further expand Solve[]'s ability

1. bring domain check back into solve.eval, so that things like Abs()
   can be evaluated
2. create system symbols such as System`Reals for domain check
3. refactor, moving most logics out of Solve

Author: BlankShrimp

mathics/builtin/numbers/calculus.py

@@ -10,7 +10,7 @@
 """
 
 from itertools import product
-from typing import Optional, Union
+from typing import Optional
 
 import numpy as np
 import sympy
@@ -59,23 +59,27 @@
 from mathics.core.systemsymbols import (
     SymbolAnd,
     SymbolAutomatic,
+    SymbolComplex,
     SymbolConditionalExpression,
     SymbolD,
     SymbolDerivative,
     SymbolInfinity,
     SymbolInfix,
+    SymbolInteger,
     SymbolIntegrate,
     SymbolLeft,
     SymbolLog,
     SymbolNIntegrate,
     SymbolO,
+    SymbolReal,
     SymbolRule,
     SymbolSequence,
     SymbolSeries,
     SymbolSeriesData,
     SymbolSimplify,
     SymbolUndefined,
 )
+from mathics.eval.calculus import solve_sympy
 from mathics.eval.makeboxes import format_element
 from mathics.eval.nevaluator import eval_N
 from mathics.eval.numbers.calculus.integrators import (
@@ -2210,105 +2214,38 @@ class Solve(Builtin):
     messages = {
         "eqf": "`1` is not a well-formed equation.",
         "svars": 'Equations may not give solutions for all "solve" variables.',
+        "fulldim": "The solution set contains a full-dimensional component; use Reduce for complete solution information.",
     }
 
-    # FIXME: the problem with removing the domain parameter from the outside
-    # is that the we can't make use of this information inside
-    # the evaluation method where it is may be needed.
     rules = {
-        "Solve[eqs_, vars_, Complexes]": "Solve[eqs, vars]",
-        "Solve[eqs_, vars_, Reals]": (
-            "Cases[Solve[eqs, vars], {Rule[x_,y_?RealValuedNumberQ]}]"
-        ),
-        "Solve[eqs_, vars_, Integers]": (
-            "Cases[Solve[eqs, vars], {Rule[x_,y_Integer]}]"
-        ),
+        "Solve[eqs_, vars_]": "Solve[eqs, vars, Complexes]"
     }
     summary_text = "find generic solutions for variables"
 
-    def eval(self, eqs, vars, evaluation: Evaluation):
-        "Solve[eqs_, vars_]"
+    def eval(self, eqs, vars, domain, evaluation: Evaluation):
+        "Solve[eqs_, vars_, domain_]"
 
-        vars_original = vars
-        head_name = vars.get_head_name()
+        variables = vars
+        head_name = variables.get_head_name()
         if head_name == "System`List":
-            vars = vars.elements
+            variables = variables.elements
         else:
-            vars = [vars]
-        for var in vars:
+            variables = [variables]
+        for var in variables:
             if (
                 (isinstance(var, Atom) and not isinstance(var, Symbol)) or
                 head_name in ("System`Plus", "System`Times", "System`Power") or  # noqa
                 A_CONSTANT & var.get_attributes(evaluation.definitions)
             ):
 
-                evaluation.message("Solve", "ivar", vars_original)
+                evaluation.message("Solve", "ivar", vars)
                 return
 
-        vars_sympy = [var.to_sympy() for var in vars]
-        if None in vars_sympy:
+        sympy_variables = [var.to_sympy() for var in variables]
+        if None in sympy_variables:
             evaluation.message("Solve", "ivar")
             return
-        all_var_tuples = list(zip(vars, vars_sympy))
-
-        def cut_var_dimension(expressions: Union[Expression, list[Expression]]):
-            '''delete unused variables to avoid SymPy's PolynomialError
-            : Not a zero-dimensional system in e.g. Solve[x^2==1&&z^2==-1,{x,y,z}]'''
-            if not isinstance(expressions, list):
-                expressions = [expressions]
-            subset_vars = set()
-            subset_vars_sympy = set()
-            for var, var_sympy in all_var_tuples:
-                pattern = Pattern.create(var)
-                for equation in expressions:
-                    if not equation.is_free(pattern, evaluation):
-                        subset_vars.add(var)
-                        subset_vars_sympy.add(var_sympy)
-            return subset_vars, subset_vars_sympy
-
-        def solve_sympy(equations: Union[Expression, list[Expression]]):
-            if not isinstance(equations, list):
-                equations = [equations]
-            equations_sympy = []
-            denoms_sympy = []
-            subset_vars, subset_vars_sympy = cut_var_dimension(equations)
-            for equation in equations:
-                if equation is SymbolTrue:
-                    continue
-                elif equation is SymbolFalse:
-                    return []
-                elements = equation.elements
-                for left, right in [(elements[index], elements[index + 1]) for index in range(len(elements) - 1)]:
-                    # ↑ to deal with things like a==b==c==d
-                    left = left.to_sympy()
-                    right = right.to_sympy()
-                    if left is None or right is None:
-                        return []
-                    equation_sympy = left - right
-                    equation_sympy = sympy.together(equation_sympy)
-                    equation_sympy = sympy.cancel(equation_sympy)
-                    equations_sympy.append(equation_sympy)
-                    numer, denom = equation_sympy.as_numer_denom()
-                    denoms_sympy.append(denom)
-            try:
-                results = sympy.solve(equations_sympy, subset_vars_sympy, dict=True)  # no transform_dict needed with dict=True
-                # Filter out results for which denominator is 0
-                # (SymPy should actually do that itself, but it doesn't!)
-                results = [
-                    sol
-                    for sol in results
-                    if all(sympy.simplify(denom.subs(sol)) != 0 for denom in denoms_sympy)
-                ]
-                return results
-            except sympy.PolynomialError:
-                # raised for e.g. Solve[x^2==1&&z^2==-1,{x,y,z}] when not deleting
-                # unused variables beforehand
-                return []
-            except NotImplementedError:
-                return []
-            except TypeError as exc:
-                if str(exc).startswith("expected Symbol, Function or Derivative"):
-                    evaluation.message("Solve", "ivar", vars_original)
+        variable_tuples = list(zip(variables, sympy_variables))
 
         def solve_recur(expression: Expression):
             '''solve And, Or and List within the scope of sympy,
@@ -2336,7 +2273,7 @@ def solve_recur(expression: Expression):
                             inequations.append(sub_condition)
                     else:
                         inequations.append(child.to_sympy())
-                solutions.extend(solve_sympy(equations))
+                solutions.extend(solve_sympy(evaluation, equations, variables, domain))
                 conditions = sympy.And(*inequations)
                 result = [sol for sol in solutions if conditions.subs(sol)]
                 return result, None if solutions else conditions
@@ -2346,7 +2283,7 @@ def solve_recur(expression: Expression):
                 conditions = []
                 for child in expression.elements:
                     if child.has_form("Equal", 2):
-                        solutions.extend(solve_sympy(child))
+                        solutions.extend(solve_sympy(evaluation, child, variables, domain))
                     elif child.get_head_name() in ('System`And', 'System`Or'):  # I don't believe List would be in here
                         sub_solution, sub_condition = solve_recur(child)
                         solutions.extend(sub_solution)
@@ -2365,8 +2302,8 @@ def solve_recur(expression: Expression):
             if conditions is not None:
                 evaluation.message("Solve", "fulldim")
         else:
-            if eqs.has_form("Equal", 2):
-                solutions = solve_sympy(eqs)
+            if eqs.get_head_name() == "System`Equal":
+                solutions = solve_sympy(evaluation, eqs, variables, domain)
             else:
                 evaluation.message("Solve", "fulldim")
                 return ListExpression(ListExpression())
@@ -2376,7 +2313,7 @@ def solve_recur(expression: Expression):
             return ListExpression(ListExpression())
 
         if any(
-            sol and any(var not in sol for var in vars_sympy) for sol in solutions
+            sol and any(var not in sol for var in sympy_variables) for sol in solutions
         ):
             evaluation.message("Solve", "svars")
 
@@ -2385,7 +2322,7 @@ def solve_recur(expression: Expression):
                 ListExpression(
                     *(
                         Expression(SymbolRule, var, from_sympy(sol[var_sympy]))
-                        for var, var_sympy in all_var_tuples
+                        for var, var_sympy in variable_tuples
                         if var_sympy in sol
                     ),
                 )

mathics/core/atoms.py

@@ -774,9 +774,9 @@ def get_sort_key(self, pattern_sort=False) -> tuple:
     def sameQ(self, other) -> bool:
         """Mathics SameQ"""
         return (
-            isinstance(other, Complex)
-            and self.real == other.real
-            and self.imag == other.imag
+            isinstance(other, Complex) and
+            self.real == other.real and
+            self.imag == other.imag
         )
 
     def round(self, d=None) -> "Complex":

mathics/core/convert/sympy.py

@@ -5,6 +5,7 @@
 Conversion to SymPy is handled directly in BaseElement descendants.
 """
 
+from collections.abc import Iterable
 from typing import Optional, Type, Union
 
 import sympy
@@ -13,9 +14,6 @@
 # Import the singleton class
 from sympy.core.numbers import S
 
-BasicSympy = sympy.Expr
-
-
 from mathics.core.atoms import (
     MATHICS3_COMPLEX_I,
     Complex,
@@ -40,6 +38,7 @@
 )
 from mathics.core.list import ListExpression
 from mathics.core.number import FP_MANTISA_BINARY_DIGITS
+from mathics.core.rules import Pattern
 from mathics.core.symbols import (
     Symbol,
     SymbolFalse,
@@ -62,16 +61,21 @@
     SymbolGreater,
     SymbolGreaterEqual,
     SymbolIndeterminate,
+    SymbolIntegers,
     SymbolLess,
     SymbolLessEqual,
     SymbolMatrixPower,
     SymbolO,
     SymbolPi,
     SymbolPiecewise,
+    SymbolReals,
     SymbolSlot,
     SymbolUnequal,
 )
 
+BasicSympy = sympy.Expr
+
+
 SymbolPrime = Symbol("Prime")
 SymbolRoot = Symbol("Root")
 SymbolRootSum = Symbol("RootSum")
@@ -130,6 +134,39 @@ def to_sympy_matrix(data, **kwargs) -> Optional[sympy.MutableDenseMatrix]:
         return None
 
 
+def apply_domain_to_symbols(symbols: Iterable[sympy.Symbol], domain) -> dict[sympy.Symbol, sympy.Symbol]:
+    """Create new sympy symbols with domain applied.
+    Return a dict maps old to new.
+    """
+    # FIXME: this substitute solution would break when Solve[Abs[x]==3, x],where x=-3 and x=3.
+    # However, substituting symbol prior to actual solving would cause sympy to have biased assumption,
+    # it would refuse to solve Abs() when symbol is in Complexes
+    result = {}
+    for symbol in symbols:
+        if domain == SymbolReals:
+            new_symbol = sympy.Symbol(repr(symbol), real=True)
+        elif domain == SymbolIntegers:
+            new_symbol = sympy.Symbol(repr(symbol), integer=True)
+        else:
+            new_symbol = symbol
+        result[symbol] = new_symbol
+    return result
+
+
+def cut_dimension(evaluation, expressions: Union[Expression, list[Expression]], symbols: Iterable[sympy.Symbol]) -> set[sympy.Symbol]:
+    '''delete unused variables to avoid SymPy's PolynomialError
+    : Not a zero-dimensional system in e.g. Solve[x^2==1&&z^2==-1,{x,y,z}]'''
+    if not isinstance(expressions, list):
+        expressions = [expressions]
+    subset = set()
+    for symbol in symbols:
+        pattern = Pattern.create(symbol)
+        for equation in expressions:
+            if not equation.is_free(pattern, evaluation):
+                subset.add(symbol)
+    return subset
+
+
 class SympyExpression(BasicSympy):
     is_Function = True
     nargs = None
@@ -363,9 +400,9 @@ def old_from_sympy(expr) -> BaseElement:
             if is_Cn_expr(name):
                 return Expression(SymbolC, Integer(int(name[1:])))
             if name.startswith(sympy_symbol_prefix):
-                name = name[len(sympy_symbol_prefix) :]
+                name = name[len(sympy_symbol_prefix):]
             if name.startswith(sympy_slot_prefix):
-                index = name[len(sympy_slot_prefix) :]
+                index = name[len(sympy_slot_prefix):]
                 return Expression(SymbolSlot, Integer(int(index)))
         elif expr.is_NumberSymbol:
             name = str(expr)
@@ -517,7 +554,7 @@ def old_from_sympy(expr) -> BaseElement:
                     *[from_sympy(arg) for arg in expr.args]
                 )
             if name.startswith(sympy_symbol_prefix):
-                name = name[len(sympy_symbol_prefix) :]
+                name = name[len(sympy_symbol_prefix):]
         args = [from_sympy(arg) for arg in expr.args]
         builtin = sympy_to_mathics.get(name)
         if builtin is not None:

mathics/core/systemsymbols.py

@@ -56,6 +56,7 @@
 SymbolCompile = Symbol("System`Compile")
 SymbolCompiledFunction = Symbol("System`CompiledFunction")
 SymbolComplex = Symbol("System`Complex")
+SymbolComplexes = Symbol("System`Complexes")
 SymbolComplexInfinity = Symbol("System`ComplexInfinity")
 SymbolCondition = Symbol("System`Condition")
 SymbolConditionalExpression = Symbol("System`ConditionalExpression")
@@ -124,6 +125,7 @@
 SymbolInfix = Symbol("System`Infix")
 SymbolInputForm = Symbol("System`InputForm")
 SymbolInteger = Symbol("System`Integer")
+SymbolIntegers = Symbol("System`Integers")
 SymbolIntegrate = Symbol("System`Integrate")
 SymbolLeft = Symbol("System`Left")
 SymbolLength = Symbol("System`Length")
@@ -200,6 +202,7 @@
 SymbolRational = Symbol("System`Rational")
 SymbolRe = Symbol("System`Re")
 SymbolReal = Symbol("System`Real")
+SymbolReals = Symbol("System`Reals")
 SymbolRealAbs = Symbol("System`RealAbs")
 SymbolRealDigits = Symbol("System`RealDigits")
 SymbolRealSign = Symbol("System`RealSign")

test/builtin/calculus/test_solve.py

@@ -43,14 +43,34 @@ def test_solve():
             "Issue #1235",
         ),
         (
-            "Solve[{x^2==4 && x < 0},{x}]",
-            "{x->-2}",
-            "",
+            "Solve[Abs[-2/3*(lambda + 2) + 8/3 + 4] == 4, lambda,Reals]",
+            "{{lambda -> 2}, {lambda -> 14}}",
+            "abs()",
         ),
         (
-            "Solve[{x^2==4 && x < 0 && x > -4},{x}]",
-            "{x->-2}",
-            "",
+            "Solve[q^3 == (20-12)/(4-3), q,Reals]",
+            "{{q -> 2}}",
+            "domain check",
+        ),
+        (
+            "Solve[x + Pi/3 == 2k*Pi + Pi/6 || x + Pi/3 == 2k*Pi + 5Pi/6, x,Reals]",
+            "{{x -> -Pi / 6 + 2 k Pi}, {x -> Pi / 2 + 2 k Pi}}",
+            "logics involved",
+        ),
+        (
+            "Solve[m - 1 == 0 && -(m + 1) != 0, m,Reals]",
+            "{{m -> 1}}",
+            "logics and constraints",
+        ),
+        (
+            "Solve[(lambda + 1)/6 == 1/(mu - 1) == lambda/4, {lambda, mu},Reals]",
+            "{{lambda -> 2, mu -> 3}}",
+            "chained equations",
+        ),
+        (
+            "Solve[2*x0*Log[x0] + x0 - 2*a*x0 == -1 && x0^2*Log[x0] - a*x0^2 + b == b - x0, {x0, a, b},Reals]",
+            "{{x0 -> 1, a -> 1}}",
+            "excess variable b",
         ),
     ):
         session.evaluate("Clear[h]; Clear[g]; Clear[f];")