Theorem Proving in Type Theory

As one aspect of the endeavor to c r e a te new i n t e l l e c t u a l t o o l s f o r mankind, we w ish to enable computers to p rove , and t o a s s i s t i n the p roo f s o f , theorems o f mathematics and ( e v e n t u a l l y ) o t h e r d i s c i p l i n e s which have ach ieved the r e q u i s i t e l o g i c a l p r e c i s i o n . For t h i s purpose, a p a r t i c u l a r l y s u i t a b l e f o rma l language i s Church 's f o r m u l a t i o n [4] o f type t heo ry w i t h A -convers ion . i n t h i s language t r a d i t i o n a l mathemat ica l n o t a t i o n s can be expressed ve ry d i r e c t l y , and the i n t u i t i v e d i s t i n c t i o n s between d i f f e r e n t types of mathemat ica l e n t i t i e s (such as numb e r s , f u n c t i o n s , and se ts o f f u n c t i o n s ) are made s y n t a c t i c a l l y e x p l i c i t .