Issue with Upvalues

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I want to introduce two variables (I call them EXt and EXtC, where "C" stands for complex conjugate) which would mimic the behavior of a phase of a complex number. For that, I use the following tags:

 EXt /: EXt EXtC := 1;
 EXtC /: EXtC EXt := 1;
 EXt /: EXt EXtC^n_ := EXtC^(n - 1);
 EXtC /: EXtC EXt^n_ := EXt^(n - 1);
 EXt /: EXt^(n_?Negative) := EXtC^(-n);
 EXtC /: EXtC^(n_?Negative) := EXt^(-n);
 EXt /: Conjugate[EXt] := EXtC; 
 EXtC /: Conjugate[EXtC] := EXt;

With that, I can simplify expressions like

EXt^2 EXt^2

i.e. when both of the variables have the same power (and this power is a number)

However, I am not able to simplify the expressions in which the powers are different. For example, I cannot simplify (i.e. make it equal to EXtC in this case),

EXt^2 EXtC^3

even with the use of FullSimplify. I tried to introduce the following tag

EXt /: EXt^n_ EXtC^m_ := EXtC^(m - n)

but soon learned (for example, from here) that the upvalue mechanism can only scan one level deep, so I expectedly get the error message

TagSetDelayed::tagpos: Tag EXt in EXt^n_ EXtC^m_ is too deep for an assigned rule to be found. >>

Any ideas on how to circumvent this restriction and implement this property?

asked 2 hours ago

user43283

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up vote
4
down vote

favorite

I want to introduce two variables (I call them EXt and EXtC, where "C" stands for complex conjugate) which would mimic the behavior of a phase of a complex number. For that, I use the following tags:

 EXt /: EXt EXtC := 1;
 EXtC /: EXtC EXt := 1;
 EXt /: EXt EXtC^n_ := EXtC^(n - 1);
 EXtC /: EXtC EXt^n_ := EXt^(n - 1);
 EXt /: EXt^(n_?Negative) := EXtC^(-n);
 EXtC /: EXtC^(n_?Negative) := EXt^(-n);
 EXt /: Conjugate[EXt] := EXtC; 
 EXtC /: Conjugate[EXtC] := EXt;

With that, I can simplify expressions like

EXt^2 EXt^2

i.e. when both of the variables have the same power (and this power is a number)

However, I am not able to simplify the expressions in which the powers are different. For example, I cannot simplify (i.e. make it equal to EXtC in this case),

EXt^2 EXtC^3

even with the use of FullSimplify. I tried to introduce the following tag

EXt /: EXt^n_ EXtC^m_ := EXtC^(m - n)

but soon learned (for example, from here) that the upvalue mechanism can only scan one level deep, so I expectedly get the error message

TagSetDelayed::tagpos: Tag EXt in EXt^n_ EXtC^m_ is too deep for an assigned rule to be found. >>

Any ideas on how to circumvent this restriction and implement this property?

asked 2 hours ago

user43283

1211

New contributor

add a commentÂ |Â

up vote
4
down vote

favorite

I want to introduce two variables (I call them EXt and EXtC, where "C" stands for complex conjugate) which would mimic the behavior of a phase of a complex number. For that, I use the following tags:

 EXt /: EXt EXtC := 1;
 EXtC /: EXtC EXt := 1;
 EXt /: EXt EXtC^n_ := EXtC^(n - 1);
 EXtC /: EXtC EXt^n_ := EXt^(n - 1);
 EXt /: EXt^(n_?Negative) := EXtC^(-n);
 EXtC /: EXtC^(n_?Negative) := EXt^(-n);
 EXt /: Conjugate[EXt] := EXtC; 
 EXtC /: Conjugate[EXtC] := EXt;

With that, I can simplify expressions like

EXt^2 EXt^2

i.e. when both of the variables have the same power (and this power is a number)

However, I am not able to simplify the expressions in which the powers are different. For example, I cannot simplify (i.e. make it equal to EXtC in this case),

EXt^2 EXtC^3

even with the use of FullSimplify. I tried to introduce the following tag

EXt /: EXt^n_ EXtC^m_ := EXtC^(m - n)

but soon learned (for example, from here) that the upvalue mechanism can only scan one level deep, so I expectedly get the error message

TagSetDelayed::tagpos: Tag EXt in EXt^n_ EXtC^m_ is too deep for an assigned rule to be found. >>

Any ideas on how to circumvent this restriction and implement this property?

asked 2 hours ago

user43283

1211

New contributor

I want to introduce two variables (I call them EXt and EXtC, where "C" stands for complex conjugate) which would mimic the behavior of a phase of a complex number. For that, I use the following tags:

 EXt /: EXt EXtC := 1;
 EXtC /: EXtC EXt := 1;
 EXt /: EXt EXtC^n_ := EXtC^(n - 1);
 EXtC /: EXtC EXt^n_ := EXt^(n - 1);
 EXt /: EXt^(n_?Negative) := EXtC^(-n);
 EXtC /: EXtC^(n_?Negative) := EXt^(-n);
 EXt /: Conjugate[EXt] := EXtC; 
 EXtC /: Conjugate[EXtC] := EXt;

With that, I can simplify expressions like

EXt^2 EXt^2

i.e. when both of the variables have the same power (and this power is a number)

However, I am not able to simplify the expressions in which the powers are different. For example, I cannot simplify (i.e. make it equal to EXtC in this case),

EXt^2 EXtC^3

even with the use of FullSimplify. I tried to introduce the following tag

EXt /: EXt^n_ EXtC^m_ := EXtC^(m - n)

but soon learned (for example, from here) that the upvalue mechanism can only scan one level deep, so I expectedly get the error message

TagSetDelayed::tagpos: Tag EXt in EXt^n_ EXtC^m_ is too deep for an assigned rule to be found. >>

Any ideas on how to circumvent this restriction and implement this property?

simplifying-expressions symbolic upvalues

asked 2 hours ago

user43283

1211

New contributor

asked 2 hours ago

user43283

1211

New contributor

asked 2 hours ago

user43283

1211

New contributor

asked 2 hours ago

user43283

1211

asked 2 hours ago

user43283

1211

New contributor

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2 Answers
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2
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How about the following?:

Clear[ext, extc]
ext /: ext[n_] extc[m_] := extc[m - n]
ext /: ext[n_] ext[m_] := ext[m + n]
extc /: extc[n_] extc[m_] := extc[m + n]
ext /: Conjugate@ext[n_] := extc[n]
extc /: Conjugate@extc[n_] := ext[n]
extc[n_?Negative] := ext[-n]
ext[n_?Negative] := extc[-n]
extc[0] = 1;

The compromises are:

EXt^n and EXtC^n are expressed as ext[n] and extc[n].

Single EXt and EXtC must be expressed as ext[1] and extc[1].

Example:

ext@2 extc@3
(* extc[1] *)

ext@1 extc@n // Conjugate
(* ext[-1 + n] *)

answered 1 hour ago

xzczd

25k467236

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up vote
2
down vote

Here's a variation of @xzczd's idea, using only a single symbol and adding formatting:

Clear[ext]
ext[n_] ext[m_] ^:= ext[n+m]
ext[n_]^m_ ^:= ext[n m]
Conjugate[ext[n_]] ^:= ext[-n]
ext[0] = 1;

MakeBoxes[ext[n_],StandardForm]:=Switch[n,
 0, "1",
 1, MakeBoxes[EXt],
 -1, MakeBoxes[EXtC],
 _Integer?Negative, With[s=-n, MakeBoxes[EXtC^s]],
 _, MakeBoxes[EXt^n]
]

For example:

EXt = ext[1]
EXtC = ext[-1]

EXt

EXtC

And:

EXt^2 EXtC^2
EXt^2 EXtC^3
EXt EXtC^n //Conjugate

1

EXtC

EXt^(-1 + n)

edited 13 mins ago

answered 19 mins ago

Carl Woll

61.8k280158

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2 Answers
2

active

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votes

2 Answers
2

active

oldest

votes

up vote
2
down vote

How about the following?:

Clear[ext, extc]
ext /: ext[n_] extc[m_] := extc[m - n]
ext /: ext[n_] ext[m_] := ext[m + n]
extc /: extc[n_] extc[m_] := extc[m + n]
ext /: Conjugate@ext[n_] := extc[n]
extc /: Conjugate@extc[n_] := ext[n]
extc[n_?Negative] := ext[-n]
ext[n_?Negative] := extc[-n]
extc[0] = 1;

The compromises are:

EXt^n and EXtC^n are expressed as ext[n] and extc[n].

Single EXt and EXtC must be expressed as ext[1] and extc[1].

Example:

ext@2 extc@3
(* extc[1] *)

ext@1 extc@n // Conjugate
(* ext[-1 + n] *)

answered 1 hour ago

xzczd

25k467236

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up vote
2
down vote

How about the following?:

Clear[ext, extc]
ext /: ext[n_] extc[m_] := extc[m - n]
ext /: ext[n_] ext[m_] := ext[m + n]
extc /: extc[n_] extc[m_] := extc[m + n]
ext /: Conjugate@ext[n_] := extc[n]
extc /: Conjugate@extc[n_] := ext[n]
extc[n_?Negative] := ext[-n]
ext[n_?Negative] := extc[-n]
extc[0] = 1;

The compromises are:

EXt^n and EXtC^n are expressed as ext[n] and extc[n].

Single EXt and EXtC must be expressed as ext[1] and extc[1].

Example:

ext@2 extc@3
(* extc[1] *)

ext@1 extc@n // Conjugate
(* ext[-1 + n] *)

answered 1 hour ago

xzczd

25k467236

add a commentÂ |Â

up vote
2
down vote

How about the following?:

Clear[ext, extc]
ext /: ext[n_] extc[m_] := extc[m - n]
ext /: ext[n_] ext[m_] := ext[m + n]
extc /: extc[n_] extc[m_] := extc[m + n]
ext /: Conjugate@ext[n_] := extc[n]
extc /: Conjugate@extc[n_] := ext[n]
extc[n_?Negative] := ext[-n]
ext[n_?Negative] := extc[-n]
extc[0] = 1;

The compromises are:

EXt^n and EXtC^n are expressed as ext[n] and extc[n].

Single EXt and EXtC must be expressed as ext[1] and extc[1].

Example:

ext@2 extc@3
(* extc[1] *)

ext@1 extc@n // Conjugate
(* ext[-1 + n] *)

answered 1 hour ago

xzczd

25k467236

How about the following?:

Clear[ext, extc]
ext /: ext[n_] extc[m_] := extc[m - n]
ext /: ext[n_] ext[m_] := ext[m + n]
extc /: extc[n_] extc[m_] := extc[m + n]
ext /: Conjugate@ext[n_] := extc[n]
extc /: Conjugate@extc[n_] := ext[n]
extc[n_?Negative] := ext[-n]
ext[n_?Negative] := extc[-n]
extc[0] = 1;

The compromises are:

EXt^n and EXtC^n are expressed as ext[n] and extc[n].

Single EXt and EXtC must be expressed as ext[1] and extc[1].

Example:

ext@2 extc@3
(* extc[1] *)

ext@1 extc@n // Conjugate
(* ext[-1 + n] *)

answered 1 hour ago

xzczd

25k467236

answered 1 hour ago

xzczd

25k467236

answered 1 hour ago

xzczd

25k467236

answered 1 hour ago

xzczd

25k467236

add a commentÂ |Â

up vote
2
down vote

Here's a variation of @xzczd's idea, using only a single symbol and adding formatting:

Clear[ext]
ext[n_] ext[m_] ^:= ext[n+m]
ext[n_]^m_ ^:= ext[n m]
Conjugate[ext[n_]] ^:= ext[-n]
ext[0] = 1;

MakeBoxes[ext[n_],StandardForm]:=Switch[n,
 0, "1",
 1, MakeBoxes[EXt],
 -1, MakeBoxes[EXtC],
 _Integer?Negative, With[s=-n, MakeBoxes[EXtC^s]],
 _, MakeBoxes[EXt^n]
]

For example:

EXt = ext[1]
EXtC = ext[-1]

EXt

EXtC

And:

EXt^2 EXtC^2
EXt^2 EXtC^3
EXt EXtC^n //Conjugate

1

EXtC

EXt^(-1 + n)

edited 13 mins ago

answered 19 mins ago

Carl Woll

61.8k280158

add a commentÂ |Â

up vote
2
down vote

Here's a variation of @xzczd's idea, using only a single symbol and adding formatting:

Clear[ext]
ext[n_] ext[m_] ^:= ext[n+m]
ext[n_]^m_ ^:= ext[n m]
Conjugate[ext[n_]] ^:= ext[-n]
ext[0] = 1;

MakeBoxes[ext[n_],StandardForm]:=Switch[n,
 0, "1",
 1, MakeBoxes[EXt],
 -1, MakeBoxes[EXtC],
 _Integer?Negative, With[s=-n, MakeBoxes[EXtC^s]],
 _, MakeBoxes[EXt^n]
]

For example:

EXt = ext[1]
EXtC = ext[-1]

EXt

EXtC

And:

EXt^2 EXtC^2
EXt^2 EXtC^3
EXt EXtC^n //Conjugate

1

EXtC

EXt^(-1 + n)

edited 13 mins ago

answered 19 mins ago

Carl Woll

61.8k280158

add a commentÂ |Â

up vote
2
down vote

Here's a variation of @xzczd's idea, using only a single symbol and adding formatting:

Clear[ext]
ext[n_] ext[m_] ^:= ext[n+m]
ext[n_]^m_ ^:= ext[n m]
Conjugate[ext[n_]] ^:= ext[-n]
ext[0] = 1;

MakeBoxes[ext[n_],StandardForm]:=Switch[n,
 0, "1",
 1, MakeBoxes[EXt],
 -1, MakeBoxes[EXtC],
 _Integer?Negative, With[s=-n, MakeBoxes[EXtC^s]],
 _, MakeBoxes[EXt^n]
]

For example:

EXt = ext[1]
EXtC = ext[-1]

EXt

EXtC

And:

EXt^2 EXtC^2
EXt^2 EXtC^3
EXt EXtC^n //Conjugate

1

EXtC

EXt^(-1 + n)

edited 13 mins ago

answered 19 mins ago

Carl Woll

61.8k280158

Here's a variation of @xzczd's idea, using only a single symbol and adding formatting:

Clear[ext]
ext[n_] ext[m_] ^:= ext[n+m]
ext[n_]^m_ ^:= ext[n m]
Conjugate[ext[n_]] ^:= ext[-n]
ext[0] = 1;

MakeBoxes[ext[n_],StandardForm]:=Switch[n,
 0, "1",
 1, MakeBoxes[EXt],
 -1, MakeBoxes[EXtC],
 _Integer?Negative, With[s=-n, MakeBoxes[EXtC^s]],
 _, MakeBoxes[EXt^n]
]

For example:

EXt = ext[1]
EXtC = ext[-1]

EXt

EXtC

And:

EXt^2 EXtC^2
EXt^2 EXtC^3
EXt EXtC^n //Conjugate

1

EXtC

EXt^(-1 + n)

edited 13 mins ago

answered 19 mins ago

Carl Woll

61.8k280158

edited 13 mins ago

answered 19 mins ago

Carl Woll

61.8k280158

answered 19 mins ago

Carl Woll

61.8k280158

answered 19 mins ago

Carl Woll

61.8k280158

add a commentÂ |Â

user43283 is a new contributor. Be nice, and check out our Code of Conduct.

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