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Add a latex macro to help escape and write multiline latex

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Danila Fedorin 2020-02-29 20:42:36 -08:00
parent 1879ba2c2b
commit 252d82469c
2 changed files with 34 additions and 6 deletions
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content/blog/10_compiler_polymorphism.md
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@ -55,9 +55,34 @@ summary of each of these rules.
Rule|Name and Description Rule|Name and Description
-----|------- -----|-------
$$\\frac{x:\\sigma \\in \\Gamma}{\\Gamma \\vdash x:\\sigma}$$| __Var__: If the variable \\(x\\) is known to have some polymorphic type \\(\\sigma\\) then an expression consisting only of that variable is of that type. {{< latex >}}
$$\\frac{\\Gamma \\vdash e\_1 : \\tau\_1 \\rightarrow \\tau\_2 \\quad \\Gamma \\vdash e\_2 : \\tau\_1}{\\Gamma \\vdash e\_1 \\; e\_2 : \\tau\_2}$$| __App__: If an expression \\(e\_1\\), which is a function from monomorphic type \\(\\tau\_1\\) to another monomorphic type \\(\\tau\_2\\), is applied to an argument \\(e\_2\\) of type \\(\\tau\_1\\), then the result is of type \\(\\tau\_2\\). \frac
$$\\frac{\\Gamma, x:\\tau \\vdash e : \\tau'}{\\Gamma \\vdash \\lambda x.e : \\tau \\rightarrow \\tau'}$$| __Abs__: If the body \\(e\\) of a lambda abstraction \\(\\lambda x.e\\) is of type \\(\\tau'\\) when \\(x\\) is of type \\(\\tau\\) then the whole lambda abstraction is of type \\(\\tau \\rightarrow \\tau'\\). {x:\sigma \in \Gamma}
$$\\frac{\\Gamma \\vdash e : \\tau \\quad \\text{matcht}(\\tau, p\_i) = b\_i \\quad \\Gamma,b\_i \\vdash e\_i : \\tau\_c}{\\Gamma \\vdash \\text{case} \\; e \\; \\text{of} \\; \\\{ (p\_1,e\_1) \\ldots (p\_n, e\_n) \\\} : \\tau\_c }$$ | __Case__: This rule is not part of Hindley-Milner, and is specific to our language. If the expression being case-analyzed is of type \\(\\tau\\) and each branch \\((p\_i, e\_i)\\) is of the same type \\(\\tau\_c\\) when the pattern \\(p\_i\\) works with type \\(\\tau\\) producing extra bindings \\(b\_i\\), the whole case expression is of type \\(\\tau\_c\\). {\Gamma \vdash x:\sigma}
$$\\frac{\\Gamma \\vdash e : \\sigma \\quad \\sigma' \\sqsubseteq \\sigma}{\\Gamma \\vdash e : \\sigma'}$$| __Inst (New)__: If type \\(\\sigma'\\) is an instantiation of type \\(\\sigma\\) then an expression of type \\(\\sigma\\) is also an expression of type \\(\\sigma'\\). {{< /latex >}}| __Var__: If the variable \\(x\\) is known to have some polymorphic type \\(\\sigma\\) then an expression consisting only of that variable is of that type.
$$\\frac{\\Gamma \\vdash e : \\sigma \\quad \\alpha \\not \\in \\text{free}(\\Gamma)}{\\Gamma \\vdash e : \\forall a . \\sigma}$$| __Gen (New)__: If an expression has a type with free variables, this rule allows us generalize it to allow all possible types to be used for these free variables. {{< latex >}}
\frac
{\Gamma \vdash e_1 : \tau_1 \rightarrow \tau_2 \quad \Gamma \vdash e_2 : \tau_1}
{\Gamma \vdash e_1 \; e_2 : \tau_2}
{{< /latex >}}| __App__: If an expression \\(e\_1\\), which is a function from monomorphic type \\(\\tau\_1\\) to another monomorphic type \\(\\tau\_2\\), is applied to an argument \\(e\_2\\) of type \\(\\tau\_1\\), then the result is of type \\(\\tau\_2\\).
{{< latex >}}
\frac
{\Gamma, x:\tau \vdash e : \tau'}
{\Gamma \vdash \lambda x.e : \tau \rightarrow \tau'}
{{< /latex >}}| __Abs__: If the body \\(e\\) of a lambda abstraction \\(\\lambda x.e\\) is of type \\(\\tau'\\) when \\(x\\) is of type \\(\\tau\\) then the whole lambda abstraction is of type \\(\\tau \\rightarrow \\tau'\\).
{{< latex >}}
\frac
{\Gamma \vdash e : \tau \quad \text{matcht}(\tau, p_i) = b_i
\quad \Gamma,b_i \vdash e_i : \tau_c}
{\Gamma \vdash \text{case} \; e \; \text{of} \;
\{ (p_1,e_1) \ldots (p_n, e_n) \} : \tau_c }
{{< /latex >}}| __Case__: This rule is not part of Hindley-Milner, and is specific to our language. If the expression being case-analyzed is of type \\(\\tau\\) and each branch \\((p\_i, e\_i)\\) is of the same type \\(\\tau\_c\\) when the pattern \\(p\_i\\) works with type \\(\\tau\\) producing extra bindings \\(b\_i\\), the whole case expression is of type \\(\\tau\_c\\).
{{< latex >}}
\frac{\Gamma \vdash e : \sigma \quad \sigma' \sqsubseteq \sigma}
{\Gamma \vdash e : \sigma'}
{{< /latex >}}| __Inst (New)__: If type \\(\\sigma'\\) is an instantiation of type \\(\\sigma\\) then an expression of type \\(\\sigma\\) is also an expression of type \\(\\sigma'\\).
{{< latex >}}
\frac
{\Gamma \vdash e : \sigma \quad \alpha \not \in \text{free}(\Gamma)}
{\Gamma \vdash e : \forall \alpha . \sigma}
{{< /latex >}}| __Gen (New)__: If an expression has a type with free variables, this rule allows us generalize it to allow all possible types to be used for these free variables.

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themes/vanilla/layouts/shortcodes/latex.html Normal file
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$$
{{ .Inner }}
$$