An absolute value cannot equal a negative (#812)

BREAKING-CHANGES.md

@@ -48,6 +48,7 @@ read first.
 | **silent** | `k/(a x^2 + c)` and `k/sqrt(a x^2 + c)` for symbolic `a` | `NaN` | a piecewise on the sign of the discriminant |
 | loud | `Compile` over a missing variable | `KeyNotFoundException` | `UncompilableNodeException` |
 | loud | the target frameworks | `net7.0;netstandard2.0` | `netstandard2.0;net8.0;net10.0` |
+| **silent** | `abs(x) = c` for a negative `c` | a set of non-solutions | the empty set |
 
 ---
 

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

@@ -259,11 +259,17 @@ partial record Absf
             // abs(f(x)) = value
             // f(x) = value * e ^ (i * n)
             // x = f(x).InvertNode(value * e ^ (i * n), x)
+            //
+            // value * e ^ (i * r) has modulus |value|, not value, so it solves the equation
+            // only where the two agree -- that is, where value is real and not negative.
+            // Without that condition abs(x) = -1 came back as { -e ^ (i * r) provided r in RR },
+            // whose members have modulus 1 and solve nothing.
+            // https://github.com/asc-community/AngouriMath/issues/812
             private protected override IEnumerable<Entity> InvertNode(Entity value, Entity x)
             {
                 var @var = Variable.CreateUnique(value + Argument, "r");
                 return Argument.Invert(value * MathS.e.Pow(MathS.i * @var), x)
-                    .Select(c => c.Provided(@var.In(MathS.Sets.R)));
+                    .Select(c => c.Provided(@var.In(MathS.Sets.R) & new GreaterOrEqualf(value, 0)));
             }
         }
 

Sources/Tests/UnitTests/Common/SolverRegressionTest.cs

@@ -199,5 +199,49 @@ public void NewtonSolverDoesNotLeakAnUncompilableNodeException()
             // The omega constant: the root of x = -ln(x).
             Assert.Equal(0.5671432904097838, root.EvalNumerical().RealPart.EDecimal.ToDouble(), 9);
         }
+
+        // abs(x) = a was inverted to the circle a * e ^ (i * r), which has modulus |a| and
+        // so only solves the equation where a is real and not negative. The condition was
+        // missing, so a negative right-hand side came back with a set of non-solutions
+        // rather than with nothing. https://github.com/asc-community/AngouriMath/issues/812
+        [Theory]
+        [InlineData("abs(x) = -1")]
+        [InlineData("abs(x) = -3/2")]
+        [InlineData("abs(x) + 1 = 0")]
+        public void Issue812_AnAbsoluteValueCannotEqualANegative(string equation)
+        {
+            var solutions = (Entity.Set.FiniteSet)equation.ToEntity().Solve("x");
+            Assert.Empty(solutions);
+        }
+
+        // The guard must not cost the answers that were right. Each solution is checked by
+        // substituting it back, with the free parameter of the circle given a value, rather
+        // than by comparing the printed form.
+        [Theory]
+        [InlineData("3", 3.0)]
+        [InlineData("1/2", 0.5)]
+        [InlineData("0", 0.0)]
+        public void Issue812_ANonNegativeRightHandSideStillSolves(string rhs, double modulus)
+        {
+            var solutions = (Entity.Set.FiniteSet)$"abs(x) = {rhs}".ToEntity().Solve("x");
+            var solution = Assert.Single(solutions);
+            var parameter = solution.Vars.Single(v => v.Name != "x");
+            foreach (var point in new[] { 0.0, 1.0, -2.5 })
+            {
+                var candidate = solution.Substitute(parameter, point);
+                while (candidate is Entity.Providedf(var inner, _)) candidate = inner;
+                Assert.Equal(modulus,
+                    MathS.Abs(candidate).EvalNumerical().RealPart.EDecimal.ToDouble(), 9);
+            }
+        }
+
+        // A symbolic right-hand side cannot be decided either way, so the condition is
+        // carried rather than resolved -- which is what #318 asked this inversion to do.
+        [Fact]
+        public void Issue812_ASymbolicRightHandSideCarriesTheCondition()
+        {
+            var printed = "abs(x) = a".ToEntity().Solve("x").Stringize();
+            Assert.Contains("a >= 0", printed);
+        }
     }
 }