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MaplePrimes Announcement

Introducing VerifyTools

by: aroche 305 Product: Maple

August 21 2024

6 2

VerifyTools is a package that has been available in Maple for roughly 24 years, but until now it has never been documented, as it was originally intended for internal use only. Documentation for it will be included in the next release of Maple. Here is a preview:

VerifyTools is similar to the TypeTools package. A type is essentially a predicate that a single expression can either satisfy or not. Analogously, a verification is a predicate that applies to a pair of expressions, comparing them. Just as types can be combined to produce compound types, verifications can also be combined to produce compund verifications. New types can be created, retrieved, queried, or deleted using the commands AddType, GetType (or GetTypes), Exists, and RemoveType, respectively. Similarly in the VerifyTools package we can create, retrieve, query or delete verifications using AddVerification, GetVerification (or GetVerifications), Exists, and RemoveVerification.

The package command VerifyTools:-Verify is also available as the top-level Maple command verify which should already be familiar to expert Maple users. Similarly, the command VerifyTools:-IsVerification is also available as a type, that is,

VerifyTools:-IsVerification(ver);

will return the same as

type(ver, 'verification');

The following examples show what can be done with these commands. Note that in each example where the Verify command is used, it is equivalent to the top-level Maple command verify. (Also note that VerifyTools commands shown below will be slightly different compared to the Maple2024 version):

>	with(VerifyTools):

Suppose we want to create a verification which will checks that the length of a result has not increased compared to the expected result. We can do this using the AddVerification command:

>	AddVerification(length_not_increased, (a, b) -> evalb(length(a) <= length(b)));

First, we can check the existence of our new verification and get its value:

>	Exists(length_not_increased);

>	GetVerification(length_not_increased);

For named verifications, IsVerification is equivalent to Exists (since names are only recognized as verifications if an entry exists for them in the verification database):

>	IsVerification(length_not_increased);

On the other hand, a nontrivial structured verification can be checked with IsVerification,

>	IsVerification(boolean = length_not_increased);

whereas Exists only accepts names:

>	Exists(boolean = length_not_increased);

Error, invalid input: VerifyTools:-Exists expects its 1st argument, x, to be of type symbol, but received boolean = length_not_increased

The preceding command using Exists is also equivalent to the following type call:

>	type(boolean = length_not_increased, verification);

Now, let's use the new verification:

>	Verify(x + 1/x, (x^2 + 1)/x, length_not_increased);

>	Verify((x^2 + 1)/x, x + 1/x, length_not_increased);

Finally, let's remove the verification:

>	RemoveVerification(length_not_increased);

>	Exists(length_not_increased);

>	GetVerification(length_not_increased);

Error, (in VerifyTools:-GetVerification) length_not_increased is not a recognized verification

GetVerifications returns the list of all verifications known to the system:

>	GetVerifications();

Download VerificationTools_blogpost.mw

Austin Roche
Software Architect
Mathematical Software
Maplesoft

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One of the first topics we tackled was the Fourier series. We used Maple to illustrate how the Fourier series approximates a given function as more terms are added. We explored this through both static plots and interactive animations.

To help the student understand the behavior of different types of functions, we defined piecewise functions using Maple's piecewise command. This allowed us to model functions that behave differently over various intervals, such as the following cubic function exercise

Maple's Explore command was an effective tool for creating an interactive learning environment. We used it to create sliders that allowed the student to vary parameters, such as the number of terms in a Fourier series, and see the immediate impact on the plot.

" F(x):={[[-1,-1<x<0],[1,0<x<1]]; "

proc (x) options operator, arrow, function_assign; piecewise(-1 < x and x < 0, -1, 0 < x and x < 1, 1) end proc

(1)

L := 2; a__0 := (int(F(x), x = -(1/2)*L .. (1/2)*L))/L

(2)

proc (n) options operator, arrow; 2*(int(F(x)*cos(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(3)

proc (n) options operator, arrow; 2*(int(F(x)*sin(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(4)

" #` Fourier series function` `F__fourier`(x,N):=`a__0`+(∑)(`a__n`(n)&lowast;cos(2 * n * Pi * x / L) +`b__n`(n)&lowast;sin(2* n * Pi * x / L));"

proc (x, N) options operator, arrow, function_assign; a__0+sum(a__n(n)*cos(2*n*Pi*x/L)+b__n(n)*sin(2*n*Pi*x/L), n = 1 .. N) end proc

(5)

Explore(plot([piecewise(-3 < x and x < -1, F(x+2), -1 < x and x < 1, F(x), 1 < x and x < 3, F(x-2)), F__fourier(x, N)], x = -3 .. 3, color = [blue, purple], numpoints = 200), N = 1 .. 40, title = "Fourier Series Approximation with N Terms")

$" #` Define the piecewise function` F(x):={[[0,-1<x<0],[x^(2),0<x<1]]; "$

proc (x) options operator, arrow, function_assign; piecewise(-1 < x and x < 0, 0, 0 < x and x < 1, x^2) end proc

(6)

L := 2; a__0 := (int(F(x), x = -(1/2)*L .. (1/2)*L))/L

(7)

proc (n) options operator, arrow; 2*(int(F(x)*cos(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(8)

proc (n) options operator, arrow; 2*(int(F(x)*sin(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(9)

" #` Fourier series function` `F__fourier`(x,N):=`a__0`+(∑)(`a__n`(n)&lowast;cos(2 * n * Pi * x / L) +`b__n`(n)&lowast;sin(2* n * Pi * x / L));"

proc (x, N) options operator, arrow, function_assign; a__0+sum(a__n(n)*cos(2*n*Pi*x/L)+b__n(n)*sin(2*n*Pi*x/L), n = 1 .. N) end proc

(10)

" #` Define the piecewise function` F(x):={[[x+2,-2<x<0],[2-2 x,0<x<2]]; "

proc (x) options operator, arrow, function_assign; piecewise(-2 < x and x < 0, x+2, 0 < x and x < 2, 2-2*x) end proc

(11)

L := 2; a__0 := (int(F(x), x = -(1/2)*L .. (1/2)*L))/L

(12)

proc (n) options operator, arrow; 2*(int(F(x)*cos(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(13)

proc (n) options operator, arrow; 2*(int(F(x)*sin(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(14)

" #` Fourier series function` `F__fourier`(x,N):=`a__0`+(∑)(`a__n`(n)&lowast;cos(2 * n * Pi * x / L) +`b__n`(n)&lowast;sin(2* n * Pi * x / L));"

proc (x, N) options operator, arrow, function_assign; a__0+sum(a__n(n)*cos(2*n*Pi*x/L)+b__n(n)*sin(2*n*Pi*x/L), n = 1 .. N) end proc

(15)

proc (x) options operator, arrow; piecewise(-1 < x and x < 1, x-x^3, 0) end proc

(16)

L := 2; a__0 := (int(F(x), x = -(1/2)*L .. (1/2)*L))/L

(17)

proc (n) options operator, arrow; 2*(int(F(x)*cos(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(18)

proc (n) options operator, arrow; 2*(int(F(x)*sin(2*n*Pi*x/L), x = -(1/2)*L .. (1/2)*L))/L end proc

(19)

b__n(n)

-4*(n^2*Pi^2*sin(n*Pi)+3*cos(n*Pi)*Pi*n-3*sin(n*Pi))/(n^4*Pi^4)

(20)

" #` Fourier series function` `F__fourier`(x,N):=`a__0`+(∑)(`a__n`(n)&lowast;cos(2 * n * Pi * x / L) +`b__n`(n)&lowast;sin(2* n * Pi * x / L)); #` Plot the Fourier series approximation` p8:=plot([`F__fourier`(x,40)],x = -3.. 3 ,numpoints=200, color=[red]) :"