## Journal of Symbolic Logic

### The Logic of First Order Intuitionistic Type Theory with Weak Sigma- Elimination

M. D. G. Swaen

#### Abstract

Via the formulas-as-types embedding certain extensions of Heyting Arithmetic can be represented in intuitionistic type theories. In this paper we discuss the embedding of $\omega$-sorted Heyting Arithmetic $\mathbf{HA}^\omega$ into a type theory $\mathbf{WL}$, that can be described as Troelstra's system $\mathbf{ML}^1_0$ with so-called weak $\Sigma$-elimination rules. By syntactical means it is proved that a formula is derivable in $\mathbf{HA}^\omega$ if and only if its corresponding type in $\mathbf{WL}$ is inhabited. Analogous results are proved for Diller's so-called restricted system and for a type theory based on predicate logic instead of arithmetic.

#### Article information

Source
J. Symbolic Logic, Volume 56, Issue 2 (1991), 467-483.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743651

Mathematical Reviews number (MathSciNet)
MR1133079

Zentralblatt MATH identifier
0736.03023

JSTOR