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product_type [2014/03/30 19:41] nikolaj |
product_type [2014/04/01 14:13] nikolaj |
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| ===== Product type ===== | ===== Product type ===== | ||
| ==== Type ==== | ==== Type ==== | ||
| - | | @#55CCEE: context | @#55CCEE: $X, Y, Z$ ... type | | + | | @#55CCEE: context | @#55CCEE: $X, Y$ ... type | |
| | @#FF8866: rule | @#FF8866: ${\large\frac{a\ :\ X\hspace{1cm}b\ :\ Y}{\langle a,b\rangle\ :\ X\times Y}}(\times\mathcal I)$ | | | @#FF8866: rule | @#FF8866: ${\large\frac{a\ :\ X\hspace{1cm}b\ :\ Y}{\langle a,b\rangle\ :\ X\times Y}}(\times\mathcal I)$ | | ||
| | @#FF8866: rule | @#FF8866: ${\large\frac{p :\ X\times Y}{\pi_{\mathcal l}(p)\ :\ X\hspace{1cm}\pi_{\mathcal r}(p)\ :\ Y}}(\times\mathcal E)$ | | | @#FF8866: rule | @#FF8866: ${\large\frac{p :\ X\times Y}{\pi_{\mathcal l}(p)\ :\ X\hspace{1cm}\pi_{\mathcal r}(p)\ :\ Y}}(\times\mathcal E)$ | | ||
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| Depending on what we want to do with the type system, we might just replace expressions of computational equivalence $\rightsquigarrow$ by equality '$=$'. This symbol denotes a term rewriting, i.e. a computations alla | Depending on what we want to do with the type system, we might just replace expressions of computational equivalence $\rightsquigarrow$ by equality '$=$'. This symbol denotes a term rewriting, i.e. a computations alla | ||
| - | $\lambda x.\,\sin\,x\ \rightsquigarrow_\eta\ \sin,$ | + | $\left(\lambda x.\,3+x\right)\,4\ \rightsquigarrow_\beta\ 3+4,$ |
| or | or | ||
| - | $\left(\lambda x.\,3+x\right)\,4\ \rightsquigarrow_\beta\ 3+4.$ | + | $\lambda x.\,\sin\,x\ \rightsquigarrow_\eta\ \sin$ |
| + | == Logic == | ||
| + | |||
| + | Note how the rules reflect the logical "and". | ||
| ==== Parents ==== | ==== Parents ==== | ||
| === Related === | === Related === | ||
| - | [[Type theory]] | + | [[Type]] |
