nLab
dependent sum natural deduction - table

type theorycategory theory
syntaxsemantics
natural deductionuniversal construction
dependent sum typedependent sum
type formationX:Typex:XA(x):Type( x:XA(x)):Type\frac{\vdash\: X \colon Type \;\;\;\;\; x \colon X \;\vdash\; A(x)\colon Type}{\vdash \; \left(\sum_{x \colon X} A\left(x\right)\right) \colon Type}
term introductionx:Xa:A(x)(x,a): x:XA(x)\frac{x \colon X \;\vdash\; a \colon A(x)}{\vdash (x,a) \colon \sum_{x' \colon X} A\left(x'\right) }
term eliminationt:( x:XA(x))p 1(t):Xp 2(t):A(p 1(t))\frac{\vdash\; t \colon \left(\sum_{x \colon X} A\left(x\right)\right)}{\vdash\; p_1(t) \colon X\;\;\;\;\; \vdash\; p_2(t) \colon A(p_1(t))}
computation rulep 1(x,a)=xp 2(x,a)=ap_1(x,a) = x\;\;\;\; p_2(x,a) = a
Revised on December 10, 2013 02:46:22 by Andrew Stacey (78.91.22.168)