nLab product natural deduction - table

	type theory	category theory
	syntax	semantics
	natural deduction	universal construction
	product type	product
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	$p_1(a,b) = a\;\;\; p_2(a,b) = b$	$\array{ && Q \\ & {}^{\mathllap{a}}\swarrow &\downarrow_{(a,b)}& \searrow^{\mathrlap{b}} \\ A &\stackrel{p_1}{\leftarrow}& A \times B& \stackrel{p_2}{\to} & B}$