# nLab product natural deduction - table

type theorycategory theory
syntaxsemantics
natural deductionuniversal construction
product typeproduct
type formation$\frac{\vdash \;A \colon Type \;\;\;\;\; \vdash \;B \colon Type}{\vdash A \times B \colon Type}$$A,B \in \mathcal{C} \Rightarrow A \times B \in \mathcal{C}$
term introduction$\frac{\vdash\; a \colon A\;\;\;\;\; \vdash\; b \colon B}{ \vdash \; (a,b) \colon A \times B}$$\array{ && Q\\ & {}^{\mathllap{a}}\swarrow &\downarrow_{\mathrlap{(a,b)}}& \searrow^{\mathrlap{b}}\\ A &&A \times B&& B }$
term elimination$\frac{\vdash\; t \colon A \times B}{\vdash\; p_1(t) \colon A} \;\;\;\;\;\frac{\vdash\; t \colon A \times B}{\vdash\; p_2(t) \colon B}$$\array{ && Q\\ &&\downarrow^{t} && \\ A &\stackrel{p_1}{\leftarrow}& A \times B &\stackrel{p_2}{\to}& B}$
computation rule$p_1(a,b) = a\;\;\; p_2(a,b) = b$$\array{ && Q \\ & {}^{\mathllap{a}}\swarrow &\downarrow_{(a,b)}& \searrow^{\mathrlap{b}} \\ A &\stackrel{p_1}{\leftarrow}& A \times B& \stackrel{p_2}{\to} & B}$
Revised on September 29, 2012 02:40:58 by Mike Shulman (192.16.204.218)