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Funcons-beta : Patterns.cbs | PLAIN | PDF

OUTLINE

Patterns

\[\begin{align*} [ \ \KEY{Datatype} \quad & \NAMEREF{patterns} \\ \KEY{Funcon} \quad & \NAMEREF{pattern} \\ \KEY{Funcon} \quad & \NAMEREF{pattern-any} \\ \KEY{Funcon} \quad & \NAMEREF{pattern-bind} \\ \KEY{Funcon} \quad & \NAMEREF{pattern-type} \\ \KEY{Funcon} \quad & \NAMEREF{pattern-else} \\ \KEY{Funcon} \quad & \NAMEREF{pattern-unite} \\ \KEY{Funcon} \quad & \NAMEREF{match} \\ \KEY{Funcon} \quad & \NAMEREF{match-loosely} \\ \KEY{Funcon} \quad & \NAMEREF{case-match} \\ \KEY{Funcon} \quad & \NAMEREF{case-match-loosely} \\ \KEY{Funcon} \quad & \NAMEREF{case-variant-value} \ ] \end{align*}\]

General patterns are simple patterns or structured patterns. Matching a pattern to a value either computes an environment or fails.

Simple patterns are constructed from abstractions whose bodies depend on a given value, and whose executions either compute environments or fail.

Structured patterns are composite values whose components may include simple patterns as well as other values.

Matching a structured value to a structured pattern is similar to assigning a structured value to a structured variable, with simple pattern components matching component values analogously to simple variable components assigned component values.

Note that patterns match only values, not (empty or proper) sequences.

\[\begin{align*} \KEY{Meta-variables} \quad & \VAR{T}, \VAR{T}' <: \NAMEHYPER{../..}{Value-Types}{values} \end{align*}\]

Simple patterns

\[\begin{align*} \KEY{Datatype} \quad \NAMEDECL{patterns} \ ::= \ & \NAMEDECL{pattern}( \_ : \NAMEHYPER{../.}{Generic}{abstractions} ( \NAMEHYPER{../..}{Value-Types}{values} \TO \NAMEHYPER{../../../Computations/Normal}{Binding}{environments} )) \end{align*}\]

\(\SHADE{\NAMEREF{patterns}}\) is the type of simple patterns that can match values of a particular type.

\(\SHADE{\NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{X} ) )}\) constructs a pattern with dynamic bindings, and \(\SHADE{\NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{closure} ( \VAR{X} ) )}\) computes a pattern with static bindings. However, there is no difference between dynamic and static bindings when the pattern is matched in the same scope where it is constructed.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{pattern-any} : \TO \NAMEREF{patterns} \\&\quad \leadsto \NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEHYPER{../../Composite}{Maps}{map} ( \ ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{pattern-any}}\) matches any value, computing the empty environment.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{pattern-bind}( \VAR{I} : \NAMEHYPER{../../../Computations/Normal}{Binding}{identifiers}) : \TO \NAMEREF{patterns} \\&\quad \leadsto \NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEHYPER{../../../Computations/Normal}{Binding}{bind-value} ( \VAR{I}, \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{pattern-bind} ( \VAR{I} )}\) matches any value, computing the environment binding \(\SHADE{\VAR{I}}\) to that value.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{pattern-type}( \VAR{T} ) : \TO \NAMEREF{patterns} \\&\quad \leadsto \NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEHYPER{../../../Computations/Normal}{Flowing}{if-true-else} ( \NAMEHYPER{../..}{Value-Types}{is-in-type} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given}, \VAR{T} ), \NAMEHYPER{../../Composite}{Maps}{map} ( \ ), \NAMEHYPER{../../../Computations/Abnormal}{Failing}{fail} ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{pattern-type} ( \VAR{T} )}\) matches any value of type \(\SHADE{\VAR{T}}\), computing the empty environment.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{pattern-else}( \_ : \NAMEHYPER{../..}{Value-Types}{values}, \_ : \NAMEHYPER{../..}{Value-Types}{values}) : \TO \NAMEREF{patterns} \\ \KEY{Rule} \quad & \NAMEREF{pattern-else} ( \VAR{P}\SUB{1} : \NAMEHYPER{../..}{Value-Types}{values}, \VAR{P}\SUB{2} : \NAMEHYPER{../..}{Value-Types}{values} ) \leadsto \\&\quad \NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEHYPER{../../../Computations/Abnormal}{Failing}{else} ( \NAMEREF{match} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given}, \VAR{P}\SUB{1} ), \NAMEREF{match} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given}, \VAR{P}\SUB{2} ) ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{pattern-else} ( \VAR{P}\SUB{1}, \VAR{P}\SUB{2} )}\) matches all values matched by \(\SHADE{\VAR{P}\SUB{1}}\) or by \(\SHADE{\VAR{P}\SUB{2}}\). If a value matches \(\SHADE{\VAR{P}\SUB{1}}\), that match gives the computed environment; if a value does not match \(\SHADE{\VAR{P}\SUB{1}}\) but matches \(\SHADE{\VAR{P}\SUB{2}}\), that match gives the computed environment; otherwise the match fails.

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{pattern-unite}( \_ : \NAMEHYPER{../..}{Value-Types}{values}, \_ : \NAMEHYPER{../..}{Value-Types}{values}) : \TO \NAMEREF{patterns} \\ \KEY{Rule} \quad & \NAMEREF{pattern-unite} ( \VAR{P}\SUB{1} : \NAMEHYPER{../..}{Value-Types}{values}, \VAR{P}\SUB{2} : \NAMEHYPER{../..}{Value-Types}{values} ) \leadsto \\&\quad \NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \NAMEHYPER{../../../Computations/Normal}{Binding}{collateral} ( \NAMEREF{match} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given}, \VAR{P}\SUB{1} ), \NAMEREF{match} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given}, \VAR{P}\SUB{2} ) ) ) ) \end{align*}\]

\(\SHADE{\NAMEREF{pattern-unite} ( \VAR{P}\SUB{1}, \VAR{P}\SUB{2} )}\) matches all values matched by both \(\SHADE{\VAR{P}\SUB{1}}\) and \(\SHADE{\VAR{P}\SUB{2}}\), then uniting the computed environments, which fails if the domains of the environments overlap.

Pattern matching

\[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{match}( \_ : \NAMEHYPER{../..}{Value-Types}{values}, \_ : \NAMEHYPER{../..}{Value-Types}{values}) : \TO \NAMEHYPER{../../../Computations/Normal}{Binding}{environments} \end{align*}\]

\(\SHADE{\NAMEREF{match} ( \VAR{V}, \VAR{P} )}\) takes a (potentially structured) value \(\SHADE{\VAR{V}}\) and a (potentially structured) pattern \(\SHADE{\VAR{P}}\). Provided that the structure and all components of \(\SHADE{\VAR{P}}\) exactly match the structure and corresponding components of \(\SHADE{\VAR{V}}\), the environments computed by the simple pattern matches are united.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{match} ( \VAR{V} : \NAMEHYPER{../..}{Value-Types}{values}, \NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{X} ) ) ) \leadsto \NAMEHYPER{../../../Computations/Normal}{Giving}{give} ( \VAR{V}, \VAR{X} ) \\ \KEY{Rule} \quad & \RULE{ & \VAR{I}\SUB{2} \neq \STRING{pattern} }{ & \NAMEREF{match} ( \\&\quad \NAMEHYPER{../../Composite}{Datatypes}{datatype-value} ( \VAR{I}\SUB{1} : \NAMEHYPER{../../../Computations/Normal}{Binding}{identifiers}, \VAR{V}\SUB{1}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ), \\&\quad \NAMEHYPER{../../Composite}{Datatypes}{datatype-value} ( \VAR{I}\SUB{2} : \NAMEHYPER{../../../Computations/Normal}{Binding}{identifiers}, \VAR{V}\SUB{2}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ) ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{sequential} ( \\&\quad\quad \NAMEHYPER{../../../Computations/Abnormal}{Failing}{check-true} ( \NAMEHYPER{../..}{Value-Types}{is-equal} ( \VAR{I}\SUB{1}, \VAR{I}\SUB{2} ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Abnormal}{Failing}{check-true} ( \NAMEHYPER{../..}{Value-Types}{is-equal} ( \NAMEHYPER{../../Composite}{Sequences}{length} \ \VAR{V}\SUB{1}\STAR, \NAMEHYPER{../../Composite}{Sequences}{length} \ \VAR{V}\SUB{2}\STAR ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Normal}{Binding}{collateral} ( \\&\quad\quad\quad \NAMEHYPER{../../../Computations/Normal}{Giving}{interleave-map} ( \\&\quad\quad\quad\quad \NAMEREF{match} ( \NAMEHYPER{../../Composite}{Tuples}{tuple-elements} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ), \\&\quad\quad\quad\quad \NAMEHYPER{../../Composite}{Tuples}{tuple-zip} ( \NAMEHYPER{../../Composite}{Tuples}{tuple} ( \VAR{V}\SUB{1}\STAR ), \NAMEHYPER{../../Composite}{Tuples}{tuple} ( \VAR{V}\SUB{2}\STAR ) ) ) ) ) } \\ \KEY{Rule} \quad & \RULE{ & \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{2} ) == \{ \ \} }{ & \NAMEREF{match} ( \VAR{M}\SUB{1} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ), \VAR{M}\SUB{2} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ) ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{if-true-else} ( \NAMEHYPER{../..}{Value-Types}{is-equal} ( \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{1} ), \{ \ \} ), \NAMEHYPER{../../Composite}{Maps}{map} ( \ ), \NAMEHYPER{../../../Computations/Abnormal}{Failing}{fail} ) } \\ \KEY{Rule} \quad & \RULE{ & \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{2} ) \neq \{ \ \}\\& \NAMEHYPER{../../Composite}{Sets}{some-element} ( \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{2} ) ) \leadsto \VAR{K} }{ & \NAMEREF{match} ( \VAR{M}\SUB{1} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ), \VAR{M}\SUB{2} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ) ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{if-true-else} ( \\&\quad\quad \NAMEHYPER{../../Composite}{Sets}{is-in-set} ( \VAR{K}, \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{1} ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Normal}{Binding}{collateral} ( \\&\quad\quad\quad \NAMEREF{match} ( \NAMEHYPER{../../Composite}{Maps}{map-lookup} ( \VAR{M}\SUB{1}, \VAR{K} ), \NAMEHYPER{../../Composite}{Maps}{map-lookup} ( \VAR{M}\SUB{2}, \VAR{K} ) ), \\&\quad\quad\quad \NAMEREF{match} ( \NAMEHYPER{../../Composite}{Maps}{map-delete} ( \VAR{M}\SUB{1}, \{ \VAR{K} \} ), \NAMEHYPER{../../Composite}{Maps}{map-delete} ( \VAR{M}\SUB{2}, \{ \VAR{K} \} ) ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Abnormal}{Failing}{fail} ) } \\ \KEY{Rule} \quad & \RULE{ & \VAR{P} : \mathop{\sim} ( \NAMEHYPER{../../Composite}{Datatypes}{datatype-values} \mid \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ) ) }{ & \NAMEREF{match} ( \VAR{V} : \NAMEHYPER{../..}{Value-Types}{values}, \VAR{P} : \NAMEHYPER{../..}{Value-Types}{values} ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{if-true-else} ( \NAMEHYPER{../..}{Value-Types}{is-equal} ( \VAR{V}, \VAR{P} ), \NAMEHYPER{../../Composite}{Maps}{map} ( \ ), \NAMEHYPER{../../../Computations/Abnormal}{Failing}{fail} ) } \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{match-loosely}( \_ : \NAMEHYPER{../..}{Value-Types}{values}, \_ : \NAMEHYPER{../..}{Value-Types}{values}) : \TO \NAMEHYPER{../../../Computations/Normal}{Binding}{environments} \end{align*}\]

\(\SHADE{\NAMEREF{match-loosely} ( \VAR{V}, \VAR{P} )}\) takes a (potentially structured) value \(\SHADE{\VAR{V}}\) and a (potentially structured) pattern \(\SHADE{\VAR{P}}\). Provided that the structure and all components of \(\SHADE{\VAR{P}}\) loosely match the structure and corresponding components of \(\SHADE{\VAR{V}}\), the environments computed by the simple pattern matches are united.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{match-loosely} ( \VAR{V} : \NAMEHYPER{../..}{Value-Types}{values}, \NAMEREF{pattern} ( \NAMEHYPER{../.}{Generic}{abstraction} ( \VAR{X} ) ) ) \leadsto \NAMEHYPER{../../../Computations/Normal}{Giving}{give} ( \VAR{V}, \VAR{X} ) \\ \KEY{Rule} \quad & \RULE{ & \VAR{I}\SUB{2} \neq \STRING{pattern} }{ & \NAMEREF{match-loosely} ( \\&\quad \NAMEHYPER{../../Composite}{Datatypes}{datatype-value} ( \VAR{I}\SUB{1} : \NAMEHYPER{../../../Computations/Normal}{Binding}{identifiers}, \VAR{V}\SUB{1}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ), \\&\quad \NAMEHYPER{../../Composite}{Datatypes}{datatype-value} ( \VAR{I}\SUB{2} : \NAMEHYPER{../../../Computations/Normal}{Binding}{identifiers}, \VAR{V}\SUB{2}\STAR : \NAMEHYPER{../..}{Value-Types}{values}\STAR ) ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{sequential} ( \\&\quad\quad \NAMEHYPER{../../../Computations/Abnormal}{Failing}{check-true} ( \NAMEHYPER{../..}{Value-Types}{is-equal} ( \VAR{I}\SUB{1}, \VAR{I}\SUB{2} ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Abnormal}{Failing}{check-true} ( \NAMEHYPER{../..}{Value-Types}{is-equal} ( \NAMEHYPER{../../Composite}{Sequences}{length} \ \VAR{V}\SUB{1}\STAR, \NAMEHYPER{../../Composite}{Sequences}{length} \ \VAR{V}\SUB{2}\STAR ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Normal}{Binding}{collateral} ( \\&\quad\quad\quad \NAMEHYPER{../../../Computations/Normal}{Giving}{interleave-map} ( \\&\quad\quad\quad\quad \NAMEREF{match-loosely} ( \NAMEHYPER{../../Composite}{Tuples}{tuple-elements} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ), \\&\quad\quad\quad\quad \NAMEHYPER{../../Composite}{Tuples}{tuple-zip} ( \NAMEHYPER{../../Composite}{Tuples}{tuple} ( \VAR{V}\SUB{1}\STAR ), \NAMEHYPER{../../Composite}{Tuples}{tuple} ( \VAR{V}\SUB{2}\STAR ) ) ) ) ) } \\ \KEY{Rule} \quad & \RULE{ & \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{2} ) == \{ \ \} }{ & \NAMEREF{match-loosely} ( \VAR{M}\SUB{1} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ), \VAR{M}\SUB{2} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ) ) \leadsto \NAMEHYPER{../../Composite}{Maps}{map} ( \ ) } \\ \KEY{Rule} \quad & \RULE{ & \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{2} ) \neq \{ \ \}\\& \NAMEHYPER{../../Composite}{Sets}{some-element} ( \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{2} ) ) \leadsto \VAR{K} }{ & \NAMEREF{match-loosely} ( \VAR{M}\SUB{1} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ), \VAR{M}\SUB{2} : \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ) ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{if-true-else} ( \\&\quad\quad \NAMEHYPER{../../Composite}{Sets}{is-in-set} ( \VAR{K}, \NAMEHYPER{../../Composite}{Maps}{dom} ( \VAR{M}\SUB{1} ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Normal}{Binding}{collateral} ( \\&\quad\quad\quad \NAMEREF{match-loosely} ( \NAMEHYPER{../../Composite}{Maps}{map-lookup} ( \VAR{M}\SUB{1}, \VAR{K} ), \NAMEHYPER{../../Composite}{Maps}{map-lookup} ( \VAR{M}\SUB{2}, \VAR{K} ) ), \\&\quad\quad\quad \NAMEREF{match-loosely} ( \NAMEHYPER{../../Composite}{Maps}{map-delete} ( \VAR{M}\SUB{1}, \{ \VAR{K} \} ), \NAMEHYPER{../../Composite}{Maps}{map-delete} ( \VAR{M}\SUB{2}, \{ \VAR{K} \} ) ) ), \\&\quad\quad \NAMEHYPER{../../../Computations/Abnormal}{Failing}{fail} ) } \\ \KEY{Rule} \quad & \RULE{ & \VAR{P} : \mathop{\sim} ( \NAMEHYPER{../../Composite}{Datatypes}{datatype-values} \mid \NAMEHYPER{../../Composite}{Maps}{maps} ( \_, \_ ) ) }{ & \NAMEREF{match-loosely} ( \VAR{DV} : \NAMEHYPER{../..}{Value-Types}{values}, \VAR{P} : \NAMEHYPER{../..}{Value-Types}{values} ) \leadsto \\&\quad \NAMEHYPER{../../../Computations/Normal}{Flowing}{if-true-else} ( \NAMEHYPER{../..}{Value-Types}{is-equal} ( \VAR{DV}, \VAR{P} ), \NAMEHYPER{../../Composite}{Maps}{map} ( \ ), \NAMEHYPER{../../../Computations/Abnormal}{Failing}{fail} ) } \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{case-match}( \_ : \NAMEHYPER{../..}{Value-Types}{values}, \_ : \TO \VAR{T}') : \TO \VAR{T}' \end{align*}\]

\(\SHADE{\NAMEREF{case-match} ( \VAR{P}, \VAR{X} )}\) matches \(\SHADE{\VAR{P}}\) exactly to the given value. If the match succeeds, the computed bindings have scope \(\SHADE{\VAR{X}}\).

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{case-match} ( \VAR{P} : \NAMEHYPER{../..}{Value-Types}{values}, \VAR{X} ) \leadsto \NAMEHYPER{../../../Computations/Normal}{Binding}{scope} ( \NAMEREF{match} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given}, \VAR{P} ), \VAR{X} ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{case-match-loosely}( \_ : \NAMEHYPER{../..}{Value-Types}{values}, \_ : \TO \VAR{T}') : \TO \VAR{T}' \end{align*}\]

\(\SHADE{\NAMEREF{case-match} ( \VAR{P}, \VAR{X} )}\) matches \(\SHADE{\VAR{P}}\) loosely to the given value. If the match succeeds, the computed bindings have scope \(\SHADE{\VAR{X}}\).

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{case-match-loosely} ( \VAR{P} : \NAMEHYPER{../..}{Value-Types}{values}, \VAR{X} ) \leadsto \NAMEHYPER{../../../Computations/Normal}{Binding}{scope} ( \NAMEREF{match-loosely} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given}, \VAR{P} ), \VAR{X} ) \end{align*}\] \[\begin{align*} \KEY{Funcon} \quad & \NAMEDECL{case-variant-value}( \_ : \NAMEHYPER{../../../Computations/Normal}{Binding}{identifiers}) : \TO \NAMEHYPER{../..}{Value-Types}{values} \end{align*}\]

\(\SHADE{\NAMEREF{case-variant-value} ( \VAR{I} )}\) matches values of variant \(\SHADE{\VAR{I}}\), then giving the value contained in the variant.

\[\begin{align*} \KEY{Rule} \quad & \NAMEREF{case-variant-value} ( \VAR{I} : \NAMEHYPER{../../../Computations/Normal}{Binding}{identifiers} ) \leadsto \\&\quad \NAMEREF{case-match} ( \NAMEHYPER{../../Composite}{Variants}{variant} ( \VAR{I}, \NAMEREF{pattern-any} ), \NAMEHYPER{../../Composite}{Variants}{variant-value} ( \NAMEHYPER{../../../Computations/Normal}{Giving}{given} ) ) \end{align*}\]

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