FHElium

fhelium.compile.passes ​

Built-in partial transforms for captured tensor and CKKS operations.

AggregateRotationSumsPass ​

class View source

python
AggregateRotationSumsPass(combine_rescale: bool = True, name: str = 'aggregate-rotation-sums')

Replace compatible rotation/add trees with a shared QP quotient.

Including this pass is the caller's choice to change intermediate CKKS rounding. Matching uses SSA values and represented state. Independent intermediate users are preserved. Unknown effects, caller assignments, mismatched tables/state or partial required facts leave the region unchanged.

combine_rescale also recognizes uniformly rescaled inputs and divides the aggregate by P times the complete dropped Q group. Otherwise each original input rescale remains before the aggregate key switching.

Measured on RTX PRO 6000 Blackwell Server Edition: 128x128 @ 128x8 BSGS matrices, baby step 8, N=16384, Q=8, P=1, int64 residues, scale 2^40. Latencies include device synchronization, using resident inputs; values are medians of three process medians, with 30 samples per process.

Execution / implementationMatrixBaseline (ms)Aggregate (ms)
CUDA Graph / default fusionPT x CT9.9478.539
CUDA Graph / default fusionCT x CT17.90116.444
No Graph / whole operationPT x CT12.70211.835
No Graph / whole operationCT x CT21.13225.383

Both matrix workloads improved with Graph replay. Without Graph, PT x CT improved, but CT x CT became 20.1% slower.

Attributes ​

NameTypeDefault/value
combine_rescaleboolTrue
namestr'aggregate-rotation-sums'

run ​

method

python
def run(compilation): ...

AssignCkksDepthsPass ​

class View source

python
AssignCkksDepthsPass(entry_depth: int | None = None)

Assign depths and align add/sub joins after rescale placement.

Existing rescale nodes advance one public depth. At add/sub fan-in, only the lower-depth consumer edge is advanced through a real ModSwitchOp. Repeated application does not duplicate either transition.

Attributes ​

NameTypeDefault/value
entry_depthint | NoneNone
namestrfield(default='assign-ckks-depths', init=False)

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

AssignCkksScalesPass ​

class View source

python
AssignCkksScalesPass(entry_scale: float | None = None)

Propagate actual scales through an already scheduled CKKS Program.

The pass updates represented scale metadata while preserving the arithmetic operations. Add/sub joins require equal binary64 scales. Rescale divides the input scale by config.rescale_divisor(depth), the product of the dropped Q group.

Attributes ​

NameTypeDefault/value
entry_scalefloat | NoneNone
namestrfield(default='assign-ckks-scales', init=False)

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

AssignImplementationsPass ​

class View source

python
AssignImplementationsPass(selections: Mapping[str, str] = field(default_factory=dict), overwrite: bool = False, name: str = 'assign-implementations')

Attach requested implementation names to selected operation classes.

selections maps registered textual operation names to implementation names. Matching uses full-name equality. Unselected operations remain unchanged. The resulting attributes constrain subsequent Backend resolution, which checks implementation availability.

Attributes ​

NameTypeDefault/value
selectionsMapping[str, str]field(default_factory=dict)
overwriteboolFalse
namestr'assign-implementations'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Annotate matching operation names and report each assignment.

AssignNttImplementationPass ​

class View source

python
AssignNttImplementationPass(implementation: str | None = None, overwrite: bool = False, ntt_backend: str | None = None, algorithm: str | None = None, group_width: int | None = None, radix: int | None = None, name: str = 'assign-ntt-implementation')

Assign a complete schedule or partial algorithm constraints.

Algorithm constraints apply to logical NTT operations and CKKS domain transitions. A concrete implementation applies to logical NTT operations; place this pass after CKKS lowering when assigning that implementation. Unspecified fields remain available to subsequent selection passes.

Attributes ​

NameTypeDefault/value
implementationstr | NoneNone
overwriteboolFalse
ntt_backendstr | NoneNone
algorithmstr | NoneNone
group_widthint | NoneNone
radixint | NoneNone
namestr'assign-ntt-implementation'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

EliminateDeadValuesPass ​

class View source

python
EliminateDeadValuesPass(name: str = 'eliminate-dead-values')

Erase unused operations according to registered effects.

The pass traverses known structured regions while preserving terminators and result interfaces. Unknown operations or regions, random draws, mutation, and opaque effects remain liveness roots. Removal requires purity registered for the operation's concrete class; an inherited Pure trait alone is insufficient.

Attributes ​

NameTypeDefault/value
namestr'eliminate-dead-values'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Walk blocks in reverse dependency order and remove dead producers.

EmitBackendPythonPass ​

class View source

python
EmitBackendPythonPass(registry: OperationImplementationRegistry | None = None, entry_point: str = 'generated_backend', name: str = 'emit-backend-python')

Publish direct Backend Python for the Program at this pass position.

Attributes ​

NameTypeDefault/value
registryOperationImplementationRegistry | NoneNone
entry_pointstr'generated_backend'
namestr'emit-backend-python'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

EmitEagerPythonPass ​

class View source

python
EmitEagerPythonPass(entry_point: str = 'generated_eager', name: str = 'emit-eager-python')

Publish Eager-style Python for the CKKS Program at this pass position.

Attributes ​

NameTypeDefault/value
entry_pointstr'generated_eager'
namestr'emit-eager-python'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

FuseOperationsPass ​

class View source

python
FuseOperationsPass(implementations: Sequence[FusionImplementation], min_ops: int = 2, name: str = 'fuse-operations')

Group pure SSA dependencies using caller-supplied Backend support.

Independent pure work can be scheduled around a supported region. Effects, opaque operations and caller implementation assignments preserve their order. Implementations determine operation, parameter and layout support.

Attributes ​

NameTypeDefault/value
implementationsSequence[FusionImplementation]
min_opsint2
namestr'fuse-operations'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

InitializeResourceBindingsPass ​

class View source

python
InitializeResourceBindingsPass(resources: ResourceBindings = field(default_factory=ResourceBindings), name: str = 'initialize-resource-bindings')

Replace resource state left by an earlier build with this build's base.

Attributes ​

NameTypeDefault/value
resourcesResourceBindingsfield(default_factory=ResourceBindings)
namestr'initialize-resource-bindings'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

InsertMultiplyNttTransitionsPass ​

class View source

python
InsertMultiplyNttTransitionsPass(name: str = 'insert-multiply-ntt-transitions')

Insert typed CKKS NTT transitions at registered logical multiplies.

The transformed operand type records NTT/Montgomery state, so repeated application is idempotent without a marker attribute. Unrealized casts preserve visible type-changing edges from logical values to CKKS values.

Attributes ​

NameTypeDefault/value
namestr'insert-multiply-ntt-transitions'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Insert missing transitions or return a legal no-op report.

InsertPlaintextPreparationPass ​

class View source

python
InsertPlaintextPreparationPass(name: str = 'insert-plaintext-preparation')

Insert typed plaintext preparation at registered mixed logical ops.

Existing CKKS plaintext operands make repeated application a legal no-op. The visible unrealized cast on the ciphertext edge records the still-open logical-to-CKKS type conversion without mutating its producer.

Attributes ​

NameTypeDefault/value
namestr'insert-plaintext-preparation'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Prepare supported public operands across all function blocks.

InsertRelinearizationPass ​

class View source

python
InsertRelinearizationPass(name: str = 'insert-relinearization')

Relinearize each CT3 multiplication before its first non-transition use.

Attributes ​

NameTypeDefault/value
namestr'insert-relinearization'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

InsertRescalePass ​

class View source

python
InsertRescalePass(name: str = 'insert-rescale')

Rescale each multiplication before its first non-transition use.

Attributes ​

NameTypeDefault/value
namestr'insert-rescale'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

LateRelinearizationPass ​

class View source

python
LateRelinearizationPass(name: str = 'late-relinearization')

Coalesce compatible CT3 add/sub/negate regions before relinearization.

Attributes ​

NameTypeDefault/value
namestr'late-relinearization'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

LateRescalePass ​

class View source

python
LateRescalePass(name: str = 'late-rescale')

Coalesce compatible add/sub/negate regions before rescaling.

Attributes ​

NameTypeDefault/value
namestr'late-rescale'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

LinkProgramPass ​

class View source

python
LinkProgramPass(name: str = 'link-backend-program')

Match all external references and produce the Program executable.

Attributes ​

NameTypeDefault/value
namestr'link-backend-program'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

LowerCkksToRnsNttPass ​

class View source

python
LowerCkksToRnsNttPass(selections: Mapping[str, str] = field(default_factory=dict), preserve: frozenset[str] = frozenset(), registry: CkksLoweringRegistry = DEFAULT_CKKS_LOWERINGS)

Apply caller-selected CKKS-to-RNS/NTT lowering definitions.

Attributes ​

NameTypeDefault/value
selectionsMapping[str, str]field(default_factory=dict)
preservefrozenset[str]frozenset()
registryCkksLoweringRegistryDEFAULT_CKKS_LOWERINGS
namestrfield(default='lower-ckks-to-rns-ntt', init=False)

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

LowerLogicalToCkksPass ​

class View source

python
LowerLogicalToCkksPass(name: str = 'lower-logical-to-ckks')

Lower registered logical arithmetic through visible typed CKKS edges.

Unrealized conversion casts preserve mixed-level result types for consumers that have not been lowered. Locally unresolved patterns remain legal no-ops and leave their original operations unchanged with diagnostics.

Attributes ​

NameTypeDefault/value
namestr'lower-logical-to-ckks'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Lower locally ready registered operations and report blockers.

LowerMessagePlaintextPreparationPass ​

class View source

python
LowerMessagePlaintextPreparationPass(name: str = 'lower-message-plaintext-preparation')

Expand concrete message preparation into visible CKKS operations.

Depth, scale, ordered prime ids, and modulus basis must already be assigned by a caller-selected CKKS scheduling pass. Missing state is an error rather than a request for this pass to choose a depth or scale.

Attributes ​

NameTypeDefault/value
namestr'lower-message-plaintext-preparation'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

LowerSemanticToLogicalPass ​

class View source

python
LowerSemanticToLogicalPass(name: str = 'lower-semantic-to-logical')

Classify registered encrypted semantic operations by operand roles.

Other dialects, public-only arithmetic, malformed operations, and unknown value roles remain structurally intact so partial mixed-level Programs stay valid inputs and outputs of the pass.

Attributes ​

NameTypeDefault/value
namestr'lower-semantic-to-logical'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Lower matching operation classes and report unchanged candidates.

LowerSpecializedCollectivesPass ​

class View source

python
LowerSpecializedCollectivesPass(lower_ciphertext_add: bool = True, name: str = 'lower-specialized-collectives')

Replace selected specialized collectives with generic region forms.

The pass exposes local combine arithmetic without proving associativity, rank uniformity, cross-rank ordering, or deadlock freedom. Callers remain free to preserve specialized operations for whole-operation providers.

Attributes ​

NameTypeDefault/value
lower_ciphertext_addboolTrue
namestr'lower-specialized-collectives'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Expose CKKS add as the combine region of ciphertext all-reduce.

MaterializeResourcesPass ​

class View source

python
MaterializeResourcesPass(materializer: ResourceMaterializer, name: str = 'materialize-backend-resources')

Create missing constructible resources once for the whole Program.

Attributes ​

NameTypeDefault/value
materializerResourceMaterializer
namestr'materialize-backend-resources'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

ResolveMaterialRequirementsPass ​

class View source

python
ResolveMaterialRequirementsPass(registry: OperationImplementationRegistry | None = None, key_layouts: Mapping[str, Mapping[str, object]] = field(default_factory=dict), name: str = 'resolve-material-requirements')

Resolve key layouts and implementation material operands across a Program.

The registry defaults to the Compilation workspace. key_layouts adds caller layout declarations keyed by material symbol or label. Existing operands and compatible metadata are retained; insufficient facts leave the corresponding requirements unresolved.

Attributes ​

NameTypeDefault/value
registryOperationImplementationRegistry | NoneNone
key_layoutsMapping[str, Mapping[str, object]]field(default_factory=dict)
namestr'resolve-material-requirements'

run ​

method

python
def run(compilation: Compilation) -> PassResult: ...

ResolveBackendOperationsPass ​

class View source

python
ResolveBackendOperationsPass(registry: OperationImplementationRegistry, in_place: bool = False, name: str = 'resolve-backend-operations')

Require Backend support and build the Program dispatch table.

Attributes ​

NameTypeDefault/value
registryOperationImplementationRegistry
in_placeboolFalse
namestr'resolve-backend-operations'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

ResolveRotationKeyOperandsPass ​

class View source

python
ResolveRotationKeyOperandsPass()

Replace logical rolls with CKKS rotations bound to named key resources.

The pass normalizes each step using the CKKS slot count and names its resource rotation-key:<step>. It reads CkksConfig from the Compile workspace and records the resolved coefficient automorphism on the key operand. It writes symbolic references only; key creation and live binding remain caller responsibilities.

Attributes ​

NameTypeDefault/value
namestrfield(default='resolve-rotation-key-operands', init=False)

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

ResolveTensorPlaceholdersPass ​

class View source

python
ResolveTensorPlaceholdersPass(name: str = 'resolve-tensor-placeholders')

Bind Program Tensor references from a symbol-to-Tensor table.

A partial table resolves the symbols it contains and leaves other references unchanged. Linking checks the references still required for execution.

Attributes ​

NameTypeDefault/value
namestr'resolve-tensor-placeholders'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

ReuseIntermediatesPass ​

class View source

python
ReuseIntermediatesPass(name: str = 'reuse-intermediates')

Share internal pure computations while preserving observable output storage.

Reuse requires the same concrete operation, ordered SSA operands, result types, attributes and properties. Numerical materials are ordinary operands and must therefore match by SSA identity. A mutation, random draw, unknown operation, region or implementation assignment ends a local reuse run.

Shared results must reach only fresh out-of-place readers, possibly through views. Direct returns, escaping views and unknown or assigned consumers prevent sharing. This preserves independently observable result storage without changing arithmetic order, rounding or value-state transitions.

Attributes ​

NameTypeDefault/value
namestr'reuse-intermediates'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

RotationHoistingPass ​

class View source

python
RotationHoistingPass(max_group_size: int | None = None, name: str = 'rotation-hoisting')

Share preparation among rotations of the same ciphertext data.

A rotation has the form Finish(c0, Prepare(c1, tables), key, step). Rotations with the same input SSA value, parameter operands, and compatible attributes can share Prepare despite interleaved consumers or rotations of other inputs. The emitted RotateManyOp retains each key operand, result type, and rotation step. Its common output domain and execution attributes must agree across the group.

Matching and placement stay inside a block's known-pure effect interval. Unknown operations, effects, control flow, and assigned implementations end that interval; known nested regions are processed independently. Every group is placed after its computed operands and before its first result use. Material references may move within the interval, but producer computations are not speculated or reordered.

max_group_size bounds the number of results produced together. Grouping can extend result lifetimes.

Attributes ​

NameTypeDefault/value
max_group_sizeint | NoneNone
namestr'rotation-hoisting'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Identify compatible rotations and place legal multi-result groups.

SelectExecutionLoweringsPass ​

class View source

python
SelectExecutionLoweringsPass(implementations: OperationImplementationRegistry, fusion_implementations: Sequence[FusionImplementation] = (), lowerings: CkksLoweringRegistry = DEFAULT_CKKS_LOWERINGS, allow_partial_fusion: bool = True, name: str = 'select-execution-lowerings', execution_cost: Callable[[Operation, Sequence[Operation]], tuple[float, float] | None] | None = None)

Keep applicable whole operations or expose a supported fusion expression.

A supplied implementation assignment is local to its operation. Unassigned operations without whole-operation coverage require lowering. Where both routes exist, an isolated lowering probe is accepted only when a fusion implementation supports the complete expression. With partial fusion enabled, supported subregions may instead be combined with independent implementations for the remaining operations.

execution_cost may compare the whole operation with the isolated lowered candidate after this pass's fusion scheduling. It returns (whole, lowered) costs in one caller-selected unit, or None when the comparison is unknown. Lowering is selected only for a strictly smaller supplied cost; ties retain the whole operation. Missing estimates retain the fusion-coverage decision. Implementation assignments always take precedence over this callback.

Attributes ​

NameTypeDefault/value
implementationsOperationImplementationRegistry
fusion_implementationsSequence[FusionImplementation]()
loweringsCkksLoweringRegistryDEFAULT_CKKS_LOWERINGS
allow_partial_fusionboolTrue
namestr'select-execution-lowerings'
execution_costCallable[[Operation, Sequence[Operation]], tuple[float, float] | None] | NoneNone

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

SelectNttImplementationsPass ​

class View source

python
SelectNttImplementationsPass(registry: OperationImplementationRegistry | None = None, name: str = 'select-ntt-implementations')

Ask the selected NTT implementation to complete missing schedule choices.

Caller assignments and supplied Tensor operands remain authoritative. Missing transform tables become ordinary material references, whose data must be supplied separately. Operations with insufficient selection facts remain unchanged and report why their choice was deferred.

Attributes ​

NameTypeDefault/value
registryOperationImplementationRegistry | NoneNone
namestr'select-ntt-implementations'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

SvgGraphDirection ​

type alias View source

python
SvgGraphDirection = Literal['TB', 'BT', 'LR', 'RL']

SvgGraphError ​

class View source

python
SvgGraphError()

Bases: RuntimeError

Report that Program SVG rendering could not complete.

SvgGraphField ​

type alias View source

python
SvgGraphField = Literal['name', 'opcode', 'role', 'operands', 'result_types', 'attributes', 'num_users']

SvgGraphOutput ​

class View source

python
SvgGraphOutput(path: Path, temporary: bool)

Record the SVG file produced by a visualization pass.

Attributes ​

NameTypeDefault/value
pathPath
temporarybool

SvgGraphPresentation ​

class View source

python
SvgGraphPresentation(*, fields: Collection[SvgGraphField] | None=None, attribute_names: Collection[str] | None=None, attribute_preview_chars: int | None=180, theme: SvgGraphTheme | None=None)

Select and format operation details for an SVG graph.

Subclasses can override select_attributes for filtering, format_attribute_value for one-value display policy, operation_sections for complete operation-row transformation, or operation_tooltip for aggregate hover details. These methods must not mutate their operation or Program.

select_attributes ​

method

python
def select_attributes(operation: Operation) -> tuple[tuple[str, Attribute], ...]: ...

Return sorted non-internal attributes selected for operation.

format_attribute_value ​

method

python
def format_attribute_value(operation: Operation, name: str, attribute: Attribute) -> str: ...

Format one selected attribute for its character-limited preview.

operation_sections ​

method

python
def operation_sections(context: SvgOperationContext) -> tuple[SvgNodeSection, ...]: ...

Return all displayed record rows for one operation.

Override this complete-row hook to reorder, rename, remove, or inject derived sections while reusing super().operation_sections(context).

operation_tooltip ​

method

python
def operation_tooltip(context: SvgOperationContext, sections: Collection[SvgNodeSection]) -> str | None: ...

Return one aggregate node tooltip from already-rendered sections.

SvgGraphTheme ​

class View source

python
SvgGraphTheme(*, operation_palette: Collection[str] | None=None, operation_colors: Mapping[str, str] | None=None, operation_color_key: Callable[[SvgOperationContext], str] | None=None, canvas_color: str='transparent', input_fill_color: str='#DCE4FF', output_fill_color: str='#F3D89D', node_font_color: str='#181B26', node_stroke_color: str='#596178', edge_color: str='#7B8193')

Define colors and operation color classification for an SVG graph.

An operation first receives a key from operation_color_key. A matching operation_colors entry wins; otherwise the key is deterministically mapped into operation_palette. Equal keys under one theme receive equal colors. None selects the FHElium default palette or color-key function. operation_colors=None uses the default constant color, while an empty mapping removes it. Theme configuration is read-only and unhashable because a caller-supplied color-key callable does not define a general cache identity.

Attributes ​

NameTypeDefault/value
operation_palettetuple[str, ...]
operation_colorsMapping[str, str]
operation_color_keyCallable[[SvgOperationContext], str]
canvas_colorstr
input_fill_colorstr
output_fill_colorstr
node_font_colorstr
node_stroke_colorstr
edge_colorstr

operation_fill_color ​

method

python
def operation_fill_color(context: SvgOperationContext) -> str: ...

Return the configured or stable palette color.

SvgGraphVisualizationPass ​

class View source

python
SvgGraphVisualizationPass(output_path: str | PathLike[str] | None=None, *, overwrite: bool=False, entry: str='main', presentation: SvgGraphPresentation | None=None, rank_direction: SvgGraphDirection='TB', name: str='visualize-svg')

Render one Program function to a local or temporary SVG file.

output_path selects a caller-owned local path. When it is omitted, the pass creates a unique directory under the operating system's temporary location. In both cases, SvgGraphOutput records the produced file in the Compilation workspace.

Attributes ​

NameTypeDefault/value
output_pathPath | None
overwritebool
entrystr
presentationSvgGraphPresentation | None
rank_directionSvgGraphDirection
namestr

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

Render the current Program and publish its output path.

SvgNodeSection ​

class View source

python
SvgNodeSection(name: str, value: str, tooltip: str | None = None)

Represent one labeled record row inside an SVG operation node.

value participates in Graphviz layout. tooltip optionally adds complete or alternate details for the containing node's aggregate hover text; Graphviz does not attach a separate tooltip to each record cell.

Attributes ​

NameTypeDefault/value
namestr
valuestr
tooltipstr | NoneNone

SvgOperationContext ​

class View source

python
SvgOperationContext(operation: Operation, result_names: tuple[str, ...], operand_names: tuple[str, ...])

Provide stable operation data to an SVG presentation policy.

Attributes ​

NameTypeDefault/value
operationOperation
result_namestuple[str, ...]
operand_namestuple[str, ...]

opcode ​

property

python
opcode: str

Return the registered or dynamic operation name.

ValidateExecutionRepresentationsPass ​

class View source

python
ValidateExecutionRepresentationsPass(name: str = 'validate-execution-representations')

Require concrete legal representations at fixed executable ABIs.

Attributes ​

NameTypeDefault/value
namestr'validate-execution-representations'

run ​

method

python
def run(compilation: 'Compilation') -> PassResult: ...

backend_linking_pipeline ​

function View source

python
def backend_linking_pipeline(backend: OperationBackend, *, in_place: bool=False) -> Pipeline: ...

Return the standard Backend linking sequence as an editable Pipeline.

default_svg_operation_color_key ​

function View source

python
def default_svg_operation_color_key(context: SvgOperationContext) -> str: ...

Return the default stable operation color key.

Operation names define ordinary keys. Preserved torch.call operations additionally include call kind and target so custom classifiers can delegate without reproducing that rule.