Simplifying simple Sqrt inequality

Question

I'm trying to get Mathematica to Simplify

Sqrt[x^2 + y^2] < 2

to

x^2 + y^2 < 4

Among other things I've tried based on other's questions and answers

FullSimplify[Sqrt[x^2 + y^2] < 2, {x, y} \[Element] Reals, ComplexityFunction -> LeafCount]
(* Sqrt[x^2 + y^2] < 2 *)

FullSimplify[Refine[Reduce[(Sqrt[x^2 + y^2] < 2), (x | y) \[Element] Reals]],ComplexityFunction -> LeafCount]
(* -Sqrt[4 - x^2] < y < Sqrt[4 - x^2] *)

Any suggestions? Thanks in advance.

Answer 1

It appears the Mathematica doesn't have a function to automatically Simplify a relational expression by squaring both sides.

However, one can accomplish this by providing a TransformationFunction to Simplify.

For example:

tf1[a_ < b_] := a^2 < b^2
Simplify[Sqrt[x^2 + y^2] < 2, TransformationFunctions ->{tf1, Automatic}]
(* x^2 + y^2 < 4 *)

However, tf1 only works for a LessThan relation.

Simplify[Sqrt[x^2 + y^2] > 2, TransformationFunctions -> {tf1, Automatic}]
(* Sqrt[x^2 + y^2] > 2 *)

But this can be fixed by adding an Alternatives pattern of relations. (Learned from Simplify equations with pattern assumptions )

relationOperators = Equal | Unequal | Greater | Less | GreaterEqual | LessEqual

tf2[l_~(relOp : relationOperators)~r_] := (l^2)~relOp~(r^2)

Table[Simplify[Sqrt[x^2 + y^2]~op~ 2, TransformationFunctions -> {tf2, Automatic}], {op, Level[relationOperators, -1]}]

(* {x^2 + y^2 == 4, x^2 + y^2 != 4, x^2 + y^2 > 4, x^2 + y^2 < 4, x^2 + y^2 >= 4, x^2 + y^2 <= 4} *)

Additionally, TransformationFunction can be expanded to only apply under the Condition that operands are Reals or relation is Equal or Unequal. (Learned from How to select TransformationFunctions based on Assumptions made when using Simplify? )

tf3[l_~(relOp : relationOperators)~r_] /; Simplify[(l \[Element] Reals) && (r \[Element] Reals) || relOp == Equal || relOp == Unequal]  := (l^2)~relOp~(r^2)

Table[Simplify[Sqrt[x^2 + y^2]~i~ 2, TransformationFunctions -> {tf3, Automatic}], {i, Level[relationOperators, -1]}]
(* {x^2 + y^2 == 4, x^2 + y^2 != 4, Sqrt[x^2 + y^2] > 2, Sqrt[x^2 + y^2] < 2, Sqrt[x^2 + y^2] >= 2, Sqrt[x^2 + y^2] <= 2} *)

Assuming[{x, y} \[Element] Reals, Table[Simplify[Sqrt[x^2 + y^2]~i~ 2, TransformationFunctions -> {tf3, Automatic}], {i, Level[relationOperators, -1]}]]
(* {x^2 + y^2 == 4, x^2 + y^2 != 4, x^2 + y^2 > 4, x^2 + y^2 < 4, x^2 + y^2 >= 4, x^2 + y^2 <= 4} *)