Next: About this document
CS486 Spring 1997
Lecture 7 Tuesday, Feb. 11, 1997
- Deduction Theorem for Hilbert systems and Generalized Deduction Theorem
- Define
- Deduction Theorem
- Generalization
- Relating Gentzen systems to Hilbert systems (Constructive Proof Transformations)
- Kleene axioms
- Translating tableau proof to Hilbert proof
- Corollary: completeness of Hilbert systems
- Compactness-Smullyan Chapter III
- definition
- König's lemma
- tableau-based proof of compactness
- Discuss assignments
Generalized Deduction Theorem
If we define 
to mean that there is a Hilbert system deduction of 
from 
, then the sequent rules are derivable. That is
(i) If 
then 
and
if 
and 
then 
(ii) If 
and 
then 
and
if 
then 
(iii) If 
then 
and
if 
and 
then 
Proof
We show how to transform the given Hilbert deduction into the derived ones by complete induction on the size of the Hilbert proof.
(i) Base case: assume 
in one step.
(a)
is one of 
Axiom 5
by MP
(b) is an axiom
same as (a)
(c) is 
prove 
as follows
Axiom 5
Axiom 6
by MP
Axiom 5
by MP
Induction cases: Assume that for all Hilbert deductions of length 
the theorem holds in
case 
. Let be a deduction of length 
. Consider how was deduced:
(a)same as base case
(b)same as base case
(c)same as base case
(d)by MP from a statement 
and 
Then
follows from induction hypothesis
follows from induction hypothesis
by Axiom 6
by MP
by MP
end of case
The other cases are left for the reader.
Qed
Gentzen Tableau to Hilbert System Transformation Theorem (Gentzen_to_HilbertThm)
If 
(in Refinement Logic), then is provable from 
in the Hilbert system 
(i.e. with Kleene's axioms).