math_spec.program

The program: what a file declares, with names resolved and shapes fixed.

The second public state, and the one a consumer reads. A :class:Program is every declaration a file makes and no data at all; :func:~math_spec.lowering.to_program is the only thing that builds one, so nothing here re-checks a hand-built one.

Node and declaration classes are matched with isinstance. The rules a node's structure does not show are :func:children and :func:fan_in; the questions over the walk are :func:walk and the filters beside it. A resolved where arrives as a :class:Mask. Frozen dataclasses only — no execution logic, and nothing imported from a consumer. How a consumer reads one: docs/reference/language/reading.md.

Check = Increasing | Curved | AtLeastTwo | Contiguous module-attribute

ConnectiveWhereNode = NotNode | AndNode | OrNode module-attribute

ConstraintSense = ComparisonOperator module-attribute

Derivation = MaskOf | FirstOf | LastOf module-attribute

DimensionDtype = _model.DimensionDtype module-attribute

ExpressionNode = Constant | Parameter | Variable | Dual | Negate | Add | Multiply | Power | Divide | Sum | GroupSum | At | Translate | Window | Cases module-attribute

FanIn = Literal['one-to-one', 'many-to-one', 'one-to-many'] module-attribute

ObjectiveSense = Literal['minimize', 'maximize'] module-attribute

ParameterDtype = _model.ParameterDtype module-attribute

PredicateOperator = Literal['<=', '>=', '==', '!=', '<', '>'] module-attribute

QUADRATIC_POSITIONS = frozenset(get_args(QuadraticPosition)) module-attribute

QuadraticPosition = Literal['objective', 'constraint'] module-attribute

TypedPredicateNode = ParameterComparisonNode | ParameterDefinedNode | VariableDefinedNode | DimensionComparisonNode | DimensionPositionNode | LookupComparisonNode | LookupPairComparisonNode | LookupDefinedNode module-attribute

VariableAbsence = _model.VariableAbsence module-attribute

VariableType = _model.VariableDomain module-attribute

WhereNode = BooleanLiteralNode | DimensionPositionNode | ParameterDefinedNode | VariableDefinedNode | ParameterComparisonNode | DimensionComparisonNode | LookupComparisonNode | LookupPairComparisonNode | LookupDefinedNode | NotNode | AndNode | OrNode module-attribute

Add(left, right) dataclass

Bases: Expression

left instance-attribute

right instance-attribute

AndNode(left, right) dataclass

left instance-attribute

right instance-attribute

At(operand, over, coordinate, into) dataclass

Bases: Expression

Read operand through a lookup — the adjoint of :class:GroupSum.

Same mapping table, walked the other way: GroupSum consumes over and produces into, this consumes into and produces over. The join fans out, many over labels sharing one into tuple.

coordinate instance-attribute

into instance-attribute

operand instance-attribute

over instance-attribute

AtLeastTwo(over, mask) dataclass

Each curve has at least two breakpoints — every position along over, or those mask admits.

mask instance-attribute

over instance-attribute

BooleanLiteralNode(value) dataclass

value instance-attribute

Cases(regions) dataclass

Bases: Expression

A value defined by region — exactly one region applies at each coordinate.

The regions are disjoint and total, so a consumer adds them rather than ranking them. Not a shape operator: every region spans the dims the expression does.

regions instance-attribute

Constant(value) dataclass

Bases: Expression

A scalar constant.

value instance-attribute

ConstraintDeclaration(dims, lhs, sense, rhs, where=None) dataclass

lhs sense rhs for each coord combination of dims.

Either side may carry variables and constants alike; which side a consumer gathers them onto is its own arrangement and not stated here. where masks out coord combinations (row absence, like variables).

dims instance-attribute

lhs instance-attribute

rhs instance-attribute

sense instance-attribute

where = None class-attribute instance-attribute

Contiguous(mask, values) dataclass

mask admits one consecutive run of at least one breakpoint per curve.

mask instance-attribute

values instance-attribute

Curved(x, y, over, curvature) dataclass

y over x bends, along over, the way curvature says.

That is the shape the method is exact for. either is the hull's weaker condition: any single bend, so only a mixed curve fails it.

curvature instance-attribute

over instance-attribute

x instance-attribute

y instance-attribute

DimensionComparisonNode(name, op, value) dataclass

Compare a dimension's own coordinates against a literal.

name instance-attribute

op instance-attribute

value instance-attribute

DimensionDeclaration(lookups=(), dtype='str') dataclass

A dimension and the lookups its labels carry, of both kinds.

dtype = 'str' class-attribute instance-attribute

lookups = () class-attribute instance-attribute

maps property

Every map over the dimension, targeted and label-space alike.

What binding needs a relation for: both kinds are read by a where and both arrive the same way, and only the targeted ones have a label set to be checked against.

targets property

Each targeted map over the dimension, to the dimension its values are labels of.

The question every consumer of a by= asks, and asked here so it has one answer: an operator grouping through a lookup names the target as the dim it lands on, and a partition array is named for it so an amount declared over the group's own dim can be read through it.

DimensionPositionNode(name, op, position, by=None) dataclass

Compare where a row sits along a dimension against a position — position(snapshot) == 0.

Both sides are integers, negative counting from the end. With by the position is counted within each group the lookup makes.

by = None class-attribute instance-attribute

name instance-attribute

op instance-attribute

position instance-attribute

Divide(numerator, divisor) dataclass

Bases: Expression

Quotient numerator / divisor, the divisor variable-free wherever the math reads it (math_spec.degree).

divisor instance-attribute

numerator instance-attribute

Dual(constraint) dataclass

Bases: Expression

A constraint's dual — its shadow price, read after the solve.

Stands only under an :class:ExpressionDeclaration the math never reads: the loader refuses dual() anywhere a solver ingests. One value per coordinate of the named constraint's own foreach frame, which is what :func:fan_in answers one-to-one for — the leaf reshapes nothing, like a parameter.

constraint instance-attribute

Expression() dataclass

Base class for expressions over variables and parameters.

The degree rules (math_spec.degree) hold on every tree the math reads — :attr:Program.expressions, a bound, and a named expression that is in_math — affine but where a :class:QuadraticPosition admits a :class:Multiply of two variable-carrying operands. A :class:ExpressionDeclaration the math never reads is held to none of them. No node records which tree it stands in.

ExpressionDeclaration(expression, in_math) dataclass

A named quantity — one the math reads, or one only read back after a solve.

in_math where the objective or a constraint inlines it, directly or through another entry or a macro; its body then stands inside :attr:Program.expressions and is held to the degree rules where it is read. Otherwise nothing a solver sees contains it: it is a reported quantity, its body held to no degree, the one place a :class:Dual may stand. A bound and a where name no entry, so neither decides this.

expression instance-attribute

in_math instance-attribute

FirstOf(block, mask) dataclass

A bool parameter marking, per curve, the first breakpoint mask admits.

block instance-attribute

mask instance-attribute

Footprint(quadratic, variable_types, sos_types, shapes) dataclass

Which of the language's constructs one program uses.

A subset, never the whole: an empty field says this program does not use the construct.

ATTRIBUTE	DESCRIPTION
quadratic	Each position a product of two variable-carrying operands stands in; empty is affine throughout. TYPE: frozenset[QuadraticPosition]
variable_types	Every domain declared. TYPE: frozenset[VariableType]
sos_types	The order of each special-ordered set declared. TYPE: frozenset[Literal[1, 2]]
shapes	Every expression node kind that appears. TYPE: frozenset[type[ExpressionNode]]

quadratic instance-attribute

shapes instance-attribute

sos_types instance-attribute

variable_types instance-attribute

GroupSum(operand, over, coordinate, into) dataclass

Bases: Expression

Sum operand through coordinates declared on dim over.

coordinate names lookups carried by dim over whose values are labels of the matching dim in into; the result replaces over with all of them. The two tuples are the same length and their order pairs them: several coordinates are one grouping into a product of targets, consumed in a single join.

coordinate instance-attribute

into instance-attribute

operand instance-attribute

over instance-attribute

Increasing(parameter, over) dataclass

parameter is strictly increasing along over within each curve — the x-axis a method sorts by.

over instance-attribute

parameter instance-attribute

LastOf(block, mask) dataclass

Its sibling for the last breakpoint.

block instance-attribute

mask instance-attribute

LookupComparisonNode(name, over, op, value) dataclass

Compare a lookup's values against a literal — period_of == 2030.

over is the dimension the lookup maps out of.

name instance-attribute

op instance-attribute

over instance-attribute

value instance-attribute

LookupDeclaration

Bases: NamedTuple

One declared lookup over a dimension, of either kind.

Exactly one of target and dtype is set. A targeted lookup's values are labels of target, checked for containment once the dim tables exist — which keeps a mistyped label from silently dropping its terms in the join that places them — and it is what sum(by=) lands terms on. A label space owns its values, typed by dtype the way a dimension's labels are: it is read for selection and rendering, and resolution refuses to group into one, so no expression node reaches it.

dtype = None class-attribute instance-attribute

name instance-attribute

target instance-attribute

LookupDefinedNode(name, over) dataclass

True where the named lookup has a value — a null says the label belongs to no group.

name instance-attribute

over instance-attribute

LookupPairComparisonNode(name, other, over, op) dataclass

Compare two lookups over one dimension — from != to, row by row on that dimension's table.

name instance-attribute

op instance-attribute

other instance-attribute

over instance-attribute

Mask(root) dataclass

A resolved where and the questions the language answers about it.

root is the predicate a consumer dispatches on with isinstance; every question below is derived from it. Construction folds, so a boolean literal stands at the root or nowhere in it, and refuses an unresolved tree.

ATTRIBUTE	DESCRIPTION
root	The resolved predicate the mask restricts rows by, folded. TYPE: WhereNode

atoms cached property

The mask's leaves, connectives removed — the one walk the other questions read.

Held rather than re-walked: construction takes this walk anyway, to refuse an unresolved tree, and a mask cannot change afterwards.

conjuncts property

The predicates the mask joins with AND — its AND spine flattened, stopping at an OR or a NOT.

dims property

The dims the mask is read at — the union of what each leaf carries.

Empty for a mask over nothing but literals. Read off the leaves, which resolution stamped with their declarations' dims, so a predicate built from resolved pieces answers exactly as a declaration's own does.

names_read property

The parameters, lookups and variables the mask names.

root instance-attribute

MaskOf(block, values) dataclass

A bool parameter true wherever values has a row.

The mask a points: naming one of the block's own breakpoints derives: the curve runs as far as its values do. values is the name the file wrote, so a refusal about the mask can say it.

block instance-attribute

values instance-attribute

Multiply(left, right) dataclass

Bases: Expression

Product of two operands.

Affine where at least one factor is variable-free; degree 2 where neither is, which math_spec.degree admits in a :data:QuadraticPosition alone.

left instance-attribute

right instance-attribute

Negate(operand) dataclass

Bases: Expression

operand instance-attribute

NotNode(operand) dataclass

operand instance-attribute

ObjectiveDeclaration(sense, expression) dataclass

Objective — scalar, every reduction in it one the file wrote.

expression instance-attribute

sense instance-attribute

OrNode(left, right) dataclass

left instance-attribute

right instance-attribute

Parameter(name) dataclass

Bases: Expression

A parameter reference — contributes to the constant part.

name instance-attribute

ParameterComparisonNode(name, op, value, dims) dataclass

Compare a parameter against a literal, element-wise.

dims instance-attribute

name instance-attribute

op instance-attribute

value instance-attribute

ParameterDeclaration(dims, dtype='float', derivation=None) dataclass

Shape declaration; data is bound at execution time by name.

dtype is what the declaration claims the values are, and a consumer binding data refuses a column that is not it — so the declaration is what is read, rather than whatever the column happens to hold.

derivation = None class-attribute instance-attribute

dims instance-attribute

dtype = 'float' class-attribute instance-attribute

ParameterDefinedNode(name, dims) dataclass

True wherever the named parameter is non-null and finite.

dims is the parameter's own, copied off the declaration during resolution; every leaf below that names a declaration carries its dims (or over) the same way.

dims instance-attribute

name instance-attribute

PiecewiseDeclaration(over, method, breakpoints, checks) dataclass

A piecewise: block, kept as the facts a consumer binding its data reads.

The expansion lowered the links into constraints and emitted the parameters it needs — each of those says how it is filled, on its own :attr:ParameterDeclaration.derivation. What is left here is the curve and what the block assumes of it.

ATTRIBUTE	DESCRIPTION
over	The breakpoint dimension. TYPE: str
method	How the weights are restricted. TYPE: PiecewiseMethod
breakpoints	The links' values parameters, in link order. TYPE: tuple[str, ...]
checks	What the block assumes of the numbers, each carrying its own subjects, for the consumer holding them to check. TYPE: tuple[Check, ...]

breakpoints instance-attribute

checks instance-attribute

method instance-attribute

over instance-attribute

Power(base, exponent) dataclass

Bases: Expression

base ** exponent, both variable-free wherever the math reads it.

The language refuses a variable anywhere under it (math_spec.degree), so in the program a solver sees it is degree 0 and folds to one number per coordinate like any other parameter arithmetic.

base instance-attribute

exponent instance-attribute

Program(*, parameters, variables, constraints, objective, dimensions=MappingProxyType({}), sos=MappingProxyType({}), piecewise=MappingProxyType({}), named_expressions=MappingProxyType({})) dataclass

A complete declarative description of a mathematical program, with no data in it.

Every group of declarations is keyed by the name the file wrote, in the order it wrote them, and is read-only: the mappings are wrapped at construction, so a consumer cannot rewrite what another consumer reads. A whole program is not hashable — the declarations and expression nodes inside it are, which is what dedup and memoisation ask for.

constraints instance-attribute

dimensions = MappingProxyType({}) class-attribute instance-attribute

expressions property

Every expression a row is built from — the objective and both sides of each constraint.

A :attr:named_expressions entry builds no row and is not among them.

footprint cached property

Which constructs this program uses — walked once, then held.

lookups property

Every lookup in the program, targeted and label-space alike, with the dimension it is over.

named_expressions = MappingProxyType({}) class-attribute instance-attribute

objective instance-attribute

parameters instance-attribute

piecewise = MappingProxyType({}) class-attribute instance-attribute

sos = MappingProxyType({}) class-attribute instance-attribute

variables instance-attribute

dimension(name)

Source code in src/math_spec/program.py

def dimension(self, name: str) -> DimensionDeclaration:
    return _declared(self.dimensions, name, 'dimension')

parameter(name)

Source code in src/math_spec/program.py

def parameter(self, name: str) -> ParameterDeclaration:
    return _declared(self.parameters, name, 'parameter')

variable(name)

Source code in src/math_spec/program.py

def variable(self, name: str) -> VariableDeclaration:
    return _declared(self.variables, name, 'variable')

Region(when, value) dataclass

One region of a :class:Cases: where it applies, and the value there.

when is stated on every region; the one the file wrote as otherwise: carries the negation of the others.

value instance-attribute

when instance-attribute

SosDeclaration(variable, over, sos_type, big_m=None) dataclass

One special-ordered set per coordinate of the variable's foreach minus over.

The only declaration that adds neither a column nor a row: it names columns a consumer already has and says what may be nonzero among them. Which dims those are is the variable's own foreach and is read from it: a copy here would be a second home for a fact (:meth:Program.variable).

big_m caps the linking coefficient a consumer without the concept reformulates with, and is None where the variable's own upper bound is the only cap.

big_m = None class-attribute instance-attribute

over instance-attribute

sos_type instance-attribute

variable instance-attribute

Sum(operand, over) dataclass

Bases: Expression

Sum operand over the named dims, removing them from the result.

operand instance-attribute

over instance-attribute

Translate(operand, dimension, offset, wrap, fill=None, partition=None) dataclass

Bases: Expression

Re-index along one dimension: the result at t is operand at t - offset.

wrap is edge='wrap' in the file: periodic, and stated on every node. fill is what an acyclic shift leaves behind: None leaves the vacated positions absent, so the row drops; a number makes them present and contribute it. Always None under wrap.

offset is an integer, or the name of an integer parameter that does not depend on dimension and carries its sign in the values.

partition names a lookup over dimension, and the translation then happens inside each group it makes: the neighbour is the one before in the same group, the edge is the group's, and a wrap closes each group onto itself. A coordinate the lookup sends nowhere reaches nothing.

dimension instance-attribute

fill = None class-attribute instance-attribute

offset instance-attribute

operand instance-attribute

partition = None class-attribute instance-attribute

wrap instance-attribute

Variable(name) dataclass

Bases: Expression

A variable reference — one term per existing variable row.

name instance-attribute

VariableDeclaration(dims, where=None, lower=(lambda: Constant(float('-inf')))(), upper=(lambda: Constant(float('inf')))(), variable_type='continuous', absence='undefined') dataclass

absence = 'undefined' class-attribute instance-attribute

dims instance-attribute

lower = field(default_factory=lambda: Constant(float('-inf'))) class-attribute instance-attribute

upper = field(default_factory=lambda: Constant(float('inf'))) class-attribute instance-attribute

variable_type = 'continuous' class-attribute instance-attribute

where = None class-attribute instance-attribute

VariableDefinedNode(name, dims) dataclass

True at the coordinates where the named variable exists.

dims instance-attribute

name instance-attribute

Window(operand, dimension, width, wrap, partition=None) dataclass

Bases: Expression

Sum operand over a trailing window along one dimension.

The result at t is the sum of the operand at every position from t - width + 1 through t, so a width of 1 is the operand itself. The dimension survives: this replicates terms onto the positions that can see them rather than reducing anything away.

width is a whole number, or the name of an integer parameter when the window differs per entity — a minimum up time, a rolling budget, a delivery horizon. A named width may not depend on the dimension being summed over.

wrap says whether the window reaches around the start of the axis instead of stopping short at it, and is stated on every node.

partition names a lookup over that dimension, and the window then stops at each group's edge. Positions are counted inside the group, so a coordinate the lookup places nowhere reaches nothing — not even itself.

dimension instance-attribute

operand instance-attribute

partition = None class-attribute instance-attribute

width instance-attribute

wrap instance-attribute

carries_variable(expression)

Whether a variable appears anywhere under expression.

Source code in src/math_spec/program.py

def carries_variable(expression: ExpressionNode) -> bool:
    """Whether a variable appears anywhere under *expression*."""
    return any(isinstance(node, Variable) for node in walk(expression))

check_message(block, pw, check)

The sentence a consumer raises when the data bound to block fails check.

The language's own wording, so every consumer refuses in the same words; a consumer appends what it saw.

Source code in src/math_spec/program.py

def check_message(block: str, pw: PiecewiseDeclaration, check: Check) -> str:
    """The sentence a consumer raises when the data bound to *block* fails *check*.

    The language's own wording, so every consumer refuses in the same words;
    a consumer appends what it saw.
    """
    ctx = f"piecewise '{block}'"
    match check:
        case Increasing(parameter, over):
            return (
                f"{ctx}: method: {pw.method} requires strictly increasing breakpoints in '{parameter}' along '{over}'"
            )
        case Curved(x, y, over, curvature):
            shape = 'a single bend' if curvature == 'either' else f'a {curvature} curve'
            return (
                f"{ctx}: method: {pw.method} is exact only for {shape}, and '{y}' over '{x}' along "
                f"'{over}' is not one, so the answer is wrong rather than loose. Use method: adjacency "
                f'or sos2, which take a curve of any shape.'
            )
        case AtLeastTwo():
            return (
                f'{ctx}: method: lp needs at least two breakpoints per curve — the method *is* its segment '
                f'lines, so a curve with no segment states nothing and leaves the bounded link on its own '
                f'bound. Use method: adjacency, sos2 or convex, which pin it to the points it does have.'
            )
        case Contiguous(mask, values):
            return (
                f"{ctx}: points: '{values if values is not None else mask}' must mark a consecutive run of at "
                f'least one breakpoint per curve — the chord row joins a breakpoint to the one before it, and '
                f"the domain rows sit on the curve's own first and last."
            )
        case _:
            assert_never(check)

children(expression)

The sub-expressions of expression — what every walk recurses through.

Source code in src/math_spec/program.py

def children(expression: ExpressionNode) -> tuple[ExpressionNode, ...]:
    """The sub-expressions of *expression* — what every walk recurses through."""
    if isinstance(expression, Negate):
        return (expression.operand,)
    if isinstance(expression, (Add, Multiply)):
        return (expression.left, expression.right)
    if isinstance(expression, Divide):
        return (expression.numerator, expression.divisor)
    if isinstance(expression, (Sum, GroupSum, At, Translate, Window)):
        return (expression.operand,)
    if isinstance(expression, Cases):
        return tuple(region.value for region in expression.regions)
    return ()

divisor_parameters(*expressions)

Every parameter named anywhere in a divisor under expressions.

Source code in src/math_spec/program.py

def divisor_parameters(*expressions: ExpressionNode) -> frozenset[str]:
    """Every parameter named anywhere in a divisor under *expressions*."""
    return frozenset().union(*(parameters_of(q.divisor) for q in quotients(*expressions)))

fan_in(expression)

How expression's output rows relate to its input slots.

For the absence rules, both classes other than 'one-to-one' sum several input slots into an output row.

Source code in src/math_spec/program.py

def fan_in(expression: ExpressionNode) -> FanIn:
    """How *expression*'s output rows relate to its input slots.

    For the absence rules, both classes other than ``'one-to-one'`` sum
    several input slots into an output row.
    """
    if isinstance(expression, (Sum, GroupSum)):
        return 'many-to-one'
    if isinstance(expression, Window):
        return 'one-to-many'
    if isinstance(
        expression,
        (Constant, Parameter, Variable, Dual, Negate, Add, Multiply, Power, Divide, At, Translate, Cases),
    ):
        return 'one-to-one'
    assert_never(expression)

is_quadratic(expression)

Whether expression contains a product of two variable-carrying operands.

A structural question over the program, and unrelated consumers ask it — what a solver must support, which declarations to build last, whether this form can be represented at all — so it is answered once here beside the other walks rather than once per consumer in its own terms.

Whether a degree may be written is the language's verdict, and this is not a second opinion on it: by the time a program exists the question is which shape the expression has, and the program is what is in hand to answer it.

Source code in src/math_spec/program.py

def is_quadratic(expression: ExpressionNode) -> bool:
    """Whether *expression* contains a product of two variable-carrying operands.

    A structural question over the program, and unrelated consumers ask it —
    what a solver must support, which declarations to build last, whether this
    form can be represented at all — so it is answered once here beside the
    other walks rather than once per consumer in its own terms.

    Whether a degree *may be written* is the language's verdict, and this is
    not a second opinion on it: by the time a program exists the question is
    which shape the expression has, and the program is what is in hand to
    answer it.
    """
    return any(
        isinstance(node, Multiply) and all(carries_variable(side) for side in (node.left, node.right))
        for node in walk(expression)
    )

parameters_of(*expressions)

Every parameter named anywhere under expressions.

Source code in src/math_spec/program.py

def parameters_of(*expressions: ExpressionNode) -> frozenset[str]:
    """Every parameter named anywhere under *expressions*."""
    return frozenset(node.name for node in walk(*expressions) if isinstance(node, Parameter))

quotients(*expressions)

Every division under expressions, each kept whole.

The divisor and the numerator answer different questions and one consumer needs them paired: a divisor is judged against the rows the declaration builds narrowed by the variables in its own numerator, which the flat :func:divisor_parameters cannot say.

Source code in src/math_spec/program.py

def quotients(*expressions: ExpressionNode) -> tuple[Divide, ...]:
    """Every division under *expressions*, each kept whole.

    The divisor and the numerator answer different questions and one consumer
    needs them paired: a divisor is judged against the rows the declaration
    builds *narrowed by the variables in its own numerator*, which the flat
    :func:`divisor_parameters` cannot say.
    """
    return tuple(node for node in walk(*expressions) if isinstance(node, Divide))

variables_of(*expressions)

Every variable named anywhere under expressions.

Source code in src/math_spec/program.py

def variables_of(*expressions: ExpressionNode) -> frozenset[str]:
    """Every variable named anywhere under *expressions*."""
    return frozenset(node.name for node in walk(*expressions) if isinstance(node, Variable))

walk(*expressions)

Every node under expressions, each expression itself included, parents first.

The traversal every question about a program is a filter of — which names it mentions, whether a variable stands under it, which divisions it contains. One generator rather than that five-line recursion once per question: how a program is traversed is one fact, so a node kind :func:children learns to descend into reaches every caller at once rather than the callers that remembered.

Source code in src/math_spec/program.py

def walk(*expressions: ExpressionNode) -> Iterator[ExpressionNode]:
    """Every node under *expressions*, each expression itself included, parents first.

    The traversal every *question* about a program is a filter of — which names
    it mentions, whether a variable stands under it, which divisions it
    contains. One generator rather than that five-line recursion once per
    question: how a program is traversed is one fact, so a node kind
    :func:`children` learns to descend into reaches every caller at once
    rather than the callers that remembered.
    """
    for expression in expressions:
        yield expression
        yield from walk(*children(expression))