... −◦ C −◦ H (∃x.B x) −◦ H ≡ ∀x.(B(x) −◦ H) (B ⊕ C) −◦ H ≡ (B −◦ H) & (C −◦ H) (! B) −◦ H ≡ B ⇒ H 1 −◦ H ≡ H. (In the equivalence involving ∃, x is not free in H.) As in Church’s Simple Theory of Types =-=[Chu40]-=-, both terms and formulas are built using a simply typed λ-calculus. We assume the usual rules of α, β, and η-conversion and we identify terms and formulas up to α-conversion. A term is λ-normal if it...

...each i = 1, . . . , m, the formula Gi is a body of this clause. If n = 0, then we write the head as simply ⊥ and say that the head is empty. A flat clause is essentially a clause of the LinLog system =-=[And92]-=- except that heads of flat clauses may be empty. A flat Forum formula is logically equivalent to a formula in uncurried form, namely, a formula of the form ∀¯y(B −◦ A1 . · · · . An) . . where n ≥ 0, ¯...

...ng language [MNPS91]. Forum has been used to specify a number of computation systems, ranging from objectoriented languages [DM95], imperative programming features [Mil96,Chi95], and a RISC processor =-=[Chi95]-=-. In this paper, we use Forum as a specification language for sequent calculus proof systems. For this purpose, we work often within a weaker fragment of Forum, called Flat Forum. 2.1 Flat Forum A for...

...n⌋ has a Forum proof (n ≥ 0). A number of other proof systems have been specified in Forum using this particular style of encoding. For example, Gentzen’s LK and LJ [Gen69], linear logic, LKQ and LKT =-=[DJS95]-=-, an optimization of LJ [LSS93,Dyc92], and Girard’s LU [Gir93]. 3.6 Advantages of such encodings The encoding of an object-level proof system as Forum clauses has certain advantages over encoding them...