--- title: "UpSetDelayed" language: "en" type: "Symbol" summary: "lhs ^:= rhs assigns rhs to be the delayed value of lhs, and associates the assignment with symbols that occur at level one in lhs." keywords: - associated assignment - associated definition - delayed assignment - generic programming - type based programming - object programming canonical_url: "https://reference.wolfram.com/language/ref/UpSetDelayed.html" source: "Wolfram Language Documentation" related_guides: - title: "Assignments" link: "https://reference.wolfram.com/language/guide/Assignments.en.md" related_workflows: - title: "Clear Definitions for Symbols and Functions" link: "https://reference.wolfram.com/language/workflow/ClearDefinitionsForSymbolsAndFunctions.en.md" - title: "Find All Defined Functions" link: "https://reference.wolfram.com/language/workflow/FindAllDefinedFunctions.en.md" related_functions: - title: "UpSet" link: "https://reference.wolfram.com/language/ref/UpSet.en.md" - title: "TagSetDelayed" link: "https://reference.wolfram.com/language/ref/TagSetDelayed.en.md" - title: "SetDelayed" link: "https://reference.wolfram.com/language/ref/SetDelayed.en.md" - title: "UpValues" link: "https://reference.wolfram.com/language/ref/UpValues.en.md" - title: "Unset" link: "https://reference.wolfram.com/language/ref/Unset.en.md" - title: "ClearAll" link: "https://reference.wolfram.com/language/ref/ClearAll.en.md" related_tutorials: - title: "Associating Definitions with Different Symbols" link: "https://reference.wolfram.com/language/tutorial/TransformationRulesAndDefinitions.en.md#6972" --- # UpSetDelayed (^:=) lhs ^:= rhs assigns rhs to be the delayed value of lhs, and associates the assignment with symbols that occur at level one in lhs. --- ## Examples (13) ### Basic Examples (1) ```wl In[1]:= f[g[x_]] ^:= fg[x] In[2]:= {f[g[2]], f[h[2]]} Out[2]= {fg[2], f[h[2]]} ``` ### Scope (6) #### Left-Hand Sides (5) An expression with a delayed definition is evaluated every time it is used: ```wl In[1]:= rand[int] ^:= RandomInteger[] In[2]:= Table[rand[int], {5}] Out[2]= {0, 1, 0, 0, 0} ``` --- Make definitions for special and general cases using immediate and delayed assignments: ```wl In[1]:= f[h[0]] ^= h0; f[h[x_]] ^:= 2f[h[x - 1]] In[2]:= f[h[10]] Out[2]= 1024 h0 ``` --- The definition can be associated with a symbol appearing as an argument of the left side: ```wl In[1]:= area[sq, s_] ^:= s ^ 2 In[2]:= area[sq, 2] Out[2]= 4 ``` --- The definition can be associated with the head of an argument of the left side: ```wl In[1]:= area[sq[s_]] ^:= s ^ 2 In[2]:= area[sq[3]] Out[2]= 9 ``` --- If several symbols appear in the left-hand side, upvalues for all of them are defined: ```wl In[1]:= method[g[x_], h[y_]] ^:= fgh[x, y] In[2]:= Definition[g] Out[2]= g/:method[g[x_], h[y_]] := fgh[x, y] In[3]:= Definition[h] Out[3]= h/:method[g[x_], h[y_]] := fgh[x, y] ``` #### Different Kinds of Values (1) ^:= defines upvalues: ```wl In[1]:= f[g[x_]] ^:= f1[x] f[h[x_]] := f2[x] In[2]:= {UpValues[g], DownValues[f]} Out[2]= {{HoldPattern[f[g[x_]]] :> f1[x]}, {HoldPattern[f[h[x_]]] :> f2[x]}} ``` ### Properties & Relations (6) `` ^:= `` defines upvalues in the same way as using a tag does: ```wl In[1]:= g/:f[g[x_]] := f1[x] In[2]:= f[h[x_]] ^:= f2[x] In[3]:= {UpValues[g], UpValues[h]} Out[3]= {{HoldPattern[f[g[x_]]] :> f1[x]}, {HoldPattern[f[h[x_]]] :> f2[x]}} ``` --- A tag defines only one upvalue; `` ^:= `` makes definitions for all symbols: ```wl In[1]:= g/:f1[g[x_], h[y_]] := gh[x y] In[2]:= f2[g[x_], h[y_]] ^:= gh[x y] In[3]:= {UpValues[g], UpValues[h]} Out[3]= {{HoldPattern[f1[g[x_], h[y_]]] :> gh[x y], HoldPattern[f2[g[x_], h[y_]]] :> gh[x y]}, {HoldPattern[f2[g[x_], h[y_]]] :> gh[x y]}} ``` --- The right side of an immediate definition is evaluated when the definition is made: ```wl In[1]:= rand[int] ^= RandomInteger[]; In[2]:= {rand[int], rand[int]} Out[2]= {0, 0} ``` The right side of a delayed definition is evaluated each time the definition is used: ```wl In[3]:= rand[real] ^:= RandomReal[] In[4]:= {rand[real], rand[real]} Out[4]= {0.409393, 0.730688} ``` --- Definitions with the same left side overwrite earlier ones: ```wl In[1]:= f[h[x_]] ^:= f1[x] In[2]:= f[h[x_]] ^:= f2[x] In[3]:= Definition[h] Out[3]= f[h[x_]] ^:= f2[x] ``` --- ``Definition`` prints definitions associated with a symbol: ```wl In[1]:= a_mod + b_mod ^:= modPlus[a, b] In[2]:= Definition[mod] Out[2]= a_mod + b_mod ^:= modPlus[a, b] ``` ``Information`` prints various information about a symbol, including any definitions: ```wl In[3]:= ? mod Cell$$4157`mod a_mod + b_mod ^:= modPlus[a, b] ``` ``UpValues`` returns a list of rules corresponding to any upvalues defined: ```wl In[4]:= UpValues[mod] Out[4]= {HoldPattern[a_mod + b_mod] :> modPlus[a, b]} ``` --- Delayed assignment introduces a scope that is not affected by global variables: ```wl In[1]:= x = 5; In[2]:= f[h[x_]] ^:= x ^ 2 In[3]:= {f[h[2]], x} Out[3]= {4, 5} ``` Immediate assignment does not introduce a scope: ```wl In[4]:= g[h[x_]] ^= x ^ 2; In[5]:= {g[h[2]], x} Out[5]= {25, 5} ``` ## See Also * [`UpSet`](https://reference.wolfram.com/language/ref/UpSet.en.md) * [`TagSetDelayed`](https://reference.wolfram.com/language/ref/TagSetDelayed.en.md) * [`SetDelayed`](https://reference.wolfram.com/language/ref/SetDelayed.en.md) * [`UpValues`](https://reference.wolfram.com/language/ref/UpValues.en.md) * [`Unset`](https://reference.wolfram.com/language/ref/Unset.en.md) * [`ClearAll`](https://reference.wolfram.com/language/ref/ClearAll.en.md) ## Tech Notes * [Associating Definitions with Different Symbols](https://reference.wolfram.com/language/tutorial/TransformationRulesAndDefinitions.en.md#6972) ## Related Guides * [`Assignments`](https://reference.wolfram.com/language/guide/Assignments.en.md) ## Related Workflows * [Clear Definitions for Symbols and Functions](https://reference.wolfram.com/language/workflow/ClearDefinitionsForSymbolsAndFunctions.en.md) * [Find All Defined Functions](https://reference.wolfram.com/language/workflow/FindAllDefinedFunctions.en.md) ## History * Introduced in 1988 (1.0)