SCHEMA mathematical_functions_schema;
-- This file constitutes document WG12 N921
-- Master document: ISO 10303-50:2001
-- EXPRESS last modified: 2001-09-07

REFERENCE FROM ISO13584_generic_expressions_schema     -- ISO 13584-20
  (binary_generic_expression,
   environment,
   generic_expression,
   generic_literal,
   generic_variable,
   multiple_arity_generic_expression,
   simple_generic_expression,
   unary_generic_expression,
   variable_semantics);

REFERENCE FROM ISO13584_expressions_schema             -- ISO 13584-20
  (abs_function                   AS abs_expression,
   acos_function                  AS acos_expression,
   and_expression,
   asin_function                  AS asin_expression,
   atan_function                  AS atan_expression,
   binary_boolean_expression,
   binary_function_call           AS binary_numeric_call_expression,
   binary_numeric_expression,
   boolean_defined_function       AS boolean_defined_expression,
   boolean_expression,
   boolean_literal,
   boolean_variable,
   comparison_equal,
   comparison_expression,
   comparison_greater,
   comparison_greater_equal,
   comparison_less,
   comparison_less_equal,
   comparison_not_equal,
   concat_expression,
   cos_function                   AS cos_expression,
   defined_function               AS defined_expression,
   div_expression,
   equals_expression,
   exp_function                   AS exp_expression,
   expression,
   format_function                AS format_expression,
   index_expression,
   int_literal,
   int_numeric_variable,
   int_value_function             AS int_value_expression,
   integer_defined_function       AS integer_defined_expression,
   interval_expression,
   length_function                AS length_expression,
   like_expression,
   literal_number,
   log_function                   AS log_expression,
   log10_function                 AS log10_expression,
   log2_function                  AS log2_expression,
   maximum_function               AS maximum_expression,
   minimum_function               AS minimum_expression,
   minus_expression,
   minus_function                 AS unary_minus_expression,
   mod_expression,
   mult_expression,
   multiple_arity_boolean_expression,
   multiple_arity_function_call   AS multiple_arity_numeric_call_expression,
   multiple_arity_numeric_expression,
   not_expression,
   numeric_defined_function       AS numeric_defined_expression,
   numeric_expression,
   numeric_variable,
   odd_function                   AS odd_expression,
   or_expression,
   plus_expression,
   power_expression,
   real_defined_function          AS real_defined_expression,
   real_literal,
   real_numeric_variable,
   simple_boolean_expression,
   simple_numeric_expression,
   simple_string_expression,
   sin_function                   AS sin_expression,
   slash_expression,
   sql_mappable_defined_function  AS sql_mappable_defined_expression,
   square_root_function           AS square_root_expression,
   string_defined_function        AS string_defined_expression,
   string_expression,
   string_literal,
   string_variable,
   substring_expression,
   tan_function                   AS tan_expression,
   unary_boolean_expression,
   unary_function_call            AS unary_numeric_call_expression,
   unary_numeric_expression,
   value_function                 AS value_expression,
   variable,
   xor_expression);

REFERENCE FROM support_resource_schema                 -- ISO 10303-41
  (label,
   text);

REFERENCE FROM external_reference_schema               -- ISO 10303-41
  (externally_defined_item);

REFERENCE FROM geometry_schema                         -- ISO 10303-42
  (curve,
   dimension_of,
   point,
   surface,
   volume);

CONSTANT
  schema_prefix : STRING := 'MATHEMATICAL_FUNCTIONS_SCHEMA.';
  the_integers        : elementary_space := make_elementary_space(es_integers);
  the_reals           : elementary_space := make_elementary_space(es_reals);
  the_complex_numbers : elementary_space := make_elementary_space(es_complex_numbers);
  the_numbers         : elementary_space := make_elementary_space(es_numbers);
  the_logicals        : elementary_space := make_elementary_space(es_logicals);
  the_booleans        : elementary_space := make_elementary_space(es_booleans);
  the_strings         : elementary_space := make_elementary_space(es_strings);
  the_binarys         : elementary_space := make_elementary_space(es_binarys);
  the_maths_spaces    : elementary_space := make_elementary_space(es_maths_spaces);
  the_generics        : elementary_space := make_elementary_space(es_generics);
  the_empty_space : finite_space := make_finite_space([]);
  the_nonnegative_reals         : real_interval_from_min :=
    make_real_interval_from_min(0.0, closed);
  the_zero_one_interval         : finite_real_interval := make_finite_real_interval(
    0.0, closed, 1.0, closed);
  the_zero_pi_interval          : finite_real_interval := make_finite_real_interval(
    0.0, closed, pi, closed);
  the_neg1_one_interval         : finite_real_interval := make_finite_real_interval(
    -1.0, closed, 1.0, closed);
  the_neghalfpi_halfpi_interval : finite_real_interval := make_finite_real_interval(
    -0.5*pi, closed, 0.5*pi, closed);
  the_negpi_pi_interval         : finite_real_interval := make_finite_real_interval(
    -pi, open, pi, closed);
  the_zero_tuple_space : listed_product_space := make_listed_product_space([]);
  the_tuples           : extended_tuple_space := make_extended_tuple_space(
    the_zero_tuple_space, the_generics);
  the_integer_tuples   : extended_tuple_space := make_extended_tuple_space(
    the_zero_tuple_space, the_integers);
  the_real_tuples      : extended_tuple_space := make_extended_tuple_space(
    the_zero_tuple_space, the_reals);
  the_complex_tuples   : extended_tuple_space := make_extended_tuple_space(
    the_zero_tuple_space, the_complex_numbers);
  the_empty_maths_tuple      : maths_tuple := [];
  the_empty_maths_value      : maths_value := the_empty_maths_tuple;
  the_empty_atom_based_tuple : atom_based_tuple := [];
  the_empty_atom_based_value : atom_based_value := the_empty_atom_based_tuple;
END_CONSTANT;
TYPE nonnegative_integer = INTEGER;
WHERE nonnegativity: SELF >= 0;
END_TYPE;
TYPE positive_integer = nonnegative_integer;
WHERE positivity: SELF > 0;
END_TYPE;
TYPE zero_or_one = nonnegative_integer;
WHERE in_range: (SELF = 0) OR (SELF = 1);
END_TYPE;
TYPE one_or_two = positive_integer;
WHERE in_range: (SELF = 1) OR (SELF = 2);
END_TYPE;
TYPE maths_number = NUMBER;
END_TYPE;
TYPE maths_real = REAL;
END_TYPE;
TYPE maths_integer = INTEGER;
END_TYPE;
TYPE maths_logical = LOGICAL;
END_TYPE;
TYPE maths_boolean = BOOLEAN;
END_TYPE;
TYPE maths_string = STRING;
END_TYPE;
TYPE maths_binary = BINARY;
END_TYPE;
TYPE maths_simple_atom = SELECT
  (maths_number,
   maths_real,
   maths_number,
   maths_logical,
   maths_boolean,
   maths_string,
   maths_binary);
END_TYPE;
TYPE maths_atom = SELECT
  (maths_simple_atom,
   maths_enum_atom);
END_TYPE;
TYPE atom_based_tuple = LIST OF atom_based_value;
END_TYPE;
TYPE atom_based_value = SELECT
  (maths_atom,
   atom_based_tuple);
END_TYPE;
TYPE maths_tuple = LIST [0:?] OF maths_value;
END_TYPE;
TYPE maths_value = SELECT
  (atom_based_value,
   maths_tuple,
   generic_expression);
WHERE
  constancy: NOT ('GENERIC_EXPRESSION' IN stripped_typeof(SELF)) OR
             expression_is_constant(SELF);
END_TYPE;
TYPE maths_expression = SELECT
  (atom_based_value,
   maths_tuple,
   generic_expression);
END_TYPE;
TYPE maths_function_select = SELECT
  (maths_function,
   elementary_function_enumerators);
END_TYPE;
TYPE input_selector = positive_integer;
END_TYPE;
TYPE elementary_space_enumerators = ENUMERATION OF
  (es_numbers,
   es_complex_numbers,
   es_reals,
   es_integers,
   es_logicals,
   es_booleans,
   es_strings,
   es_binarys,
   es_maths_spaces,
   es_maths_functions,
   es_generics);
END_TYPE;
TYPE ordering_type = ENUMERATION OF
  (by_rows,
   by_columns);
END_TYPE;
TYPE lower_upper = ENUMERATION OF
  (lower,
   upper);
END_TYPE;
TYPE symmetry_type = ENUMERATION OF
  (identity,
   skew,
   hermitian,
   skew_hermitian);
END_TYPE;
TYPE elementary_function_enumerators = ENUMERATION OF
  (ef_and, ef_or, ef_not, ef_xor,
   ef_negate_i, ef_add_i, ef_subtract_i, ef_multiply_i, ef_divide_i, ef_mod_i,
   ef_exponentiate_i, ef_eq_i, ef_ne_i, ef_gt_i, ef_lt_i, ef_ge_i, ef_le_i,
   ef_abs_i, ef_max_i, ef_min_i, ef_if_i,
   ef_negate_r, ef_reciprocal_r, ef_add_r, ef_subtract_r, ef_multiply_r,
   ef_divide_r, ef_mod_r, ef_exponentiate_r, ef_exponentiate_ri,
   ef_eq_r, ef_ne_r, ef_gt_r, ef_lt_r, ef_ge_r, ef_le_r, ef_abs_r,
   ef_max_r, ef_min_r, ef_acos_r, ef_asin_r, ef_atan2_r, ef_cos_r, ef_exp_r,
   ef_ln_r, ef_log2_r, ef_log10_r, ef_sin_r, ef_sqrt_r, ef_tan_r, ef_if_r,
   ef_form_c, ef_rpart_c, ef_ipart_c,
   ef_negate_c, ef_reciprocal_c, ef_add_c, ef_subtract_c, ef_multiply_c,
   ef_divide_c, ef_exponentiate_c, ef_exponentiate_ci, ef_eq_c, ef_ne_c,
   ef_conjugate_c, ef_abs_c, ef_arg_c, ef_cos_c, ef_exp_c, ef_ln_c, ef_sin_c,
   ef_sqrt_c, ef_tan_c, ef_if_c,
   ef_subscript_s, ef_eq_s, ef_ne_s, ef_gt_s, ef_lt_s, ef_ge_s, ef_le_s,
   ef_subsequence_s, ef_concat_s, ef_size_s, ef_format, ef_value, ef_like, ef_if_s,
   ef_subscript_b, ef_eq_b, ef_ne_b, ef_gt_b, ef_lt_b, ef_ge_b, ef_le_b,
   ef_subsequence_b, ef_concat_b, ef_size_b, ef_if_b,
   ef_subscript_t, ef_eq_t, ef_ne_t, ef_concat_t, ef_size_t,
   ef_entuple, ef_detuple, ef_insert, ef_remove, ef_if_t,
   ef_sum_it, ef_product_it,
   ef_add_it, ef_subtract_it, ef_scalar_mult_it, ef_dot_prod_it,
   ef_sum_rt, ef_product_rt,
   ef_add_rt, ef_subtract_rt, ef_scalar_mult_rt, ef_dot_prod_rt, ef_norm_rt,
   ef_sum_ct, ef_product_ct,
   ef_add_ct, ef_subtract_ct, ef_scalar_mult_ct, ef_dot_prod_ct, ef_norm_ct,
   ef_if, ef_ensemble, ef_member_of);
END_TYPE;
TYPE open_closed = ENUMERATION OF
  (open,
   closed);
END_TYPE;
TYPE space_constraint_type = ENUMERATION OF
  (sc_equal,
   sc_subspace,
   sc_member);
END_TYPE;
TYPE repackage_options = ENUMERATION OF
  (ro_nochange,
   ro_wrap_as_tuple,
   ro_unwrap_tuple);
END_TYPE;
TYPE extension_options = ENUMERATION OF
  (eo_none,
   eo_cont,
   eo_cont_right,
   eo_cont_left);
END_TYPE;
TYPE maths_enum_atom = SELECT
  (elementary_space_enumerators,
   ordering_type,
   lower_upper,
   symmetry_type,
   elementary_function_enumerators,
   open_closed,
   space_constraint_type,
   repackage_options,
   extension_options);
END_TYPE;
TYPE dotted_express_identifier = STRING;
WHERE syntax: dotted_identifiers_syntax(SELF);
END_TYPE;
TYPE express_identifier = dotted_express_identifier;
WHERE syntax: dot_count(SELF) = 0;
END_TYPE;
TYPE product_space = SELECT
  (uniform_product_space,
   listed_product_space);
END_TYPE;
TYPE tuple_space = SELECT
  (product_space,
   extended_tuple_space);
END_TYPE;
TYPE maths_space_or_function = SELECT
  (maths_space,
   maths_function);
END_TYPE;
TYPE real_interval = SELECT
  (real_interval_from_min,
   real_interval_to_max,
   finite_real_interval,
   elementary_space);
WHERE
  WR1: NOT ('ELEMENTARY_SPACE' IN stripped_typeof(SELF)) OR
    (SELF\elementary_space.space_id = es_reals);
END_TYPE;
ENTITY quantifier_expression
  ABSTRACT SUPERTYPE
  SUBTYPE OF (multiple_arity_generic_expression);
  variables : LIST [1:?] OF UNIQUE generic_variable;
WHERE
  WR1: SIZEOF (QUERY (vrbl <* variables | NOT (vrbl IN
       SELF\multiple_arity_generic_expression.operands))) = 0;
  WR2: SIZEOF (QUERY (vrbl <* variables | NOT ((schema_prefix +
       'BOUND_VARIABLE_SEMANTICS') IN TYPEOF (vrbl.interpretation.semantics)))) = 0;
END_ENTITY;
ENTITY dependent_variable_definition
  SUBTYPE OF (unary_generic_expression);
  name        : label;
  description : text;
END_ENTITY;
ENTITY bound_variable_semantics
  SUBTYPE OF (variable_semantics);
END_ENTITY;
ENTITY free_variable_semantics
  SUBTYPE OF (variable_semantics);
END_ENTITY;
ENTITY complex_number_literal
  SUBTYPE OF (generic_literal);
  real_part : REAL;
  imag_part : REAL;
END_ENTITY;
ENTITY logical_literal
  SUBTYPE OF (generic_literal);
  lit_value : LOGICAL;
END_ENTITY;
ENTITY binary_literal
  SUBTYPE OF (generic_literal);
  lit_value : BINARY;
END_ENTITY;
ENTITY maths_enum_literal
  SUBTYPE OF (generic_literal);
  lit_value : maths_enum_atom;
END_ENTITY;
ENTITY real_tuple_literal
  SUBTYPE OF (generic_literal);
  lit_value : LIST [1:?] OF REAL;
END_ENTITY;
ENTITY integer_tuple_literal
  SUBTYPE OF (generic_literal);
  lit_value : LIST [1:?] OF INTEGER;
END_ENTITY;
ENTITY atom_based_literal
  SUBTYPE OF (generic_literal);
  lit_value : atom_based_value;
END_ENTITY;
ENTITY maths_tuple_literal
  SUBTYPE OF (generic_literal);
  lit_value : LIST OF maths_value;
END_ENTITY;
ENTITY maths_variable
  SUBTYPE OF (generic_variable);
  values_space : maths_space;
  name         : label;
WHERE
  WR1: expression_is_constant(values_space);
END_ENTITY;
ENTITY maths_real_variable
  SUBTYPE OF (maths_variable, real_numeric_variable);
WHERE
  WR1: subspace_of_es(SELF\maths_variable.values_space,es_reals);
END_ENTITY;
ENTITY maths_integer_variable
  SUBTYPE OF (maths_variable, int_numeric_variable);
WHERE
  WR1: subspace_of_es(SELF\maths_variable.values_space,es_integers);
END_ENTITY;
ENTITY maths_boolean_variable
  SUBTYPE OF (maths_variable, boolean_variable);
WHERE
  WR1: subspace_of_es(SELF\maths_variable.values_space,es_booleans);
END_ENTITY;
ENTITY maths_string_variable
  SUBTYPE OF (maths_variable, string_variable);
WHERE
  WR1: subspace_of_es(SELF\maths_variable.values_space,es_strings);
END_ENTITY;
ENTITY function_application
  SUBTYPE OF (multiple_arity_generic_expression);
  func      : maths_function_select;
  arguments : LIST [1:?] OF maths_expression;
DERIVE
  SELF\multiple_arity_generic_expression.operands : LIST [2:?] OF generic_expression
    := [convert_to_maths_function(func)] + convert_to_operands(arguments);
WHERE
  WR1: function_applicability(func, arguments);
END_ENTITY;
ENTITY maths_space
  ABSTRACT SUPERTYPE OF (ONEOF (elementary_space,
                                finite_integer_interval,
                                integer_interval_from_min,
                                integer_interval_to_max,
                                finite_real_interval,
                                real_interval_from_min,
                                real_interval_to_max,
                                cartesian_complex_number_region,
                                polar_complex_number_region,
                                finite_space,
                                uniform_product_space,
                                listed_product_space,
                                extended_tuple_space,
                                function_space))
  SUBTYPE OF (generic_expression);
END_ENTITY;
ENTITY elementary_space
  SUBTYPE OF (maths_space, generic_literal);
  space_id : elementary_space_enumerators;
END_ENTITY;
ENTITY finite_integer_interval
  SUBTYPE OF (maths_space, generic_literal);
  min  : INTEGER;
  max  : INTEGER;
DERIVE
  size : positive_integer := max - min + 1;
WHERE
  WR1: min <= max;
END_ENTITY;
ENTITY integer_interval_from_min
  SUBTYPE OF (maths_space, generic_literal);
  min : INTEGER;
END_ENTITY;
ENTITY integer_interval_to_max
  SUBTYPE OF (maths_space, generic_literal);
  max : INTEGER;
END_ENTITY;
ENTITY finite_real_interval
  SUBTYPE OF (maths_space, generic_literal);
  min         : REAL;
  min_closure : open_closed;
  max         : REAL;
  max_closure : open_closed;
WHERE
  WR1: min < max;
END_ENTITY;
ENTITY real_interval_from_min
  SUBTYPE OF (maths_space, generic_literal);
  min         : REAL;
  min_closure : open_closed;
END_ENTITY;
ENTITY real_interval_to_max
  SUBTYPE OF (maths_space, generic_literal);
  max         : REAL;
  max_closure : open_closed;
END_ENTITY;
ENTITY cartesian_complex_number_region
  SUBTYPE OF (maths_space, generic_literal);
  real_constraint : real_interval;
  imag_constraint : real_interval;
WHERE
  WR1: min_exists(real_constraint) OR max_exists(real_constraint) OR
       min_exists(imag_constraint) OR max_exists(imag_constraint);
END_ENTITY;
ENTITY polar_complex_number_region
  SUBTYPE OF (maths_space, generic_literal);
  centre               : complex_number_literal;
  distance_constraint  : real_interval;
  direction_constraint : finite_real_interval;
WHERE
  WR1: min_exists(distance_constraint) AND (real_min(distance_constraint) >= 0.0);
  WR2: {-PI <= direction_constraint.min < PI};
  WR3: direction_constraint.max - direction_constraint.min <= 2.0*PI;
  WR4: (direction_constraint.max - direction_constraint.min < 2.0*PI) OR
       (direction_constraint.min_closure = open);
  WR5: (direction_constraint.max - direction_constraint.min < 2.0*PI) OR
       (direction_constraint.max_closure = open) OR
       (direction_constraint.min = -PI);
  WR6: (real_min(distance_constraint) > 0.0) OR max_exists(distance_constraint) OR
       (direction_constraint.max - direction_constraint.min < 2.0*PI) OR
       (direction_constraint.max_closure = open);
END_ENTITY;
ENTITY finite_space
  SUBTYPE OF (maths_space, generic_literal);
  members : SET OF maths_value;
WHERE
  WR1: VALUE_UNIQUE(members);
  WR2: SIZEOF (QUERY (expr <* QUERY (member <* members |
       'ISO13584_GENERIC_EXPRESSIONS_SCHEMA.GENERIC_EXPRESSION' IN TYPEOF (member))
       | NOT expression_is_constant(expr))) = 0;
  WR3: no_cyclic_space_reference(SELF, []);
END_ENTITY;
ENTITY uniform_product_space
  SUBTYPE OF (maths_space, generic_literal);
  base     : maths_space;
  exponent : positive_integer;
WHERE
  WR1: expression_is_constant(base);
  WR2: no_cyclic_space_reference(SELF, []);
  WR3: base <> the_empty_space;
END_ENTITY;
ENTITY listed_product_space
  SUBTYPE OF (maths_space, generic_literal);
  factors : LIST OF maths_space;
WHERE
  WR1: SIZEOF (QUERY (space <* factors |
       NOT (expression_is_constant(space)))) = 0;
  WR2: no_cyclic_space_reference(SELF, []);
  WR3: NOT (the_empty_space IN factors);
END_ENTITY;
ENTITY extended_tuple_space
  SUBTYPE OF (maths_space, generic_literal);
  base     : product_space;
  extender : maths_space;
WHERE
  WR1: expression_is_constant(base) AND
       expression_is_constant(extender);
  WR2: no_cyclic_space_reference(SELF, []);
  WR3: extender <> the_empty_space;
END_ENTITY;
ENTITY function_space
  SUBTYPE OF (maths_space, generic_literal);
  domain_constraint : space_constraint_type;
  domain_argument  : maths_space;
  range_constraint : space_constraint_type;
  range_argument   : maths_space;
WHERE
  WR1: expression_is_constant(domain_argument) AND
       expression_is_constant(range_argument);
  WR2: (domain_argument <> the_empty_space) AND
       (range_argument <> the_empty_space);
  WR3: (domain_constraint <> sc_member) OR NOT
       member_of(the_empty_space,domain_argument);
  WR4: (range_constraint <> sc_member) OR NOT
       member_of(the_empty_space,range_argument);
  WR5: NOT (any_space_satisfies(domain_constraint,domain_argument) AND
       any_space_satisfies(range_constraint,range_argument));
END_ENTITY;
ENTITY maths_function
  ABSTRACT SUPERTYPE OF (ONEOF (finite_function,
                                constant_function,
                                selector_function,
                                elementary_function,
                                restriction_function,
                                repackaging_function,
                                reindexed_array_function,
                                series_composed_function,
                                parallel_composed_function,
                                explicit_table_function,
                                homogeneous_linear_function,
                                general_linear_function,
                                b_spline_basis,
                                b_spline_function,
                                rationalize_function,
                                partial_derivative_function,
                                definite_integral_function,
                                abstracted_expression_function,
                                expression_denoted_function,
                                imported_point_function,
                                imported_curve_function,
                                imported_surface_function,
                                imported_volume_function,
                                application_defined_function))
  SUBTYPE OF (generic_expression);
DERIVE
  domain : tuple_space := derive_function_domain(SELF);
  range : tuple_space := derive_function_range(SELF);
END_ENTITY;
ENTITY finite_function
  SUBTYPE OF (maths_function, generic_literal);
  pairs : SET [1:?] OF LIST [2:2] OF maths_value;
WHERE
  WR1: VALUE_UNIQUE(list_selected_components(pairs, 1));
END_ENTITY;
ENTITY constant_function
  SUBTYPE OF (maths_function, generic_literal);
  sole_output      : maths_value;
  source_of_domain : maths_space_or_function;
WHERE
  WR1: no_cyclic_domain_reference(source_of_domain, [SELF]);
  WR2: expression_is_constant(domain_from(source_of_domain));
END_ENTITY;
ENTITY selector_function
  SUBTYPE OF (maths_function, generic_literal);
  selector : input_selector;
  source_of_domain : maths_space_or_function;
WHERE
  WR1: no_cyclic_domain_reference(source_of_domain, [SELF]);
  WR2: expression_is_constant(domain_from(source_of_domain));
END_ENTITY;
ENTITY elementary_function
  SUBTYPE OF (maths_function, generic_literal);
  func_id : elementary_function_enumerators;
END_ENTITY;
ENTITY restriction_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_space;
END_ENTITY;
ENTITY repackaging_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  input_repack    : repackage_options;
  output_repack   : repackage_options;
  selected_output : nonnegative_integer;
WHERE
  WR1: (input_repack <> ro_wrap_as_tuple) OR
       ((space_dimension(operand.domain) = 1) AND
         ((schema_prefix + 'TUPLE_SPACE') IN TYPEOF (factor1(operand.domain))));
  WR2: (output_repack <> ro_unwrap_tuple) OR
       ((space_dimension(operand.range) = 1) AND
         ((schema_prefix + 'TUPLE_SPACE') IN TYPEOF (factor1(operand.range))));
  WR3: selected_output <= space_dimension( repackage(
       operand.range, output_repack));
END_ENTITY;
ENTITY reindexed_array_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  starting_indices : LIST [1:?] OF INTEGER;
WHERE
  WR1: function_is_array(SELF\unary_generic_expression.operand);
  WR2: SIZEOF(starting_indices) = SIZEOF(shape_of_array(
       SELF\unary_generic_expression.operand));
END_ENTITY;
ENTITY series_composed_function
  SUBTYPE OF (maths_function, multiple_arity_generic_expression);
  SELF\multiple_arity_generic_expression.operands : LIST [2:?] of maths_function;
WHERE
  WR1: composable_sequence(SELF\multiple_arity_generic_expression.operands);
END_ENTITY;
ENTITY parallel_composed_function
  SUBTYPE OF (maths_function, multiple_arity_generic_expression);
  source_of_domain  : maths_space_or_function;
  prep_functions : LIST [1:?] OF maths_function;
  final_function : maths_function_select;
DERIVE
  SELF\multiple_arity_generic_expression.operands : LIST [2:?] of generic_expression
    := convert_to_operands_prcmfn(source_of_domain, prep_functions, final_function);
WHERE
  WR1: no_cyclic_domain_reference(source_of_domain, [SELF]);
  WR2: expression_is_constant(domain_from(source_of_domain));
  WR3: parallel_composed_function_domain_check(domain_from(source_of_domain),
       prep_functions);
  WR4: parallel_composed_function_composability_check(prep_functions, final_function);
END_ENTITY;
ENTITY explicit_table_function
  ABSTRACT SUPERTYPE OF (ONEOF (listed_real_data,
                                listed_integer_data,
                                listed_logical_data,
                                listed_string_data,
                                listed_complex_number_data,
                                listed_data,
                                externally_listed_data,
                                linearized_table_function,
                                basic_sparse_matrix))
  SUBTYPE OF (maths_function);
  index_base : zero_or_one;
  shape      : LIST [1:?] OF positive_integer;
END_ENTITY;
ENTITY listed_real_data
  SUBTYPE OF (explicit_table_function, generic_literal);
  values : LIST [1:?] OF REAL;
DERIVE
  self\explicit_table_function.shape : LIST [1:?] OF positive_integer :=
    [SIZEOF (values)];
END_ENTITY;
ENTITY listed_integer_data
  SUBTYPE OF (explicit_table_function, generic_literal);
  values : LIST [1:?] OF INTEGER;
DERIVE
  self\explicit_table_function.shape : LIST [1:?] OF positive_integer :=
    [SIZEOF (values)];
END_ENTITY;
ENTITY listed_logical_data
  SUBTYPE OF(explicit_table_function, generic_literal);
  values : LIST [1:?] OF LOGICAL;
DERIVE
  self\explicit_table_function.shape : LIST [1:?] OF positive_integer :=
    [SIZEOF (values)];
END_ENTITY;
ENTITY listed_string_data
  SUBTYPE OF (explicit_table_function, generic_literal);
  values : LIST [1:?] OF STRING;
DERIVE
  self\explicit_table_function.shape : LIST [1:?] OF positive_integer :=
    [SIZEOF (values)];
END_ENTITY;
ENTITY listed_complex_number_data
  SUBTYPE OF (explicit_table_function, generic_literal);
  values : LIST [2:?] OF REAL;
DERIVE
  self\explicit_table_function.shape : LIST [1:?] OF positive_integer :=
    [SIZEOF (values)/2];
WHERE
  WR1: NOT ODD (SIZEOF (values));
END_ENTITY;
ENTITY listed_data
  SUBTYPE OF (explicit_table_function, generic_literal);
  values      : LIST [1:?] OF maths_value;
  value_range : maths_space;
DERIVE
  SELF\explicit_table_function.shape : LIST [1:?] OF positive_integer :=
    [SIZEOF (values)];
WHERE
  WR1: expression_is_constant(value_range);
  WR2: SIZEOF (QUERY (val <* values | NOT (member_of( val, value_range)))) = 0;
END_ENTITY;
ENTITY externally_listed_data
  SUBTYPE OF (explicit_table_function, generic_literal, externally_defined_item);
  value_range : maths_space;
WHERE
  WR1: expression_is_constant(value_range);
END_ENTITY;
ENTITY linearized_table_function
  SUPERTYPE OF (ONEOF (standard_table_function,
                       regular_table_function,
                       triangular_matrix,
                       symmetric_matrix,
                       banded_matrix))
  SUBTYPE OF (explicit_table_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  first  : integer;
DERIVE
  source : maths_function := SELF\unary_generic_expression.operand;
WHERE
  WR1: function_is_1d_array(source);
  WR2: member_of(first, source.domain);
END_ENTITY;
ENTITY standard_table_function
  SUBTYPE OF (linearized_table_function);
  order : ordering_type;
WHERE
  WR1: extremal_position_check(SELF);
END_ENTITY;
ENTITY regular_table_function
  SUBTYPE OF (linearized_table_function);
  increments : LIST [1:?] OF INTEGER;
WHERE
  WR1: SIZEOF (increments) = SIZEOF (self\explicit_table_function.shape);
  WR2: extremal_position_check(self);
END_ENTITY;
ENTITY triangular_matrix
  SUBTYPE OF (linearized_table_function);
  default_entry : maths_value;
  lo_up         : lower_upper;
  order         : ordering_type;
WHERE
  WR1: SIZEOF (SELF\explicit_table_function.shape) = 2;
  WR2: member_of(default_entry, SELF\maths_function.range);
END_ENTITY;
ENTITY strict_triangular_matrix
  SUBTYPE OF (triangular_matrix);
  main_diagonal_value : maths_value;
END_ENTITY;
ENTITY symmetric_matrix
  SUBTYPE OF (linearized_table_function);
  symmetry : symmetry_type;
  triangle : lower_upper;
  order    : ordering_type;
WHERE
  WR1: SIZEOF (SELF\explicit_table_function.shape) = 2;
  WR2: SELF\explicit_table_function.shape[1] =
       SELF\explicit_table_function.shape[2];
  WR3: NOT (symmetry = skew) OR (
       (space_dimension(SELF\linearized_table_function.source.range) = 1) AND
        subspace_of_es(factor1(SELF\linearized_table_function.source.range),
        es_numbers));
  WR4: NOT ((symmetry = hermitian) OR (symmetry = skew_hermitian)) OR (
       (space_dimension(SELF\linearized_table_function.source.range) = 1) AND
        subspace_of_es(factor1(SELF\linearized_table_function.source.range),
        es_complex_numbers));
END_ENTITY;
ENTITY symmetric_banded_matrix
  SUBTYPE OF (symmetric_matrix);
  default_entry : maths_value;
  above         : nonnegative_integer;
WHERE
  WR1: member_of(default_entry,
       factor1(SELF\linearized_table_function.source.range));
END_ENTITY;
ENTITY banded_matrix
  SUBTYPE OF (linearized_table_function);
  default_entry : maths_value;
  below         : integer;
  above         : integer;
  order         : ordering_type;
WHERE
  WR1: SIZEOF (self\explicit_table_function.shape) = 2;
  WR2: -below <= above;
  WR3: member_of(default_entry,
       factor1(SELF\linearized_table_function.source.range));
END_ENTITY;
ENTITY basic_sparse_matrix
  SUBTYPE OF (explicit_table_function, multiple_arity_generic_expression);
  SELF\multiple_arity_generic_expression.operands : LIST [3:3] OF maths_function;
  default_entry : maths_value;
  order : ordering_type;
DERIVE
  index : maths_function := SELF\multiple_arity_generic_expression.operands[1];
  loc   : maths_function := SELF\multiple_arity_generic_expression.operands[2];
  val   : maths_function := SELF\multiple_arity_generic_expression.operands[3];
WHERE
  WR1: function_is_1d_table(index);
  WR2: function_is_1d_table(loc);
  WR3: function_is_1d_table(val);
  WR4: check_sparse_index_domain(index.domain, index_base, shape, order);
  WR5: check_sparse_index_to_loc(index.range, loc.domain);
  WR6: loc.domain = val.domain;
  WR7: check_sparse_loc_range(loc.range, index_base, shape, order);
  WR8: member_of(default_entry, val.range);
END_ENTITY;
ENTITY homogeneous_linear_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  sum_index : one_or_two;
DERIVE
  mat       : maths_function := SELF\unary_generic_expression.operand;
WHERE
  WR1: function_is_2d_table(mat);
  WR2: (space_dimension(mat.range) = 1) AND
       subspace_of_es(factor1(mat.range),es_numbers);
END_ENTITY;
ENTITY general_linear_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  sum_index : one_or_two;
DERIVE
  mat       : maths_function := SELF\unary_generic_expression.operand;
WHERE
  WR1: function_is_2d_table(mat);
  WR2: (space_dimension(mat.range) = 1) AND
       subspace_of_es(factor1(mat.range),es_numbers);
END_ENTITY;
ENTITY b_spline_basis
  SUBTYPE OF (maths_function, generic_literal);
  degree         : nonnegative_integer;
  repeated_knots : LIST [2:?] OF REAL;
DERIVE
  order          : positive_integer := degree + 1;
  num_basis      : positive_integer := SIZEOF (repeated_knots) - order;
WHERE
  WR1: num_basis >= order;
  WR2: nondecreasing(repeated_knots);
  WR3: repeated_knots[order] < repeated_knots[num_basis+1];
END_ENTITY;
ENTITY b_spline_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  basis : LIST [1:?] OF b_spline_basis;
DERIVE
  coef  : maths_function := SELF\unary_generic_expression.operand;
WHERE
  WR1: function_is_table(coef);
  WR2: (space_dimension(coef.range) = 1) AND
       (number_superspace_of(factor1(coef.range)) = the_reals);
  WR3: SIZEOF (basis) <=
       SIZEOF (shape_of_array(coef));
  WR4: compare_basis_and_coef(basis, coef);
END_ENTITY;
ENTITY rationalize_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
DERIVE
  fun : maths_function := SELF\unary_generic_expression.operand;
WHERE
  WR1: (space_dimension(fun.domain) = 1) AND (space_dimension(fun.range) = 1);
  WR2: number_tuple_subspace_check(factor1(fun.range));
  WR3: space_dimension(factor1(fun.range)) > 1;
END_ENTITY;
ENTITY partial_derivative_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  d_variables : LIST [1:?] OF input_selector;
  extension : extension_options;
DERIVE
  derivand : maths_function := SELF\unary_generic_expression.operand;
WHERE
  WR1: space_is_continuum (derivand.range);
  WR2: partial_derivative_check (derivand.domain, d_variables);
END_ENTITY;
ENTITY partial_derivative_expression
  SUBTYPE OF (unary_generic_expression);
  d_variables : LIST [1:?] OF maths_variable;
  extension : extension_options;
DERIVE
  derivand : generic_expression := SELF\unary_generic_expression.operand;
WHERE
  WR1: has_values_space (derivand);
  WR2: space_is_continuum (values_space_of (derivand));
  WR3: SIZEOF (QUERY (vbl <* d_variables | (NOT subspace_of (values_space_of (vbl),
    the_reals)) AND (NOT subspace_of (values_space_of (vbl), the_complex_numbers))
    )) = 0;
END_ENTITY;
ENTITY definite_integral_function
  SUBTYPE OF (maths_function, unary_generic_expression);
  SELF\unary_generic_expression.operand : maths_function;
  variable_of_integration : input_selector;
  lower_limit_neg_infinity : BOOLEAN;
  upper_limit_pos_infinity : BOOLEAN;
DERIVE
  integrand : maths_function := SELF\unary_generic_expression.operand;
WHERE
  WR1: space_is_continuum (integrand.range);
  WR2: definite_integral_check (integrand.domain, variable_of_integration,
    lower_limit_neg_infinity, upper_limit_pos_infinity);
END_ENTITY;
ENTITY definite_integral_expression
  SUBTYPE OF (quantifier_expression);
  lower_limit_neg_infinity : BOOLEAN;
  upper_limit_pos_infinity : BOOLEAN;
DERIVE
  integrand : generic_expression
    := SELF\multiple_arity_generic_expression.operands[1];
  variable_of_integration : maths_variable
    := SELF\multiple_arity_generic_expression.operands[2];
  SELF\quantifier_expression.variables : LIST [1:1] OF UNIQUE generic_variable
    := [variable_of_integration];
WHERE
  WR1: has_values_space (integrand);
  WR2: space_is_continuum (values_space_of (integrand));
  WR3: definite_integral_expr_check (SELF\multiple_arity_generic_expression.operands,
    lower_limit_neg_infinity, upper_limit_pos_infinity);
END_ENTITY;
ENTITY abstracted_expression_function
  SUBTYPE OF (maths_function, quantifier_expression);
DERIVE
  SELF\quantifier_expression.variables : LIST [1:?] OF UNIQUE generic_variable :=
    remove_first(SELF\multiple_arity_generic_expression.operands);
  expr : generic_expression := SELF\multiple_arity_generic_expression.operands[1];
WHERE
  WR1: SIZEOF (QUERY ( operand <*
       SELF\multiple_arity_generic_expression.operands | NOT (
       has_values_space( operand)))) = 0;
END_ENTITY;
ENTITY expression_denoted_function
  SUBTYPE OF (maths_function, unary_generic_expression);
DERIVE
  expr : generic_expression := SELF\unary_generic_expression.operand;
WHERE
  WR1: (schema_prefix + 'FUNCTION_SPACE') IN TYPEOF (values_space_of(expr));
END_ENTITY;
ENTITY imported_point_function
  SUBTYPE OF (maths_function, generic_literal);
  geometry : point;
END_ENTITY;
ENTITY imported_curve_function
  SUBTYPE OF (maths_function, generic_literal);
  geometry          : curve;
  parametric_domain : tuple_space;
WHERE
  WR1: expression_is_constant(parametric_domain);
END_ENTITY;
ENTITY imported_surface_function
  SUBTYPE OF (maths_function, generic_literal);
  geometry          : surface;
  parametric_domain : tuple_space;
WHERE
  WR1: expression_is_constant(parametric_domain);
END_ENTITY;
ENTITY imported_volume_function
  SUBTYPE OF (maths_function, generic_literal);
  geometry          : volume;
  parametric_domain : tuple_space;
WHERE
  WR1: expression_is_constant(parametric_domain);
END_ENTITY;
ENTITY application_defined_function
  SUBTYPE OF (maths_function);
  explicit_domain : tuple_space;
  explicit_range  : tuple_space;
  parameters      : LIST OF maths_value;
WHERE
  WR1: expression_is_constant(explicit_domain);
  WR2: expression_is_constant(explicit_range);
END_ENTITY;
ENTITY mathematical_description;
  described  : maths_expression;
  describing : STRING;
  encoding   : label;
END_ENTITY;
FUNCTION all_members_of_es(sv : SET OF maths_value;
                           es : elementary_space_enumerators) : LOGICAL;
  CONSTANT
    base_types : SET OF STRING := ['NUMBER','COMPLEX_NUMBER_LITERAL','REAL',
      'INTEGER','LOGICAL','BOOLEAN','STRING','BINARY','MATHS_SPACE',
      'MATHS_FUNCTION','LIST','ELEMENTARY_SPACE_ENUMERATORS','ORDERING_TYPE',
      'LOWER_UPPER','SYMMETRY_TYPE','ELEMENTARY_FUNCTION_ENUMERATORS',
      'OPEN_CLOSED','SPACE_CONSTRAINT_TYPE','REPACKAGE_OPTIONS',
      'EXTENSION_OPTIONS'];
  END_CONSTANT;
  LOCAL
    v : maths_value;
    key_type : STRING := '';
    types : SET OF STRING;
    ge : generic_expression;
    cum : LOGICAL := TRUE;
    vspc : maths_space;
  END_LOCAL;
  IF NOT EXISTS (sv) OR NOT EXISTS (es) THEN  RETURN (FALSE);  END_IF;
  CASE es OF
  es_numbers :         key_type := 'NUMBER';
  es_complex_numbers : key_type := 'COMPLEX_NUMBER_LITERAL';
  es_reals :           key_type := 'REAL';
  es_integers :        key_type := 'INTEGER';
  es_logicals :        key_type := 'LOGICAL';
  es_booleans :        key_type := 'BOOLEAN';
  es_strings :         key_type := 'STRING';
  es_binarys :         key_type := 'BINARY';
  es_maths_spaces :    key_type := 'MATHS_SPACE';
  es_maths_functions : key_type := 'MATHS_FUNCTION';
  es_generics :        RETURN (TRUE);
  END_CASE;
  REPEAT i := 1 TO SIZEOF (sv);
    IF NOT EXISTS (sv[i]) THEN  RETURN (FALSE);  END_IF;
    v := simplify_maths_value(sv[i]);
    types := stripped_typeof(v);
    IF key_type IN types THEN  SKIP;  END_IF;
    IF (es = es_numbers) AND ('COMPLEX_NUMBER_LITERAL' IN types) THEN  SKIP;  END_IF;
    IF SIZEOF (base_types * types) > 0 THEN  RETURN (FALSE);  END_IF;
    -- Must be a generic_expression which doesn't simplify and which is not a
    -- complex_number_literal, maths_space, or maths_function.
    ge := v;
    IF has_values_space(ge) THEN
      vspc := values_space_of(ge);
      IF NOT subspace_of_es(vspc,es) THEN
        IF NOT compatible_spaces(vspc,make_elementary_space(es)) THEN
          RETURN (FALSE);
        END_IF;
        cum := UNKNOWN;
      END_IF;
    ELSE
      cum := UNKNOWN;
    END_IF;
    IF cum = FALSE THEN  RETURN (FALSE);  END_IF;
  END_REPEAT;
  RETURN (cum);
END_FUNCTION;  -- all_members_of_es
FUNCTION any_space_satisfies(sc  : space_constraint_type;
                             spc : maths_space) : BOOLEAN;
  LOCAL
    spc_id : elementary_space_enumerators;
  END_LOCAL;
  IF (sc = sc_equal) OR NOT ('ELEMENTARY_SPACE' IN stripped_typeof(spc)) THEN
    RETURN (FALSE);
  END_IF;
  spc_id := spc\elementary_space.space_id;
  IF sc = sc_subspace THEN
    RETURN (bool(spc_id = es_generics));
  END_IF;
  IF sc = sc_member THEN
    RETURN (bool((spc_id = es_generics) OR (spc_id = es_maths_spaces)));
  END_IF;
  -- Should be unreachable.
  RETURN (?);
END_FUNCTION;  -- any_space_satisfies
FUNCTION assoc_product_space(ts1, ts2 : tuple_space) : tuple_space;
  LOCAL
    types1 : SET OF STRING := stripped_typeof (ts1);
    types2 : SET OF STRING := stripped_typeof (ts2);
    up1, up2 : uniform_product_space := make_uniform_product_space(the_reals,1);
    lp1, lp2, lps : listed_product_space := the_zero_tuple_space;
    et1, et2, ets : extended_tuple_space := the_tuples;
    use_up1, use_up2, use_lp1, use_lp2 : BOOLEAN;
    factors : LIST OF maths_space := [];
    tspace : tuple_space;
  END_LOCAL;
  -- Identify type of first operand
  IF 'UNIFORM_PRODUCT_SPACE' IN types1 THEN
    up1 := ts1;  use_up1 := true;  use_lp1 := false;
  ELSE
    IF 'LISTED_PRODUCT_SPACE' IN types1 THEN
      lp1 := ts1;  use_up1 := false;  use_lp1 := true;
    ELSE
      IF NOT ('EXTENDED_TUPLE_SPACE' IN types1) THEN
        -- Unreachable when this function was written.
        RETURN (?);
      END_IF;
      et1 := ts1;  use_up1 := false;  use_lp1 := false;
    END_IF;
  END_IF;
  -- Identify type of second operand
  IF 'UNIFORM_PRODUCT_SPACE' IN types2 THEN
    up2 := ts2;  use_up2 := true;  use_lp2 := false;
  ELSE
    IF 'LISTED_PRODUCT_SPACE' IN types2 THEN
      lp2 := ts2;  use_up2 := false;  use_lp2 := true;
    ELSE
      IF NOT ('EXTENDED_TUPLE_SPACE' IN types2) THEN
        -- Unreachable when this function was written.
        RETURN (?);
      END_IF;
      et2 := ts2;  use_up2 := false;  use_lp2 := false;
    END_IF;
  END_IF;
  -- Construction for each combination of cases
  IF use_up1 THEN
    IF use_up2 THEN
      IF up1.base = up2.base THEN
        tspace := make_uniform_product_space(up1.base, up1.exponent + up2.exponent);
      ELSE
        factors := [up1.base : up1.exponent, up2.base : up2.exponent];
        tspace := make_listed_product_space(factors);
      END_IF;
    ELSE
      IF use_lp2 THEN
        -- Avoid compiler confusion by breaking into two lines.
        factors := [up1.base : up1.exponent];
        factors := factors + lp2.factors;
        tspace := make_listed_product_space(factors);
      ELSE
        tspace := assoc_product_space(up1, et2.base);
        tspace := make_extended_tuple_space(tspace, et2.extender);
      END_IF;
    END_IF;
  ELSE
    IF use_lp1 THEN
      IF use_up2 THEN
        -- Avoid compiler confusion by breaking into two lines.
        factors := [up2.base : up2.exponent];
        factors := lp1.factors + factors;
        tspace := make_listed_product_space(factors);
      ELSE
        IF use_lp2 THEN
          tspace := make_listed_product_space(lp1.factors + lp2.factors);
        ELSE
          tspace := assoc_product_space(lp1, et2.base);
          tspace := make_extended_tuple_space(tspace, et2.extender);
        END_IF;
      END_IF;
    ELSE
      IF use_up2 THEN
        IF et1.extender = up2.base THEN
          tspace := assoc_product_space(et1.base, up2);
          tspace := make_extended_tuple_space(tspace, et1.extender);
        ELSE
          -- No subtype is available to represent this cartesian product.
          RETURN (?);
        END_IF;
      ELSE
        IF use_lp2 THEN
          factors := lp2.factors;
          REPEAT i := 1 TO SIZEOF (factors);
            IF et1.extender <> factors[i] THEN
              -- No subtype available to represent this cartesian product.
              RETURN (?);
            END_IF;
          END_REPEAT;
          tspace := assoc_product_space(et1.base, lp2);
          tspace := make_extended_tuple_space(tspace, et1.extender);
        ELSE
          IF et1.extender = et2.extender THEN
            -- Next line may assign indeterminate (?) to tspace.
            tspace := assoc_product_space(et1, et2.base);
          ELSE
            -- No subtype available to represent this cartesian product.
            RETURN (?);
          END_IF;
        END_IF;
      END_IF;
    END_IF;
  END_IF;
  RETURN (tspace);
END_FUNCTION;  -- assoc_product_space
FUNCTION atan2(y, x : REAL) : REAL;
  LOCAL
    r : REAL;
  END_LOCAL;
  IF (y = 0.0) AND (x = 0.0) THEN  RETURN (?);  END_IF;
  r := atan(y,x);
  IF x < 0.0 THEN
    IF y < 0.0 THEN  r := r - PI;
    ELSE             r := r + PI;  END_IF;
  END_IF;
  RETURN (r);
END_FUNCTION;  -- atan2
FUNCTION bool(lgcl: LOGICAL) : BOOLEAN;
  IF NOT EXISTS (lgcl) THEN  RETURN (FALSE);  END_IF;
  IF lgcl <> TRUE      THEN  RETURN (FALSE);  END_IF;
  RETURN (TRUE);
END_FUNCTION;  -- bool
FUNCTION check_sparse_index_domain(idxdom : tuple_space;
                                   base   : zero_or_one;
                                   shape  : LIST [1:?] OF positive_integer;
                                   order  : ordering_type) : BOOLEAN;
  LOCAL
    mthspc : maths_space;
    interval : finite_integer_interval;
    i : INTEGER;
  END_LOCAL;
  mthspc := factor1(idxdom);
  -- A consequence of WR1 of basic_sparse_matrix is that here we need only
  -- consider the case that mthspc is a finite integer interval and is the only
  -- factor space of idxdom.
  interval := mthspc;
  IF order = by_rows THEN  i := 1;  ELSE  i := 2;  END_IF;
  RETURN (bool((interval.min <= base) AND (interval.max >= base + shape[i])));
  -- The index function is evaluated at (base+shape[i]) when determining the
  -- upper search bound for entries of the last row or column, respectively.
END_FUNCTION;  -- check_sparse_index_domain;
FUNCTION check_sparse_loc_range(locrng : tuple_space;
                                base   : zero_or_one;
                                shape  : LIST [1:?] OF positive_integer;
                                order  : ordering_type) : BOOLEAN;
  LOCAL
    mthspc : maths_space;
    interval : finite_integer_interval;
    i : INTEGER;
  END_LOCAL;
  IF space_dimension(locrng) <> 1 THEN  RETURN (FALSE);  END_IF;
  mthspc := factor1(locrng);
  IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (mthspc)) THEN
    RETURN (FALSE);
  END_IF;
  interval := mthspc;
  IF order = by_rows THEN  i := 2;  ELSE  i := 1;  END_IF;
  RETURN (bool((interval.min >= base) AND (interval.max <= base + shape[i] - 1)));
END_FUNCTION;  -- check_sparse_loc_range;
FUNCTION check_sparse_index_to_loc(index_range, loc_domain : tuple_space) : BOOLEAN;
  LOCAL
    temp : maths_space;
    idx_rng_itvl, loc_dmn_itvl : finite_integer_interval;
  END_LOCAL;
  temp := factor1 (index_range);
  IF (schema_prefix + 'TUPLE_SPACE') IN TYPEOF (temp) THEN
    temp := factor1 (temp);
  END_IF;
  IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp)) THEN
    RETURN (FALSE);
  END_IF;
  idx_rng_itvl := temp;
  temp := factor1 (loc_domain);
  IF (schema_prefix + 'TUPLE_SPACE') IN TYPEOF (temp) THEN
    temp := factor1 (temp);
  END_IF;
  IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp)) THEN
    RETURN (FALSE);
  END_IF;
  loc_dmn_itvl := temp;
  RETURN (bool((loc_dmn_itvl.min <= idx_rng_itvl.min) AND
    (idx_rng_itvl.max <= loc_dmn_itvl.max+1)));
END_FUNCTION;  -- check_sparse_index_to_loc
FUNCTION compare_basis_and_coef(basis : LIST [1:?] OF b_spline_basis;
                                coef  : maths_function) : BOOLEAN;
  LOCAL
    shape : LIST OF positive_integer;
  END_LOCAL;
  IF NOT EXISTS (basis) OR NOT EXISTS (coef) THEN  RETURN (FALSE);  END_IF;
  shape := shape_of_array(coef);
  IF NOT EXISTS (shape) THEN  RETURN (FALSE);  END_IF;
  IF SIZEOF (shape) < SIZEOF (basis) THEN  RETURN (FALSE);  END_IF;
  REPEAT i := 1 TO SIZEOF (basis);
    IF (basis[i].num_basis = shape[i]) <> TRUE THEN  RETURN (FALSE);  END_IF;
  END_REPEAT;
  RETURN (TRUE);
END_FUNCTION;  -- compare_basis_and_coef
FUNCTION compare_list_and_value(lv : LIST OF GENERIC:G;
                                op : elementary_function_enumerators;
                                v  : GENERIC:G) : BOOLEAN;
  IF NOT EXISTS (lv) OR NOT EXISTS (op) OR NOT EXISTS (v) THEN
    RETURN (FALSE);
  END_IF;
  REPEAT i := 1 TO SIZEOF (lv);
    IF NOT compare_values(lv[i], op, v) THEN
      RETURN (FALSE);
    END_IF;
  END_REPEAT;
  RETURN (TRUE);
END_FUNCTION;  -- compare_list_and_value
FUNCTION compare_values(v1 : GENERIC:G;
                        op : elementary_function_enumerators;
                        v2 : GENERIC:G) : BOOLEAN;
  -- This algorithm assumes a comparison between "incompatible" types will
  -- produce the indeterminate value (or UNKNOWN?).
  LOCAL
    logl : LOGICAL := UNKNOWN;
  END_LOCAL;
  IF NOT EXISTS (v1) OR NOT EXISTS (op) OR NOT EXISTS (v2) THEN
    RETURN (FALSE);
  END_IF;
  CASE op OF
  ef_eq_i : logl := (v1 = v2);
  ef_ne_i : logl := (v1 <> v2);
  ef_gt_i : logl := (v1 > v2);
  ef_lt_i : logl := (v1 < v2);
  ef_ge_i : logl := (v1 >= v2);
  ef_le_i : logl := (v1 <= v2);
  END_CASE;
  IF EXISTS (logl) THEN
    IF logl = TRUE THEN  RETURN (TRUE);  END_IF;
  END_IF;
  RETURN (FALSE);
END_FUNCTION;  -- compare_values
FUNCTION compatible_complex_number_regions(sp1, sp2 : maths_space) : BOOLEAN;
  LOCAL
    typenames : SET OF string := stripped_typeof (sp1);
    crgn1, crgn2 : cartesian_complex_number_region;
    prgn1, prgn2, prgn1c2, prgn2c1 : polar_complex_number_region;
    sp1_is_crgn, sp2_is_crgn : BOOLEAN;
  END_LOCAL;
  IF 'CARTESIAN_COMPLEX_NUMBER_REGION' IN typenames THEN
    sp1_is_crgn := TRUE;
    crgn1 := sp1;
  ELSE
    IF 'POLAR_COMPLEX_NUMBER_REGION' IN typenames THEN
      sp1_is_crgn := FALSE;
      prgn1 := sp1;
    ELSE
      -- Improper usage: Default response is to assume compatibility.
      RETURN (TRUE);
    END_IF;
  END_IF;
  typenames := stripped_typeof (sp2);
  IF 'CARTESIAN_COMPLEX_NUMBER_REGION' IN typenames THEN
    sp2_is_crgn := TRUE;
    crgn2 := sp2;
  ELSE
    IF 'POLAR_COMPLEX_NUMBER_REGION' IN typenames THEN
      sp2_is_crgn := FALSE;
      prgn2 := sp2;
    ELSE
      -- Improper usage: Default response is to assume compatibility.
      RETURN (TRUE);
    END_IF;
  END_IF;
  IF sp1_is_crgn AND sp2_is_crgn THEN
    -- two cartesian regions
    RETURN (compatible_intervals(crgn1.real_constraint, crgn2.real_constraint)
      AND compatible_intervals(crgn1.imag_constraint, crgn2.imag_constraint));
  END_IF;
  IF NOT sp1_is_crgn AND NOT sp2_is_crgn AND
    (prgn1.centre.real_part = prgn2.centre.real_part) AND
    (prgn1.centre.imag_part = prgn2.centre.imag_part) THEN
    -- two polar regions with common centre
    IF NOT compatible_intervals(prgn1.distance_constraint,
      prgn2.distance_constraint) THEN
      RETURN (FALSE);
    END_IF;
    IF compatible_intervals(prgn1.direction_constraint,
      prgn2.direction_constraint) THEN
      RETURN (TRUE);
    END_IF;
    -- Deal with direction ambiguity by 2 pi.
    IF (prgn1.direction_constraint.max > PI) AND (prgn2.direction_constraint.max < PI)
      THEN
      RETURN (compatible_intervals(prgn2.direction_constraint,
        make_finite_real_interval(-PI,open,prgn1.direction_constraint.max-2.0*PI,
        prgn1.direction_constraint.max_closure)));
    END_IF;
    IF (prgn2.direction_constraint.max > PI) AND (prgn1.direction_constraint.max < PI)
      THEN
      RETURN (compatible_intervals(prgn1.direction_constraint,
        make_finite_real_interval(-PI,open,prgn2.direction_constraint.max-2.0*PI,
        prgn2.direction_constraint.max_closure)));
    END_IF;
    RETURN (FALSE);
  END_IF;
  -- Make do with imperfect tests for remaining cases.
  IF sp1_is_crgn AND NOT sp2_is_crgn THEN
    crgn2 := enclose_pregion_in_cregion(prgn2);
    prgn1 := enclose_cregion_in_pregion(crgn1,prgn2.centre);
    RETURN (compatible_complex_number_regions(crgn1,crgn2)
      AND compatible_complex_number_regions(prgn1,prgn2));
  END_IF;
  IF NOT sp1_is_crgn AND sp2_is_crgn THEN
    crgn1 := enclose_pregion_in_cregion(prgn1);
    prgn2 := enclose_cregion_in_pregion(crgn2,prgn1.centre);
    RETURN (compatible_complex_number_regions(crgn1,crgn2)
      AND compatible_complex_number_regions(prgn1,prgn2));
  END_IF;
  -- Two polar regions with different centres
  prgn1c2 := enclose_pregion_in_pregion(prgn1,prgn2.centre);
  prgn2c1 := enclose_pregion_in_pregion(prgn2,prgn1.centre);
  RETURN (compatible_complex_number_regions(prgn1,prgn2c1)
    AND compatible_complex_number_regions(prgn1c2,prgn2));
END_FUNCTION;  -- compatible_complex_number_regions
FUNCTION compatible_es_values(esval1, esval2 : elementary_space_enumerators) : BOOLEAN;
  LOCAL
    esval1_is_numeric, esval2_is_numeric : LOGICAL;
  END_LOCAL;
  IF (esval1 = esval2) OR (esval1 = es_generics) OR (esval2 = es_generics) THEN
    RETURN (TRUE);
  END_IF;
  esval1_is_numeric := (esval1 >= es_numbers) AND (esval1 <= es_integers);
  esval2_is_numeric := (esval2 >= es_numbers) AND (esval2 <= es_integers);
  IF (esval1_is_numeric AND (esval2 = es_numbers)) OR
    (esval2_is_numeric AND (esval1 = es_numbers)) THEN
    RETURN (TRUE);
  END_IF;
  IF esval1_is_numeric XOR esval2_is_numeric THEN
    RETURN (FALSE);
  END_IF;
  IF ((esval1 = es_logicals) AND (esval2 = es_booleans)) OR
    ((esval1 = es_booleans) AND (esval2 = es_logicals)) THEN
    RETURN (TRUE);
  END_IF;
  -- All other cases are incompatible
  RETURN (FALSE);
END_FUNCTION;  -- compatible_es_values
FUNCTION compatible_intervals(sp1, sp2 : maths_space) : BOOLEAN;
  LOCAL
    amin, amax : REAL;
  END_LOCAL;
  IF min_exists(sp1) AND max_exists(sp2) THEN
    amin := real_min(sp1);  amax := real_max(sp2);
    IF amin > amax THEN  RETURN (FALSE);  END_IF;
    IF amin = amax THEN
      RETURN (min_included(sp1) AND max_included(sp2));
    END_IF;
  END_IF;
  IF min_exists(sp2) AND max_exists(sp1) THEN
    amin := real_min(sp2);  amax := real_max(sp1);
    IF amin > amax THEN  RETURN (FALSE);  END_IF;
    IF amin = amax THEN
      RETURN (min_included(sp2) AND max_included(sp1));
    END_IF;
  END_IF;
  RETURN (TRUE);
END_FUNCTION;  -- compatible_intervals
FUNCTION compatible_spaces(sp1, sp2 : maths_space) : BOOLEAN;
  LOCAL
    types1 : SET OF STRING := stripped_typeof (sp1);
    types2 : SET OF STRING := stripped_typeof (sp2);
    lgcl : LOGICAL := UNKNOWN;
    m, n : INTEGER;
    s1, s2 : maths_space;
  END_LOCAL;
  IF 'FINITE_SPACE' IN types1 THEN
    REPEAT i := 1 TO SIZEOF (sp1\finite_space.members);
      lgcl := member_of(sp1\finite_space.members[i], sp2);
      IF lgcl <> FALSE THEN
        RETURN (TRUE);
      END_IF;
    END_REPEAT;
    RETURN (FALSE);
  END_IF;
  IF 'FINITE_SPACE' IN types2 THEN
    REPEAT i := 1 TO SIZEOF (sp2\finite_space.members);
      lgcl := member_of(sp2\finite_space.members[i], sp1);
      IF lgcl <> FALSE THEN
        RETURN (TRUE);
      END_IF;
    END_REPEAT;
    RETURN (FALSE);
  END_IF;
  IF 'ELEMENTARY_SPACE' IN types1 THEN
    IF sp1\elementary_space.space_id = es_generics THEN
      RETURN (TRUE);
    END_IF;
    IF 'ELEMENTARY_SPACE' IN types2 THEN
      RETURN (compatible_es_values(sp1\elementary_space.space_id,
        sp2\elementary_space.space_id));
    END_IF;
    IF ('FINITE_INTEGER_INTERVAL' IN types2) OR
      ('INTEGER_INTERVAL_FROM_MIN' IN types2) OR
      ('INTEGER_INTERVAL_TO_MAX' IN types2) THEN
      RETURN (compatible_es_values(sp1\elementary_space.space_id, es_integers));
    END_IF;
    IF ('FINITE_REAL_INTERVAL' IN types2) OR
      ('REAL_INTERVAL_FROM_MIN' IN types2) OR
      ('REAL_INTERVAL_TO_MAX' IN types2) THEN
      RETURN (compatible_es_values(sp1\elementary_space.space_id, es_reals));
    END_IF;
    IF ('CARTESIAN_COMPLEX_NUMBER_REGION' IN types2) OR
      ('POLAR_COMPLEX_NUMBER_REGION' IN types2) THEN
      RETURN (compatible_es_values(sp1\elementary_space.space_id, es_complex_numbers));
    END_IF;
    IF 'TUPLE_SPACE' IN types2 THEN
      RETURN (FALSE);
    END_IF;
    IF 'FUNCTION_SPACE' IN types2 THEN
      RETURN (bool(sp1\elementary_space.space_id = es_maths_functions));
    END_IF;
    -- Should be unreachable.
    RETURN (TRUE);
  END_IF;
  IF 'ELEMENTARY_SPACE' IN types2 THEN
    IF sp2\elementary_space.space_id = es_generics THEN
      RETURN (TRUE);
    END_IF;
    IF ('FINITE_INTEGER_INTERVAL' IN types1) OR
      ('INTEGER_INTERVAL_FROM_MIN' IN types1) OR
      ('INTEGER_INTERVAL_TO_MAX' IN types1) THEN
      RETURN (compatible_es_values(sp2\elementary_space.space_id, es_integers));
    END_IF;
    IF ('FINITE_REAL_INTERVAL' IN types1) OR
      ('REAL_INTERVAL_FROM_MIN' IN types1) OR
      ('REAL_INTERVAL_TO_MAX' IN types1) THEN
      RETURN (compatible_es_values(sp2\elementary_space.space_id, es_reals));
    END_IF;
    IF ('CARTESIAN_COMPLEX_NUMBER_REGION' IN types1) OR
      ('POLAR_COMPLEX_NUMBER_REGION' IN types1) THEN
      RETURN (compatible_es_values(sp2\elementary_space.space_id, es_complex_numbers));
    END_IF;
    IF 'TUPLE_SPACE' IN types1 THEN
      RETURN (FALSE);
    END_IF;
    IF 'FUNCTION_SPACE' IN types1 THEN
      RETURN (bool(sp2\elementary_space.space_id = es_maths_functions));
    END_IF;
    -- Should be unreachable.
    RETURN (TRUE);
  END_IF;
  IF subspace_of_es(sp1,es_integers) THEN  -- Note that sp1 finite already handled.
    IF subspace_of_es(sp2,es_integers) THEN  -- Note that sp2 finite already handled.
      RETURN (compatible_intervals(sp1,sp2));
    END_IF;
    RETURN (FALSE);
  END_IF;
  IF subspace_of_es(sp2,es_integers) THEN
    RETURN (FALSE);
  END_IF;
  IF subspace_of_es(sp1,es_reals) THEN  -- Note that sp1 finite already handled.
    IF subspace_of_es(sp2,es_reals) THEN  -- Note that sp2 finite already handled.
      RETURN (compatible_intervals(sp1,sp2));
    END_IF;
    RETURN (FALSE);
  END_IF;
  IF subspace_of_es(sp2,es_reals) THEN
    RETURN (FALSE);
  END_IF;
  IF subspace_of_es(sp1,es_complex_numbers) THEN  -- Note sp1 finite already handled.
    IF subspace_of_es(sp2,es_complex_numbers) THEN  -- Note sp2 finite already handled.
      RETURN (compatible_complex_number_regions(sp1,sp2));
    END_IF;
    RETURN (FALSE);
  END_IF;
  IF subspace_of_es(sp2,es_complex_numbers) THEN
    RETURN (FALSE);
  END_IF;
  IF 'UNIFORM_PRODUCT_SPACE' IN types1 THEN
    IF 'UNIFORM_PRODUCT_SPACE' IN types2 THEN
      IF sp1\uniform_product_space.exponent <> sp2\uniform_product_space.exponent THEN
        RETURN (FALSE);
      END_IF;
      RETURN (compatible_spaces(sp1\uniform_product_space.base,
        sp2\uniform_product_space.base));
    END_IF;
    IF 'LISTED_PRODUCT_SPACE' IN types2 THEN
      n := SIZEOF (sp2\listed_product_space.factors);
      IF sp1\uniform_product_space.exponent <> n THEN
        RETURN (FALSE);
      END_IF;
      REPEAT i := 1 TO n;
        IF NOT compatible_spaces(sp1\uniform_product_space.base,
          sp2\listed_product_space.factors[i]) THEN
          RETURN (FALSE);
        END_IF;
      END_REPEAT;
      RETURN (TRUE);
    END_IF;
    IF 'EXTENDED_TUPLE_SPACE' IN types2 THEN
      m := sp1\uniform_product_space.exponent;
      n := space_dimension(sp2\extended_tuple_space.base);
      IF m < n THEN
        RETURN (FALSE);
      END_IF;
      IF m = n THEN
        RETURN (compatible_spaces(sp1, sp2\extended_tuple_space.base));
      END_IF;
      RETURN (compatible_spaces(sp1, assoc_product_space(
        sp2\extended_tuple_space.base, make_uniform_product_space(
        sp2\extended_tuple_space.extender, m - n))));
    END_IF;
    IF 'FUNCTION_SPACE' IN types2 THEN
      RETURN (FALSE);
    END_IF;
    -- Should be unreachable.
    RETURN (TRUE);
  END_IF;
  IF 'LISTED_PRODUCT_SPACE' IN types1 THEN
    n := SIZEOF (sp1\listed_product_space.factors);
    IF 'UNIFORM_PRODUCT_SPACE' IN types2 THEN
      IF n <> sp2\uniform_product_space.exponent THEN
        RETURN (FALSE);
      END_IF;
      REPEAT i := 1 TO n;
        IF NOT compatible_spaces(sp2\uniform_product_space.base,
          sp1\listed_product_space.factors[i]) THEN
          RETURN (FALSE);
        END_IF;
      END_REPEAT;
      RETURN (TRUE);
    END_IF;
    IF 'LISTED_PRODUCT_SPACE' IN types2 THEN
      IF n <> SIZEOF (sp2\listed_product_space.factors) THEN
        RETURN (FALSE);
      END_IF;
      REPEAT i := 1 TO n;
        IF NOT compatible_spaces(sp1\listed_product_space.factors[i],
          sp2\listed_product_space.factors[i]) THEN
          RETURN (FALSE);
        END_IF;
      END_REPEAT;
      RETURN (TRUE);
    END_IF;
    IF 'EXTENDED_TUPLE_SPACE' IN types2 THEN
      m := space_dimension(sp2\extended_tuple_space.base);
      IF n < m THEN
        RETURN (FALSE);
      END_IF;
      IF n = m THEN
        RETURN (compatible_spaces(sp1, sp2\extended_tuple_space.base));
      END_IF;
      RETURN (compatible_spaces(sp1, assoc_product_space(
        sp2\extended_tuple_space.base, make_uniform_product_space(
        sp2\extended_tuple_space.extender, n - m))));
    END_IF;
    IF (schema_prefix + 'FUNCTION_SPACE') IN types2 THEN
      RETURN (FALSE);
    END_IF;
    -- Should be unreachable.
    RETURN (TRUE);
  END_IF;
  IF 'EXTENDED_TUPLE_SPACE' IN types1 THEN
    IF ('UNIFORM_PRODUCT_SPACE' IN types2) OR
      ('LISTED_PRODUCT_SPACE' IN types2) THEN
      RETURN (compatible_spaces(sp2, sp1));
    END_IF;
    IF 'EXTENDED_TUPLE_SPACE' IN types2 THEN
      IF NOT compatible_spaces(sp1\extended_tuple_space.extender,
        sp2\extended_tuple_space.extender) THEN
        RETURN (FALSE);
      END_IF;
      n := space_dimension(sp1\extended_tuple_space.base);
      m := space_dimension(sp2\extended_tuple_space.base);
      IF n < m THEN
        RETURN (compatible_spaces(assoc_product_space(sp1\extended_tuple_space.base,
          make_uniform_product_space(sp1\extended_tuple_space.extender, m - n)),
          sp2\extended_tuple_space.base));
      END_IF;
      IF n = m THEN
        RETURN (compatible_spaces(sp1\extended_tuple_space.base,
          sp2\extended_tuple_space.base));
      END_IF;
      IF n > m THEN
        RETURN (compatible_spaces(sp1\extended_tuple_space.base,
          assoc_product_space(sp2\extended_tuple_space.base,
          make_uniform_product_space(sp2\extended_tuple_space.extender, n - m))));
      END_IF;
    END_IF;
    IF 'FUNCTION_SPACE' IN types2 THEN
      RETURN (FALSE);
    END_IF;
    -- Should be unreachable.
    RETURN (TRUE);
  END_IF;
  IF 'FUNCTION_SPACE' IN types1 THEN
    IF 'FUNCTION_SPACE' IN types2 THEN
      s1 := sp1\function_space.domain_argument;
      s2 := sp2\function_space.domain_argument;
      CASE sp1\function_space.domain_constraint OF
      sc_equal : BEGIN
        CASE sp2\function_space.domain_constraint OF
        sc_equal : lgcl := subspace_of(s1, s2) AND subspace_of(s2, s1);
        sc_subspace : lgcl := subspace_of(s1, s2);
        sc_member : lgcl := member_of(s1, s2);
        END_CASE;
        END;
      sc_subspace :BEGIN
        CASE sp2\function_space.domain_constraint OF
        sc_equal : lgcl := subspace_of(s2, s1);
        sc_subspace : lgcl := compatible_spaces(s1, s2);
        sc_member : lgcl := UNKNOWN;
        END_CASE;
        END;
      sc_member :BEGIN
        CASE sp2\function_space.domain_constraint OF
        sc_equal : lgcl := member_of(s2, s1);
        sc_subspace : lgcl := UNKNOWN;
        sc_member : lgcl := compatible_spaces(s1, s2);
        END_CASE;
        END;
      END_CASE;
      IF lgcl = FALSE THEN
        RETURN (FALSE);
      END_IF;
      s1 := sp1\function_space.range_argument;
      s2 := sp2\function_space.range_argument;
      CASE sp1\function_space.range_constraint OF
      sc_equal : BEGIN
        CASE sp2\function_space.range_constraint OF
        sc_equal : lgcl := subspace_of(s1, s2) AND subspace_of(s2, s1);
        sc_subspace : lgcl := subspace_of(s1, s2);
        sc_member : lgcl := member_of(s1, s2);
        END_CASE;
        END;
      sc_subspace :BEGIN
        CASE sp2\function_space.range_constraint OF
        sc_equal : lgcl := subspace_of(s2, s1);
        sc_subspace : lgcl := compatible_spaces(s1, s2);
        sc_member : lgcl := UNKNOWN;
        END_CASE;
        END;
      sc_member :BEGIN
        CASE sp2\function_space.range_constraint OF
        sc_equal : lgcl := member_of(s2, s1);
        sc_subspace : lgcl := UNKNOWN;
        sc_member : lgcl := compatible_spaces(s1, s2);
        END_CASE;
        END;
      END_CASE;
      IF lgcl = FALSE THEN
        RETURN (FALSE);
      END_IF;
      RETURN (TRUE);
    END_IF;
    -- Should be unreachable.
    RETURN (TRUE);
  END_IF;
  -- Should be unreachable.
  RETURN (TRUE);
END_FUNCTION;  -- compatible_spaces
FUNCTION composable_sequence(operands : LIST [2:?] OF maths_function) : BOOLEAN;
  REPEAT i := 1 TO SIZEOF (operands) - 1;
    IF NOT compatible_spaces (operands[i].range, operands[i+1].domain) THEN
      RETURN (FALSE);
    END_IF;
  END_REPEAT;
  RETURN (TRUE);
END_FUNCTION;  -- composable_sequence
FUNCTION convert_to_literal(val : maths_atom) : generic_literal;
  LOCAL
    types : SET OF STRING := TYPEOF (val);
  END_LOCAL;
  IF 'INTEGER' IN types THEN  RETURN (make_int_literal (val));      END_IF;
  IF 'REAL'    IN types THEN  RETURN (make_real_literal (val));     END_IF;
  IF 'BOOLEAN' IN types THEN  RETURN (make_boolean_literal (val));  END_IF;
  IF 'STRING'  IN types THEN  RETURN (make_string_literal (val));   END_IF;
  IF 'LOGICAL' IN types THEN  RETURN (make_logical_literal (val));  END_IF;
  IF 'BINARY'  IN types THEN  RETURN (make_binary_literal (val));   END_IF;
  IF (schema_prefix + 'MATHS_ENUM_ATOM') IN types THEN
    RETURN (make_maths_enum_literal (val));
  END_IF;
  -- Should be unreachable
  RETURN (?);
END_FUNCTION;  -- convert_to_literal
FUNCTION convert_to_maths_function(func : maths_function_select) : maths_function;
  LOCAL
    efenum : elementary_function_enumerators;
    mthfun : maths_function;
  END_LOCAL;
  IF (schema_prefix + 'MATHS_FUNCTION') IN TYPEOF (func) THEN
    mthfun := func;
  ELSE
    efenum := func;
    mthfun := make_elementary_function (efenum);
  END_IF;
  RETURN (mthfun);
END_FUNCTION;  -- convert_to_maths_function
FUNCTION convert_to_maths_value(val : GENERIC:G) : maths_value;
  LOCAL
    types : SET OF STRING := TYPEOF (val);
    ival  : maths_integer;
    rval  : maths_real;
    nval  : maths_number;
    tfval : maths_boolean;
    lval  : maths_logical;
    sval  : maths_string;
    bval  : maths_binary;
    tval  : maths_tuple := the_empty_maths_tuple;
    mval  : maths_value;
  END_LOCAL;
  IF (schema_prefix + 'MATHS_VALUE') IN types THEN  RETURN (val);  END_IF;
  IF 'INTEGER' IN types THEN  ival := val;   RETURN (ival);   END_IF;
  IF 'REAL'    IN types THEN  rval := val;   RETURN (rval);   END_IF;
  IF 'NUMBER'  IN types THEN  nval := val;   RETURN (nval);   END_IF;
  IF 'BOOLEAN' IN types THEN  tfval := val;  RETURN (tfval);  END_IF;
  IF 'LOGICAL' IN types THEN  lval := val;   RETURN (lval);   END_IF;
  IF 'STRING'  IN types THEN  sval := val;   RETURN (sval);   END_IF;
  IF 'BINARY'  IN types THEN  bval := val;   RETURN (bval);   END_IF;
  IF 'LIST' IN types THEN
    REPEAT i := 1 TO SIZEOF (val);
      mval := convert_to_maths_value (val[i]);
      IF NOT EXISTS (mval) THEN  RETURN (?);  END_IF;
      INSERT (tval, mval, i-1);
    END_REPEAT;
    RETURN (tval);
  END_IF;
  RETURN (?);
END_FUNCTION;  -- convert_to_maths_value
FUNCTION convert_to_operand(val : maths_value) : generic_expression;
  LOCAL
    types  : SET OF STRING := stripped_typeof (val);
  END_LOCAL;
  -- Use intermediate variables of appropriate declared types to help the compilers.
  IF 'GENERIC_EXPRESSION' IN types THEN  RETURN (val);  END_IF;
  IF 'MATHS_ATOM' IN types THEN  RETURN (convert_to_literal (val));  END_IF;
  IF 'ATOM_BASED_VALUE' IN types THEN  RETURN (make_atom_based_literal(val));  END_IF;
  IF 'MATHS_TUPLE' IN types THEN  RETURN (make_maths_tuple_literal(val));  END_IF;
  -- Should be unreachable
  RETURN (?);
END_FUNCTION;  -- convert_to_operand
FUNCTION convert_to_operands(values : AGGREGATE OF maths_value)
                            : LIST OF generic_expression;
  LOCAL
    operands : LIST OF generic_expression := [];
    loc : INTEGER := 0;
  END_LOCAL;
  IF NOT EXISTS (values) THEN  RETURN (?);  END_IF;
  REPEAT i := LOINDEX (values) TO HIINDEX (values);
    INSERT (operands, convert_to_operand (values[i]), loc);
    loc := loc + 1;
  END_REPEAT;
  RETURN (operands);
END_FUNCTION;  -- convert_to_operands
FUNCTION convert_to_operands_prcmfn(srcdom  : maths_space_or_function;
                                    prepfun : LIST OF maths_function;
                                    finfun  : maths_function_select)
                                   : LIST [2:?] OF generic_expression;
  LOCAL
    operands : LIST OF generic_expression := [];
  END_LOCAL;
  INSERT (operands, srcdom, 0);
  REPEAT i := 1 TO SIZEOF (prepfun);
    INSERT (operands, prepfun[i], i);
  END_REPEAT;
  INSERT (operands, convert_to_maths_function (finfun), SIZEOF (prepfun)+1);
  RETURN (operands);
END_FUNCTION;  -- convert_to_operands_prcmfn
FUNCTION definite_integral_check(domain   : tuple_space;
                                 vrblint  : input_selector;
                                 lowerinf : BOOLEAN;
                                 upperinf : BOOLEAN) : BOOLEAN;
  LOCAL
    domn : tuple_space := domain;
    fspc : maths_space;
    dim : nonnegative_integer;
    k : positive_integer;
  END_LOCAL;
  IF (space_dimension (domain) = 1) AND ((schema_prefix + 'TUPLE_SPACE') IN
    TYPEOF (factor1 (domain))) THEN
    domn := factor1 (domain);
  END_IF;
  dim := space_dimension (domn);
  k := vrblint;
  IF k > dim THEN  RETURN (FALSE);  END_IF;
  fspc := factor_space (domn, k);
  IF NOT ((schema_prefix + 'REAL_INTERVAL') IN TYPEOF (fspc)) THEN
    RETURN (FALSE);
  END_IF;
  IF lowerinf AND min_exists (fspc) THEN  RETURN (FALSE);  END_IF;
  IF upperinf AND max_exists (fspc) THEN  RETURN (FALSE);  END_IF;
  RETURN (TRUE);
END_FUNCTION;  -- definite_integral_check
FUNCTION definite_integral_expr_check(operands : LIST [2:?] OF generic_expression;
                                      lowerinf : BOOLEAN;
                                      upperinf : BOOLEAN) : BOOLEAN;
  LOCAL
    nops : INTEGER := 2;
    vspc : maths_space;
    dim : nonnegative_integer;
    k : positive_integer;
    bspc : maths_space;
  END_LOCAL;
  IF NOT lowerinf THEN  nops := nops + 1;  END_IF;
  IF NOT upperinf THEN  nops := nops + 1;  END_IF;
  IF SIZEOF (operands) <> nops THEN  RETURN (FALSE);  END_IF;
  IF NOT ('GENERIC_VARIABLE' IN stripped_typeof(operands[2])) THEN
    RETURN (FALSE);
  END_IF;
  IF NOT has_values_space (operands[2]) THEN  RETURN (FALSE);  END_IF;
  vspc := values_space_of (operands[2]);
  IF NOT ('REAL_INTERVAL' IN stripped_typeof(vspc)) THEN  RETURN (FALSE);  END_IF;
  IF lowerinf THEN
    IF min_exists (vspc) THEN  RETURN (FALSE);  END_IF;
    k := 3;
  ELSE
    IF NOT has_values_space (operands[3]) THEN  RETURN (FALSE);  END_IF;
    bspc := values_space_of (operands[3]);
    IF NOT compatible_spaces (bspc, vspc) THEN  RETURN (FALSE);  END_IF;
    k := 4;
  END_IF;
  IF upperinf THEN
    IF max_exists (vspc) THEN  RETURN (FALSE);  END_IF;
  ELSE
    IF NOT has_values_space (operands[k]) THEN  RETURN (FALSE);  END_IF;
    bspc := values_space_of (operands[k]);
    IF NOT compatible_spaces (bspc, vspc) THEN  RETURN (FALSE);  END_IF;
  END_IF;
  RETURN (TRUE);
END_FUNCTION;  -- definite_integral_expr_check
FUNCTION derive_definite_integral_domain(igrl : definite_integral_function)
                                        : tuple_space;

  -- Internal utility function:
  FUNCTION process_product_space(spc         : product_space;
                                 idx, prefix : INTEGER;
                                 vdomn       : maths_space) : product_space;
    LOCAL
      uspc : uniform_product_space;
      expnt : INTEGER;
      factors : LIST OF maths_space;
    END_LOCAL;
    IF (schema_prefix + 'UNIFORM_PRODUCT_SPACE') IN TYPEOF (spc) THEN
      uspc := spc;
      expnt := uspc.exponent + prefix;
      IF idx <= uspc.exponent THEN  expnt := expnt - 1;  END_IF;
      IF expnt = 0 THEN
        RETURN (make_listed_product_space([]));
      ELSE
        RETURN (make_uniform_product_space(uspc.base,expnt));
      END_IF;
    ELSE
      factors := spc\listed_product_space.factors;
      IF idx <= SIZEOF (factors) THEN  REMOVE (factors, idx);  END_IF;
      IF prefix > 0 THEN
        INSERT (factors, vdomn, 0);
        IF prefix > 1 THEN  INSERT (factors, vdomn, 0);  END_IF;
      END_IF;
      RETURN (make_listed_product_space(factors));
    END_IF;
  END_FUNCTION;  -- process_product_space

  -- Resume body of derive_definite_integral_domain function
  LOCAL
    idomn : tuple_space := igrl.integrand.domain;
    types : SET OF STRING := TYPEOF (idomn);
    idx : INTEGER := igrl.variable_of_integration;
    tupled : BOOLEAN := bool(((space_dimension(idomn) = 1) AND
                             ((schema_prefix + 'TUPLE_SPACE') IN types)));
    prefix : INTEGER := 0;
    espc : extended_tuple_space;
    vdomn : maths_space;
  END_LOCAL;
  IF tupled THEN
    idomn := factor1(idomn);
    types := TYPEOF (idomn);
  END_IF;
  IF igrl.lower_limit_neg_infinity THEN  prefix := prefix + 1;  END_IF;
  IF igrl.upper_limit_pos_infinity THEN  prefix := prefix + 1;  END_IF;
  vdomn := factor_space(idomn,idx);
  IF (schema_prefix + 'EXTENDED_TUPLE_SPACE') IN types THEN
    espc := idomn;
    idomn := make_extended_tuple_space(process_product_space(espc.base,idx,
      prefix,vdomn),espc.extender);
  ELSE
    idomn := process_product_space(idomn,idx,prefix,vdomn);
  END_IF;
  IF tupled THEN  RETURN (one_tuples_of(idomn));
  ELSE            RETURN (idomn);                 END_IF;
END_FUNCTION;  -- derive_definite_integral_domain
FUNCTION derive_elementary_function_domain(ef_val : elementary_function_enumerators)
                                          : tuple_space;
  IF NOT EXISTS (ef_val) THEN  RETURN (?);  END_IF;
  CASE ef_val OF
  ef_and : RETURN (make_extended_tuple_space (the_zero_tuple_space, the_logicals));
  ef_or : RETURN (make_extended_tuple_space (the_zero_tuple_space, the_logicals));
  ef_not : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_xor : RETURN (make_uniform_product_space (the_logicals, 2));
  ef_negate_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_add_i : RETURN (the_integer_tuples);
  ef_subtract_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_multiply_i : RETURN (the_integer_tuples);
  ef_divide_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_mod_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_exponentiate_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_eq_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_ne_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_gt_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_lt_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_ge_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_le_i : RETURN (make_uniform_product_space (the_integers, 2));
  ef_abs_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_if_i : RETURN (make_listed_product_space ([the_logicals, the_integers,
    the_integers]));
  ef_negate_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_reciprocal_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_add_r : RETURN (the_real_tuples);
  ef_subtract_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_multiply_r : RETURN (the_real_tuples);
  ef_divide_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_mod_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_exponentiate_r : RETURN (make_listed_product_space ([the_nonnegative_reals,
    the_reals]));
  ef_exponentiate_ri : RETURN (make_listed_product_space ([the_reals, the_integers]));
  ef_eq_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_ne_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_gt_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_lt_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_ge_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_le_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_abs_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_acos_r : RETURN (make_uniform_product_space (the_neg1_one_interval, 1));
  ef_asin_r : RETURN (make_uniform_product_space (the_neg1_one_interval, 1));
  ef_atan2_r : RETURN (make_uniform_product_space (the_reals, 2));
  ef_cos_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_exp_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_ln_r : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_log2_r : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_log10_r : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_sin_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_sqrt_r : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_tan_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_if_r : RETURN (make_listed_product_space ([the_logicals, the_reals, the_reals]));
  ef_negate_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_reciprocal_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_add_c : RETURN (the_complex_tuples);
  ef_subtract_c : RETURN (make_uniform_product_space (the_complex_numbers, 2));
  ef_multiply_c : RETURN (the_complex_tuples);
  ef_divide_c : RETURN (make_uniform_product_space (the_complex_numbers, 2));
  ef_exponentiate_c : RETURN (make_uniform_product_space (the_complex_numbers, 2));
  ef_exponentiate_ci : RETURN (make_listed_product_space ([the_complex_numbers,
    the_integers]));
  ef_eq_c : RETURN (make_uniform_product_space (the_complex_numbers, 2));
  ef_ne_c : RETURN (make_uniform_product_space (the_complex_numbers, 2));
  ef_conjugate_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_abs_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_arg_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_cos_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_exp_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_ln_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_sin_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_sqrt_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_tan_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_if_c : RETURN (make_listed_product_space ([the_logicals, the_complex_numbers,
    the_complex_numbers]));
  ef_subscript_s : RETURN (make_listed_product_space ([the_strings, the_integers]));
  ef_eq_s : RETURN (make_uniform_product_space (the_strings, 2));
  ef_ne_s : RETURN (make_uniform_product_space (the_strings, 2));
  ef_gt_s : RETURN (make_uniform_product_space (the_strings, 2));
  ef_lt_s : RETURN (make_uniform_product_space (the_strings, 2));
  ef_ge_s : RETURN (make_uniform_product_space (the_strings, 2));
  ef_le_s : RETURN (make_uniform_product_space (the_strings, 2));
  ef_subsequence_s : RETURN (make_listed_product_space ([the_strings, the_integers,
    the_integers]));
  ef_concat_s : RETURN (make_extended_tuple_space (the_zero_tuple_space, the_strings));
  ef_size_s : RETURN (make_uniform_product_space (the_strings, 1));
  ef_format : RETURN (make_listed_product_space ([the_numbers, the_strings]));
  ef_value : RETURN (make_uniform_product_space (the_strings, 1));
  ef_like : RETURN (make_uniform_product_space (the_strings, 2));
  ef_if_s : RETURN (make_listed_product_space ([the_logicals, the_strings,
    the_strings]));
  ef_subscript_b : RETURN (make_listed_product_space ([the_binarys, the_integers]));
  ef_eq_b : RETURN (make_uniform_product_space (the_binarys, 2));
  ef_ne_b : RETURN (make_uniform_product_space (the_binarys, 2));
  ef_gt_b : RETURN (make_uniform_product_space (the_binarys, 2));
  ef_lt_b : RETURN (make_uniform_product_space (the_binarys, 2));
  ef_ge_b : RETURN (make_uniform_product_space (the_binarys, 2));
  ef_le_b : RETURN (make_uniform_product_space (the_binarys, 2));
  ef_subsequence_b : RETURN (make_listed_product_space ([the_binarys, the_integers,
    the_integers]));
  ef_concat_b : RETURN (make_extended_tuple_space (the_zero_tuple_space, the_binarys));
  ef_size_b : RETURN (make_uniform_product_space (the_binarys, 1));
  ef_if_b : RETURN (make_listed_product_space ([the_logicals, the_binarys,
    the_binarys]));
  ef_subscript_t : RETURN (make_listed_product_space ([the_tuples, the_integers]));
  ef_eq_t : RETURN (make_uniform_product_space (the_tuples, 2));
  ef_ne_t : RETURN (make_uniform_product_space (the_tuples, 2));
  ef_concat_t : RETURN (make_extended_tuple_space (the_zero_tuple_space, the_tuples));
  ef_size_t : RETURN (make_uniform_product_space (the_tuples, 1));
  ef_entuple : RETURN (the_tuples);
  ef_detuple : RETURN (make_uniform_product_space (the_generics, 1));
  ef_insert : RETURN (make_listed_product_space ([the_tuples, the_generics,
    the_integers]));
  ef_remove : RETURN (make_listed_product_space ([the_tuples, the_integers]));
  ef_if_t : RETURN (make_listed_product_space ([the_logicals, the_tuples,
    the_tuples]));
  ef_sum_it : RETURN (make_uniform_product_space (the_integer_tuples, 1));
  ef_product_it : RETURN (make_uniform_product_space (the_integer_tuples, 1));
  ef_add_it : RETURN (make_extended_tuple_space (the_integer_tuples,
    the_integer_tuples));
  ef_subtract_it : RETURN (make_uniform_product_space (the_integer_tuples, 2));
  ef_scalar_mult_it : RETURN (make_listed_product_space ([the_integers,
    the_integer_tuples]));
  ef_dot_prod_it : RETURN (make_uniform_product_space (the_integer_tuples, 2));
  ef_sum_rt : RETURN (make_uniform_product_space (the_real_tuples, 1));
  ef_product_rt : RETURN (make_uniform_product_space (the_real_tuples, 1));
  ef_add_rt : RETURN (make_extended_tuple_space (the_real_tuples, the_real_tuples));
  ef_subtract_rt : RETURN (make_uniform_product_space (the_real_tuples, 2));
  ef_scalar_mult_rt : RETURN (make_listed_product_space ([the_reals,
    the_real_tuples]));
  ef_dot_prod_rt : RETURN (make_uniform_product_space (the_real_tuples, 2));
  ef_norm_rt : RETURN (make_uniform_product_space (the_real_tuples, 1));
  ef_sum_ct : RETURN (make_uniform_product_space (the_complex_tuples, 1));
  ef_product_ct : RETURN (make_uniform_product_space (the_complex_tuples, 1));
  ef_add_ct : RETURN (make_extended_tuple_space (the_complex_tuples,
    the_complex_tuples));
  ef_subtract_ct : RETURN (make_uniform_product_space (the_complex_tuples, 2));
  ef_scalar_mult_ct : RETURN (make_listed_product_space ([the_complex_numbers,
    the_complex_tuples]));
  ef_dot_prod_ct : RETURN (make_uniform_product_space (the_complex_tuples, 2));
  ef_norm_ct : RETURN (make_uniform_product_space (the_complex_tuples, 1));
  ef_if : RETURN (make_listed_product_space ([the_logicals, the_generics,
    the_generics]));
  ef_ensemble : RETURN (the_tuples);
  ef_member_of : RETURN (make_listed_product_space ([the_generics, the_maths_spaces]));
  OTHERWISE : RETURN (?);
  END_CASE;
END_FUNCTION;  -- derive_elementary_function_domain
FUNCTION derive_elementary_function_range(ef_val : elementary_function_enumerators)
                                         : tuple_space;
  IF NOT EXISTS (ef_val) THEN  RETURN (?);  END_IF;
  CASE ef_val OF
  ef_and : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_or : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_not : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_xor : RETURN (make_uniform_product_space (the_logicals, 2));
  ef_negate_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_add_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_subtract_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_multiply_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_divide_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_mod_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_exponentiate_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_eq_i : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ne_i : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_gt_i : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_lt_i : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ge_i : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_le_i : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_abs_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_if_i : RETURN (make_uniform_product_space (the_integers, 1));
  ef_negate_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_reciprocal_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_add_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_subtract_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_multiply_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_divide_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_mod_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_exponentiate_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_exponentiate_ri : RETURN (make_uniform_product_space (the_reals, 1));
  ef_eq_r : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ne_r : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_gt_r : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_lt_r : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ge_r : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_le_r : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_abs_r : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_acos_r : RETURN (make_uniform_product_space (the_zero_pi_interval, 1));
  ef_asin_r : RETURN (make_uniform_product_space (the_neghalfpi_halfpi_interval, 1));
  ef_atan2_r : RETURN (make_uniform_product_space (the_negpi_pi_interval, 1));
  ef_cos_r : RETURN (make_uniform_product_space (the_neg1_one_interval, 1));
  ef_exp_r : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_ln_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_log2_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_log10_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_sin_r : RETURN (make_uniform_product_space (the_neg1_one_interval, 1));
  ef_sqrt_r : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_tan_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_if_r : RETURN (make_uniform_product_space (the_reals, 1));
  ef_negate_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_reciprocal_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_add_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_subtract_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_multiply_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_divide_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_exponentiate_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_exponentiate_ci : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_eq_c : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ne_c : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_conjugate_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_abs_c : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_arg_c : RETURN (make_uniform_product_space (the_negpi_pi_interval, 1));
  ef_cos_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_exp_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_ln_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_sin_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_sqrt_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_tan_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_if_c : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_subscript_s : RETURN (make_uniform_product_space (the_strings, 1));
  ef_eq_s : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ne_s : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_gt_s : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_lt_s : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ge_s : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_le_s : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_subsequence_s : RETURN (make_uniform_product_space (the_strings, 1));
  ef_concat_s : RETURN (make_uniform_product_space (the_strings, 1));
  ef_size_s : RETURN (make_uniform_product_space (the_integers, 1));
  ef_format : RETURN (make_uniform_product_space (the_strings, 1));
  ef_value : RETURN (make_uniform_product_space (the_reals, 1));
  ef_like : RETURN (make_uniform_product_space (the_booleans, 1));
  ef_if_s : RETURN (make_uniform_product_space (the_strings, 1));
  ef_subscript_b : RETURN (make_uniform_product_space (the_binarys, 1));
  ef_eq_b : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ne_b : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_gt_b : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_lt_b : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ge_b : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_le_b : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_subsequence_b : RETURN (make_uniform_product_space (the_binarys, 1));
  ef_concat_b : RETURN (make_uniform_product_space (the_binarys, 1));
  ef_size_b : RETURN (make_uniform_product_space (the_integers, 1));
  ef_if_b : RETURN (make_uniform_product_space (the_binarys, 1));
  ef_subscript_t : RETURN (make_uniform_product_space (the_generics, 1));
  ef_eq_t : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_ne_t : RETURN (make_uniform_product_space (the_logicals, 1));
  ef_concat_t : RETURN (make_uniform_product_space (the_tuples, 1));
  ef_size_t : RETURN (make_uniform_product_space (the_integers, 1));
  ef_entuple : RETURN (make_uniform_product_space (the_tuples, 1));
  ef_detuple : RETURN (the_tuples);
  ef_insert : RETURN (make_uniform_product_space (the_tuples, 1));
  ef_remove : RETURN (make_uniform_product_space (the_tuples, 1));
  ef_if_t : RETURN (make_uniform_product_space (the_tuples, 1));
  ef_sum_it : RETURN (make_uniform_product_space (the_integers, 1));
  ef_product_it : RETURN (make_uniform_product_space (the_integers, 1));
  ef_add_it : RETURN (make_uniform_product_space (the_integer_tuples, 1));
  ef_subtract_it : RETURN (make_uniform_product_space (the_integer_tuples, 1));
  ef_scalar_mult_it : RETURN (make_uniform_product_space (the_integer_tuples, 1));
  ef_dot_prod_it : RETURN (make_uniform_product_space (the_integers, 1));
  ef_sum_rt : RETURN (make_uniform_product_space (the_reals, 1));
  ef_product_rt : RETURN (make_uniform_product_space (the_reals, 1));
  ef_add_rt : RETURN (make_uniform_product_space (the_real_tuples, 1));
  ef_subtract_rt : RETURN (make_uniform_product_space (the_real_tuples, 1));
  ef_scalar_mult_rt : RETURN (make_uniform_product_space (the_real_tuples, 1));
  ef_dot_prod_rt : RETURN (make_uniform_product_space (the_reals, 1));
  ef_norm_rt : RETURN (make_uniform_product_space (the_reals, 1));
  ef_sum_ct : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_product_ct : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_add_ct : RETURN (make_uniform_product_space (the_complex_tuples, 1));
  ef_subtract_ct : RETURN (make_uniform_product_space (the_complex_tuples, 1));
  ef_scalar_mult_ct : RETURN (make_uniform_product_space (the_complex_tuples, 1));
  ef_dot_prod_ct : RETURN (make_uniform_product_space (the_complex_numbers, 1));
  ef_norm_ct : RETURN (make_uniform_product_space (the_nonnegative_reals, 1));
  ef_if : RETURN (make_uniform_product_space (the_generics, 1));
  ef_ensemble : RETURN (make_uniform_product_space (the_maths_spaces, 1));
  ef_member_of : RETURN (make_uniform_product_space (the_logicals, 1));
  OTHERWISE : RETURN (?);
  END_CASE;
END_FUNCTION;  -- derive_elementary_function_range
FUNCTION derive_finite_function_domain(pairs : SET [1:?] OF LIST [2:2] OF maths_value)
                                      : tuple_space;
  LOCAL
    result : SET OF maths_value := [];
  END_LOCAL;
-- An ambiguity in ISO 10303-11:1994 pages 99-101 leaves the result of the following
-- three lines ambiguous in those cases where an operand is simultaneously a member
-- of the base type and the aggregate type.
-- REPEAT i := 1 TO SIZEOF (pairs);
--   result := result + pairs[i][1];
-- END_REPEAT;
-- The next line unions an empty set and the desired list to get the desired set.
  result := result + list_selected_components (pairs, 1);
  RETURN (one_tuples_of (make_finite_space (result)));
END_FUNCTION;  -- derive_finite_function_domain
FUNCTION derive_finite_function_range(pairs : SET [1:?] OF LIST [2:2] OF maths_value)
                                     : tuple_space;
  LOCAL
    result : SET OF maths_value := [];
  END_LOCAL;
-- An ambiguity in ISO 10303-11:1994 pages 99-101 leaves the result of the following
-- three lines ambiguous in those cases where an operand is simultaneously a member
-- of the base type and the aggregate type.
-- REPEAT i := 1 TO SIZEOF (pairs);
--   result := result + pairs[i][2];
-- END_REPEAT;
-- The next line unions an empty set and the desired list to get the desired set.
  result := result + list_selected_components (pairs, 2);
  RETURN (one_tuples_of (make_finite_space (result)));
END_FUNCTION;  -- derive_finite_function_range
FUNCTION derive_function_domain(func : maths_function) : tuple_space;
  LOCAL
    typenames : SET OF STRING := stripped_typeof(func);
    tspace : tuple_space := make_listed_product_space ([]);
    shape : LIST OF positive_integer;
    sidxs  : LIST OF INTEGER := [0];
    itvl   : finite_integer_interval;
    factors : LIST OF finite_integer_interval := [];
    is_uniform : BOOLEAN := TRUE;
  END_LOCAL;
  IF 'FINITE_FUNCTION' IN typenames THEN
    RETURN (derive_finite_function_domain (func\finite_function.pairs));
  END_IF;
  IF 'CONSTANT_FUNCTION' IN typenames THEN
    RETURN (domain_from (func\constant_function.source_of_domain));
  END_IF;
  IF 'SELECTOR_FUNCTION' IN typenames THEN
    RETURN (domain_from (func\selector_function.source_of_domain));
  END_IF;
  IF 'ELEMENTARY_FUNCTION' IN typenames THEN
    RETURN (derive_elementary_function_domain (func\elementary_function.func_id));
  END_IF;
  IF 'RESTRICTION_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (func\restriction_function.operand));
  END_IF;
  IF 'REPACKAGING_FUNCTION' IN typenames THEN
    IF func\repackaging_function.input_repack = ro_nochange THEN
      RETURN (func\repackaging_function.operand.domain);
    END_IF;
    IF func\repackaging_function.input_repack = ro_wrap_as_tuple THEN
      RETURN (factor1 (func\repackaging_function.operand.domain));
    END_IF;
    IF func\repackaging_function.input_repack = ro_unwrap_tuple THEN
      RETURN (one_tuples_of (func\repackaging_function.operand.domain));
    END_IF;
    -- Unreachable, as there is no other possible value for input_repack.
    RETURN (?);
  END_IF;
  IF 'REINDEXED_ARRAY_FUNCTION' IN typenames THEN
    shape := shape_of_array(func\unary_generic_expression.operand);
    sidxs := func\reindexed_array_function.starting_indices;
    REPEAT i := 1 TO SIZEOF (shape);
      itvl := make_finite_integer_interval (sidxs[i], sidxs[i]+shape[i]-1);
      INSERT (factors, itvl, i-1);
      IF shape[i] <> shape[1] THEN  is_uniform := FALSE;  END_IF;
    END_REPEAT;
    IF is_uniform THEN
      RETURN (make_uniform_product_space (factors[1], SIZEOF (shape)));
    END_IF;
    RETURN (make_listed_product_space (factors));
  END_IF;
  IF 'SERIES_COMPOSED_FUNCTION' IN typenames THEN
    RETURN (func\series_composed_function.operands[1].domain);
  END_IF;
  IF 'PARALLEL_COMPOSED_FUNCTION' IN typenames THEN
    RETURN (domain_from (func\parallel_composed_function.source_of_domain));
  END_IF;
  IF 'EXPLICIT_TABLE_FUNCTION' IN typenames THEN
    shape := func\explicit_table_function.shape;
    sidxs[1] := func\explicit_table_function.index_base;
    REPEAT i := 1 TO SIZEOF (shape);
      itvl := make_finite_integer_interval (sidxs[1], sidxs[1]+shape[i]-1);
      INSERT (factors, itvl, i-1);
      IF shape[i] <> shape[1] THEN  is_uniform := FALSE;  END_IF;
    END_REPEAT;
    IF is_uniform THEN
      RETURN (make_uniform_product_space (factors[1], SIZEOF (shape)));
    END_IF;
    RETURN (make_listed_product_space (factors));
  END_IF;
  IF 'HOMOGENEOUS_LINEAR_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space
      (factor1 (func\homogeneous_linear_function.mat.range),
      func\homogeneous_linear_function.mat\explicit_table_function.shape
      [func\homogeneous_linear_function.sum_index])));
  END_IF;
  IF 'GENERAL_LINEAR_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space
      (factor1 (func\general_linear_function.mat.range),
      func\general_linear_function.mat\explicit_table_function.shape
      [func\general_linear_function.sum_index] - 1)));
  END_IF;
  IF 'B_SPLINE_BASIS' IN typenames THEN
    RETURN (one_tuples_of (make_finite_real_interval
      (func\b_spline_basis.repeated_knots[func\b_spline_basis.order], closed,
      func\b_spline_basis.repeated_knots[func\b_spline_basis.num_basis+1], closed)));
  END_IF;
  IF 'B_SPLINE_FUNCTION' IN typenames THEN
    REPEAT i := 1 TO SIZEOF (func\b_spline_function.basis);
      tspace := assoc_product_space (tspace, func\b_spline_function.basis[i].domain);
    END_REPEAT;
    RETURN (one_tuples_of (tspace));
  END_IF;
  IF 'RATIONALIZE_FUNCTION' IN typenames THEN
    RETURN (func\rationalize_function.fun.domain);
  END_IF;
  IF 'PARTIAL_DERIVATIVE_FUNCTION' IN typenames THEN
    RETURN (func\partial_derivative_function.derivand.domain);
  END_IF;
  IF 'DEFINITE_INTEGRAL_FUNCTION' IN typenames THEN
    RETURN (derive_definite_integral_domain(func));
  END_IF;
  IF 'ABSTRACTED_EXPRESSION_FUNCTION' IN typenames THEN
    REPEAT i := 1 TO SIZEOF (func\abstracted_expression_function.variables);
      tspace := assoc_product_space (tspace, one_tuples_of (values_space_of
        (func\abstracted_expression_function.variables[i])));
    END_REPEAT;
    RETURN (tspace);
  END_IF;
  IF 'EXPRESSION_DENOTED_FUNCTION' IN typenames THEN
    RETURN (values_space_of (func\expression_denoted_function.expr)\function_space.
      domain_argument);
  END_IF;
  IF 'IMPORTED_POINT_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_listed_product_space ([])));
  END_IF;
  IF 'IMPORTED_CURVE_FUNCTION' IN typenames THEN
    RETURN (func\imported_curve_function.parametric_domain);
  END_IF;
  IF 'IMPORTED_SURFACE_FUNCTION' IN typenames THEN
    RETURN (func\imported_surface_function.parametric_domain);
  END_IF;
  IF 'IMPORTED_VOLUME_FUNCTION' IN typenames THEN
    RETURN (func\imported_volume_function.parametric_domain);
  END_IF;
  IF 'APPLICATION_DEFINED_FUNCTION' IN typenames THEN
    RETURN (func\application_defined_function.explicit_domain);
  END_IF;
  -- Unreachable, as no other subtypes of maths_function are permissible without
  -- first modifying this function to account for them.
  RETURN (?);
END_FUNCTION;  -- derive_function_domain
FUNCTION derive_function_range(func : maths_function) : tuple_space;
  LOCAL
    typenames : SET OF STRING := stripped_typeof(func);
    tspace : tuple_space := make_listed_product_space ([]);
    m, n : nonnegative_integer := 0;
  END_LOCAL;
  IF 'FINITE_FUNCTION' IN typenames THEN
    RETURN (derive_finite_function_range (func\finite_function.pairs));
  END_IF;
  IF 'CONSTANT_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_finite_space ([func\constant_function.sole_output])));
  END_IF;
  IF 'SELECTOR_FUNCTION' IN typenames THEN
    tspace := func.domain;
    IF (space_dimension(tspace) = 1) AND ((schema_prefix + 'TUPLE_SPACE') IN
      TYPEOF (tspace)) THEN
      tspace := factor1 (tspace);
    END_IF;
    RETURN (one_tuples_of (factor_space (tspace, func\selector_function.selector)));
  END_IF;
  IF 'ELEMENTARY_FUNCTION' IN typenames THEN
    RETURN (derive_elementary_function_range (func\elementary_function.func_id));
  END_IF;
  IF 'RESTRICTION_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (func\restriction_function.operand));
  END_IF;
  IF 'REPACKAGING_FUNCTION' IN typenames THEN
    tspace := func\repackaging_function.operand.range;
    IF func\repackaging_function.output_repack = ro_wrap_as_tuple THEN
      tspace := one_tuples_of (tspace);
    END_IF;
    IF func\repackaging_function.output_repack = ro_unwrap_tuple THEN
      tspace := factor1 (tspace);
    END_IF;
    IF func\repackaging_function.selected_output > 0 THEN
      tspace := one_tuples_of (factor_space (tspace,
        func\repackaging_function.selected_output));
    END_IF;
    RETURN (tspace);
  END_IF;
  IF 'REINDEXED_ARRAY_FUNCTION' IN typenames THEN
    RETURN (func\unary_generic_expression.operand\maths_function.range);
  END_IF;
  IF 'SERIES_COMPOSED_FUNCTION' IN typenames THEN
    RETURN (func\series_composed_function.operands[SIZEOF
      (func\series_composed_function.operands)].range);
  END_IF;
  IF 'PARALLEL_COMPOSED_FUNCTION' IN typenames THEN
    RETURN (func\parallel_composed_function.final_function.range);
  END_IF;
  IF 'EXPLICIT_TABLE_FUNCTION' IN typenames THEN
    IF 'LISTED_REAL_DATA' IN typenames THEN
      RETURN (one_tuples_of (the_reals));
    END_IF;
    IF 'LISTED_INTEGER_DATA' IN typenames THEN
      RETURN (one_tuples_of (the_integers));
    END_IF;
    IF 'LISTED_LOGICAL_DATA' IN typenames THEN
      RETURN (one_tuples_of (the_logicals));
    END_IF;
    IF 'LISTED_STRING_DATA' IN typenames THEN
      RETURN (one_tuples_of (the_strings));
    END_IF;
    IF 'LISTED_COMPLEX_NUMBER_DATA' IN typenames THEN
      RETURN (one_tuples_of (the_complex_numbers));
    END_IF;
    IF 'LISTED_DATA' IN typenames THEN
      RETURN (one_tuples_of (func\listed_data.value_range));
    END_IF;
    IF 'EXTERNALLY_LISTED_DATA' IN typenames THEN
      RETURN (one_tuples_of (func\externally_listed_data.value_range));
    END_IF;
    IF 'LINEARIZED_TABLE_FUNCTION' IN typenames THEN
      RETURN (func\linearized_table_function.source.range);
    END_IF;
    IF 'BASIC_SPARSE_MATRIX' IN typenames THEN
      RETURN (func\basic_sparse_matrix.val.range);
    END_IF;
    -- Unreachable, as no other subtypes of explicit_table_function are permissible
    -- without first modifying this function to account for them.
    RETURN (?);
  END_IF;
  IF 'HOMOGENEOUS_LINEAR_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space
      (factor1 (func\homogeneous_linear_function.mat.range),
      func\homogeneous_linear_function.mat\explicit_table_function.shape
      [3 - func\homogeneous_linear_function.sum_index])));
  END_IF;
  IF 'GENERAL_LINEAR_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space
      (factor1 (func\general_linear_function.mat.range),
      func\general_linear_function.mat\explicit_table_function.shape
      [3 - func\general_linear_function.sum_index])));
  END_IF;
  IF 'B_SPLINE_BASIS' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space (the_reals,
      func\b_spline_basis.num_basis)));
  END_IF;
  IF 'B_SPLINE_FUNCTION' IN typenames THEN
    tspace := factor1 (func\b_spline_function.coef.domain);
    m := SIZEOF (func\b_spline_function.basis);
    n := space_dimension (tspace);
    IF m = n THEN
      RETURN (one_tuples_of (the_reals));
    END_IF;
    IF m = n - 1 THEN
      RETURN (one_tuples_of (make_uniform_product_space (the_reals,
        factor_space (tspace, n)\finite_integer_interval.size)));
    END_IF;
    tspace := extract_factors (tspace, m+1, n);
    RETURN (one_tuples_of (make_function_space (sc_equal, tspace, sc_subspace,
      number_superspace_of (func\b_spline_function.coef.range))));
  END_IF;
  IF 'RATIONALIZE_FUNCTION' IN typenames THEN
    tspace := factor1 (func\rationalize_function.fun.range);
    n := space_dimension (tspace);
    RETURN (one_tuples_of (make_uniform_product_space (number_superspace_of (
      factor1 (tspace)), n-1)));
  END_IF;
  IF 'PARTIAL_DERIVATIVE_FUNCTION' IN typenames THEN
    RETURN (drop_numeric_constraints (
      func\partial_derivative_function.derivand.range));
  END_IF;
  IF 'DEFINITE_INTEGRAL_FUNCTION' IN typenames THEN
    RETURN (drop_numeric_constraints (
      func\definite_integral_function.integrand.range));
  END_IF;
  IF 'ABSTRACTED_EXPRESSION_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of(values_space_of(func\abstracted_expression_function.expr)));
  END_IF;
  IF 'EXPRESSION_DENOTED_FUNCTION' IN typenames THEN
    RETURN (values_space_of (func\expression_denoted_function.expr)\function_space.
      range_argument);
  END_IF;
  IF 'IMPORTED_POINT_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space (the_reals,
      dimension_of (func\imported_point_function.geometry))));
  END_IF;
  IF 'IMPORTED_CURVE_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space (the_reals,
      dimension_of (func\imported_curve_function.geometry))));
  END_IF;
  IF 'IMPORTED_SURFACE_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space (the_reals,
      dimension_of (func\imported_surface_function.geometry))));
  END_IF;
  IF 'IMPORTED_VOLUME_FUNCTION' IN typenames THEN
    RETURN (one_tuples_of (make_uniform_product_space (the_reals,
      dimension_of (func\imported_volume_function.geometry))));
  END_IF;
  IF 'APPLICATION_DEFINED_FUNCTION' IN typenames THEN
    RETURN (func\application_defined_function.explicit_range);
  END_IF;
  -- Unreachable, as no other subtypes of maths_function are permissible without
  -- first modifying this function to account for them.
  RETURN (?);
END_FUNCTION;  -- derive_function_range
FUNCTION domain_from(ref : maths_space_or_function) : tuple_space;
  LOCAL
    typenames : SET OF STRING := stripped_typeof(ref);
    func      : maths_function;
  END_LOCAL;
  IF NOT EXISTS (ref) THEN  RETURN (?);  END_IF;
  IF 'TUPLE_SPACE' IN typenames THEN  RETURN (ref);                  END_IF;
  IF 'MATHS_SPACE' IN typenames THEN  RETURN (one_tuples_of (ref));  END_IF;
  func := ref;
  IF 'CONSTANT_FUNCTION' IN typenames THEN
    RETURN (domain_from (func\constant_function.source_of_domain));
  END_IF;
  IF 'SELECTOR_FUNCTION' IN typenames THEN
    RETURN (domain_from (func\selector_function.source_of_domain));
  END_IF;
  IF 'PARALLEL_COMPOSED_FUNCTION' IN typenames THEN
    RETURN (domain_from (func\parallel_composed_function.source_of_domain));
  END_IF;
  RETURN (func.domain);
END_FUNCTION;  -- domain_from
FUNCTION dot_count(str : STRING) : INTEGER;
  LOCAL
    n : INTEGER := 0;
  END_LOCAL;
  REPEAT i := 1 TO LENGTH (str);
    IF str[i] = '.' THEN  n := n + 1;  END_IF;
  END_REPEAT;
  RETURN (n);
END_FUNCTION;  -- dot_count
FUNCTION dotted_identifiers_syntax(str : STRING) : BOOLEAN;
  LOCAL
    k : positive_integer;
    m : positive_integer;
  END_LOCAL;
  IF NOT EXISTS (str) THEN  RETURN (FALSE);  END_IF;
  k := parse_express_identifier (str, 1);
  IF k = 1 THEN  RETURN (FALSE);  END_IF;
  REPEAT WHILE k <= LENGTH (str);
    IF (str[k] <> '.') OR (k = LENGTH (str)) THEN  RETURN (FALSE);  END_IF;
    m := parse_express_identifier (str, k+1);
    IF m = k + 1 THEN  RETURN (FALSE);  END_IF;
    k := m;
  END_REPEAT;
  RETURN (TRUE);
END_FUNCTION;  -- dotted_identifiers_syntax
FUNCTION drop_numeric_constraints(spc : maths_space) : maths_space;
  LOCAL
    typenames : SET OF STRING := stripped_typeof(spc);
    tspc : listed_product_space;
    factors : LIST OF maths_space := [];
    xspc : extended_tuple_space;
  END_LOCAL;
  IF 'UNIFORM_PRODUCT_SPACE' IN typenames THEN
    RETURN (make_uniform_product_space (drop_numeric_constraints (
      spc\uniform_product_space.base), spc\uniform_product_space.exponent));
  END_IF;
  IF 'LISTED_PRODUCT_SPACE' IN typenames THEN
    tspc := spc;
    REPEAT i := 1 TO SIZEOF (tspc.factors);
      INSERT (factors, drop_numeric_constraints (tspc.factors[i]), i-1);
    END_REPEAT;
    RETURN (make_listed_product_space (factors));
  END_IF;
  IF 'EXTENDED_TUPLE_SPACE' IN typenames THEN
    xspc := spc;
    RETURN (make_extended_tuple_space (drop_numeric_constraints (xspc.base),
      drop_numeric_constraints (xspc.extender)));
  END_IF;
  IF subspace_of_es (spc, es_numbers) THEN
    RETURN (number_superspace_of (spc));
  END_IF;
  RETURN (spc);
END_FUNCTION;  -- drop_numeric_constraints
FUNCTION enclose_cregion_in_pregion(crgn   : cartesian_complex_number_region;
                                    centre : complex_number_literal)
                                   : polar_complex_number_region;
  -- Find equivalent direction in range -PI < a <= PI.
  FUNCTION angle(a : REAL) : REAL;
    REPEAT WHILE a > PI;    a := a - 2.0*PI;  END_REPEAT;
    REPEAT WHILE a <= -PI;  a := a + 2.0*PI;  END_REPEAT;
    RETURN (a);
  END_FUNCTION;
  -- Determine whether a real is strictly within a real interval
  FUNCTION strictly_in(z    : REAL;
                       zitv : real_interval) : LOGICAL;
    RETURN ((NOT min_exists(zitv) OR (z > real_min(zitv))) AND
      (NOT max_exists(zitv) OR (z < real_max(zitv))));
  END_FUNCTION;
  -- Include direction in minmax collection
  PROCEDURE angle_minmax(    ab, a            : REAL;
                             a_in             : BOOLEAN;
                         VAR amin, amax       : REAL;
                         VAR amin_in, amax_in : BOOLEAN);
    a := angle(a - ab);
    IF amin = a THEN  amin_in := amin_in OR a_in;  END_IF;
    IF amin > a THEN  amin := a;  amin_in := a_in;  END_IF;
    IF amax = a THEN  amax_in := amax_in OR a_in;  END_IF;
    IF amax < a THEN  amax := a;  amax_in := a_in;  END_IF;
  END_PROCEDURE;
  -- Include distance in max collection
  PROCEDURE range_max(    r       : REAL;
                          incl    : BOOLEAN;
                      VAR rmax    : REAL;
                      VAR rmax_in : BOOLEAN);
    IF rmax = r THEN  rmax_in := rmax_in OR incl;   END_IF;
    IF rmax < r THEN  rmax := r;  rmax_in := incl;  END_IF;
  END_PROCEDURE;
  -- Include distance in min collection
  PROCEDURE range_min(    r       : REAL;
                          incl    : BOOLEAN;
                      VAR rmin    : REAL;
                      VAR rmin_in : BOOLEAN);
    IF rmin = r THEN  rmin_in := rmin_in OR incl;  END_IF;
    IF (rmin < 0.0) OR (rmin > r) THEN  rmin := r;  rmin_in := incl;  END_IF;
  END_PROCEDURE;
  LOCAL
    xitv, yitv : real_interval;
    is_xmin, is_xmax, is_ymin, is_ymax : BOOLEAN;
    xmin, xmax, ymin, ymax, xc, yc : REAL := 0.0;
    xmin_in, xmax_in, ymin_in, ymax_in : BOOLEAN := FALSE;
    rmin, rmax : REAL := -1.0;
    amin : REAL := 4.0;
    amax : REAL := -4.0;
    rmax_exists, outside : BOOLEAN := TRUE;
    rmin_in, rmax_in, amin_in, amax_in : BOOLEAN := FALSE;
    ab, a, r : REAL := 0.0;
    incl : BOOLEAN;
    ritv : real_interval;
    aitv : finite_real_interval;
    minclo, maxclo : open_closed := open;
  END_LOCAL;
  IF NOT EXISTS (crgn) OR NOT EXISTS (centre) THEN  RETURN (?);  END_IF;
  -- Extract elementary input information
  xitv := crgn.real_constraint;
  yitv := crgn.imag_constraint;
  xc := centre.real_part;
  yc := centre.imag_part;
  is_xmin := min_exists(xitv);
  is_xmax := max_exists(xitv);
  is_ymin := min_exists(yitv);
  is_ymax := max_exists(yitv);
  IF is_xmin THEN  xmin := real_min(xitv);  xmin_in := min_included(xitv);  END_IF;
  IF is_xmax THEN  xmax := real_max(xitv);  xmax_in := max_included(xitv);  END_IF;
  IF is_ymin THEN  ymin := real_min(yitv);  ymin_in := min_included(yitv);  END_IF;
  IF is_ymax THEN  ymax := real_max(yitv);  ymax_in := max_included(yitv);  END_IF;
  rmax_exists := is_xmin AND is_xmax AND is_ymin AND is_ymax;
  -- Identify base direction with respect to which all relevant directions lie
  -- within +/- 0.5*PI, or that the centre lies properly inside crgn.
  IF       is_xmin AND (xc <= xmin)  THEN  ab := 0.0;
  ELSE IF  is_ymin AND (yc <= ymin)  THEN  ab := 0.5*PI;
  ELSE IF  is_ymax AND (yc >= ymax)  THEN  ab := -0.5*PI;
  ELSE IF  is_xmax AND (xc >= xmax)  THEN  ab := PI;
  ELSE                                     outside := FALSE;
  END_IF;  END_IF;  END_IF;  END_IF;
  IF NOT outside AND NOT rmax_exists THEN
    RETURN (?);  -- No enclosing polar region exists (requires whole plane)
  END_IF;
  -- Identify any closest point on a side but not a corner.
  IF       is_xmin AND (xc <= xmin) AND strictly_in(yc,yitv) THEN
    rmin := xmin - xc;  rmin_in := xmin_in;
  ELSE IF  is_ymin AND (yc <= ymin) AND strictly_in(xc,xitv) THEN
    rmin := ymin - yc;  rmin_in := ymin_in;
  ELSE IF  is_ymax AND (yc >= ymax) AND strictly_in(xc,xitv) THEN
    rmin := yc - ymax;  rmin_in := ymax_in;
  ELSE IF  is_xmax AND (xc >= xmax) AND strictly_in(yc,yitv) THEN
    rmin := xc - xmax;  rmin_in := xmax_in;
  END_IF;  END_IF;  END_IF;  END_IF;
  IF is_xmin THEN
    IF is_ymin THEN  -- Consider lower left corner
      r := SQRT((xmin-xc)**2 + (ymin-yc)**2);
      incl := xmin_in AND ymin_in;
      IF rmax_exists THEN  range_max(r,incl,rmax,rmax_in);  END_IF;
      IF outside THEN
        IF r > 0.0 THEN
          range_min(r,incl,rmin,rmin_in);
          a := angle(atan2(ymin-yc,xmin-xc) - ab);
          IF xc = xmin THEN  incl := xmin_in;  END_IF;
          IF yc = ymin THEN  incl := ymin_in;  END_IF;
          angle_minmax(ab,a,incl,amin,amax,amin_in,amax_in);
        ELSE  -- Centre at lower left corner
          rmin := 0.0;                rmin_in := xmin_in AND ymin_in;
          amin := angle(0.0-ab);      amin_in := ymin_in;
          amax := angle(0.5*PI-ab);   amax_in := xmin_in;
        END_IF;
      END_IF;
    ELSE IF xc <= xmin THEN  -- Consider points near (xmin, -infinity)
      angle_minmax(ab,-0.5*PI,(xc=xmin) AND xmin_in,amin,amax,amin_in,amax_in);
    END_IF;  END_IF;
    IF NOT is_ymax AND (xc <= xmin) THEN  -- Consider points near (xmin, +infinity)
      angle_minmax(ab,0.5*PI,(xc=xmin) AND xmin_in,amin,amax,amin_in,amax_in);
    END_IF;
  END_IF;
  IF is_ymin THEN
    IF is_xmax THEN  -- Consider lower right corner
      r := SQRT((xmax-xc)**2 + (ymin-yc)**2);
      incl := xmax_in AND ymin_in;
      IF rmax_exists THEN  range_max(r,incl,rmax,rmax_in);  END_IF;
      IF outside THEN
        IF r > 0.0 THEN
          range_min(r,incl,rmin,rmin_in);
          a := angle(atan2(ymin-yc,xmax-xc) - ab);
          IF xc = xmax THEN  incl := xmax_in;  END_IF;
          IF yc = ymin THEN  incl := ymin_in;  END_IF;
          angle_minmax(ab,a,incl,amin,amax,amin_in,amax_in);
        ELSE  -- Centre at lower right corner
          rmin := 0.0;                rmin_in := xmax_in AND ymin_in;
          amin := angle(0.5*PI-ab);   amin_in := ymin_in;
          amax := angle(PI-ab);       amax_in := xmax_in;
        END_IF;
      END_IF;
    ELSE IF yc <= ymin THEN  -- Consider points near (+infinity, ymin)
      angle_minmax(ab,0.0,(yc=ymin) AND ymin_in,amin,amax,amin_in,amax_in);
    END_IF;  END_IF;
    IF NOT is_xmin AND (yc <= ymin) THEN  -- Consider points near (-infinity, ymin)
      angle_minmax(ab,PI,(yc=ymin) AND ymin_in,amin,amax,amin_in,amax_in);
    END_IF;
  END_IF;
  IF is_xmax THEN
    IF is_ymax THEN  -- Consider upper right corner
      r := SQRT((xmax-xc)**2 + (ymax-yc)**2);
      incl := xmax_in AND ymax_in;
      IF rmax_exists THEN  range_max(r,incl,rmax,rmax_in);  END_IF;
      IF outside THEN
        IF r > 0.0 THEN
          range_min(r,incl,rmin,rmin_in);
          a := angle(atan2(ymax-yc,xmax-xc) - ab);
          IF xc = xmax THEN  incl := xmax_in;  END_IF;
          IF yc = ymax THEN  incl := ymax_in;  END_IF;
          angle_minmax(ab,a,incl,amin,amax,amin_in,amax_in);
        ELSE  -- Centre at lower left corner
          rmin := 0.0;                rmin_in := xmax_in AND ymax_in;
          amin := angle(-PI-ab);      amin_in := ymax_in;
          amax := angle(-0.5*PI-ab);  amax_in := xmax_in;
        END_IF;
      END_IF;
    ELSE IF xc >= xmax THEN  -- Consider points near (xmax, +infinity)
      angle_minmax(ab,0.5*PI,(xc=xmax) AND xmax_in,amin,amax,amin_in,amax_in);
    END_IF;  END_IF;
    IF NOT is_ymin AND (xc >= xmax) THEN  -- Consider points near (xmax, -infinity)
      angle_minmax(ab,-0.5*PI,(xc=xmax) AND xmax_in,amin,amax,amin_in,amax_in);
    END_IF;
  END_IF;
  IF is_ymax THEN
    IF is_xmin THEN  -- Consider upper left corner
      r := SQRT((xmin-xc)**2 + (ymax-yc)**2);
      incl := xmin_in AND ymax_in;
      IF rmax_exists THEN  range_max(r,incl,rmax,rmax_in);  END_IF;
      IF outside THEN
        IF r > 0.0 THEN
          range_min(r,incl,rmin,rmin_in);
          a := angle(atan2(ymax-yc,xmin-xc) - ab);
          IF xc = xmin THEN  incl := xmin_in;  END_IF;
          IF yc = ymax THEN  incl := ymax_in;  END_IF;
          angle_minmax(ab,a,incl,amin,amax,amin_in,amax_in);
        ELSE  -- Centre at lower right corner
          rmin := 0.0;                rmin_in := xmin_in AND ymax_in;
          amin := angle(0.5*PI-ab);   amin_in := ymax_in;
          amax := angle(PI-ab);       amax_in := xmin_in;
        END_IF;
      END_IF;
    ELSE IF yc >= ymax THEN  -- Consider points near (-infinity, ymax)
      angle_minmax(ab,PI,(yc=ymax) AND ymax_in,amin,amax,amin_in,amax_in);
    END_IF;  END_IF;
    IF NOT is_xmax AND (yc >= ymax) THEN  -- Consider points near (+infinity, ymax)
      angle_minmax(ab,0.0,(yc=ymax) AND ymax_in,amin,amax,amin_in,amax_in);
    END_IF;
  END_IF;
  IF outside THEN  -- Change direction origin from ab back to zero
    amin := angle(amin+ab);
    IF amin = PI THEN  amin := -PI;  END_IF;
    amax := angle(amax+ab);
    IF amax <= amin THEN  amax := amax + 2.0*PI;  END_IF;
  ELSE
    amin := -PI;  amin_in := FALSE;
    amax := PI;   amax_in := FALSE;
  END_IF;
  IF amin_in THEN  minclo := closed;  END_IF;
  IF amax_in THEN  maxclo := closed;  END_IF;
  aitv := make_finite_real_interval(amin,minclo,amax,maxclo);
  minclo := open;
  IF rmin_in THEN  minclo := closed;  END_IF;
  IF rmax_exists THEN
    maxclo := open;
    IF rmax_in THEN  maxclo := closed;  END_IF;
    ritv := make_finite_real_interval(rmin,minclo,rmax,maxclo);
  ELSE
    ritv := make_real_interval_from_min(rmin,minclo);
  END_IF;
  RETURN (make_polar_complex_number_region(centre,ritv,aitv));
END_FUNCTION;  -- enclose_cregion_in_pregion
FUNCTION enclose_pregion_in_cregion(prgn : polar_complex_number_region)
                                   : cartesian_complex_number_region;
  PROCEDURE nearest_good_direction(acart    : REAL;
                                   aitv     : finite_real_interval;
                                   VAR a    : REAL;
                                   VAR a_in : BOOLEAN);
    a := acart;                    a_in := TRUE;
    IF      a < aitv.min THEN
      -- a+2.0*PI > aitv.min automatically!
      IF a+2.0*PI < aitv.max THEN                               RETURN;  END_IF;
      IF a+2.0*PI = aitv.max THEN  a_in := max_included(aitv);  RETURN;  END_IF;
    ELSE IF a = aitv.min THEN      a_in := min_included(aitv);  RETURN;
    ELSE IF a < aitv.max THEN                                   RETURN;
    ELSE IF a = aitv.max THEN      a_in := max_included(aitv);  RETURN;
    END_IF;  END_IF;  END_IF;  END_IF;
    IF COS(acart - aitv.max) >= COS(acart - aitv.min) THEN
      a := aitv.max;               a_in := max_included(aitv);
    ELSE
      a := aitv.min;               a_in := min_included(aitv);
    END_IF;
  END_PROCEDURE;
  LOCAL
    xc, yc, xmin, xmax, ymin, ymax : REAL := 0.0;
    ritv, xitv, yitv : real_interval;
    aitv : finite_real_interval;
    xmin_exists, xmax_exists, ymin_exists, ymax_exists : BOOLEAN;
    xmin_in, xmax_in, ymin_in, ymax_in : BOOLEAN := FALSE;
    a, r : REAL := 0.0;
    a_in : BOOLEAN := FALSE;
    min_clo, max_clo : open_closed := open;
  END_LOCAL;
  IF NOT EXISTS (prgn) THEN  RETURN (?);  END_IF;
  -- Extract elementary input data
  xc := prgn.centre.real_part;
  yc := prgn.centre.imag_part;
  ritv := prgn.distance_constraint;
  aitv := prgn.direction_constraint;
  -- Determine xmin data
  nearest_good_direction(PI,aitv,a,a_in);
  IF COS(a) >= 0.0 THEN
    xmin_exists := TRUE;
    xmin := xc + real_min(ritv)*COS(a);
    xmin_in := a_in AND (min_included(ritv) OR (COS(a) = 0.0));
  ELSE
    IF max_exists(ritv) THEN
      xmin_exists := TRUE;
      xmin := xc + real_max(ritv)*COS(a);
      xmin_in := a_in AND max_included(ritv);
    ELSE
      xmin_exists := FALSE;
    END_IF;
  END_IF;
  -- Determine xmax data
  nearest_good_direction(0.0,aitv,a,a_in);
  IF COS(a) <= 0.0 THEN
    xmax_exists := TRUE;
    xmax := xc + real_min(ritv)*COS(a);
    xmax_in := a_in AND (min_included(ritv) OR (COS(a) = 0.0));
  ELSE
    IF max_exists(ritv) THEN
      xmax_exists := TRUE;
      xmax := xc + real_max(ritv)*COS(a);
      xmax_in := a_in AND max_included(ritv);
    ELSE
      xmax_exists := FALSE;
    END_IF;
  END_IF;
  -- Determine ymin data
  nearest_good_direction(-0.5*PI,aitv,a,a_in);
  IF SIN(a) >= 0.0 THEN
    ymin_exists := TRUE;
    ymin := yc + real_min(ritv)*SIN(a);
    ymin_in := a_in AND (min_included(ritv) OR (SIN(a) = 0.0));
  ELSE
    IF max_exists(ritv) THEN
      ymin_exists := TRUE;
      ymin := yc + real_max(ritv)*SIN(a);
      ymin_in := a_in AND max_included(ritv);
    ELSE
      ymin_exists := FALSE;
    END_IF;
  END_IF;
  -- Determine ymax data
  nearest_good_direction(0.5*PI,aitv,a,a_in);
  IF SIN(a) <= 0.0 THEN
    ymax_exists := TRUE;
    ymax := yc + real_min(ritv)*SIN(a);
    ymax_in := a_in AND (min_included(ritv) OR (SIN(a) = 0.0));
  ELSE
    IF max_exists(ritv) THEN
      ymax_exists := TRUE;
      ymax := yc + real_max(ritv)*SIN(a);
      ymax_in := a_in AND max_included(ritv);
    ELSE
      ymax_exists := FALSE;
    END_IF;
  END_IF;
  -- Construct result
  IF NOT (xmin_exists OR xmax_exists OR ymin_exists OR ymax_exists) THEN
    RETURN (?);  -- No finite boundaries exist
  END_IF;
  -- Construct real_constraint
  IF xmin_exists THEN
    IF xmin_in THEN  min_clo := closed;  ELSE  min_clo := open;  END_IF;
    IF xmax_exists THEN
      IF xmax_in THEN  max_clo := closed;  ELSE  max_clo := open;  END_IF;
      xitv := make_finite_real_interval(xmin,min_clo,xmax,max_clo);
    ELSE
      xitv := make_real_interval_from_min(xmin,min_clo);
    END_IF;
  ELSE
    IF xmax_exists THEN
      IF xmax_in THEN  max_clo := closed;  ELSE  max_clo := open;  END_IF;
      xitv := make_real_interval_to_max(xmax,max_clo);
    ELSE
      xitv := the_reals;
    END_IF;
  END_IF;
  -- Construct imag_constraint
  IF ymin_exists THEN
    IF ymin_in THEN  min_clo := closed;  ELSE  min_clo := open;  END_IF;
    IF ymax_exists THEN
      IF ymax_in THEN  max_clo := closed;  ELSE  max_clo := open;  END_IF;
      yitv := make_finite_real_interval(ymin,min_clo,ymax,max_clo);
    ELSE
      yitv := make_real_interval_from_min(ymin,min_clo);
    END_IF;
  ELSE
    IF ymax_exists THEN
      IF ymax_in THEN  max_clo := closed;  ELSE  max_clo := open;  END_IF;
      yitv := make_real_interval_to_max(ymax,max_clo);
    ELSE
      yitv := the_reals;
    END_IF;
  END_IF;
  -- Construct cartesian region
  RETURN (make_cartesian_complex_number_region(xitv,yitv));
END_FUNCTION;  -- enclose_pregion_in_cregion
FUNCTION enclose_pregion_in_pregion(prgn   : polar_complex_number_region;
                                    centre : complex_number_literal)
                                   : polar_complex_number_region;
  -- Find equivalent direction in range -PI < a <= PI.
  FUNCTION angle(a : REAL) : REAL;
    REPEAT WHILE a > PI;    a := a - 2.0*PI;  END_REPEAT;
    REPEAT WHILE a <= -PI;  a := a + 2.0*PI;  END_REPEAT;
    RETURN (a);
  END_FUNCTION;
  -- Find proper limits for direction interval
  PROCEDURE angle_range(VAR amin, amax : REAL);
    amin := angle(amin);
    IF amin = PI THEN  amin := -PI;  END_IF;
    amax := angle(amax);
    IF amax <= amin THEN  amax := amax + 2.0*PI;  END_IF;
  END_PROCEDURE;
  -- Determine whether a direction is strictly within a direction interval
  FUNCTION strictly_in(a    : REAL;
                       aitv : finite_real_interval) : LOGICAL;
    a := angle(a);
    RETURN ({aitv.min < a < aitv.max} OR {aitv.min < a+2.0*PI < aitv.max});
  END_FUNCTION;
  -- Find min and max and related inclusion booleans among four candidates,
  -- using a base direction chosen to ensure the algebraic comparisons are valid.
  PROCEDURE find_aminmax(    ab,a0,a1,a2,a3  : REAL;
                             in0,in1,in2,in3 : BOOLEAN;
                         VAR amin,amax       : REAL;
                         VAR amin_in,amax_in : BOOLEAN);
    LOCAL
      a : REAL;
    END_LOCAL;
    amin := angle(a0-ab);                  amin_in := in0;
    amax := amin;                          amax_in := in0;
    a := angle(a1-ab);
    IF a = amin THEN                       amin_in := amin_in OR in1;  END_IF;
    IF a < amin THEN  amin := a;           amin_in := in1;             END_IF;
    IF a = amax THEN                       amax_in := amax_in OR in1;  END_IF;
    IF a > amax THEN  amax := a;           amax_in := in1;             END_IF;
    a := angle(a2-ab);
    IF a = amin THEN                       amin_in := amin_in OR in2;  END_IF;
    IF a < amin THEN  amin := a;           amin_in := in2;             END_IF;
    IF a = amax THEN                       amax_in := amax_in OR in2;  END_IF;
    IF a > amax THEN  amax := a;           amax_in := in2;             END_IF;
    a := angle(a3-ab);
    IF a = amin THEN                       amin_in := amin_in OR in3;  END_IF;
    IF a < amin THEN  amin := a;           amin_in := in3;             END_IF;
    IF a = amax THEN                       amax_in := amax_in OR in3;  END_IF;
    IF a > amax THEN  amax := a;           amax_in := in3;             END_IF;
    amin := amin+ab;
    amax := amax+ab;
    angle_range(amin,amax);
  END_PROCEDURE;

  LOCAL
    ritp, ritv : real_interval;
    aitp, aitv : finite_real_interval;
    xp, yp, xc, yc, rmax, rmin, amin, amax, rc, acp, apc : REAL := 0.0;
    rmax_in, rmin_in, amin_in, amax_in : BOOLEAN := FALSE;
    rmxp, rmnp, x, y, r, a, ab, r0, a0, r1, a1, r2, a2, r3, a3 : REAL := 0.0;
    in0, in1, in2, in3, inn : BOOLEAN := FALSE;
    minclo, maxclo : open_closed := open;
  END_LOCAL;
  -- Extract elementary input information
  IF NOT EXISTS (prgn) OR NOT EXISTS (centre) THEN  RETURN (?);  END_IF;
  xp := prgn.centre.real_part;
  yp := prgn.centre.imag_part;
  ritp := prgn.distance_constraint;
  aitp := prgn.direction_constraint;
  xc := centre.real_part;
  yc := centre.imag_part;
  IF (xc = xp) AND (yc = yp) THEN  RETURN (prgn);  END_IF;
  rc := SQRT((xp-xc)**2 + (yp-yc)**2);
  acp := atan2(yp-yc,xp-xc);
  apc := atan2(yc-yp,xc-xp);
  rmnp := real_min(ritp);
  -- Analyse cases by existence of max distance and direction limits
  IF max_exists(ritp) THEN
    rmxp := real_max(ritp);
    IF aitp.max - aitp.min = 2.0*PI THEN
      -- annulus or disk, with or without slot or puncture
      inn := NOT max_included(aitp);  -- slot exists;
      a := angle(aitp.min);  -- slot direction
      rmax := rc+rmxp;                    rmax_in := max_included(ritp);
      IF inn AND (acp = a) THEN  rmax_in := FALSE;  END_IF;
      IF rc > rmxp THEN
        a0 := ASIN(rmxp/rc);
        amin := angle(acp-a0);            amin_in := max_included(ritp);
        IF amin = PI THEN  amin := -PI;  END_IF;
        amax := angle(acp+a0);            amax_in := amin_in;
        IF amax < amin THEN  amax := amax + 2.0*PI;  END_IF;
        rmin := rc-rmxp;                  rmin_in := amin_in;
        IF inn THEN
          -- slotted case
          IF apc = a THEN  rmin_in := FALSE;  END_IF;
          IF angle(amin+0.5*PI) = a THEN  amin_in := FALSE;  END_IF;
          IF angle(amax-0.5*PI) = a THEN  amax_in := FALSE;  END_IF;
        END_IF;
      ELSE IF rc = rmxp THEN
        amin := angle(acp-0.5*PI);        amin_in := FALSE;
        IF amin = PI THEN  amin := -PI;  END_IF;
        amax := angle(acp+0.5*PI);        amax_in := FALSE;
        IF amax < amin THEN  amax := amax + 2.0*PI;  END_IF;
        rmin := 0.0;                      rmin_in := max_included(ritp);
        IF inn AND (apc = a) THEN  rmin_in := FALSE;  END_IF;
      ELSE IF rc > rmnp THEN
        IF inn AND (apc = a) THEN  -- in the slot
          rmin := 0.0;                    rmin_in := FALSE;
          amin := aitp.min;               amin_in := FALSE;
          amax