ∃xA), ∃xA→ ∀x(A→ B)→ B (x not free in B), A→ B → A ∧B, A ∧B → (A→ B → C)→ C.