Source : ISO 10303-50
SCHEMA mathematical_functions_schema;
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;;
cnlit : := STRING := schema_prefix + 'COMPLEX_NUMBER_LITERAL';;
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;
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;
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;
FUNCTION assoc_product_space
(ts1 : tuple_space; 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;
FUNCTION atan2
(y : REAL; 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;
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;
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;
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;
FUNCTION check_sparse_index_to_loc
(index_range : tuple_space; 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;
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;
FUNCTION compare_list_and_value
(lv : LIST OF GENERICG; op : elementary_function_enumerators; v : GENERIC) : 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;
FUNCTION compare_values
(v1 : GENERIC; op : elementary_function_enumerators; v2 : GENERIC) : 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;
FUNCTION compatible_complex_number_regions
(sp1 : maths_space; 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;
FUNCTION compatible_es_values
(esval1 : elementary_space_enumerators; 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;
FUNCTION compatible_intervals
(sp1 : maths_space; 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;
FUNCTION compatible_spaces
(sp1 : maths_space; 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;
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;
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;
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;
FUNCTION convert_to_maths_value
(val : GENERIC) : 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;
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;
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;
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;
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;
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;
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;
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;
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;
FUNCTION derive_finite_function_domain
(pairs : SET[1:?] OF LIST) : 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;
FUNCTION derive_finite_function_range
(pairs : SET[1:?] OF LIST) : 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;
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;
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;
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;
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;
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;
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;
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;
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;
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_FUNCTION;
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;
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;
FUNCTION equal_cregion_pregion
(crgn : cartesian_complex_number_region; prgn : polar_complex_number_region) : LOGICAL;
LOCAL arng, amin, xc, yc : REAL; aitv, xitv, yitv : real_interval; c_in : BOOLEAN; END_LOCAL; IF NOT EXISTS (crgn) OR NOT EXISTS (prgn) THEN RETURN (FALSE); END_IF; IF max_exists(prgn.distance_constraint) THEN RETURN (FALSE); END_IF; IF real_min(prgn.distance_constraint) <> 0.0 THEN RETURN (FALSE); END_IF; c_in := min_included(prgn.distance_constraint); aitv := prgn.direction_constraint; amin := aitv.min; arng := aitv.max - amin; xc := prgn.centre.real_part; yc := prgn.centre.imag_part; xitv := crgn.real_constraint; yitv := crgn.imag_constraint; IF arng = 0.5*PI THEN IF amin = 0.0 THEN -- quadrant to upper right RETURN (NOT max_exists(xitv) AND NOT max_exists(yitv) AND min_exists(xitv) AND min_exists(yitv) AND (real_min(xitv) = xc) AND (real_min(yitv) = yc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND min_included(xitv) AND min_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND max_included(aitv) AND min_included(xitv) AND NOT min_included(yitv)) OR (NOT c_in AND min_included(aitv) AND NOT max_included(aitv) AND NOT min_included(xitv) AND min_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT min_included(xitv) AND NOT min_included(yitv)))); END_IF; IF amin = 0.5*PI THEN -- quadrant to upper left RETURN (max_exists(xitv) AND NOT max_exists(yitv) AND NOT min_exists(xitv) AND min_exists(yitv) AND (real_max(xitv) = xc) AND (real_min(yitv) = yc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND max_included(xitv) AND min_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND max_included(aitv) AND max_included(xitv) AND NOT min_included(yitv)) OR (NOT c_in AND min_included(aitv) AND NOT max_included(aitv) AND NOT max_included(xitv) AND min_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT max_included(xitv) AND NOT min_included(yitv)))); END_IF; IF amin = -PI THEN -- quadrant to lower left RETURN (max_exists(xitv) AND max_exists(yitv) AND NOT min_exists(xitv) AND NOT min_exists(yitv) AND (real_max(xitv) = xc) AND (real_max(yitv) = yc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND max_included(xitv) AND max_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND max_included(aitv) AND max_included(xitv) AND NOT max_included(yitv)) OR (NOT c_in AND min_included(aitv) AND NOT max_included(aitv) AND NOT max_included(xitv) AND max_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT max_included(xitv) AND NOT max_included(yitv)))); END_IF; IF amin = -0.5*PI THEN -- quadrant to lower right RETURN (NOT max_exists(xitv) AND max_exists(yitv) AND min_exists(xitv) AND NOT min_exists(yitv) AND (real_min(xitv) = xc) AND (real_max(yitv) = yc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND min_included(xitv) AND max_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND max_included(aitv) AND min_included(xitv) AND NOT max_included(yitv)) OR (NOT c_in AND min_included(aitv) AND NOT max_included(aitv) AND NOT min_included(xitv) AND max_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT min_included(xitv) AND NOT max_included(yitv)))); END_IF; END_IF; IF arng = PI THEN IF amin = 0.0 THEN -- upper half space RETURN (NOT max_exists(xitv) AND NOT max_exists(yitv) AND NOT min_exists(xitv) AND min_exists(yitv) AND (real_min(yitv) = yc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND min_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT min_included(yitv)))); END_IF; IF amin = 0.5*PI THEN -- left half space RETURN (max_exists(xitv) AND NOT max_exists(yitv) AND NOT min_exists(xitv) AND NOT min_exists(yitv) AND (real_max(xitv) = xc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND max_included(xitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT max_included(xitv)))); END_IF; IF amin = -PI THEN -- lower half space RETURN (NOT max_exists(xitv) AND max_exists(yitv) AND NOT min_exists(xitv) AND NOT min_exists(yitv) AND (real_max(yitv) = yc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND max_included(yitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT max_included(yitv)))); END_IF; IF amin = -0.5*PI THEN -- right half space RETURN (NOT max_exists(xitv) AND NOT max_exists(yitv) AND min_exists(xitv) AND NOT min_exists(yitv) AND (real_min(xitv) = xc) AND ((c_in AND min_included(aitv) AND max_included(aitv) AND min_included(xitv)) OR (NOT c_in AND NOT min_included(aitv) AND NOT max_included(aitv) AND NOT min_included(xitv)))); END_IF; END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION equal_maths_functions
(fun1 : maths_function; fun2 : maths_function) : LOGICAL;
LOCAL cum : LOGICAL; END_LOCAL; IF fun1 = fun2 THEN RETURN (TRUE); END_IF; cum := equal_maths_spaces(fun1.domain,fun2.domain); IF cum = FALSE THEN RETURN (FALSE); END_IF; cum := cum AND equal_maths_spaces(fun1.range,fun2.range); IF cum = FALSE THEN RETURN (FALSE); END_IF; -- A lot of further analysis is possible, but not required. RETURN (UNKNOWN);
END_FUNCTION;
FUNCTION equal_maths_spaces
(spc1 : maths_space; spc2 : maths_space) : LOGICAL;
LOCAL spc1types : SET OF STRING := stripped_typeof(spc1); spc2types : SET OF STRING := stripped_typeof(spc2); set1, set2 : SET OF maths_value; cum : LOGICAL := TRUE; base : maths_space; expnt : INTEGER; factors : LIST OF maths_space; factors2 : LIST OF maths_space; fs1, fs2 : function_space; cum2 : LOGICAL; END_LOCAL; IF spc1 = spc2 THEN RETURN (TRUE); END_IF; -- Consider cases where it is not yet certain that spc1 <> spc2. IF 'FINITE_SPACE' IN spc1types THEN set1 := spc1\finite_space.members; IF 'FINITE_SPACE' IN spc2types THEN -- Members may have different but equivalent representations and in -- different orders. May also have disguised repeats in same set of members. set2 := spc2\finite_space.members; REPEAT i := 1 TO SIZEOF (set1); cum := cum AND member_of (set1[i], spc2); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; IF cum = TRUE THEN REPEAT i := 1 TO SIZEOF (set2); cum := cum AND member_of (set2[i], spc1); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; END_IF; RETURN (cum); END_IF; IF 'FINITE_INTEGER_INTERVAL' IN spc2types THEN set2 := []; REPEAT i := spc2\finite_integer_interval.min TO spc2\finite_integer_interval.max; set2 := set2 + [i]; END_REPEAT; RETURN (equal_maths_spaces(spc1,make_finite_space(set2))); END_IF; END_IF; IF ('FINITE_INTEGER_INTERVAL' IN spc1types) AND ('FINITE_SPACE' IN spc2types) THEN set1 := []; REPEAT i := spc1\finite_integer_interval.min TO spc1\finite_integer_interval.max; set1 := set1 + [i]; END_REPEAT; RETURN (equal_maths_spaces(make_finite_space(set1),spc2)); END_IF; IF ('CARTESIAN_COMPLEX_NUMBER_REGION' IN spc1types) AND ('POLAR_COMPLEX_NUMBER_REGION' IN spc2types) THEN -- Quadrants and half spaces have two representations RETURN (equal_cregion_pregion(spc1,spc2)); END_IF; IF ('POLAR_COMPLEX_NUMBER_REGION' IN spc1types) AND ('CARTESIAN_COMPLEX_NUMBER_REGION' IN spc2types) THEN -- Quadrants and half spaces have two representations RETURN (equal_cregion_pregion(spc2,spc1)); END_IF; IF 'UNIFORM_PRODUCT_SPACE' IN spc1types THEN base := spc1\uniform_product_space.base; expnt := spc1\uniform_product_space.exponent; IF 'UNIFORM_PRODUCT_SPACE' IN spc2types THEN IF expnt <> spc2\uniform_product_space.exponent THEN RETURN (FALSE); END_IF; RETURN (equal_maths_spaces(base,spc2\uniform_product_space.base)); END_IF; IF 'LISTED_PRODUCT_SPACE' IN spc2types THEN factors := spc2\listed_product_space.factors; IF expnt <> SIZEOF (factors) THEN RETURN (FALSE); END_IF; REPEAT i := 1 TO SIZEOF (factors); cum := cum AND equal_maths_spaces(base,factors[i]); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; END_IF; IF 'LISTED_PRODUCT_SPACE' IN spc1types THEN factors := spc1\listed_product_space.factors; IF 'UNIFORM_PRODUCT_SPACE' IN spc2types THEN IF spc2\uniform_product_space.exponent <> SIZEOF (factors) THEN RETURN (FALSE); END_IF; base := spc2\uniform_product_space.base; REPEAT i := 1 TO SIZEOF (factors); cum := cum AND equal_maths_spaces(base,factors[i]); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; IF 'LISTED_PRODUCT_SPACE' IN spc2types THEN factors2 := spc2\listed_product_space.factors; IF SIZEOF (factors) <> SIZEOF (factors2) THEN RETURN (FALSE); END_IF; REPEAT i := 1 TO SIZEOF (factors); cum := cum AND equal_maths_spaces(factors[i],factors2[i]); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; END_IF; IF ('EXTENDED_TUPLE_SPACE' IN spc1types) AND ('EXTENDED_TUPLE_SPACE' IN spc2types) THEN RETURN (equal_maths_spaces(spc1\extended_tuple_space.extender, spc2\extended_tuple_space.extender) AND equal_maths_spaces( spc1\extended_tuple_space.base, spc2\extended_tuple_space.base)); END_IF; IF ('FUNCTION_SPACE' IN spc1types) AND ('FUNCTION_SPACE' IN spc2types) THEN fs1 := spc1; fs2 := spc2; IF fs1.domain_constraint <> fs2.domain_constraint THEN IF (fs1.domain_constraint = sc_equal) OR (fs2.domain_constraint = sc_equal) THEN RETURN (FALSE); END_IF; IF (fs1.domain_constraint <> sc_subspace) THEN fs1 := spc2; fs2 := spc1; END_IF; IF (fs1.domain_constraint <> sc_subspace) OR (fs2.domain_constraint <> sc_member) THEN -- Safety check. Should be unreachable. RETURN (UNKNOWN); END_IF; IF any_space_satisfies(fs1.domain_constraint,fs1.domain_argument) <> any_space_satisfies(fs2.domain_constraint,fs2.domain_argument) THEN RETURN (FALSE); END_IF; IF NOT ('FINITE_SPACE' IN stripped_typeof(fs2.domain_argument)) THEN RETURN (FALSE); END_IF; IF SIZEOF (['FINITE_SPACE','FINITE_INTEGER_INTERVAL'] * stripped_typeof(fs1.domain_argument)) = 0 THEN RETURN (FALSE); END_IF; -- Remaining cases too complex. RETURN (UNKNOWN); END_IF; cum := equal_maths_spaces(fs1.domain_argument,fs2.domain_argument); IF cum = FALSE THEN RETURN (FALSE); END_IF; IF fs1.range_constraint <> fs2.range_constraint THEN IF (fs1.range_constraint = sc_equal) OR (fs2.range_constraint = sc_equal) THEN RETURN (FALSE); END_IF; IF (fs1.range_constraint <> sc_subspace) THEN fs1 := spc2; fs2 := spc1; END_IF; IF (fs1.range_constraint <> sc_subspace) OR (fs2.range_constraint <> sc_member) THEN -- Safety check. Should be unreachable. RETURN (UNKNOWN); END_IF; IF any_space_satisfies(fs1.range_constraint,fs1.range_argument) <> any_space_satisfies(fs2.range_constraint,fs2.range_argument) THEN RETURN (FALSE); END_IF; IF NOT ('FINITE_SPACE' IN stripped_typeof(fs2.range_argument)) THEN RETURN (FALSE); END_IF; IF SIZEOF (['FINITE_SPACE','FINITE_INTEGER_INTERVAL'] * stripped_typeof(fs1.range_argument)) = 0 THEN RETURN (FALSE); END_IF; -- Remaining cases too complex. RETURN (UNKNOWN); END_IF; cum := cum AND equal_maths_spaces(fs1.range_argument,fs2.range_argument); RETURN (cum); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION equal_maths_values
(val1 : maths_value; val2 : maths_value) : LOGICAL;
FUNCTION mem_of_vs(val1, val2 : maths_value) : LOGICAL; IF NOT has_values_space(val2) THEN RETURN (UNKNOWN); END_IF; IF NOT member_of(val1,values_space_of(val2)) THEN RETURN (FALSE); END_IF; RETURN (UNKNOWN);
END_FUNCTION;
FUNCTION es_subspace_of_es
(es1 : elementary_space_enumerators; es2 : elementary_space_enumerators) : BOOLEAN;
IF NOT EXISTS (es1) OR NOT EXISTS (es2) THEN RETURN (FALSE); END_IF; IF es1 = es2 THEN RETURN (TRUE); END_IF; IF es2 = es_generics THEN RETURN (TRUE); END_IF; IF (es1 = es_booleans) AND (es2 = es_logicals) THEN RETURN (TRUE); END_IF; IF (es2 = es_numbers) AND ((es1 = es_complex_numbers) OR (es1 = es_reals) OR (es1 = es_integers)) THEN RETURN (TRUE); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION expression_is_constant
(expr : generic_expression) : BOOLEAN;
RETURN (bool(SIZEOF (free_variables_of (expr)) = 0));
END_FUNCTION;
FUNCTION extract_factors
(tspace : tuple_space; m : INTEGER; n : INTEGER) : tuple_space;
LOCAL tsp : tuple_space := the_zero_tuple_space; END_LOCAL; REPEAT i := m TO n; tsp := assoc_product_space (tsp, factor_space (tspace, i)); END_REPEAT; RETURN (tsp);
END_FUNCTION;
FUNCTION extremal_position_check
(fun : linearized_table_function) : BOOLEAN;
LOCAL source_domain : maths_space; source_interval : finite_integer_interval; index : INTEGER := 1; base : INTEGER; shape : LIST OF positive_integer; ndim : positive_integer; slo, shi : INTEGER; sublo : LIST OF INTEGER := []; subhi : LIST OF INTEGER := []; END_LOCAL; IF NOT EXISTS (fun) THEN RETURN (FALSE); END_IF; source_domain := factor1 (fun.source.domain); IF (schema_prefix + 'TUPLE_SPACE') IN TYPEOF (source_domain) THEN source_domain := factor1 (source_domain); END_IF; IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (source_domain)) THEN RETURN (FALSE); END_IF; source_interval := source_domain; base := fun\explicit_table_function.index_base; shape := fun\explicit_table_function.shape; IF (schema_prefix + 'STANDARD_TABLE_FUNCTION') IN TYPEOF (fun) THEN REPEAT j := 1 TO SIZEOF (shape); index := index * shape[j]; END_REPEAT; index := fun.first + index - 1; RETURN (bool({source_interval.min <= index <= source_interval.max})); END_IF; IF (schema_prefix + 'REGULAR_TABLE_FUNCTION') IN TYPEOF (fun) THEN ndim := SIZEOF (fun\explicit_table_function.shape); REPEAT j:= 1 TO ndim; slo := base; shi := base + shape[j] - 1; IF fun\regular_table_function.increments[j] >= 0 THEN INSERT (sublo, slo, j-1); INSERT (subhi, shi, j-1); ELSE INSERT (sublo, shi, j-1); INSERT (subhi, slo, j-1); END_IF; END_REPEAT; index := regular_indexing (sublo, base, shape, fun\regular_table_function.increments, fun.first); IF NOT ({source_interval.min <= index <= source_interval.max}) THEN RETURN (FALSE); END_IF; index := regular_indexing (subhi, base, shape, fun\regular_table_function.increments, fun.first); IF NOT ({source_interval.min <= index <= source_interval.max}) THEN RETURN (FALSE); END_IF; RETURN (TRUE); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION factor1
(tspace : tuple_space) : maths_space;
LOCAL typenames : SET OF STRING := TYPEOF (tspace); END_LOCAL; IF (schema_prefix + 'UNIFORM_PRODUCT_SPACE') IN typenames THEN RETURN (tspace\uniform_product_space.base); END_IF; IF (schema_prefix + 'LISTED_PRODUCT_SPACE') IN typenames THEN RETURN (tspace\listed_product_space.factors[1]); -- This path could return the indeterminate value if the list is empty. -- This is the correct result for this case. END_IF; IF (schema_prefix + 'EXTENDED_TUPLE_SPACE') IN typenames THEN RETURN (factor1 (tspace\extended_tuple_space.base)); END_IF; -- Should not be reachable. RETURN (?);
END_FUNCTION;
FUNCTION factor_space
(tspace : tuple_space; idx : positive_integer) : maths_space;
LOCAL typenames : SET OF STRING := TYPEOF (tspace); END_LOCAL; IF (schema_prefix + 'UNIFORM_PRODUCT_SPACE') IN typenames THEN IF idx <= tspace\uniform_product_space.exponent THEN RETURN (tspace\uniform_product_space.base); END_IF; RETURN (?); END_IF; IF (schema_prefix + 'LISTED_PRODUCT_SPACE') IN typenames THEN IF idx <= SIZEOF (tspace\listed_product_space.factors) THEN RETURN (tspace\listed_product_space.factors[idx]); END_IF; RETURN (?); END_IF; IF (schema_prefix + 'EXTENDED_TUPLE_SPACE') IN typenames THEN IF idx <= space_dimension (tspace\extended_tuple_space.base) THEN RETURN (factor_space (tspace\extended_tuple_space.base, idx)); END_IF; RETURN (tspace\extended_tuple_space.extender); END_IF; -- Should not be reachable. RETURN (?);
END_FUNCTION;
FUNCTION free_variables_of
(expr : generic_expression) : SET OF generic_variable;
LOCAL typenames : SET OF STRING := stripped_typeof(expr); result : SET OF generic_variable := []; exprs : LIST OF generic_expression := []; END_LOCAL; IF 'GENERIC_LITERAL' IN typenames THEN RETURN (result); END_IF; IF 'GENERIC_VARIABLE' IN typenames THEN result := result + expr; RETURN (result); END_IF; IF 'QUANTIFIER_EXPRESSION' IN typenames THEN exprs := QUERY (ge <* expr\multiple_arity_generic_expression.operands | NOT (ge IN expr\quantifier_expression.variables)); REPEAT i := 1 TO SIZEOF (exprs); result := result + free_variables_of (exprs[i]); END_REPEAT; REPEAT i := 1 TO SIZEOF (expr\quantifier_expression.variables); result := result - expr\quantifier_expression.variables[i]; END_REPEAT; RETURN (result); END_IF; IF 'UNARY_GENERIC_EXPRESSION' IN typenames THEN RETURN (free_variables_of (expr\unary_generic_expression.operand)); END_IF; IF 'BINARY_GENERIC_EXPRESSION' IN typenames THEN result := free_variables_of (expr\binary_generic_expression.operands[1]); RETURN (result + free_variables_of (expr\binary_generic_expression.operands[2])); END_IF; IF 'MULTIPLE_ARITY_GENERIC_EXPRESSION' IN typenames THEN REPEAT i := 1 TO SIZEOF (expr\multiple_arity_generic_expression.operands); result := result + free_variables_of ( expr\multiple_arity_generic_expression.operands[i]); END_REPEAT; RETURN (result); END_IF; -- In this case the subtype shall not contain any variable (see IP1 in -- generic_expression). RETURN (result);
END_FUNCTION;
FUNCTION function_applicability
(func : maths_function_select; arguments : LIST[1:?] OF maths_value) : BOOLEAN;
LOCAL domain : tuple_space := convert_to_maths_function(func).domain; domain_types : SET OF STRING := TYPEOF (domain); narg : positive_integer := SIZEOF (arguments); arg : generic_expression; END_LOCAL; IF (schema_prefix + 'PRODUCT_SPACE') IN domain_types THEN IF space_dimension (domain) <> narg THEN RETURN (FALSE); END_IF; ELSE IF (schema_prefix + 'EXTENDED_TUPLE_SPACE') IN domain_types THEN IF space_dimension (domain) > narg THEN RETURN (FALSE); END_IF; ELSE RETURN (FALSE); -- Should be unreachable END_IF; END_IF; REPEAT i := 1 TO narg; arg := convert_to_operand (arguments[i]); IF NOT has_values_space (arg) THEN RETURN (FALSE); END_IF; IF NOT compatible_spaces (factor_space (domain, i), values_space_of (arg)) THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE);
END_FUNCTION;
FUNCTION function_is_1d_array
(func : maths_function) : BOOLEAN;
LOCAL temp : maths_space; END_LOCAL; IF NOT EXISTS (func) THEN RETURN (FALSE); END_IF; IF space_dimension (func.domain) <> 1 THEN RETURN (FALSE); END_IF; temp := factor1 (func.domain); IF (schema_prefix + 'PRODUCT_SPACE') IN TYPEOF (temp) THEN IF space_dimension (temp) <> 1 THEN RETURN (FALSE); END_IF; temp := factor1 (temp); END_IF; IF (schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp) THEN RETURN (TRUE); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION function_is_1d_table
(func : maths_function) : BOOLEAN;
LOCAL temp : maths_space; itvl : finite_integer_interval; END_LOCAL; IF NOT EXISTS (func) THEN RETURN (FALSE); END_IF; IF space_dimension (func.domain) <> 1 THEN RETURN (FALSE); END_IF; temp := factor1 (func.domain); IF (schema_prefix + 'PRODUCT_SPACE') IN TYPEOF (temp) THEN IF space_dimension (temp) <> 1 THEN RETURN (FALSE); END_IF; temp := factor1 (temp); END_IF; IF (schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp) THEN itvl := temp; RETURN (bool((itvl.min = 0) OR (itvl.min = 1))); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION function_is_2d_table
(func : maths_function) : BOOLEAN;
LOCAL temp : maths_space; pspace : product_space; itvl1, itvl2 : finite_integer_interval; END_LOCAL; IF NOT EXISTS (func) THEN RETURN (FALSE); END_IF; IF space_dimension (func.domain) <> 1 THEN RETURN (FALSE); END_IF; temp := factor1 (func.domain); IF NOT ('PRODUCT_SPACE' IN stripped_typeof(temp)) THEN RETURN (FALSE); END_IF; pspace := temp; IF space_dimension (pspace) <> 2 THEN RETURN (FALSE); END_IF; temp := factor1 (pspace); IF NOT ('FINITE_INTEGER_INTERVAL' IN stripped_typeof(temp)) THEN RETURN (FALSE); END_IF; itvl1 := temp; temp := factor_space (pspace, 2); IF NOT ('FINITE_INTEGER_INTERVAL' IN stripped_typeof(temp)) THEN RETURN (FALSE); END_IF; itvl2 := temp; RETURN (bool((itvl1.min = itvl2.min) AND ((itvl1.min = 0) OR (itvl1.min = 1))));
END_FUNCTION;
FUNCTION function_is_array
(func : maths_function) : BOOLEAN;
LOCAL tspace : tuple_space; temp : maths_space; END_LOCAL; IF NOT EXISTS (func) THEN RETURN (FALSE); END_IF; tspace := func.domain; IF (space_dimension (tspace) = 1) AND ((schema_prefix + 'TUPLE_SPACE') IN TYPEOF (factor1 (tspace))) THEN tspace := factor1 (tspace); END_IF; IF NOT ((schema_prefix + 'PRODUCT_SPACE') IN TYPEOF (tspace)) THEN RETURN (FALSE); END_IF; REPEAT i := 1 TO space_dimension (tspace); temp := factor_space (tspace, i); IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp)) THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE);
END_FUNCTION;
FUNCTION function_is_table
(func : maths_function) : BOOLEAN;
LOCAL tspace : tuple_space; temp : maths_space; base : INTEGER; END_LOCAL; IF NOT EXISTS (func) THEN RETURN (FALSE); END_IF; tspace := func.domain; IF (space_dimension (tspace) = 1) AND ((schema_prefix + 'TUPLE_SPACE') IN TYPEOF (factor1 (tspace))) THEN tspace := factor1 (tspace); END_IF; IF NOT ((schema_prefix + 'PRODUCT_SPACE') IN TYPEOF (tspace)) THEN RETURN (FALSE); END_IF; temp := factor1 (tspace); IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp)) THEN RETURN (FALSE); END_IF; base := temp\finite_integer_interval.min; IF (base <> 0) AND (base <> 1) THEN RETURN (FALSE); END_IF; REPEAT i := 2 TO space_dimension (tspace); temp := factor_space (tspace, i); IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp)) THEN RETURN (FALSE); END_IF; IF temp\finite_integer_interval.min <> base THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE);
END_FUNCTION;
FUNCTION has_values_space
(expr : generic_expression) : BOOLEAN;
LOCAL typenames : SET OF STRING := stripped_typeof (expr); END_LOCAL; IF 'EXPRESSION' IN typenames THEN RETURN (bool(('NUMERIC_EXPRESSION' IN typenames) OR ('STRING_EXPRESSION' IN typenames) OR ('BOOLEAN_EXPRESSION' IN typenames))); END_IF; IF 'MATHS_FUNCTION' IN typenames THEN RETURN (TRUE); END_IF; IF 'FUNCTION_APPLICATION' IN typenames THEN RETURN (TRUE); END_IF; IF 'MATHS_SPACE' IN typenames THEN RETURN (TRUE); END_IF; IF 'MATHS_VARIABLE' IN typenames THEN RETURN (TRUE); END_IF; IF 'DEPENDENT_VARIABLE_DEFINITION' IN typenames THEN RETURN (has_values_space (expr\unary_generic_expression.operand)); END_IF; IF 'COMPLEX_NUMBER_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'LOGICAL_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'BINARY_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'MATHS_ENUM_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'REAL_TUPLE_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'INTEGER_TUPLE_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'ATOM_BASED_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'MATHS_TUPLE_LITERAL' IN typenames THEN RETURN (TRUE); END_IF; IF 'PARTIAL_DERIVATIVE_EXPRESSION' IN typenames THEN RETURN (TRUE); END_IF; IF 'DEFINITE_INTEGRAL_EXPRESSION' IN typenames THEN RETURN (TRUE); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION list_selected_components
(aggr : LIST OF LIST; k : positive_integer) : LIST OF maths_value;
LOCAL result : LIST OF maths_value := []; j : INTEGER := 0; END_LOCAL; REPEAT i := LOINDEX (aggr) TO HIINDEX (aggr); IF k <= SIZEOF (aggr[i]) THEN INSERT (result, aggr[i][k], j); j := j + 1; END_IF; END_REPEAT; RETURN (result);
END_FUNCTION;
FUNCTION make_abstracted_expression_function
(operands : LIST[2:?] OF generic_expression) : abstracted_expression_function;
RETURN (abstracted_expression_function() || maths_function() || generic_expression() || quantifier_expression (remove_first (operands)) -- derived || multiple_arity_generic_expression (operands) );
END_FUNCTION;
FUNCTION make_atom_based_literal
(lit_value : atom_based_value) : atom_based_literal;
RETURN (atom_based_literal (lit_value) || generic_literal() || simple_generic_expression() || generic_expression() );
END_FUNCTION;
FUNCTION make_b_spline_basis
(degree : nonnegative_integer; repeated_knots : LIST[2:?] OF REAL) : b_spline_basis;
RETURN (b_spline_basis (degree, repeated_knots) || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_b_spline_function
(coef : maths_function; bases : LIST[1:?] OF b_spline_basis) : b_spline_function;
RETURN (b_spline_function (bases) || maths_function() || generic_expression() || unary_generic_expression (coef) );
END_FUNCTION;
FUNCTION make_banded_matrix
(index_base : zero_or_one; shape : LIST[1:?] OF positive_integer; source : maths_function; first : INTEGER; default_entry : maths_value; below : INTEGER; above : INTEGER; order : ordering_type) : banded_matrix;
RETURN (banded_matrix (default_entry, below, above, order) || linearized_table_function (first) || explicit_table_function (index_base, shape) || maths_function() || generic_expression() || unary_generic_expression (source) );
END_FUNCTION;
FUNCTION make_basic_sparse_matrix
(index_base : zero_or_one; shape : LIST[1:?] OF positive_integer; operands : LIST[3:3] OF maths_function; default_entry : maths_value; order : ordering_type) : basic_sparse_matrix;
RETURN (basic_sparse_matrix (default_entry, order) || explicit_table_function (index_base, shape) || maths_function() || generic_expression() || multiple_arity_generic_expression (operands) );
END_FUNCTION;
FUNCTION make_binary_literal
(lit_value : BINARY) : binary_literal;
RETURN (binary_literal (lit_value) || generic_literal() || simple_generic_expression() || generic_expression() );
END_FUNCTION;
FUNCTION make_boolean_literal
(lit_value : BOOLEAN) : boolean_literal;
RETURN (boolean_literal (lit_value) || simple_boolean_expression() || boolean_expression() || expression() || generic_expression() || simple_generic_expression() || generic_literal() );
END_FUNCTION;
FUNCTION make_cartesian_complex_number_region
(real_constraint : real_interval; imag_constraint : real_interval) : cartesian_complex_number_region;
RETURN (cartesian_complex_number_region (real_constraint, imag_constraint) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_complex_number_literal
(rpart : REAL; ipart : REAL) : complex_number_literal;
RETURN (complex_number_literal (rpart, ipart) || generic_literal() || simple_generic_expression() || generic_expression() );
END_FUNCTION;
FUNCTION make_constant_function
(sole_value : maths_value; src_of_domn : maths_space_or_function) : constant_function;
RETURN (constant_function (sole_value, src_of_domn) || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_cos_expression
(operand : numeric_expression) : cos_expression;
RETURN (cos_expression() || unary_numeric_call_expression() || unary_numeric_expression() || numeric_expression() || expression() || generic_expression() || unary_generic_expression (operand) );
END_FUNCTION;
FUNCTION make_definite_integral_expression
(operands : LIST[2:4] OF generic_expression; loinf : BOOLEAN; upinf : BOOLEAN) : definite_integral_expression;
RETURN (definite_integral_expression (loinf, upinf) || quantifier_expression ([operands[2]]) || multiple_arity_generic_expression (operands) || generic_expression() );
END_FUNCTION;
FUNCTION make_definite_integral_function
(integrand : maths_function; varintg : input_selector; loinf : BOOLEAN; upinf : BOOLEAN) : definite_integral_function;
RETURN (definite_integral_function (varintg, loinf, upinf) || maths_function() || generic_expression() || unary_generic_expression (integrand) );
END_FUNCTION;
FUNCTION make_elementary_function
(func_id : elementary_function_enumerators) : elementary_function;
RETURN (elementary_function (func_id) || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_elementary_space
(space_id : elementary_space_enumerators) : elementary_space;
RETURN (elementary_space (space_id) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_environment
(varbl : generic_variable; sem : variable_semantics) : environment;
RETURN (environment (varbl, sem) );
END_FUNCTION;
FUNCTION make_expression_denoted_function
(expression : generic_expression) : expression_denoted_function;
RETURN (expression_denoted_function() || maths_function() || generic_expression() || unary_generic_expression (expression) );
END_FUNCTION;
FUNCTION make_extended_tuple_space
(base : product_space; extender : maths_space) : extended_tuple_space;
RETURN (extended_tuple_space (base, extender) || maths_space () || generic_expression() || generic_literal () || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_finite_function
(pairs : SET[1:?] OF LIST) : finite_function;
RETURN (finite_function (pairs) || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_finite_integer_interval
(min : INTEGER; max : INTEGER) : finite_integer_interval;
RETURN (finite_integer_interval (min, max) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_finite_real_interval
(min : REAL; minclo : open_closed; max : REAL; maxclo : open_closed) : finite_real_interval;
RETURN (finite_real_interval (min, minclo, max, maxclo) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_finite_space
(members : SET OF maths_value) : finite_space;
RETURN (finite_space (members) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_function_application
(afunction : maths_function_select; arguments : LIST[1:?] OF maths_value) : function_application;
RETURN (function_application (afunction, arguments) || multiple_arity_generic_expression (convert_to_maths_function (afunction) + convert_to_operands (arguments)) -- derived || generic_expression() );
END_FUNCTION;
FUNCTION make_function_space
(domain_constraint : space_constraint_type; domain_argument : maths_space; range_constraint : space_constraint_type; range_argument : maths_space) : function_space;
RETURN (function_space (domain_constraint, domain_argument, range_constraint, range_argument) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_general_linear_function
(mat : maths_function; sum_index : one_or_two) : general_linear_function;
RETURN (general_linear_function (sum_index) || maths_function() || generic_expression() || unary_generic_expression (mat) );
END_FUNCTION;
FUNCTION make_int_literal
(lit_value : INTEGER) : int_literal;
RETURN (int_literal () || literal_number(lit_value) || simple_numeric_expression() || numeric_expression() || expression() || generic_expression() || simple_generic_expression() || generic_literal() );
END_FUNCTION;
FUNCTION make_integer_interval_from_min
(min : INTEGER) : integer_interval_from_min;
RETURN (integer_interval_from_min (min) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_listed_complex_number_data
(index_base : zero_or_one; values : LIST[2:?] OF REAL) : listed_complex_number_data;
RETURN (listed_complex_number_data (values) || explicit_table_function (index_base, [SIZEOF (values)/2]) -- 2nd derived || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_listed_data
(index_base : zero_or_one; values : LIST[2:?] OF maths_value; value_range : maths_space) : listed_data;
RETURN (listed_data (values, value_range) || explicit_table_function (index_base, [SIZEOF (values)]) -- 2nd derived || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_listed_integer_data
(index_base : zero_or_one; values : LIST[1:?] OF INTEGER) : listed_integer_data;
RETURN (listed_integer_data (values) || explicit_table_function (index_base, [SIZEOF (values)]) -- 2nd derived || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_listed_product_space
(factors : LIST OF maths_space) : listed_product_space;
RETURN (listed_product_space (factors) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_listed_real_data
(index_base : zero_or_one; values : LIST[1:?] OF REAL) : listed_real_data;
RETURN (listed_real_data (values) || explicit_table_function (index_base, [SIZEOF (values)]) -- 2nd derived || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_logical_literal
(lit_value : LOGICAL) : logical_literal;
RETURN (logical_literal (lit_value) || generic_literal() || simple_generic_expression() || generic_expression() );
END_FUNCTION;
FUNCTION make_maths_enum_literal
(lit_value : maths_enum_atom) : maths_enum_literal;
RETURN (maths_enum_literal (lit_value) || generic_literal() || simple_generic_expression() || generic_expression() );
END_FUNCTION;
FUNCTION make_maths_real_variable
(values_space : maths_space; name : label) : maths_real_variable;
RETURN (maths_real_variable() || maths_variable (values_space, name) || generic_variable() || simple_generic_expression() || generic_expression() || real_numeric_variable() || numeric_variable() || variable() );
END_FUNCTION;
FUNCTION make_maths_tuple_literal
(lit_value : LIST OF maths_value) : maths_tuple_literal;
RETURN (maths_tuple_literal (lit_value) || generic_literal() || simple_generic_expression() || generic_expression() );
END_FUNCTION;
FUNCTION make_mult_expression
(operands : LIST[2:?] OF generic_expression) : mult_expression;
RETURN (mult_expression() || multiple_arity_numeric_expression() || numeric_expression() || expression() || generic_expression() || multiple_arity_generic_expression (operands) );
END_FUNCTION;
FUNCTION make_parallel_composed_function
(srcdom : maths_space_or_function; prepfuncs : LIST[2:?] OF maths_function; finfunc : maths_function_select) : parallel_composed_function;
RETURN (parallel_composed_function (srcdom, prepfuncs, finfunc) || maths_function() || generic_expression() || multiple_arity_generic_expression (convert_to_operands_prcmfn ( srcdom, prepfuncs, finfunc)) ); -- derived
END_FUNCTION;
FUNCTION make_partial_derivative_expression
(derivand : generic_expression; dvars : LIST[1:?] OF maths_variable; extend : extension_options) : partial_derivative_expression;
RETURN (partial_derivative_expression (dvars, extend) || unary_generic_expression (derivand) || generic_expression() );
END_FUNCTION;
FUNCTION make_partial_derivative_function
(derivand : maths_function; dvars : LIST[1:?] OF input_selector; extend : extension_options) : partial_derivative_function;
RETURN (partial_derivative_function (dvars, extend) || maths_function() || generic_expression() || unary_generic_expression (derivand) );
END_FUNCTION;
FUNCTION make_polar_complex_number_region
(centre : complex_number_literal; dis_constraint : real_interval; dir_constraint : finite_real_interval) : polar_complex_number_region;
RETURN (polar_complex_number_region (centre, dis_constraint, dir_constraint) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_rationalize_function
(fun : maths_function) : rationalize_function;
RETURN (rationalize_function() || maths_function() || generic_expression() || unary_generic_expression (fun) );
END_FUNCTION;
FUNCTION make_real_interval_from_min
(min : REAL; minclo : open_closed) : real_interval_from_min;
RETURN (real_interval_from_min (min, minclo) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_real_interval_to_max
(max : REAL; maxclo : open_closed) : real_interval_to_max;
RETURN (real_interval_to_max (max, maxclo) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_real_literal
(lit_value : REAL) : real_literal;
RETURN (real_literal () || literal_number(lit_value) || simple_numeric_expression() || numeric_expression() || expression() || generic_expression() || simple_generic_expression() || generic_literal() );
END_FUNCTION;
FUNCTION make_regular_table_function
(index_base : zero_or_one; shape : LIST[1:?] OF positive_integer; operand : maths_function; first : INTEGER; increments : LIST[1:?] OF INTEGER) : regular_table_function;
RETURN (regular_table_function (increments) || linearized_table_function (first) || explicit_table_function (index_base, shape) || maths_function() || generic_expression() || unary_generic_expression (operand) );
END_FUNCTION;
FUNCTION make_reindexed_array_function
(func : maths_function; start_idxs : LIST[1:?] OF INTEGER) : reindexed_array_function;
RETURN (reindexed_array_function(start_idxs) || maths_function() || generic_expression() || unary_generic_expression (func) );
END_FUNCTION;
FUNCTION make_repackaging_function
(operand : maths_function; input_repack : repackage_options; output_repack : repackage_options; selected_output : nonnegative_integer) : repackaging_function;
RETURN (repackaging_function (input_repack, output_repack, selected_output) || maths_function() || generic_expression() || unary_generic_expression (operand) );
END_FUNCTION;
FUNCTION make_selector_function
(selector : input_selector; src_of_domn : maths_space_or_function) : selector_function;
RETURN (selector_function (selector, src_of_domn) || maths_function() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION make_series_composed_function
(functions : LIST[2:?] OF maths_function) : series_composed_function;
RETURN (series_composed_function() || maths_function() || generic_expression() || multiple_arity_generic_expression (functions) );
END_FUNCTION;
FUNCTION make_sin_expression
(operand : numeric_expression) : sin_expression;
RETURN (sin_expression() || unary_numeric_call_expression() || unary_numeric_expression() || numeric_expression() || expression() || generic_expression() || unary_generic_expression (operand) );
END_FUNCTION;
FUNCTION make_standard_table_function
(index_base : zero_or_one; shape : LIST[1:?] OF positive_integer; operand : maths_function; first : INTEGER; order : ordering_type) : standard_table_function;
RETURN (standard_table_function (order) || linearized_table_function (first) || explicit_table_function (index_base, shape) || maths_function() || generic_expression() || unary_generic_expression (operand) );
END_FUNCTION;
FUNCTION make_strict_triangular_matrix
(index_base : zero_or_one; shape : LIST[1:?] OF positive_integer; source : maths_function; first : INTEGER; default_entry : maths_value; lo_up : lower_upper; order : ordering_type; main_diagonal_value : maths_value) : strict_triangular_matrix;
RETURN (strict_triangular_matrix (main_diagonal_value) || triangular_matrix (default_entry, lo_up, order) || linearized_table_function (first) || explicit_table_function (index_base, shape) || maths_function() || generic_expression() || unary_generic_expression (source) );
END_FUNCTION;
FUNCTION make_string_literal
(lit_value : STRING) : string_literal;
RETURN (string_literal (lit_value) || simple_string_expression() || string_expression() || expression() || generic_expression() || simple_generic_expression() || generic_literal() );
END_FUNCTION;
FUNCTION make_unary_minus_expression
(operand : numeric_expression) : unary_minus_expression;
RETURN (unary_minus_expression() || unary_numeric_call_expression() || unary_numeric_expression() || numeric_expression() || expression() || generic_expression() || unary_generic_expression (operand) );
END_FUNCTION;
FUNCTION make_uniform_product_space
(base : maths_space; exponent : positive_integer) : uniform_product_space;
RETURN (uniform_product_space (base, exponent) || maths_space() || generic_expression() || generic_literal() || simple_generic_expression() );
END_FUNCTION;
FUNCTION max_exists
(spc : maths_space) : BOOLEAN;
LOCAL types : SET OF STRING := TYPEOF (spc); END_LOCAL; RETURN (bool(((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN types) OR ((schema_prefix + 'INTEGER_INTERVAL_TO_MAX') IN types) OR ((schema_prefix + 'FINITE_REAL_INTERVAL') IN types) OR ((schema_prefix + 'REAL_INTERVAL_TO_MAX') IN types)));
END_FUNCTION;
FUNCTION max_included
(spc : maths_space) : BOOLEAN;
LOCAL types : SET OF STRING := TYPEOF (spc); END_LOCAL; IF ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN types) OR ((schema_prefix + 'INTEGER_INTERVAL_TO_MAX') IN types) THEN RETURN (TRUE); END_IF; IF ((schema_prefix + 'FINITE_REAL_INTERVAL') IN types) THEN RETURN (bool(spc\finite_real_interval.max_closure = closed)); END_IF; IF ((schema_prefix + 'REAL_INTERVAL_TO_MAX') IN types) THEN RETURN (bool(spc\real_interval_to_max.max_closure = closed)); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION member_of
(val : GENERIC; spc : maths_space) : LOGICAL;
-- Trivial function introduced to avoid NIST Fedex compiler error FUNCTION fedex(val : AGGREGATE OF GENERIC:X; i : INTEGER) : GENERIC:X; RETURN (val[i]);
END_FUNCTION;
FUNCTION min_exists
(spc : maths_space) : BOOLEAN;
LOCAL types : SET OF STRING := TYPEOF (spc); END_LOCAL; RETURN (bool(((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN types) OR ((schema_prefix + 'INTEGER_INTERVAL_FROM_MIN') IN types) OR ((schema_prefix + 'FINITE_REAL_INTERVAL') IN types) OR ((schema_prefix + 'REAL_INTERVAL_FROM_MIN') IN types)));
END_FUNCTION;
FUNCTION min_included
(spc : maths_space) : BOOLEAN;
LOCAL types : SET OF STRING := TYPEOF (spc); END_LOCAL; IF ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN types) OR ((schema_prefix + 'INTEGER_INTERVAL_FROM_MIN') IN types) THEN RETURN (TRUE); END_IF; IF ((schema_prefix + 'FINITE_REAL_INTERVAL') IN types) THEN RETURN (bool(spc\finite_real_interval.min_closure = closed)); END_IF; IF ((schema_prefix + 'REAL_INTERVAL_FROM_MIN') IN types) THEN RETURN (bool(spc\real_interval_from_min.min_closure = closed)); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION no_cyclic_domain_reference
(ref : maths_space_or_function; used : SET OF maths_function) : BOOLEAN;
LOCAL typenames : SET OF STRING := TYPEOF (ref); func : maths_function; END_LOCAL; IF (NOT EXISTS (ref)) OR (NOT EXISTS (used)) THEN RETURN (FALSE); END_IF; IF (schema_prefix + 'MATHS_SPACE') IN typenames THEN RETURN (TRUE); END_IF; func := ref; IF func IN used THEN RETURN (FALSE); END_IF; IF (schema_prefix + 'CONSTANT_FUNCTION') IN typenames THEN RETURN (no_cyclic_domain_reference (func\constant_function.source_of_domain, used + [func])); END_IF; IF (schema_prefix + 'SELECTOR_FUNCTION') IN typenames THEN RETURN (no_cyclic_domain_reference (func\selector_function.source_of_domain, used + [func])); END_IF; IF (schema_prefix + 'PARALLEL_COMPOSED_FUNCTION') IN typenames THEN RETURN (no_cyclic_domain_reference ( func\parallel_composed_function.source_of_domain, used + [func])); END_IF; RETURN (TRUE);
END_FUNCTION;
FUNCTION no_cyclic_space_reference
(spc : maths_space; refs : SET OF maths_space) : BOOLEAN;
LOCAL types : SET OF STRING; refs_plus : SET OF maths_space; END_LOCAL; IF (spc IN refs) THEN RETURN (FALSE); END_IF; types := TYPEOF (spc); refs_plus := refs + spc; IF (schema_prefix + 'FINITE_SPACE') IN types THEN RETURN (bool(SIZEOF (QUERY (sp <* QUERY (mem <* spc\finite_space.members | (schema_prefix + 'MATHS_SPACE') IN TYPEOF (mem)) | NOT no_cyclic_space_reference (sp, refs_plus))) = 0)); END_IF; IF (schema_prefix + 'UNIFORM_PRODUCT_SPACE') IN types THEN RETURN (no_cyclic_space_reference (spc\uniform_product_space.base, refs_plus)); END_IF; IF (schema_prefix + 'LISTED_PRODUCT_SPACE') IN types THEN RETURN (bool(SIZEOF (QUERY (fac <* spc\listed_product_space.factors | NOT no_cyclic_space_reference (fac, refs_plus))) = 0)); END_IF; IF (schema_prefix + 'EXTENDED_TUPLE_SPACE') IN types THEN RETURN (no_cyclic_space_reference (spc\extended_tuple_space.base, refs_plus) AND no_cyclic_space_reference (spc\extended_tuple_space.extender, refs_plus)); END_IF; -- spc contains no references to other spaces RETURN (TRUE);
END_FUNCTION;
FUNCTION nondecreasing
(lr : LIST OF REAL) : BOOLEAN;
IF NOT EXISTS (lr) THEN RETURN (FALSE); END_IF; REPEAT j := 2 TO SIZEOF (lr); IF lr[j] < lr[j-1] THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE);
END_FUNCTION;
FUNCTION number_superspace_of
(spc : maths_space) : elementary_space;
IF subspace_of_es(spc,es_integers) THEN RETURN (the_integers); END_IF; IF subspace_of_es(spc,es_reals) THEN RETURN (the_reals); END_IF; IF subspace_of_es(spc,es_complex_numbers) THEN RETURN (the_complex_numbers); END_IF; IF subspace_of_es(spc,es_numbers) THEN RETURN (the_numbers); END_IF; RETURN (?);
END_FUNCTION;
FUNCTION number_tuple_subspace_check
(spc : maths_space) : LOGICAL;
LOCAL types : SET OF STRING := stripped_typeof(spc); factors : LIST OF maths_space; cum : LOGICAL := TRUE; END_LOCAL; IF 'UNIFORM_PRODUCT_SPACE' IN types THEN RETURN (subspace_of_es(spc\uniform_product_space.base,es_numbers)); END_IF; IF 'LISTED_PRODUCT_SPACE' IN types THEN factors := spc\listed_product_space.factors; REPEAT i := 1 TO SIZEOF (factors); cum := cum AND subspace_of_es(factors[i],es_numbers); END_REPEAT; RETURN (cum); END_IF; IF 'EXTENDED_TUPLE_SPACE' IN types THEN cum := subspace_of_es(spc\extended_tuple_space.extender,es_numbers); cum := cum AND number_tuple_subspace_check(spc\extended_tuple_space.base); RETURN (cum); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION one_tuples_of
(spc : maths_space) : tuple_space;
RETURN (make_uniform_product_space (spc, 1));
END_FUNCTION;
FUNCTION parallel_composed_function_composability_check
(funcs : LIST OF maths_function; final : maths_function_select) : BOOLEAN;
LOCAL tplsp : tuple_space := the_zero_tuple_space; finfun : maths_function := convert_to_maths_function (final); END_LOCAL; REPEAT i := 1 TO SIZEOF (funcs); tplsp := assoc_product_space (tplsp, funcs[i].range); END_REPEAT; RETURN (compatible_spaces (tplsp, finfun.domain));
END_FUNCTION;
FUNCTION parallel_composed_function_domain_check
(comdom : tuple_space; funcs : LIST OF maths_function) : BOOLEAN;
REPEAT i := 1 TO SIZEOF (funcs); IF NOT (compatible_spaces (comdom, funcs[i].domain)) THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE);
END_FUNCTION;
FUNCTION parse_express_identifier
(s : STRING; i : positive_integer) : positive_integer;
LOCAL k : positive_integer; END_LOCAL; k := i; IF i <= LENGTH (s) THEN IF (s[i] LIKE '@') THEN REPEAT UNTIL (k > LENGTH (s)) OR ((s[k] <> '_') AND NOT (s[k] LIKE '@') AND NOT (s[k] LIKE '#')); k := k + 1; END_REPEAT; END_IF; END_IF; RETURN (k);
END_FUNCTION;
FUNCTION partial_derivative_check
(domain : tuple_space; d_vars : LIST[1:?] OF input_selector) : BOOLEAN;
LOCAL domn : tuple_space := domain; fspc : maths_space; dim : INTEGER; k : 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); REPEAT i := 1 TO SIZEOF (d_vars); k := d_vars[i]; IF k > dim THEN RETURN (FALSE); END_IF; fspc := factor_space (domn, k); IF (NOT subspace_of_es (fspc,es_reals)) AND (NOT subspace_of_es (fspc,es_complex_numbers)) THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE);
END_FUNCTION;
FUNCTION real_max
(spc : maths_space) : REAL;
LOCAL types : SET OF STRING := TYPEOF (spc); END_LOCAL; IF ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN types) THEN RETURN (spc\finite_integer_interval.max); END_IF; IF ((schema_prefix + 'INTEGER_INTERVAL_TO_MAX') IN types) THEN RETURN (spc\integer_interval_to_max.max); END_IF; IF ((schema_prefix + 'FINITE_REAL_INTERVAL') IN types) THEN RETURN (spc\finite_real_interval.max); END_IF; IF ((schema_prefix + 'REAL_INTERVAL_TO_MAX') IN types) THEN RETURN (spc\real_interval_to_max.max); END_IF; RETURN (?);
END_FUNCTION;
FUNCTION real_min
(spc : maths_space) : REAL;
LOCAL types : SET OF STRING := TYPEOF (spc); END_LOCAL; IF ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN types) THEN RETURN (spc\finite_integer_interval.min); END_IF; IF ((schema_prefix + 'INTEGER_INTERVAL_FROM_MIN') IN types) THEN RETURN (spc\integer_interval_from_min.min); END_IF; IF ((schema_prefix + 'FINITE_REAL_INTERVAL') IN types) THEN RETURN (spc\finite_real_interval.min); END_IF; IF ((schema_prefix + 'REAL_INTERVAL_FROM_MIN') IN types) THEN RETURN (spc\real_interval_from_min.min); END_IF; RETURN (?);
END_FUNCTION;
FUNCTION regular_indexing
(sub : LIST OF INTEGER; base : zero_or_one; shape : LIST[1:?] OF positive_integer; inc : LIST[1:?] OF INTEGER; first : INTEGER) : INTEGER;
LOCAL k : INTEGER; index : INTEGER; END_LOCAL; IF NOT EXISTS (sub) OR NOT EXISTS (base) OR NOT EXISTS (shape) OR NOT EXISTS (inc) OR NOT EXISTS (first) THEN RETURN (?); END_IF; IF (SIZEOF (sub) <> SIZEOF (inc)) OR (SIZEOF (sub) <> SIZEOF (shape)) THEN RETURN (?); END_IF; index := first; REPEAT j := 1 TO SIZEOF (sub); IF NOT EXISTS (sub[j]) OR NOT EXISTS (inc[j]) THEN RETURN (?); END_IF; k := sub[j] - base; IF NOT ({0 <= k < shape[j]}) THEN RETURN (?); END_IF; index := index + k*inc[j]; END_REPEAT; RETURN (index);
END_FUNCTION;
FUNCTION remove_first
(alist : LIST OF GENERICGEN) : LIST OF GENERICGEN;
LOCAL blist : LIST OF GENERIC:GEN := alist; END_LOCAL; IF SIZEOF (blist) > 0 THEN REMOVE (blist, 1); END_IF; RETURN (blist);
END_FUNCTION;
FUNCTION repackage
(tspace : tuple_space; repckg : repackage_options) : tuple_space;
CASE repckg OF ro_nochange : RETURN (tspace); ro_wrap_as_tuple : RETURN (one_tuples_of (tspace)); ro_unwrap_tuple : RETURN (factor1 (tspace)); OTHERWISE : RETURN (?); END_CASE;
END_FUNCTION;
FUNCTION shape_of_array
(func : maths_function) : LIST OF positive_integer;
LOCAL tspace : tuple_space; temp : maths_space; result : LIST OF positive_integer := []; END_LOCAL; IF (schema_prefix + 'EXPLICIT_TABLE_FUNCTION') IN TYPEOF (func) THEN RETURN (func\explicit_table_function.shape); END_IF; tspace := func.domain; IF (space_dimension (tspace) = 1) AND ((schema_prefix + 'TUPLE_SPACE') IN TYPEOF (factor1 (tspace))) THEN tspace := factor1 (tspace); END_IF; REPEAT i := 1 TO space_dimension (tspace); temp := factor_space (tspace, i); IF NOT ((schema_prefix + 'FINITE_INTEGER_INTERVAL') IN TYPEOF (temp)) THEN RETURN (?); END_IF; INSERT (result, temp\finite_integer_interval.size, i-1); END_REPEAT; RETURN (result);
END_FUNCTION;
FUNCTION simplify_function_application
(expr : function_application) : maths_value;
FUNCTION ctmv(x : GENERIC:G) : maths_value; RETURN (convert_to_maths_value(x));
END_FUNCTION;
FUNCTION makec
(x : REAL; y : REAL) : complex_number_literal;
RETURN (make_complex_number_literal(x,y));
END_FUNCTION;
FUNCTION good_t
(v : maths_value; tn : STRING) : BOOLEAN;
LOCAL tpl : LIST OF maths_value; END_LOCAL; IF 'LIST' IN TYPEOF (v) THEN tpl := v; REPEAT i := 1 TO SIZEOF (tpl); IF NOT (tn IN TYPEOF (tpl[i])) THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION simplify_generic_expression
(expr : generic_expression) : maths_value;
FUNCTION restore_unary(expr : unary_generic_expression; opnd : generic_expression) : generic_expression; expr.operand := opnd; RETURN (expr);
END_FUNCTION;
FUNCTION restore_binary
(expr : binary_generic_expression; opd1 : generic_expression; opd2 : generic_expression) : generic_expression;
expr.operands[1] := opd1; expr.operands[2] := opd2; RETURN (expr);
END_FUNCTION;
FUNCTION restore_mulary
(expr : multiple_arity_generic_expression; ops : LIST OF generic_expression) : generic_expression;
expr.operands := ops; RETURN (expr);
END_FUNCTION;
FUNCTION make_number_literal
(nmb : NUMBER) : generic_literal;
IF 'INTEGER' IN TYPEOF (nmb) THEN RETURN (make_int_literal(nmb)); END_IF; RETURN (make_real_literal(nmb));
END_FUNCTION;
FUNCTION simplify_maths_space
(spc : maths_space) : maths_space;
LOCAL stypes : SET OF STRING := stripped_typeof (spc); sset : SET OF maths_value; zset : SET OF maths_value := []; zval : maths_value; zspc : maths_space; zallint : BOOLEAN := TRUE; zint, zmin, zmax : INTEGER; factors : LIST OF maths_space; zfactors : LIST OF maths_space := []; rspc : maths_space; END_LOCAL; IF 'FINITE_SPACE' IN stypes THEN sset := spc\finite_space.members; REPEAT i := 1 TO SIZEOF (sset); zval := simplify_maths_value(sset[i]); zset := zset + [zval]; IF zallint AND ('INTEGER' IN TYPEOF (zval)) THEN zint := zval; IF i = 1 THEN zmin := zint; zmax := zint; ELSE IF zint < zmin THEN zmin := zint; END_IF; IF zint > zmax THEN zmax := zint; END_IF; END_IF; ELSE zallint := FALSE; END_IF; END_REPEAT; IF zallint AND (SIZEOF(zset) = zmax-zmin+1) THEN RETURN (make_finite_integer_interval(zmin,zmax)); END_IF; RETURN (make_finite_space(zset)); END_IF; IF 'UNIFORM_PRODUCT_SPACE' IN stypes THEN zspc := simplify_maths_space(spc\uniform_product_space.base); RETURN (make_uniform_product_space(zspc,spc\uniform_product_space.exponent)); END_IF; IF 'LISTED_PRODUCT_SPACE' IN stypes THEN factors := spc\listed_product_space.factors; REPEAT i := 1 TO SIZEOF (factors); INSERT (zfactors, simplify_maths_space(factors[i]), i-1); END_REPEAT; RETURN (make_listed_product_space(zfactors)); END_IF; IF 'EXTENDED_TUPLE_SPACE' IN stypes THEN zspc := simplify_maths_space(spc\extended_tuple_space.base); rspc := simplify_maths_space(spc\extended_tuple_space.extender); RETURN (make_extended_tuple_space(zspc,rspc)); END_IF; IF 'FUNCTION_SPACE' IN stypes THEN zspc := simplify_maths_space(spc\function_space.domain_argument); rspc := simplify_maths_space(spc\function_space.range_argument); RETURN (make_function_space(spc\function_space.domain_constraint,zspc, spc\function_space.range_constraint,rspc)); END_IF; RETURN (spc);
END_FUNCTION;
FUNCTION simplify_maths_value
(val : maths_value) : maths_value;
LOCAL vtypes : SET OF STRING := stripped_typeof(val); vlist : LIST OF maths_value; nlist : LIST OF maths_value := []; END_LOCAL; IF 'GENERIC_EXPRESSION' IN vtypes THEN RETURN (simplify_generic_expression(val)); END_IF; IF 'LIST' IN vtypes THEN vlist := val; REPEAT i := 1 TO SIZEOF (vlist); INSERT (nlist, simplify_maths_value(vlist[i]), i-1); END_REPEAT; RETURN (convert_to_maths_value(nlist)); END_IF; RETURN (val);
END_FUNCTION;
FUNCTION singleton_member_of
(spc : maths_space) : maths_value;
LOCAL types : SET OF STRING := stripped_typeof (spc); END_LOCAL; IF 'FINITE_SPACE' IN types THEN IF SIZEOF (spc\finite_space.members) = 1 THEN RETURN (spc\finite_space.members[1]); END_IF; RETURN (?); END_IF; IF 'FINITE_INTEGER_INTERVAL' IN types THEN IF spc\finite_integer_interval.size = 1 THEN RETURN (spc\finite_integer_interval.min); END_IF; RETURN (?); END_IF; RETURN (?);
END_FUNCTION;
FUNCTION space_dimension
(tspace : tuple_space) : nonnegative_integer;
LOCAL types : SET OF STRING := TYPEOF (tspace); END_LOCAL; IF (schema_prefix + 'UNIFORM_PRODUCT_SPACE') IN types THEN RETURN (tspace\uniform_product_space.exponent); END_IF; IF (schema_prefix + 'LISTED_PRODUCT_SPACE') IN types THEN RETURN (SIZEOF (tspace\listed_product_space.factors)); END_IF; IF (schema_prefix + 'EXTENDED_TUPLE_SPACE') IN types THEN -- In the case of an extended_tuple_space, the minimum dimension is returned. RETURN (space_dimension (tspace\extended_tuple_space.base)); END_IF; -- Should be unreachable RETURN (?);
END_FUNCTION;
FUNCTION space_is_continuum
(space : maths_space) : BOOLEAN;
LOCAL typenames : SET OF STRING := TYPEOF (space); factors : LIST OF maths_space; END_LOCAL; IF NOT EXISTS (space) THEN RETURN (FALSE); END_IF; IF subspace_of_es(space,es_reals) OR subspace_of_es(space,es_complex_numbers) THEN RETURN (TRUE); END_IF; IF (schema_prefix + 'UNIFORM_PRODUCT_SPACE') IN typenames THEN RETURN (space_is_continuum(space\uniform_product_space.base)); END_IF; IF (schema_prefix + 'LISTED_PRODUCT_SPACE') IN typenames THEN factors := space\listed_product_space.factors; IF SIZEOF(factors) = 0 THEN RETURN (FALSE); END_IF; REPEAT i := 1 TO SIZEOF (factors); IF NOT space_is_continuum(factors[i]) THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (TRUE); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION space_is_singleton
(spc : maths_space) : BOOLEAN;
LOCAL types : SET OF STRING := stripped_typeof (spc); END_LOCAL; IF 'FINITE_SPACE' IN types THEN RETURN (bool(SIZEOF (spc\finite_space.members) = 1)); END_IF; IF 'FINITE_INTEGER_INTERVAL' IN types THEN RETURN (bool(spc\finite_integer_interval.size = 1)); END_IF; RETURN (FALSE);
END_FUNCTION;
FUNCTION stripped_typeof
(arg : GENERIC) : SET OF STRING;
LOCAL types : SET OF STRING := TYPEOF (arg); stypes : SET OF STRING := []; n : INTEGER := LENGTH (schema_prefix); END_LOCAL; REPEAT i := 1 TO SIZEOF (types); IF types[i][1:n] = schema_prefix THEN stypes := stypes + [types[i][n+1:LENGTH(types[i])]]; ELSE stypes := stypes + [types[i]]; END_IF; END_REPEAT; RETURN (stypes);
END_FUNCTION;
FUNCTION subspace_of
(space1 : maths_space; space2 : maths_space) : LOGICAL;
LOCAL spc1 : maths_space := simplify_maths_space(space1); spc2 : maths_space := simplify_maths_space(space2); types1 : SET OF STRING := stripped_typeof (spc1); types2 : SET OF STRING := stripped_typeof (spc2); lgcl, cum : LOGICAL; es_val : elementary_space_enumerators; bnd1, bnd2 : REAL; n : INTEGER; sp1, sp2 : maths_space; prgn1, prgn2 : polar_complex_number_region; aitv : finite_real_interval; END_LOCAL; IF NOT EXISTS (spc1) OR NOT EXISTS (spc2) THEN RETURN (FALSE); END_IF; IF spc2 = the_generics THEN RETURN (TRUE); END_IF; IF 'ELEMENTARY_SPACE' IN types1 THEN IF NOT ('ELEMENTARY_SPACE' IN types2) THEN RETURN (FALSE); END_IF; es_val := spc2\elementary_space.space_id; IF spc1\elementary_space.space_id = es_val THEN RETURN (TRUE); END_IF; -- Note that the cases (spc2=the_generics) and (spc1=spc2) have been handled. CASE spc1\elementary_space.space_id OF es_numbers : RETURN (FALSE); es_complex_numbers : RETURN (es_val = es_numbers); es_reals : RETURN (es_val = es_numbers); es_integers : RETURN (es_val = es_numbers); es_logicals : RETURN (FALSE); es_booleans : RETURN (es_val = es_logicals); es_strings : RETURN (FALSE); es_binarys : RETURN (FALSE); es_maths_spaces : RETURN (FALSE); es_maths_functions : RETURN (FALSE); es_generics : RETURN (FALSE); END_CASE; -- Should be unreachable. RETURN (UNKNOWN); END_IF; IF 'FINITE_INTEGER_INTERVAL' IN types1 THEN cum := TRUE; REPEAT i := spc1\finite_integer_interval.min TO spc1\finite_integer_interval.max; cum := cum AND member_of (i, spc2); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; IF 'INTEGER_INTERVAL_FROM_MIN' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN es_val := spc2\elementary_space.space_id; RETURN ((es_val = es_numbers) OR (es_val = es_integers)); END_IF; IF 'INTEGER_INTERVAL_FROM_MIN' IN types2 THEN RETURN (spc1\integer_interval_from_min.min>=spc2\integer_interval_from_min.min); END_IF; RETURN (FALSE); END_IF; IF 'INTEGER_INTERVAL_TO_MAX' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN es_val := spc2\elementary_space.space_id; RETURN ((es_val = es_numbers) OR (es_val = es_integers)); END_IF; IF 'INTEGER_INTERVAL_TO_MAX' IN types2 THEN RETURN (spc1\integer_interval_to_max.max <= spc2\integer_interval_to_max.max); END_IF; RETURN (FALSE); END_IF; IF 'FINITE_REAL_INTERVAL' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN es_val := spc2\elementary_space.space_id; RETURN ((es_val = es_numbers) OR (es_val = es_reals)); END_IF; IF ('FINITE_REAL_INTERVAL' IN types2) OR ('REAL_INTERVAL_FROM_MIN' IN types2) OR ('REAL_INTERVAL_TO_MAX' IN types2) THEN IF min_exists (spc2) THEN bnd1 := spc1\finite_real_interval.min; bnd2 := real_min (spc2); IF (bnd1 < bnd2) OR ((bnd1 = bnd2) AND min_included (spc1) AND NOT min_included (spc2)) THEN RETURN (FALSE); END_IF; END_IF; IF max_exists (spc2) THEN bnd1 := spc1\finite_real_interval.max; bnd2 := real_max (spc2); IF (bnd1 > bnd2) OR ((bnd1 = bnd2) AND max_included (spc1) AND NOT max_included (spc2)) THEN RETURN (FALSE); END_IF; END_IF; RETURN (TRUE); END_IF; RETURN (FALSE); END_IF; IF 'REAL_INTERVAL_FROM_MIN' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN es_val := spc2\elementary_space.space_id; RETURN ((es_val = es_numbers) OR (es_val = es_reals)); END_IF; IF 'REAL_INTERVAL_FROM_MIN' IN types2 THEN bnd1 := spc1\real_interval_from_min.min; bnd2 := spc2\real_interval_from_min.min; RETURN ((bnd2 < bnd1) OR ((bnd2 = bnd1) AND (min_included (spc2) OR NOT min_included (spc1)))); END_IF; RETURN (FALSE); END_IF; IF 'REAL_INTERVAL_TO_MAX' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN es_val := spc2\elementary_space.space_id; RETURN ((es_val = es_numbers) OR (es_val = es_reals)); END_IF; IF 'REAL_INTERVAL_TO_MAX' IN types2 THEN bnd1 := spc1\real_interval_to_max.max; bnd2 := spc2\real_interval_to_max.max; RETURN ((bnd2 > bnd1) OR ((bnd2 = bnd1) AND (max_included (spc2) OR NOT max_included (spc1)))); END_IF; RETURN (FALSE); END_IF; IF 'CARTESIAN_COMPLEX_NUMBER_REGION' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN es_val := spc2\elementary_space.space_id; RETURN ((es_val = es_numbers) OR (es_val = es_complex_numbers)); END_IF; IF 'CARTESIAN_COMPLEX_NUMBER_REGION' IN types2 THEN RETURN (subspace_of(spc1\cartesian_complex_number_region.real_constraint, spc2\cartesian_complex_number_region.real_constraint) AND subspace_of(spc1\cartesian_complex_number_region.imag_constraint, spc2\cartesian_complex_number_region.imag_constraint)); END_IF; IF 'POLAR_COMPLEX_NUMBER_REGION' IN types2 THEN RETURN (subspace_of(enclose_cregion_in_pregion(spc1, spc2\polar_complex_number_region.centre),spc2)); END_IF; RETURN (FALSE); END_IF; IF 'POLAR_COMPLEX_NUMBER_REGION' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN es_val := spc2\elementary_space.space_id; RETURN ((es_val = es_numbers) OR (es_val = es_complex_numbers)); END_IF; IF 'CARTESIAN_COMPLEX_NUMBER_REGION' IN types2 THEN RETURN (subspace_of(enclose_pregion_in_cregion(spc1),spc2)); END_IF; IF 'POLAR_COMPLEX_NUMBER_REGION' IN types2 THEN prgn1 := spc1; prgn2 := spc2; IF prgn1.centre = prgn2.centre THEN IF prgn2.direction_constraint.max > PI THEN aitv := make_finite_real_interval(-PI,open,prgn2.direction_constraint.max -2.0*PI,prgn2.direction_constraint.max_closure); RETURN (subspace_of(prgn1.distance_constraint,prgn2.distance_constraint) AND (subspace_of(prgn1.direction_constraint,prgn2.direction_constraint) OR subspace_of(prgn1.direction_constraint,aitv))); ELSE RETURN (subspace_of(prgn1.distance_constraint,prgn2.distance_constraint) AND subspace_of(prgn1.direction_constraint,prgn2.direction_constraint)); END_IF; END_IF; RETURN (subspace_of(enclose_pregion_in_pregion(prgn1,prgn2.centre),prgn2)); END_IF; RETURN (FALSE); END_IF; IF 'FINITE_SPACE' IN types1 THEN cum := TRUE; REPEAT i := 1 TO SIZEOF (spc1\finite_space.members); cum := cum AND member_of (spc1\finite_space.members[i], spc2); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; IF 'PRODUCT_SPACE' IN types1 THEN IF 'PRODUCT_SPACE' IN types2 THEN IF space_dimension (spc1) = space_dimension (spc2) THEN cum := TRUE; REPEAT i := 1 TO space_dimension (spc1); cum := cum AND subspace_of (factor_space(spc1,i), factor_space(spc2,i)); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; END_IF; IF 'EXTENDED_TUPLE_SPACE' IN types2 THEN IF space_dimension (spc1) >= space_dimension (spc2) THEN cum := TRUE; REPEAT i := 1 TO space_dimension (spc1); cum := cum AND subspace_of (factor_space(spc1,i), factor_space(spc2,i)); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; END_IF; RETURN (FALSE); END_IF; IF 'EXTENDED_TUPLE_SPACE' IN types1 THEN IF 'EXTENDED_TUPLE_SPACE' IN types2 THEN n := space_dimension (spc1); IF n < space_dimension (spc2) THEN n := space_dimension (spc2); END_IF; cum := TRUE; REPEAT i := 1 TO n+1; cum := cum AND subspace_of (factor_space(spc1,i), factor_space(spc2,i)); IF cum = FALSE THEN RETURN (FALSE); END_IF; END_REPEAT; RETURN (cum); END_IF; RETURN (FALSE); END_IF; IF 'FUNCTION_SPACE' IN types1 THEN IF 'ELEMENTARY_SPACE' IN types2 THEN RETURN (spc2\elementary_space.space_id = es_maths_functions); END_IF; IF 'FUNCTION_SPACE' IN types2 THEN cum := TRUE; sp1 := spc1\function_space.domain_argument; sp2 := spc2\function_space.domain_argument; CASE spc1\function_space.domain_constraint OF sc_equal : BEGIN CASE spc2\function_space.domain_constraint OF sc_equal : cum := cum AND equal_maths_spaces (sp1, sp2); sc_subspace : cum := cum AND subspace_of (sp1, sp2); sc_member : cum := cum AND member_of (sp1, sp2); END_CASE; END; sc_subspace : BEGIN CASE spc2\function_space.domain_constraint OF sc_equal : RETURN (FALSE); sc_subspace : cum := cum AND subspace_of (sp1, sp2); sc_member : BEGIN IF NOT member_of (sp1, sp2) THEN RETURN (FALSE); END_IF; cum := UNKNOWN; END; END_CASE; END; sc_member : BEGIN CASE spc2\function_space.domain_constraint OF sc_equal : cum := cum AND space_is_singleton(sp1) AND equal_maths_spaces(singleton_member_of(sp1),sp2); sc_subspace : BEGIN IF NOT member_of (sp2, sp1) THEN RETURN (FALSE); END_IF; cum := UNKNOWN; END; sc_member : cum := cum AND (subspace_of (sp1, sp2)); END_CASE; END; END_CASE; IF cum = FALSE THEN RETURN (FALSE); END_IF; sp1 := spc1\function_space.range_argument; sp2 := spc2\function_space.range_argument; CASE spc1\function_space.range_constraint OF sc_equal : BEGIN CASE spc2\function_space.range_constraint OF sc_equal : cum := cum AND equal_maths_spaces (sp1, sp2); sc_subspace : cum := cum AND subspace_of (sp1, sp2); sc_member : cum := cum AND member_of (sp1, sp2); END_CASE; END; sc_subspace : BEGIN CASE spc2\function_space.domain_constraint OF sc_equal : RETURN (FALSE); sc_subspace : cum := cum AND subspace_of (sp1, sp2); sc_member : BEGIN IF NOT member_of (sp1, sp2) THEN RETURN (FALSE); END_IF; cum := UNKNOWN; END; END_CASE; END; sc_member : BEGIN CASE spc2\function_space.domain_constraint OF sc_equal : cum := cum AND space_is_singleton(sp1) AND equal_maths_spaces(singleton_member_of(sp1),sp2); sc_subspace : BEGIN IF NOT member_of (sp2, sp1) THEN RETURN (FALSE); END_IF; cum := UNKNOWN; END; sc_member : cum := cum AND subspace_of (sp1, sp2); END_CASE; END; END_CASE; RETURN (cum); END_IF; RETURN (FALSE); END_IF; -- Should be unreachable RETURN (UNKNOWN);
END_FUNCTION;
FUNCTION subspace_of_es
(spc : maths_space; es : elementary_space_enumerators) : LOGICAL;
LOCAL types : SET OF STRING := stripped_typeof(spc); END_LOCAL; IF NOT EXISTS (spc) OR NOT EXISTS (es) THEN RETURN (FALSE); END_IF; IF 'ELEMENTARY_SPACE' IN types THEN RETURN (es_subspace_of_es(spc\elementary_space.space_id,es)); END_IF; IF 'FINITE_SPACE' IN types THEN RETURN (all_members_of_es(spc\finite_space.members,es)); END_IF; CASE es OF es_numbers : RETURN ( ('FINITE_INTEGER_INTERVAL' IN types) OR ('INTEGER_INTERVAL_FROM_MIN' IN types) OR ('INTEGER_INTERVAL_TO_MAX' IN types) OR ('FINITE_REAL_INTERVAL' IN types) OR ('REAL_INTERVAL_FROM_MIN' IN types) OR ('REAL_INTERVAL_TO_MAX' IN types) OR ('CARTESIAN_COMPLEX_NUMBER_REGION' IN types) OR ('POLAR_COMPLEX_NUMBER_REGION' IN types) ); es_complex_numbers : RETURN ( ('CARTESIAN_COMPLEX_NUMBER_REGION' IN types) OR ('POLAR_COMPLEX_NUMBER_REGION' IN types) ); es_reals : RETURN ( ('FINITE_REAL_INTERVAL' IN types) OR ('REAL_INTERVAL_FROM_MIN' IN types) OR ('REAL_INTERVAL_TO_MAX' IN types) ); es_integers : RETURN ( ('FINITE_INTEGER_INTERVAL' IN types) OR ('INTEGER_INTERVAL_FROM_MIN' IN types) OR ('INTEGER_INTERVAL_TO_MAX' IN types) ); es_logicals : RETURN (FALSE); es_booleans : RETURN (FALSE); es_strings : RETURN (FALSE); es_binarys : RETURN (FALSE); es_maths_spaces : RETURN (FALSE); es_maths_functions : RETURN ('FUNCTION_SPACE' IN types); es_generics : RETURN (TRUE); END_CASE; RETURN (UNKNOWN);
END_FUNCTION;
FUNCTION substitute
(expr : generic_expression; vars : LIST[1:?] OF generic_variable; vals : LIST[1:?] OF maths_value) : generic_expression;
LOCAL types : SET OF STRING := stripped_typeof(expr); opnds : LIST OF generic_expression; op1, op2 : generic_expression; qvars : LIST OF generic_variable; srcdom : maths_space_or_function; prpfun : LIST [1:?] OF maths_function; finfun : maths_function_select; END_LOCAL; IF SIZEOF (vars) <> SIZEOF (vals) THEN RETURN (?); END_IF; IF 'GENERIC_LITERAL' IN types THEN RETURN (expr); END_IF; IF 'GENERIC_VARIABLE' IN types THEN REPEAT i := 1 TO SIZEOF (vars); IF expr :=: vars[i] THEN RETURN (vals[i]); END_IF; END_REPEAT; RETURN (expr); END_IF; IF 'QUANTIFIER_EXPRESSION' IN types THEN qvars := expr\quantifier_expression.variables; -- Variables subject to a quantifier do not participate in this kind of -- substitution process. REPEAT i := SIZEOF (vars) TO 1 BY -1; IF vars[i] IN qvars THEN REMOVE (vars, i); REMOVE (vals, i); END_IF; END_REPEAT; opnds := expr\multiple_arity_generic_expression.operands; REPEAT i := 1 TO SIZEOF (opnds); IF NOT (opnds[i] IN qvars) THEN expr\multiple_arity_generic_expression.operands[i] := substitute(opnds[i],vars,vals); -- This technique will not work on subtypes of quantifier_expression -- which derive their operands from other attributes! END_IF; END_REPEAT; RETURN (expr); -- operands modified! END_IF; IF 'UNARY_GENERIC_EXPRESSION' IN types THEN op1 := expr\unary_generic_expression.operand; expr\unary_generic_expression.operand := substitute(op1, vars, vals); -- This technique will not work on subtypes of unary_generic_expression -- which derive their operands from other attributes! END_IF; IF 'BINARY_GENERIC_EXPRESSION' IN types THEN op1 := expr\binary_generic_expression.operands[1]; expr\binary_generic_expression.operands[1] := substitute(op1, vars, vals); op2 := expr\binary_generic_expression.operands[2]; expr\binary_generic_expression.operands[2] := substitute(op2, vars, vals); -- This technique will not work on subtypes of binary_generic_expression -- which derive their operands from other attributes! END_IF; IF 'PARALLEL_COMPOSED_FUNCTION' IN types THEN -- Subtype of multiple_arity_generic_expression which derives its operands. srcdom := expr\parallel_composed_function.source_of_domain; prpfun := expr\parallel_composed_function.prep_functions; finfun := expr\parallel_composed_function.final_function; srcdom := substitute(srcdom,vars,vals); REPEAT i := 1 TO SIZEOF (prpfun); prpfun[i] := substitute(prpfun[i],vars,vals); END_REPEAT; IF 'MATHS_FUNCTION' IN stripped_typeof(finfun) THEN finfun := substitute(finfun,vars,vals); END_IF; RETURN (make_parallel_composed_function(srcdom,prpfun,finfun)); END_IF; IF 'MULTIPLE_ARITY_GENERIC_EXPRESSION' IN types THEN opnds := expr\multiple_arity_generic_expression.operands; REPEAT i := 1 TO SIZEOF (opnds); expr\multiple_arity_generic_expression.operands[i] := substitute(opnds[i],vars,vals); -- This technique will not work on subtypes of multiple_arity_generic_ -- expression which derive their operands from other attributes! END_REPEAT; END_IF; RETURN (expr);
END_FUNCTION;
FUNCTION values_space_of
(expr : generic_expression) : maths_space;
LOCAL e_prefix : STRING := 'ISO13584_EXPRESSIONS_SCHEMA.'; typenames : SET OF STRING := TYPEOF (expr); END_LOCAL; IF (schema_prefix + 'MATHS_VARIABLE') IN typenames THEN RETURN (expr\maths_variable.values_space); END_IF; IF (e_prefix + 'EXPRESSION') IN typenames THEN IF (e_prefix + 'NUMERIC_EXPRESSION') IN typenames THEN IF expr\numeric_expression.is_int THEN IF (e_prefix + 'INT_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\int_literal.the_value])); ELSE RETURN (the_integers); END_IF; ELSE IF (e_prefix + 'REAL_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\real_literal.the_value])); ELSE RETURN (the_reals); END_IF; END_IF; END_IF; IF (e_prefix + 'BOOLEAN_EXPRESSION') IN typenames THEN IF (e_prefix + 'BOOLEAN_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\boolean_literal.the_value])); ELSE RETURN (the_booleans); END_IF; END_IF; IF (e_prefix + 'STRING_EXPRESSION') IN typenames THEN IF (e_prefix + 'STRING_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\string_literal.the_value])); ELSE RETURN (the_strings); END_IF; END_IF; RETURN (?); -- unknown subtype of expression END_IF; IF (schema_prefix + 'MATHS_FUNCTION') IN typenames THEN IF expression_is_constant (expr) THEN RETURN (make_finite_space ([expr])); ELSE RETURN (make_function_space (sc_equal, expr\maths_function.domain, sc_equal, expr\maths_function.range)); END_IF; END_IF; IF (schema_prefix + 'FUNCTION_APPLICATION') IN typenames THEN RETURN (expr\function_application.func.range); END_IF; IF (schema_prefix + 'MATHS_SPACE') IN typenames THEN IF expression_is_constant (expr) THEN RETURN (make_finite_space ([expr])); ELSE -- This case cannot occur in this version of the schema. -- When it becomes possible, the subtypes should be analysed and -- more finely defined spaces returned. RETURN (make_elementary_space (es_maths_spaces)); END_IF; END_IF; IF (schema_prefix + 'DEPENDENT_VARIABLE_DEFINITION') IN typenames THEN RETURN (values_space_of (expr\unary_generic_expression.operand)); END_IF; IF (schema_prefix + 'COMPLEX_NUMBER_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr])); END_IF; IF (schema_prefix + 'LOGICAL_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\logical_literal.lit_value])); END_IF; IF (schema_prefix + 'BINARY_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\binary_literal.lit_value])); END_IF; IF (schema_prefix + 'MATHS_ENUM_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\maths_enum_literal.lit_value])); END_IF; IF (schema_prefix + 'REAL_TUPLE_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\real_tuple_literal.lit_value])); END_IF; IF (schema_prefix + 'INTEGER_TUPLE_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\integer_tuple_literal.lit_value])); END_IF; IF (schema_prefix + 'ATOM_BASED_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\atom_based_literal.lit_value])); END_IF; IF (schema_prefix + 'MATHS_TUPLE_LITERAL') IN typenames THEN RETURN (make_finite_space ([expr\maths_tuple_literal.lit_value])); END_IF; IF (schema_prefix + 'PARTIAL_DERIVATIVE_EXPRESSION') IN typenames THEN RETURN (drop_numeric_constraints (values_space_of ( expr\partial_derivative_expression.derivand))); END_IF; IF (schema_prefix + 'DEFINITE_INTEGRAL_EXPRESSION') IN typenames THEN RETURN (drop_numeric_constraints (values_space_of ( expr\definite_integral_expression.integrand))); END_IF; RETURN (?); -- not recognized as a mathematical expression
END_FUNCTION;
PROCEDURE angle_minmax (ab : REAL; a : REAL; a_in : BOOLEAN; VAR amin : REAL; amax : REAL; VAR amin_in : BOOLEAN; 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;
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;
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;
PROCEDURE angle_range (VAR amin : REAL; 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;
PROCEDURE find_aminmax (ab : REAL; a0 : REAL; a1 : REAL; a2 : REAL; a3 : REAL; in0 : BOOLEAN; in1 : BOOLEAN; in2 : BOOLEAN; in3 : BOOLEAN; VAR amin : REAL; amax : REAL; VAR amin_in : BOOLEAN; 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;
PROCEDURE parts (c : complex_number_literal; VAR x : REAL; y : REAL) : ;
x := c.real_part; y := c.imag_part;
END_PROCEDURE;
END_SCHEMA; -- mathematical_functions_schema