Read an equality as a difference equal to zero in an equation and in a system

BREAKING-CHANGES.md

@@ -40,6 +40,26 @@ A not-equal written `<>` is read as `not (a = b)`, and printed that way.
 
 ---
 
+## Unreleased — since 2.6.0
+
+### An equality in an equation or a system is read as a difference equal to zero
+
+**Silent: wrong answers.** `MathS.SolveEquation`, `Entity.SolveEquation` and `MathS.Equations` read each equation as an
+expression equal to zero, and passed an equality `a = b` on as it stood. One equation came back as the set of x where
+`a = b` and `b = 0`, which is empty unless `b` is zero, and a system came back with no solution. An equality is now read
+as `a - b = 0`, as a statement's own solve reads it: `"x + 1 = 3".Solve("x")` was already `{ 2 }`
+([#1673](https://github.com/asc-community/AngouriMath/issues/1673)).
+
+| Input | Was (2.6.0) | Now |
+|---|---|---|
+| `MathS.SolveEquation("x + 1 = 3", "x")` | `{ x : x + 1 = 3 and 3 = 0 }` | `{ 2 }` |
+| `"x^2 = 4".SolveEquation("x")` | `{ x : x ^ 2 = 4 and 4 = 0 }` | `{ 2, -2 }` |
+| `"a * x = b".ToEntity().SolveEquation("x")` | `{ x : a * x = b and b = 0 }`, which holds only where `b = 0` | `{ b / a }` |
+| `MathS.Equations("x + y = 3", "x - y = 1").Solve("x", "y")` | `null` | `[[2, 1]]` |
+| `MathS.Equations("x + y = 3", "x - y = 1").ToString()` | `x + y = 3 = 0` and `x - y = 1 = 0` | `x + y - 3 = 0` and `x - y - 1 = 0` |
+
+---
+
 ## 2.6.0 — since 2.5.0
 
 ### A high half-odd power of a quadratic beside a power of another is reduced in exact rationals

Sources/AngouriMath/Convenience/MathS.cs

@@ -267,7 +267,7 @@ public static Integer GreatestCommonDivisor(Integer a, Integer b)
         public static EquationSystem Equations(IEnumerable<Entity> equations) => new EquationSystem(equations);
 
         /// <summary>Solves one equation over one variable</summary>
-        /// <param name="equation">An equation that is assumed to equal 0</param>
+        /// <param name="equation">An expression that is assumed to equal 0, or an equality <c>a = b</c>, read as <c>a - b = 0</c></param>
         /// <param name="var">Variable whose values we are looking for</param>
         /// <returns>A <see cref="Set"/> of possible values or intervals of values</returns>
         /// <example>

Sources/AngouriMath/Core/EquationSystem.cs

@@ -22,7 +22,8 @@ public sealed class EquationSystem : ILatexizeable
         private readonly IEnumerable<Entity> equations;
 
         /// <summary>
-        /// The equations you pass should not have an <see cref="MathS.Equality"/> node.
+        /// Each equation is an expression equal to zero, or an equality <c>a = b</c>, which is read as
+        /// <c>a - b = 0</c>, as <see cref="MathS.SolveEquation"/> reads one in a single variable.
         /// After having created a system of equations, you may solve or latexize it.
         /// </summary>
         /// <param name="equations">
@@ -36,10 +37,11 @@ public sealed class EquationSystem : ILatexizeable
         /// });
         /// </code>
         /// </example>
-        public EquationSystem(IEnumerable<Entity> equations) => this.equations = equations;
+        public EquationSystem(IEnumerable<Entity> equations) => this.equations = equations.Select(AsAnExpressionEqualToZero).ToList();
 
         /// <summary>
-        /// The equations you pass should not have an <see cref="MathS.Equality"/> node.
+        /// Each equation is an expression equal to zero, or an equality <c>a = b</c>, which is read as
+        /// <c>a - b = 0</c>, as <see cref="MathS.SolveEquation"/> reads one in a single variable.
         /// After having created a system of equations, you may solve or latexize it.
         /// </summary>
         /// <param name="equations">
@@ -53,7 +55,13 @@ public sealed class EquationSystem : ILatexizeable
         /// );
         /// </code>
         /// </example>
-        public EquationSystem(params Entity[] equations) => this.equations = equations;
+        public EquationSystem(params Entity[] equations) => this.equations = equations.Select(AsAnExpressionEqualToZero).ToList();
+
+        // An equality was passed on as it stood and the system answered that it had no solution:
+        // `x + y = 3` and `x - y = 1` gave null, where `x + y - 3` and `x - y - 1` give `[[2, 1]]`.
+        // https://github.com/asc-community/AngouriMath/issues/1673
+        private static Entity AsAnExpressionEqualToZero(Entity equation)
+            => equation is Equalsf(var left, var right) ? left - right : equation;
 
         /// <summary>
         /// Solves a system of equations, if possible.

Sources/AngouriMath/Functions/Continuous/Solvers/EquationSolver.cs

@@ -20,6 +20,12 @@ internal static class EquationSolver
         /// <summary>Solves one equation</summary>
         internal static Set Solve(Entity equation, Variable x)
         {
+            // An equality `a = b` is `a - b = 0`, as a statement's own solve reads it: passed on as it
+            // stood, `MathS.SolveEquation("x + 1 = 3", x)` was `{ x : x + 1 = 3 and 3 = 0 }`, which is
+            // empty, where "x + 1 = 3".Solve("x") is `{ 2 }`.
+            // https://github.com/asc-community/AngouriMath/issues/1673
+            if (equation is Equalsf(var left, var right))
+                equation = left - right;
             using var _ = MathS.Settings.PrecisionErrorZeroRange.Set(1e-12m);
             using var __ = MathS.Settings.FloatToRationalIterCount.Set(0);
             using var ___ = MathS.Settings.MaxExpansionTermCount.Set(50);

Sources/MCP/AngouriMath.Mcp/SelfTest.cs

@@ -75,22 +75,6 @@ private static readonly (string Name, Func<bool> StillBroken, string Documented)
     [
         ("Simplify(sqrt(x^2))", () => Value("sqrt(x^2)") != "abs(x)",
             "left as written rather than reduced to abs(x)"),
-        ("MathS.Equations does not solve a system written as equalities", () =>
-        {
-            // x + y = 3 and x - y = 1 meet at (2, 1), which the same system written in '= 0'
-            // form is solved to. A check for an exception alone would read a system that comes
-            // back with no solution as fixed. https://github.com/asc-community/AngouriMath/issues/1673
-            try
-            {
-                var solutions = MathS.Equations("x + y = 3".ToEntity(), "x - y = 1".ToEntity()).Solve("x", "y");
-                return solutions is null || solutions.RowCount != 1
-                    || solutions[0, 0].Evaled != 2 || solutions[0, 1].Evaled != 1;
-            }
-            catch (Exception)
-            {
-                return true;
-            }
-        }, "x + y = 3, x - y = 1 has no solution when written as equalities; each equation must be given in '= 0' form"),
         ("DefiniteIntegral is a first-order rule", () =>
         {
             // Halving the step count roughly doubles the error iff the rule is first order.

Sources/MCP/AngouriMath.Mcp/Tools.cs

@@ -1166,11 +1166,8 @@ private static JsonObject SolveSystem(JsonObject args)
             if (!TryParse(eq, "equations", false, out var e, out var w, out var error))
                 return error!;
 
-            // EquationSystem wants each equation in "= 0" form, and handed an equality it
-            // finds no solution, so normalise here: writing 'x + y = 3' is the natural thing
-            // for a caller to do, and it should just work.
-            // https://github.com/asc-community/AngouriMath/issues/1673
-            parsed.Add(e is Equalsf equality ? equality.Left - equality.Right : e);
+            // An equality, 'x + y = 3', is read by the library as 'x + y - 3 = 0'.
+            parsed.Add(e);
             warnings.AddRange(w);
         }
 

Sources/MCP/README.md

@@ -151,9 +151,8 @@ existing tool, then a prompt, and add a tool last. A tool's description is a rou
 documentation: say when to call it, and where a model would wrongly trust itself, say so.
 
 This is an adapter, not a second computer algebra system. Mathematics belongs in the library, where
-the terminal and the notebooks get it too, and four workarounds here wait for it:
+the terminal and the notebooks get it too, and three workarounds here wait for it:
 eigenvalues by the characteristic polynomial ([#1676](https://github.com/asc-community/AngouriMath/issues/1676)),
 a numeric comparison with a tolerance ([#1674](https://github.com/asc-community/AngouriMath/issues/1674)),
-`a = b` rewritten to `a - b` for a system ([#1673](https://github.com/asc-community/AngouriMath/issues/1673)),
 and the numeric definite integral run twice to find its good digits
 ([#1675](https://github.com/asc-community/AngouriMath/issues/1675)).

Sources/Tests/UnitTests/Algebra/SolveTest/AnEqualityReadAsADifferenceTest.cs

@@ -0,0 +1,54 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using AngouriMath.Extensions;
+using Xunit;
+using static AngouriMath.Entity.Set;
+
+namespace AngouriMath.Tests.Algebra
+{
+    /// <summary>
+    /// An equation written as an equality <c>a = b</c> is solved as <c>a - b = 0</c> where an expression
+    /// equal to zero is expected, as a statement's own solve reads it. Passed on as it stood, a system
+    /// answered that it had no solution, and <see cref="MathS.SolveEquation"/> answered a set builder
+    /// with a false condition in it, which is empty.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/1673">#1673</a>
+    /// </summary>
+    [Trait("Area", "Algebra")]
+    public sealed class AnEqualityReadAsADifferenceTest
+    {
+        [Theory]
+        [InlineData("x + y = 3", "x - y = 1")]
+        [InlineData("x + y = 3", "x - y - 1")]
+        [InlineData("x + y - 3", "x - y = 1")]
+        [InlineData("x = 3 - y", "y = x - 1")]
+        public void ASystemOfEqualitiesIsSolved(string first, string second)
+        {
+            var solution = MathS.Equations(first.ToEntity(), second.ToEntity()).Solve("x", "y");
+            Assert.NotNull(solution);
+            Assert.Equal(1, solution!.RowCount);
+            Assert.Equal(2, (int)solution[0, 0].EvalNumerical().RealPart);
+            Assert.Equal(1, (int)solution[0, 1].EvalNumerical().RealPart);
+        }
+
+        [Fact]
+        public void ASystemPrintsEachEquationAsADifferenceEqualToZero()
+            => Assert.Equal("x + y - 3 = 0\nx - y - 1 = 0", MathS.Equations("x + y = 3".ToEntity(), "x - y = 1".ToEntity()).ToString());
+
+        [Fact]
+        public void OneEqualityIsSolvedInItsVariable()
+            => Assert.Equal(new FiniteSet(2), MathS.SolveEquation("x + 1 = 3", "x"));
+
+        [Fact]
+        public void TheExtensionReadsAnEqualityTheSameWay()
+            => Assert.Equal(new FiniteSet(-2, 2), "x^2 = 4".SolveEquation("x"));
+
+        [Fact]
+        public void AnEqualityWithAParameterIsSolvedForTheVariable()
+            => Assert.Equal(new FiniteSet("b / a".ToEntity().InnerSimplified), "a * x = b".ToEntity().SolveEquation("x").InnerSimplified);
+    }
+}