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# 15-815 Automated Theorem Proving

# Lecture 5: Sequent Calculus

We introduce the sequent calculus as a means to search for normal natural
deductions in a purely bottom-up fashion. We then relate it normal natural
deductions by appropriate soundness and completeness theorems. In order to
model arbitrary natural deductions have to add the rule of cut. If we can
show that cut is redundant (whenever the premises of a cut have a cut-free
proof, then does the conclusion), we can see immediately that the system must
be consistent. Moreover, we obtain a procedure for translating an arbitrary
natural deduction to a normal one.

### Reading

- Handout 4 on
*Sequent Calculus*
(also availabe in PDF format).

### Key Concepts

- Sequents
- Right and left rules
- Exchange, weakening and contraction
- Proof by induction on the structure of a derivation
- Generalizing the induction hypothesis
- Cut

### Links

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Frank Pfenning
fp@cs