Computability and Logic

186 PROOFS AND COMPLETENESS
Unless otherwise specified, ‘derivable’ is to mean ‘derivable using (R0)–
(R8)’. All proofs should be constructive, not appealing to the soundness and
completeness theorems.
14.3 Show that if Ɣ, A, B ⇒ is derivable, then Ɣ, A & B ⇒ is derivable.
14.4 Show that if Ɣ ⇒ A, and Ɣ ⇒ B, are derivable, then Ɣ ⇒ A & B, is
derivable.
14.5 Show that if Ɣ ⇒ A(c), is derivable, then Ɣ ⇒∀xA(x), is derivable,
provided c does not appear in Ɣ, ,orA(x).
14.6 Show that if Ɣ, A(t) ⇒ is derivable, then Ɣ, ∀xA(x) ⇒ is derivable.
14.7 Show that ∀xFx& ∀ xGx is deducible from ∀x(Fx & Gx).
14.8 Show that ∀x (Fx & Gx) is deducible from ∀xFx & ∀xGx.
14.9 Show that the transitivity of identity, ∀x∀y∀z(x = y & y = z → x = z) is
demonstrable.
14.10 Show that if Ɣ, A(s) ⇒ is derivable, then Ɣ, s = t, A(t) ⇒ is derivable.
14.11 Prove the following (left) inversion lemma for disjunction: if there is a derivation
of Ɣ ⇒{(A ∨ B)}∪ using rules (R0)–(R8), then there is such a derivation
of Ɣ ⇒{A, B}∪.
14.12 Prove the following (right) inversion lemma for disjunction: if there is a derivation
of Ɣ ∪{(A ∨ B)}⇒, then there is a derivation of Ɣ ∪{A}⇒, and
there is a derivation of Ɣ ∪{B}⇒.
14.13 Consider adding one or the other of the following rules to (R0)–(R8):
(R11)
(R12)
Ɣ ∪{A}⇒
Ɣ ⇒{A}∪
.
Ɣ ⇒
Ɣ ∪{(A ∨∼A)}⇒
Ɣ ⇒ .
Show that a sequent is derivable on adding (R11) if and only if it is derivable
on adding (R12).