337 lines
6.5 KiB
ReStructuredText
337 lines
6.5 KiB
ReStructuredText
The *Spineless Tagless G-Machine*
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=================================
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WIP. This will hopefully be expanded into a thorough walkthrough of the state
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machine.
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Evaluation is complete when a single :code:`NNum` remains on the stack and the
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dump is empty.
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.. math::
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\transrule
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{ a : \nillist
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& \nillist
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& h
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\begin{bmatrix}
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a : \mathtt{NNum} \; n
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\end{bmatrix}
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& g
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}
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{ \mathtt{HALT}
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}
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Dereference an indirection passed as an argument to a function.
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.. math::
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\transrule
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{a : s & d & h
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\begin{bmatrix}
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a : \mathtt{NAp} \; a_1 \; a_2 \\
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a_2 : \mathtt{NInd} \; a_3
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\end{bmatrix} & g}
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{a : s & d & h[a : \mathtt{NAp} \; a_1 \; a_3] & g}
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Dereference an indirection on top of the stack.
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.. math::
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\transrule
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{p : s & d & h
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\begin{bmatrix}
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p : \mathtt{NInd} \; a
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\end{bmatrix} & g}
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{a : s & d & h & g}
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Perform a unary operation :math:`o(n)` with internal :code:`Prim` constructor
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:code:`O` on an argument in normal form.
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.. math::
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\transrule
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{ f : a : s
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& d
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& h
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\begin{bmatrix}
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f : \mathtt{NPrim} \; \mathtt{O} \\
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a : \mathtt{NAp} \; f \; x \\
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x : \mathtt{NNum} \; n
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\end{bmatrix}
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& g
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}
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{ a : s
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& d
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& h
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\begin{bmatrix}
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a : \mathtt{NNum} \; (o(n))
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\end{bmatrix}
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& g
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}
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Evaluate the argument of a unary operation with internal :code:`Prim`
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constructor :code:`O`.
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.. math::
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\transrule
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{ f : a : \nillist
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& d
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& h
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\begin{bmatrix}
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f : \mathtt{NPrim} \; \mathtt{O} \\
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a : \mathtt{NAp} \; f \; x
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\end{bmatrix}
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& g
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}
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{ x : \nillist
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& (f : a : \nillist) : d
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& h
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& g
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}
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Restore the stack when a sub-computation has completed.
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.. math::
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\transrule
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{ a : \nillist
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& s : d
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& h
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\begin{bmatrix}
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a : \mathtt{NNum} \; n
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\end{bmatrix}
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& g
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}
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{ s
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& d
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& h
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& g
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}
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Reduce a supercombinator and update the root with the :math:`\beta`-reduced form
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.. math::
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\transrule
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{ a_0 : a_1 : \ldots : a_n : s
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& d
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& h
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\begin{bmatrix}
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a_0 : \mathtt{NSupercomb} \; [x_1,\ldots,x_n] \; e
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\end{bmatrix}
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& g
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}
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{ a_n : s
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& d
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& h'
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& g
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\\
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& \SetCell[c=3]{c}
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\text{where } h' = \mathtt{instantiateU} \; e \; a_n \; h \; g
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}
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Perform a binary operation :math:`o(x,y)` associated with internal :code:`Prim`
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constructor :code:`O` on two :code:`NNum` s both in normal form.
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.. math::
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\transrule
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{ f : a_1 : a_2 : s
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& d
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& h
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\begin{bmatrix}
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f : \mathtt{NPrim} \; \mathtt{O} \\
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a_1 : \mathtt{NAp} \; f \; (\mathtt{NNum} \; x) \\
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a_2 : \mathtt{NAp} \; a_1 \; (\mathtt{NNum} \; y)
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\end{bmatrix}
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& g
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}
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{ a_2 : s
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& d
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& h
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\begin{bmatrix}
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a_2 : \mathtt{NNum} \; (o(x,y))
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\end{bmatrix}
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& g
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}
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In a conditional primitive, perform the reduction if the condition has been
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evaluated as True (:code:`NData 1 []`).
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.. math::
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\transrule
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{ f : a_1 : a_2 : a_3 : s
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& d
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& h
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\begin{bmatrix}
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f : \mathtt{NPrim} \; \mathtt{IfP} \\
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c : \mathtt{NPrim} \; (\mathtt{NData} \; 1 \; \nillist) \\
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a_1 : \mathtt{NAp} \; f \; c \\
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a_2 : \mathtt{NAp} \; a_1 \; x \\
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a_3 : \mathtt{NAp} \; a_2 \; y
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\end{bmatrix}
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& g
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}
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{ x : s
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& d
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& h
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& g
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}
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In a conditional primitive, perform the reduction if the condition has been
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evaluated as False (:code:`NData 0 []`).
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.. math::
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\transrule
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{ f : a_1 : a_2 : a_3 : s
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& d
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& h
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\begin{bmatrix}
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f : \mathtt{NPrim} \; \mathtt{IfP} \\
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c : \mathtt{NPrim} \; (\mathtt{NData} \; 0 \; \nillist) \\
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a_1 : \mathtt{NAp} \; f \; c \\
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a_2 : \mathtt{NAp} \; a_1 \; x \\
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a_3 : \mathtt{NAp} \; a_2 \; y
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\end{bmatrix}
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& g
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}
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{ y : s
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& d
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& h
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& g
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}
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In a conditional primitive, evaluate the condition.
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.. math::
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\transrule
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{ f : a_1 : \nillist
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& d
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& h
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\begin{bmatrix}
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f : \mathtt{NPrim} \; \mathtt{IfP} \\
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a_1 : \mathtt{NAp} \; f \; x
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\end{bmatrix}
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& g
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}
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{ x : \nillist
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& (f : a_1 : \nillist) : d
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& h
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& g
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}
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Construct :code:`NData` out of a constructor and its arguments
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.. math::
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\transrule
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{ c : a_1 : \ldots : a_n : \nillist
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& d
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& h
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\begin{bmatrix}
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c : \mathtt{NPrim} \; (\mathtt{ConP} \; t \; n) \\
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a_1 : \mathtt{NAp} \; c \; x_1 \\
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\vdots \\
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a_n : \mathtt{NAp} \; a_{n-1} \; x_n
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\end{bmatrix}
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& g
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}
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{ a_n : \nillist
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& d
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& h
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\begin{bmatrix}
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a_n : \mathtt{NData} \; t \; [x_1, \ldots, x_n]
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\end{bmatrix}
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& g
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}
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Pairs
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-----
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Evaluate the first argument if necessary
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.. math::
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\transrule
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{ c : a_1 : a_2 : \nillist
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& d
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& h
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\begin{bmatrix}
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c : \mathtt{NPrim} \; \mathtt{CasePairP} \\
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p : \mathtt{NAp} \; \_ \: \_ \\
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a_1 : \mathtt{NAp} \; c \; p \\
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a_2 : \mathtt{NAp} \; a_2 \; f
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\end{bmatrix}
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& g
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}
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{ p : \nillist
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& (a_1 : a_2 : \nillist) : d
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& h
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& g
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}
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Perform the reduction if the first argument is in normal form
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.. math::
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\transrule
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{ c : a_1 : a_2 : s
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& d
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& h
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\begin{bmatrix}
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c : \mathtt{NPrim} \; \mathtt{CasePairP} \\
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p : \mathtt{NData} \; 0 \; [x,y] \\
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a_1 : \mathtt{NAp} \; c \; p \\
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a_2 : \mathtt{NAp} \; a_1 \; f
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\end{bmatrix}
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& g
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}
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{ a_1 : a_2 : s
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& d
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& h
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\begin{bmatrix}
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a_1 : \mathtt{NAp} \; f \; x \\
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a_2 : \mathtt{NAp} \; a_1 \; y
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\end{bmatrix}
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& g
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}
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Lists
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-----
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Evaluate the scrutinee
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.. math::
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\transrule
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{ c : a_1 : a_2 : a_3 : \nillist
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& d
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& h
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\begin{bmatrix}
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c : \mathtt{NPrim} \; \mathtt{CaseListP} \\
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a_1 : \mathtt{NAp} \; c \; x
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\end{bmatrix}
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& g
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}
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{ x : \nillist
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& (a_1 : a_2 : a_3) : \nillist
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& h
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& g
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}
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If the scrutinee is :code:`Nil`, perform the appropriate reduction.
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.. math::
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\transrule
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{ c : a_1 : a_2 : a_3 : s
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& d
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& h
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\begin{bmatrix}
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c : \mathtt{NPrim} \; \mathtt{CaseListP} \\
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p : \mathtt{NData} \; 1 \; \nillist \\
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a_1 : \mathtt{NAp} \; c \; p \\
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a_2 : \mathtt{NAp} \; p \; f_\text{nil} \\
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a_3 : \mathtt{NAp} \; a_2 \; f_\text{cons}
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\end{bmatrix}
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& g
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}
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{ a_3 : s
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& d
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& h
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\begin{bmatrix}
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a_3 : \mathtt{NAp} \; f_\text{nil}
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\end{bmatrix}
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& g
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}
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