X-Git-Url: https://git.stderr.nl/gitweb?p=matthijs%2Fmaster-project%2Freport.git;a=blobdiff_plain;f=Chapters%2FNormalization.tex;h=cbb634c312d72c68872e65274973992260bc9518;hp=4f4ee15a12d6b8a4aac416d4f14bcac1fe0c090b;hb=747c2ba3d485fb5c3543e9435fbd3dff59ddb3f8;hpb=2c28bdc6ca8e883697d9640166bf17fb8a329cec diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex index 4f4ee15..cbb634c 100644 --- a/Chapters/Normalization.tex +++ b/Chapters/Normalization.tex @@ -25,7 +25,7 @@ aim to bring the core description into a simpler form, which we can subsequently translate into \small{VHDL} easily. This normal form is needed because the full core language is more expressive than \small{VHDL} in some - areas (higher order expressions, limited polymorphism using type + areas (higher-order expressions, limited polymorphism using type classes, etc.) and because core can describe expressions that do not have a direct hardware interpretation. @@ -44,16 +44,16 @@ to see if this normal form actually has the properties we would like it to have. - But, before getting into more definitions and details about this normal form, - let's try to get a feeling for it first. The easiest way to do this is by - describing the things we want to not have in a normal form. + But, before getting into more definitions and details about this normal + form, let us try to get a feeling for it first. The easiest way to do this + is by describing the things that are unwanted in the intended normal form. \startitemize \item Any \emph{polymorphism} must be removed. When laying down hardware, we - can't generate any signals that can have multiple types. All types must be + cannot generate any signals that can have multiple types. All types must be completely known to generate hardware. - \item All \emph{higher order} constructions must be removed. We can't + \item All \emph{higher-order} constructions must be removed. We cannot generate a hardware signal that contains a function, so all values, arguments and return values used must be first order. @@ -204,7 +204,7 @@ As a more complete example, consider \in{example}[ex:NormalComplete]. This example shows everything that - is allowed in normal form, except for builtin higher order functions + is allowed in normal form, except for built-in higher-order functions (like \lam{map}). The graphical version of the architecture contains a slightly simplified version, since the state tuple packing and unpacking have been left out. Instead, two separate registers are @@ -321,7 +321,7 @@ Now we have some intuition for the normal form, we can describe how we want the normal form to look like in a slightly more formal manner. The following EBNF-like description captures most of the intended structure (and - generates a subset of GHC's core format). + generates a subset of \GHC's core format). There are two things missing: Cast expressions are sometimes allowed by the prototype, but not specified here and the below @@ -365,8 +365,8 @@ \italic{userarg} := var (lvar(var)) \italic{builtinapp} := \italic{builtinfunc} | \italic{builtinapp} \italic{builtinarg} - \italic{builtinfunc} := var (bvar(var)) - \italic{builtinarg} := var (representable(var) ∧ lvar(var)) + \italic{built-infunc} := var (bvar(var)) + \italic{built-inarg} := var (representable(var) ∧ lvar(var)) | \italic{partapp} (partapp :: a -> b) | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b)) \italic{partapp} := \italic{userapp} @@ -384,8 +384,8 @@ Most function applications bound by the let expression define a component instantiation, where the input and output ports are mapped to local signals or arguments. Some of the others use a - builtin construction (\eg the \lam{case} expression) or call a - builtin function (\eg \lam{+} or \lam{map}). For these, a + built-in construction (\eg the \lam{case} expression) or call a + built-in function (\eg \lam{+} or \lam{map}). For these, a hardcoded \small{VHDL} translation is available. \section[sec:normalization:transformation]{Transformation notation} @@ -399,13 +399,13 @@ ~ -------------------------- - + ~ \stoptrans - This format desribes a transformation that applies to \lam{} and transforms it into \lam{}, assuming + This format describes a transformation that applies to \lam{} and transforms it into \lam{}, assuming that all conditions are satisfied. In this format, there are a number of placeholders in pointy brackets, most of which should be rather obvious in their meaning. Nevertheless, we will more precisely specify their meaning below: @@ -474,8 +474,8 @@ To understand this notation better, the step by step application of the η-abstraction transformation to a simple \small{ALU} will be - shown. Consider η-abstraction, described using above notation as - follows: + shown. Consider η-abstraction, which is a common transformation from + labmda calculus, described using above notation as follows: \starttrans E \lam{E :: a -> b} @@ -546,7 +546,7 @@ By now, the placeholder \lam{E} is bound to the entire expression. The placeholder \lam{x}, which occurs in the replacement template, is not bound - yet, so we need to generate a fresh binder for that. Let's use the binder + yet, so we need to generate a fresh binder for that. Let us use the binder \lam{a}. This results in the following replacement expression: \startlambda @@ -576,7 +576,7 @@ \stoplambda The transformation does not apply to this lambda abstraction, so we - look at its body. For brevity, we'll put the case expression on one line from + look at its body. For brevity, we will put the case expression on one line from now on. \startlambda @@ -598,7 +598,7 @@ function position (which makes the second condition false). In the same way the transformation does not apply to both components of this expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so - we'll skip to the components of the case expression: The scrutinee and + we will skip to the components of the case expression: The scrutinee and both alternatives. Since the opcode is not a function, it does not apply here. @@ -677,12 +677,12 @@ dictionaries, functions. \defref{representable} - A \emph{builtin function} is a function supplied by the Cλash framework, whose + A \emph{built-in function} is a function supplied by the Cλash framework, whose implementation is not valid Cλash. The implementation is of course valid Haskell, for simulation, but it is not expressable in Cλash. - \defref{builtin function} \defref{user-defined function} + \defref{built-in function} \defref{user-defined function} - For these functions, Cλash has a \emph{builtin hardware translation}, so calls + For these functions, Cλash has a \emph{built-in hardware translation}, so calls to these functions can still be translated. These are functions like \lam{map}, \lam{hwor} and \lam{length}. @@ -750,9 +750,9 @@ binder uniqueness problems in \small{GHC}. In our transformation system, maintaining this non-shadowing invariant is - a bit harder to do (mostly due to implementation issues, the prototype doesn't - use \small{GHC}'s subsitution code). Also, the following points can be - observed. + a bit harder to do (mostly due to implementation issues, the prototype + does not use \small{GHC}'s subsitution code). Also, the following points + can be observed. \startitemize \item Deshadowing does not guarantee overall uniqueness. For example, the @@ -891,6 +891,58 @@ \transexample{beta-type}{β-reduction for type abstractions}{from}{to} + \subsubsection{Unused let binding removal} + This transformation removes let bindings that are never used. + Occasionally, \GHC's desugarer introduces some unused let bindings. + + This normalization pass should really be not be necessary to get + into intended normal form (since the intended normal form + definition \refdef{intended normal form definition} does not + require that every binding is used), but in practice the + desugarer or simplifier emits some bindings that cannot be + normalized (e.g., calls to a + \hs{Control.Exception.Base.patError}) but are not used anywhere + either. To prevent the \VHDL generation from breaking on these + artifacts, this transformation removes them. + + \todo{Do not use old-style numerals in transformations} + \starttrans + letrec + a0 = E0 + \vdots + ai = Ei + \vdots + an = En + in + M \lam{ai} does not occur free in \lam{M} + ---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej}) + letrec + a0 = E0 + \vdots + ai-1 = Ei-1 + ai+1 = Ei+1 + \vdots + an = En + in + M + \stoptrans + + % And an example + \startbuffer[from] + let + x = 1 + in + 2 + \stopbuffer + + \startbuffer[to] + let + in + 2 + \stopbuffer + + \transexample{unusedlet}{Unused let binding removal}{from}{to} + \subsubsection{Empty let removal} This transformation is simple: It removes recursive lets that have no bindings (which usually occurs when unused let binding removal removes the last @@ -905,7 +957,18 @@ M \stoptrans - \todo{Example} + % And an example + \startbuffer[from] + let + in + 2 + \stopbuffer + + \startbuffer[to] + 2 + \stopbuffer + + \transexample{emptylet}{Empty let removal}{from}{to} \subsubsection[sec:normalization:simplelet]{Simple let binding removal} This transformation inlines simple let bindings, that bind some @@ -940,44 +1003,6 @@ \todo{example} - \subsubsection{Unused let binding removal} - This transformation removes let bindings that are never used. - Occasionally, \GHC's desugarer introduces some unused let bindings. - - This normalization pass should really be not be necessary to get - into intended normal form (since the intended normal form - definition \refdef{intended normal form definition} does not - require that every binding is used), but in practice the - desugarer or simplifier emits some bindings that cannot be - normalized (e.g., calls to a - \hs{Control.Exception.Base.patError}) but are not used anywhere - either. To prevent the \VHDL generation from breaking on these - artifacts, this transformation removes them. - - \todo{Don't use old-style numerals in transformations} - \starttrans - letrec - a0 = E0 - \vdots - ai = Ei - \vdots - an = En - in - M \lam{ai} does not occur free in \lam{M} - ---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej}) - letrec - a0 = E0 - \vdots - ai-1 = Ei-1 - ai+1 = Ei+1 - \vdots - an = En - in - M - \stoptrans - - \todo{Example} - \subsubsection{Cast propagation / simplification} This transform pushes casts down into the expression as far as possible. This transformation has been added to make a few @@ -1015,8 +1040,9 @@ Note that this transformation is completely optional. It is not required to get any function into intended normal form, but it does help making - the resulting VHDL output easier to read (since it removes a bunch of - components that are really boring). + the resulting VHDL output easier to read (since it removes components + that do not add any real structure, but do hide away operations and + cause extra clutter). This transform takes any top level binding generated by \GHC, whose normalized form contains only a single let binding. @@ -1054,9 +1080,9 @@ allowed in \VHDL architecture names\footnote{Technically, it is allowed to use non-alphanumerics when using extended identifiers, but it seems that none of the tooling likes - extended identifiers in filenames, so it effectively doesn't + extended identifiers in filenames, so it effectively does not work.}, so the entity would be called \quote{w7aA7f} or - something similarly unreadable and autogenerated). + something similarly meaningless and autogenerated). \subsection{Program structure} These transformations are aimed at normalizing the overall structure @@ -1161,8 +1187,9 @@ This transformation makes all non-recursive lets recursive. In the end, we want a single recursive let in our normalized program, so all non-recursive lets can be converted. This also makes other - transformations simpler: They can simply assume all lets are - recursive. + transformations simpler: They only need to be specified for recursive + let expressions (and simply will not apply to non-recursive let + expressions until this transformation has been applied). \starttrans let @@ -1337,10 +1364,10 @@ \transexample{argsimpl}{Argument simplification}{from}{to} - \subsection[sec:normalization:builtins]{Builtin functions} - This section deals with (arguments to) builtin functions. In the + \subsection[sec:normalization:built-ins]{Built-in functions} + This section deals with (arguments to) built-in functions. In the intended normal form definition\refdef{intended normal form definition} - we can see that there are three sorts of arguments a builtin function + we can see that there are three sorts of arguments a built-in function can receive. \startitemize[KR] @@ -1348,9 +1375,9 @@ common argument to any function. The argument simplification transformation described in \in{section}[sec:normalization:argsimpl] makes sure that \emph{any} representable argument to \emph{any} - function (including builtin functions) is turned into a local variable + function (including built-in functions) is turned into a local variable reference. - \item (A partial application of) a top level function (either builtin on + \item (A partial application of) a top level function (either built-in on user-defined). The function extraction transformation described in this section takes care of turning every functiontyped argument into (a partial application of) a top level function. @@ -1359,25 +1386,25 @@ transformation needed. Note that this category is exactly all expressions that are not transformed by the transformations for the previous two categories. This means that \emph{any} core expression - that is used as an argument to a builtin function will be either + that is used as an argument to a built-in function will be either transformed into one of the above categories, or end up in this categorie. In any case, the result is in normal form. \stopitemize As noted, the argument simplification will handle any representable - arguments to a builtin function. The following transformation is needed + arguments to a built-in function. The following transformation is needed to handle non-representable arguments with a function type, all other - non-representable arguments don't need any special handling. + non-representable arguments do not need any special handling. \subsubsection[sec:normalization:funextract]{Function extraction} - This transform deals with function-typed arguments to builtin + This transform deals with function-typed arguments to built-in functions. - Since builtin functions cannot be specialized (see + Since built-in functions cannot be specialized (see \in{section}[sec:normalization:specialize]) to remove the arguments, these arguments are extracted into a new global function instead. In other words, we create a new top level function that has exactly the extracted argument as its body. This greatly simplifies the - translation rules needed for builtin functions, since they only need + translation rules needed for built-in functions, since they only need to handle (partial applications of) top level functions. Any free variables occuring in the extracted arguments will become @@ -1385,14 +1412,14 @@ with a reference to the new function, applied to any free variables from the original argument. - This transformation is useful when applying higher order builtin functions + This transformation is useful when applying higher-order built-in functions like \hs{map} to a lambda abstraction, for example. In this case, the code that generates \small{VHDL} for \hs{map} only needs to handle top level functions and partial applications, not any other expression (such as lambda abstractions or even more complicated expressions). \starttrans - M N \lam{M} is (a partial aplication of) a builtin function. + M N \lam{M} is (a partial aplication of) a built-in function. --------------------- \lam{f0 ... fn} are all free local variables of \lam{N} M (x f0 ... fn) \lam{N :: a -> b} ~ \lam{N} is not a (partial application of) a top level function @@ -1472,7 +1499,7 @@ C0 v0,0 ... v0,m -> E0 \vdots Cn vn,0 ... vn,m -> En - --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder) + --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder) letrec The case expression is not an extractor case v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case \vdots @@ -1569,14 +1596,14 @@ values used in our expression representable. There are two main transformations that are applied to \emph{all} unrepresentable let bindings and function arguments. These are meant to address three - different kinds of unrepresentable values: Polymorphic values, higher - order values and literals. The transformation are described generically: - They apply to all non-representable values. However, non-representable - values that don't fall into one of these three categories will be moved - around by these transformations but are unlikely to completely - disappear. They usually mean the program was not valid in the first - place, because unsupported types were used (for example, a program using - strings). + different kinds of unrepresentable values: Polymorphic values, + higher-order values and literals. The transformation are described + generically: They apply to all non-representable values. However, + non-representable values that do not fall into one of these three + categories will be moved around by these transformations but are + unlikely to completely disappear. They usually mean the program was not + valid in the first place, because unsupported types were used (for + example, a program using strings). Each of these three categories will be detailed below, followed by the actual transformations. @@ -1600,13 +1627,13 @@ take care of exactly this. There is one case where polymorphism cannot be completely - removed: Builtin functions are still allowed to be polymorphic + removed: Built-in functions are still allowed to be polymorphic (Since we have no function body that we could properly - specialize). However, the code that generates \VHDL for builtin + specialize). However, the code that generates \VHDL for built-in functions knows how to handle this, so this is not a problem. - \subsubsection{Defunctionalization} - These transformations remove higher order expressions from our + \subsubsection[sec:normalization:defunctionalization]{Defunctionalization} + These transformations remove higher-order expressions from our program, making all values first-order. Higher order values are always introduced by lambda abstractions, none @@ -1614,7 +1641,7 @@ However, other expressions can \emph{have} a function type, when they have a lambda expression in their body. - For example, the following expression is a higher order expression + For example, the following expression is a higher-order expression that is not a lambda expression itself: \refdef{id function} @@ -1625,24 +1652,24 @@ \stoplambda The reference to the \lam{id} function shows that we can introduce a - higher order expression in our program without using a lambda + higher-order expression in our program without using a lambda expression directly. However, inside the definition of the \lam{id} function, we can be sure that a lambda expression is present. Looking closely at the definition of our normal form in \in{section}[sec:normalization:intendednormalform], we can see that - there are three possibilities for higher order values to appear in our + there are three possibilities for higher-order values to appear in our intended normal form: \startitemize[KR] \item[item:toplambda] Lambda abstractions can appear at the highest level of a top level function. These lambda abstractions introduce the arguments (input ports / current state) of the function. - \item[item:builtinarg] (Partial applications of) top level functions can appear as an - argument to a builtin function. + \item[item:built-inarg] (Partial applications of) top level functions can appear as an + argument to a built-in function. \item[item:completeapp] (Partial applications of) top level functions can appear in function position of an application. Since a partial application - cannot appear anywhere else (except as builtin function arguments), + cannot appear anywhere else (except as built-in function arguments), all partial applications are applied, meaning that all applications will become complete applications. However, since application of arguments happens one by one, in the expression: @@ -1653,7 +1680,7 @@ allowed, since it is inside a complete application. \stopitemize - We will take a typical function with some higher order values as an + We will take a typical function with some higher-order values as an example. The following function takes two arguments: a \lam{Bit} and a list of numbers. Depending on the first argument, each number in the list is doubled, or the list is returned unmodified. For the sake of @@ -1667,15 +1694,15 @@ High -> λz. z \stoplambda - This example shows a number of higher order values that we cannot + This example shows a number of higher-order values that we cannot translate to \VHDL directly. The \lam{double} binder bound in the let expression has a function type, as well as both of the alternatives of the case expression. The first alternative is a partial application of - the \lam{map} builtin function, whereas the second alternative is a + the \lam{map} built-in function, whereas the second alternative is a lambda abstraction. - To reduce all higher order values to one of the above items, a number - of transformations we've already seen are used. The η-abstraction + To reduce all higher-order values to one of the above items, a number + of transformations we have already seen are used. The η-abstraction transformation from \in{section}[sec:normalization:eta] ensures all function arguments are introduced by lambda abstraction on the highest level of a function. These lambda arguments are allowed because of @@ -1704,7 +1731,7 @@ High -> (λz. z) q \stoplambda - This propagation makes higher order values become applied (in + This propagation makes higher-order values become applied (in particular both of the alternatives of the case now have a representable type). Completely applied top level functions (like the first alternative) are now no longer invalid (they fall under @@ -1720,11 +1747,11 @@ \stoplambda As you can see in our example, all of this moves applications towards - the higher order values, but misses higher order functions bound by + the higher-order values, but misses higher-order functions bound by let expressions. The applications cannot be moved towards these values (since they can be used in multiple places), so the values will have to be moved towards the applications. This is achieved by inlining all - higher order values bound by let applications, by the + higher-order values bound by let applications, by the non-representable binding inlining transformation below. When applying it to our example, we get the following: @@ -1734,9 +1761,9 @@ High -> q \stoplambda - We've nearly eliminated all unsupported higher order values from this - expressions. The one that's remaining is the first argument to the - \lam{map} function. Having higher order arguments to a builtin + We have nearly eliminated all unsupported higher-order values from this + expressions. The one that is remaining is the first argument to the + \lam{map} function. Having higher-order arguments to a built-in function like \lam{map} is allowed in the intended normal form, but only if the argument is a (partial application) of a top level function. This is easily done by introducing a new top level function @@ -1761,10 +1788,10 @@ intended normal form. There is one case that has not been discussed yet. What if the - \lam{map} function in the example above was not a builtin function + \lam{map} function in the example above was not a built-in function but a user-defined function? Then extracting the lambda expression into a new function would not be enough, since user-defined functions - can never have higher order arguments. For example, the following + can never have higher-order arguments. For example, the following expression shows an example: \startlambda @@ -1776,13 +1803,13 @@ This example shows a function \lam{twice} that takes a function as a first argument and applies that function twice to the second argument. - Again, we've made the function monomorphic for clarity, even though + Again, we have made the function monomorphic for clarity, even though this function would be a lot more useful if it was polymorphic. The function \lam{main} uses \lam{twice} to apply a lambda epression twice. When faced with a user defined function, a body is available for that function. This means we could create a specialized version of the - function that only works for this particular higher order argument + function that only works for this particular higher-order argument (\ie, we can just remove the argument and call the specialized function without the argument). This transformation is detailed below. Applying this transformation to the example gives: @@ -1794,13 +1821,13 @@ main = λa.app' a \stoplambda - The \lam{main} function is now in normal form, since the only higher - order value there is the top level lambda expression. The new - \lam{twice'} function is a bit complex, but the entire original body of - the original \lam{twice} function is wrapped in a lambda abstraction - and applied to the argument we've specialized for (\lam{λx. x + x}) - and the other arguments. This complex expression can fortunately be - effectively reduced by repeatedly applying β-reduction: + The \lam{main} function is now in normal form, since the only + higher-order value there is the top level lambda expression. The new + \lam{twice'} function is a bit complex, but the entire original body + of the original \lam{twice} function is wrapped in a lambda + abstraction and applied to the argument we have specialized for + (\lam{λx. x + x}) and the other arguments. This complex expression can + fortunately be effectively reduced by repeatedly applying β-reduction: \startlambda twice' :: Word -> Word @@ -1824,7 +1851,7 @@ representable type: Integer literals. Cλash supports using integer literals for all three integer types supported (\hs{SizedWord}, \hs{SizedInt} and \hs{RangedWord}). This is implemented using - Haskell's \hs{Num} typeclass, which offers a \hs{fromInteger} method + Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method that converts any \hs{Integer} to the Cλash datatypes. When \GHC sees integer literals, it will automatically insert calls to @@ -1850,7 +1877,7 @@ Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not representable, and cannot be translated directly. Fortunately, there - is no need to translate them, since \lam{fromInteger} is a builtin + is no need to translate them, since \lam{fromInteger} is a built-in function that knows how to handle these values. However, this does require that the \lam{fromInteger} function is directly applied to these non-representable literal values, otherwise errors will occur. @@ -1864,7 +1891,7 @@ By inlining these let-bindings, we can ensure that unrepresentable literals bound by a let binding end up in an application of the - appropriate builtin function, where they are allowed. Since it is + appropriate built-in function, where they are allowed. Since it is possible that the application of that function is in a different function than the definition of the literal value, we will always need to specialize away any unrepresentable literals that are used as @@ -1878,7 +1905,7 @@ but to inline the binding to remove it. As we have seen in the previous sections, inlining these bindings - solves (part of) the polymorphism, higher order values and + solves (part of) the polymorphism, higher-order values and unrepresentable literals in an expression. \refdef{substitution notation} @@ -2046,8 +2073,8 @@ letrec x = M in E \stoptrans - This doesn't seem like much of an improvement, but it does get rid of - the lambda expression (and the associated higher order value), while + This does not seem like much of an improvement, but it does get rid of + the lambda expression (and the associated higher-order value), while at the same time introducing a new let binding. Since the result of every application or case expression must be bound by a let expression in the intended normal form anyway, this is probably not a problem. If @@ -2200,7 +2227,7 @@ possible proof strategies are shown below. \subsection{Graph representation} - Before looking into how to prove these properties, we'll look at + Before looking into how to prove these properties, we will look at transformation systems from a graph perspective. We will first define the graph view and then illustrate it using a simple example from lambda calculus (which is a different system than the Cλash normalization @@ -2259,7 +2286,7 @@ Of course the graph for Cλash is unbounded, since we can construct an infinite amount of Core expressions. Also, there might potentially be multiple edges between two given nodes (with different labels), though - seems unlikely to actually happen in our system. + this seems unlikely to actually happen in our system. See \in{example}[ex:TransformGraph] for the graph representation of a very simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x @@ -2278,8 +2305,9 @@ From such a graph, we can derive some properties easily: \startitemize[KR] - \item A system will \emph{terminate} if there is no path of infinite length - in the graph (this includes cycles, but can also happen without cycles). + \item A system will \emph{terminate} if there is no walk (sequence of + edges, or transformations) of infinite length in the graph (this + includes cycles, but can also happen without cycles). \item Soundness is not easily represented in the graph. \item A system is \emph{complete} if all of the nodes in the normal set have the intended normal form. The inverse (that all of the nodes outside of @@ -2294,8 +2322,8 @@ When looking at the \in{example}[ex:TransformGraph], we see that the system terminates for both the reduction and expansion systems (but note that, for - expansion, this is only true because we've limited the possible - expressions. In comlete lambda calculus, there would be a path from + expansion, this is only true because we have limited the possible + expressions. In complete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)